A novel group decision-making method for interval-valued q-rung dual hesitant fuzzy information using extended power average operator and Frank operations

Abstract

This paper advances the field of multi-attribute group decision making (MAGDM) by proposing a novel framework based on interval-valued q-rung dual hesitant fuzzy sets (IVq-RDHFSs). IVq-RDHFSs, which surpass most existing fuzzy sets, effectively represent complex fuzzy information by describing membership and non-membership degrees through interval value sets. However, prior MAGDM methods based on IVq-RDHFSs have been limited by the functions of operation rules and aggregation operators (AOs). This limitation is addressed through the construction of a new MAGDM framework, leveraging the robust Frank t-norm and t-conorm (FTT) operation and the extended power average (EPA) operator. The proposed framework features the interval-valued q-rung dual hesitant fuzzy Frank weighted extended power average (IVq-RDHFFWEPA) operator to obtain comprehensive evaluation values. The paper also introduces novel techniques for determining the weights of decision-makers and attributes. Practical applications of the proposed method are demonstrated through the assessment of desalination technology selection and rural green eco-tourism projects. Sensitivity and comparison analyses validate the superior functionality, accuracy, and flexibility of this method compared to many state-of-the-art methods. The contributions of this paper are two-fold: it develops efficient measurement techniques for IVq-RDHFSs, such as distance and weight calculation, and it introduces a comprehensive MAGDM method by integrating FTT and EPA under IVq-RDHFSs, which improves the efficiency of solving decision-making problems.

1 Introduction

Multi-attribute group decision-making (MAGDM) is an important branch of decision science and operational research, determining the wisest scheme(s) in accord with the opinions of multiple decision-makers (DMs). When faced with most decision-making problems in real life, DMs feel difficult to provide precise evaluation values, because it is impossible to obtain complete information for a limited time (Ali et al. 2021). Hence, how to tackle uncertain and fuzzy cognitive information in decision-making process is a critical research problem (Yildirim and Kuzu Yıldırım 2022). To cope with the increasingly complex decision-making environment, fuzzy set theory and its extensions (Ali 2021; Ali and Garg 2023) came into being and have aroused extensive research interest in MAGDM problems (Akram et al. 2021b, 2023; Rani and Garg 2022; Riaz and Athar Farid 2022).

Using fuzzy set theory in MAGDM is helpful to depict the vagueness of DMs’ viewpoints (Akram et al. 2022; Ali and Naeem 2023b). Pioneeringly, simulating the optimism and pessimism aspects of human cognition, Atanassov (1986) proposed the theory of intuitionistic fuzzy sets (IFSs) to handle the fuzziness of DMs’ evaluation information in the manner of membership degree (MDs) and non-memberships (NMDs). In view of the intuitiveness and efficiency of IFSs, lots of publications have studied their application in MAGDM (Akram et al. 2021c; Liu et al. 2020; Wan et al. 2019). Whereas more and more researchers pointed out the very limited ability of IFSs and investigated their extension in different situations (Ali and Mahmood 2020; Faizi et al. 2020; Muhammad et al. 2022). Especially, to reflect the original information as much as possible, Zhu et al. (2012) contributed dual hesitant fuzzy sets (DHFs) to support DMs to judge MDs and NMDs through several different possible values. Several recent publications have demonstrated the advantages of DHFs in dealing with MAGDM (Ni et al. 2022; Tang et al. 2021; Wei and Wang 2021). Afterwards, Ju et al. (2014) discovered that interval values sometimes convey judgements better than crisp numbers. Akram et al.’s (2021a) study demonstrated the effectiveness of interval values in expressing fuzzy judgement information. In Ju et al.’s (2014) study, they proposed the notion of interval-valued dual hesitant fuzzy sets (IVDHFSs), whose MD and NMD can be denoted by a set of interval values instead of precise values. Some of the latest IVDHFSs-based MAGDM methods exhibit the strong ability of IVDHFSs in expressing the complicated evaluation information (Jiang et al. 2020, 2022; Sarkar and Biswas 2021). Notably, all the above fuzzy set theories must obey the strict restriction, that is, the sum of MD and NMD should not be greater than 1. Due to this restriction, the use flexibility of IVDHFSs is dramatically discounted, even though their ability to depict fuzzy information is impressive. To fill this gap, Yager (2017) proposed the concept of generalized fuzzy sets, which was also called q-rung orthopair fuzzy sets, to relax the restriction to the qth sum of the MD and NMD is no larger than 1. Since then, many extensions of q-rung orthopair fuzzy sets have been proposed (Ali 2022, 2023; Ali and Naeem 2023a). Inspired by these, Xu et al. (2019) introduced the q-rung orthopair fuzzy sets to IVDHFSs and provided the definition of interval-valued q-rung dual hesitant fuzzy sets (IVq-RDHFSs), which requires the sum of qth power of boundary values in MD and NMD does not exceed 1. IVq-RDHFSs provide DMs with a larger space to freely express preference information, and evidently, many existing fuzzy sets are the special cases of IVq-RDHFSs, such as DHFSs (Zhu et al. 2012), IVDHFSs (Ju et al. 2014), q-rung dual hesitant fuzzy sets (q-RDHFSs) (Xu et al. 2018) and so forth. This sheds light on the overwhelming superiority of IVq-RDHFSs over these information expression tools when encountering diverse complex shifting decision issues. The IVq-RDHFSs has been successfully implemented in some MAGDM problems (Zhao et al. 2021), and their extension versions are emerging (Feng et al. 2020; Xu et al. 2020).

When addressing MAGDM, another key issue is how to aggregate fuzzy information (Shao and Zhou 2021). But unfortunately, all existing methods based on IVq-RDHFSs have two obvious weaknesses, i.e., operation rules and AOs provide very limited functions, yet they usually make an important contribution to the results of information aggregation (Wan and Yi 2015). Firstly, when looking at the operation rules, these methods are all established on the basic algebraic operation given by Xu et al. (2019). Noting that in the face of various complex problems, the demand for the universality and flexibility of decision-making methods increases sharply, while there is a lack of a generalized operation framework of IVq-RDHFSs to flexibly handle various application requirements. Prominently, FTT is one type of general t-norm (TN) and t-conorm (TC), and it is the only one that satisfies the compatibility (Frank 1978). Adopting FTT in MAGDM has the following merits: (1) (generalization) Given some specific values of $\theta$, some specific versions of AOs can be produced, because FTT can cover some other forms of TN and TC by different $\theta$, which exhibits its great flexibility; (2) (functionality) Its compatibility makes the additional parameter $\theta$ appropriate to reflect the risk preference of DMs, which makes the aggregated results more reasonable and robust. Therefore, FTT is more applicable in modeling realistic decision-making problems compared with other algorithms, such as Einstein TN and TC, Dombi TN and TC, Aczel–Alsina TN and TC, or Hamacher TN and TC. FTT has been employed to hesitant fuzzy sets (Qin et al. 2016), IFSs (Zhang et al. 2015), DHFSs (Tang et al. 2018), picture fuzzy linguistic sets (Liu and Zhang 2018), etc. And yet, to the best of our knowledge, FTT has not been studied for IVq-RDHFSs. Thus, this paper will build a series of new generalized operation rules for IVq-RDHFSs with the aid of FTT.

Secondly, as for AOs, Muirhead mean (Xu et al. 2019) and Maclaurin symmetric mean (Feng et al. 2021) operators have been applied to aggregate IVq-RDHFSs, which can consider the interrelation among input arguments. Nevertheless, they are unable to address the negative effect of extreme evaluation values on decision results. Extreme assessment is possibly “biased” or “false” data generated by DMs’ inner prejudice, which may incur serious damage to the rationality of the final outcomes. To solve this challenge, Yager (2001) presented the power average (PA) operator, which assigns smaller weights to unduly high or low values by allowing inputs to support each other. The aggregation mechanism indicates the suitability of PA operator to balance the impact of extreme values and produce more reasonable aggregated result, and thus, PA has been widely studied in MAGDM (Feng et al. 2021; Wan 2013; Wan and Dong 2015). However, the ability of PA to tackle extreme evaluation is very single. It considers extreme values to be less important by default, but in fact, for some particular applications, these extreme inputs may be sufficiently significant elements. Motivated by this, **ong et al. (2019) generalized PA to the extended power average (EPA) operator, which has stronger extreme values processing function and possesses wider application scope in real-life decision-making problems. In detail, the impressive merits of EPA are: (1) (Flexibility) The weights of input data in EPA can dynamically adjust predicated on actual needs, which permits DMs to focus more on the sufficiently important inputs; (2) (Practicability) EPA is able to model different decision problems with diverse parameters, covering situations where the unduly high or low inputs can be considered possibly unreasonable or sufficiently important. Recently, Li et al. (2021a) utilized EPA and Archimedean t-norm and t-conorm to solve MAGDM under q-RDHFSs environment. Li et al.’s study (2021a) indicates the potential of EPA in solving fuzzy decision problems. Meanwhile, since IVq-RDHFSs can capture fuzzy information more accurately than q-RDHFSs, extending EPA into IVq-RDHFSs is of great significance for more effective MAGDM processing.

To sum up, the motivation of our paper can be summarized as follows. This study aims to address two key issues in MAGDM: how to express and handle fuzzy information more accurately and how to aggregate fuzzy information effectively. Despite many studies using different fuzzy set theories to tackle these issues, there are still notable shortcomings and challenges. In particular, the existing methods based on IVq-RDHFSs exhibit significant deficiencies in terms of their operation rules and AOs. Therefore, the motivation for this study is to construct a series of new generalized operation rules for IVq-RDHFSs based on FTT and to investigate the effectiveness of the EPA operator in handling IVq-RDHFSs. The necessary of our paper can be described from the following aspects. First, the existing methods based on IVq-RDHFSs are overly simplistic in terms of their operation rules and lack a generalized operation framework capable of flexibly handling various application demands. As FTT is a general t-norm and t-conorm with good compatibility, it is necessary to establish new operation rules using FTT. Second, the existing AOs have problems in handling the negative impact of extreme evaluation values. Although the PA operator has some advantages in balancing the impact of extreme values, its ability to handle extreme evaluations is singular and cannot meet the requirements of special application scenarios. Therefore, it is necessary to extend the PA operator with the EPA operator, which has a stronger ability to handle extreme values. Lastly, as IVq-RDHFSs can capture fuzzy information more accurately than q-RDHFSs, extending the EPA to IVq-RDHFSs is significant for more effectively handling MAGDM. In summary, the motivation and necessity for this research stem from a deep understanding of the MAGDM problem and an awareness of the challenges and shortcomings faced by existing methods in addressing these issues. This study aims to enhance our ability to deal with these issues by proposing new methods and frameworks.

Predicated on the above analysis, the contributions of this paper are manifested in: (1) New operation rules of IVq-RDHFSs are established by fusing FTT into IVq-RDHFSs, which are more generalized and functional than existing algebraic rules; (2) Based on these new rules, by extending EPA into IVq-RDHFSs, some new AOs is constructed to flexibly simulate two situations that eliminate or emphasize the impact of extreme values; (3) The efficient techniques of distance and entropy weight calculation for IVq-RDHFSs is provided. More comprehensively, a new method to determine DM weight is developed; (4) A complete framework based on these proposed algorithms for solving MAGDM under IVq-RDHFSs is devised. Benefiting from the excellent IVq-RDHFSs, FTT and EPA, this framework outperforms many existing methods. (5) Two case studies are conducted to show the applicability of this method in practical decision-making problems.

We organize the remainder of this paper as follows. Section 2 reviews preliminaries. Section 3 investigates new FTT operations for IVq-RDHFSs. Section 4 provides a collection of AOs for IVq-RDHFSs inspired by EPA. Section 5 presents the methods of determining DM and attribute weights. Section 6 introduces a novel MAGDM approach. Section 7 illustrates the application and noteworthy superiorities of the proposed method by a series of numerical examples. Section 8 offers a conclusion.

2 Preliminaries

In the present section, we briefly review some fundamental notions which will be used in the following sections.

2.1 Interval-valued q-rung dual hesitant fuzzy sets

Definition 1

(Xu et al. 2019). Let X be an ordinary fixed set, an interval-valued q-rung dual hesitant fuzzy set (IVq-RDHFS) D defined on X is expressed as

$$D=\left\{\langle x,{h}_{D}\left(x\right),{g}_{D}\left(x\right)\rangle \left|x\in X\right.\right\},$$

(1)

where ${h}_{D}\left(x\right)={\bigcup }_{[{r}_{D}^{l},{r}_{D}^{u}]\in {h}_{D}\left(x\right)}\left\{\left[{r}_{D}^{l},{r}_{D}^{u}\right]\right\}$ and ${g}_{D}\left(x\right)={\bigcup }_{[{\eta }_{D}^{l},{\eta }_{D}^{u}]\in {g}_{D}\left(x\right)}\left\{\left[{\eta }_{D}^{l},{\eta }_{D}^{u}\right]\right\}$ are two sets of some interval values contained in $\left[0,1\right]$, denoting the possible MD and NMD of the element $x\in X$ to the set D, respectively, such that

$$\left[{r}_{D}^{l},{r}_{D}^{u}\right],\left[{\eta }_{D}^{l},{\eta }_{D}^{u}\right]\subset \left[{0,1}\right], and {\left({\left({r}_{D}^{u}\right)}^{+}\right)}^{q}+{\left({\left({\eta }_{D}^{u}\right)}^{+}\right)}^{q}\le 1 \left(q\ge 1\right)$$

where ${\left[{r}_{D}^{l},{r}_{D}^{u}\right]\in h}_{D}\left(x\right)$ and $\left[{\eta }_{D}^{l},{\eta }_{D}^{u}\right]\in {g}_{D}\left(x\right)$, ${\left({r}_{D}^{u}\right)}^{+}\in {h}_{D}^{+}\left(x\right)={\bigcup }_{\left[{r}_{D}^{l},{r}_{D}^{u}\right]\in {h}_{D}\left(x\right)}max\left\{{r}_{D}^{u}\right\},$ and ${\left({\eta }_{D}^{u}\right)}^{+}\in {g}_{D}^{+}\left(x\right)={\bigcup }_{[{\eta }_{D}^{l},{\eta }_{D}^{u}]\in {g}_{D}\left(x\right)}max\left\{{\eta }_{D}^{u}\right\}$ for all $x\in X$. For convenience, we call the pair $d\left(x\right)=\left\{{h}_{D}\left(x\right),{g}_{D}\left(x\right)\right\}$ an interval-valued q-rung dual hesitant fuzzy element (IVq-RDHFE), which can be denoted as $d=\left\{h,g\right\}$.

Definition 2

(Xu et al. 2019). Let ${d}_{i}=\left\{{h}_{i},{g}_{i}\right\}\left(i={1,2}\right)$ be any two IVq-RDHFEs and $\lambda$ be a positive real number, then

• 1.
  $$ \begin{aligned} d_{1} \oplus d_{2} & = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {r_{2}^{l} ,r_{2}^{u} } \right] \in h_{2} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} ,\left[ {\eta_{2}^{l} ,\eta_{2}^{u} } \right] \in g_{2} }} \left\{ {\left\{ {\left[ {\left( {\left( {r_{1}^{l} } \right)^{q} + \left( {r_{2}^{l} } \right)^{q} - \left( {r_{1}^{l} r_{2}^{l} } \right)^{q} } \right)^{1/q} ,} \right.} \right.} \right. \\ & \quad \left. {\left. {\left. {\left( {\left( {r_{1}^{u} } \right)^{q} + \left( {r_{2}^{u} } \right)^{q} - \left( {r_{1}^{u} r_{2}^{u} } \right)^{q} } \right)^{1/q} } \right]} \right\},\left\{ {\left[ {\eta_{1}^{l} \eta_{2}^{l} ,\eta_{1}^{u} \eta_{2}^{u} } \right]} \right\}} \right\}; \\ \end{aligned} $$

  (2)
• 2.
  $$ \begin{aligned} d_{1} \otimes d_{2} & = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {r_{2}^{l} ,r_{2}^{u} } \right] \in h_{2} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} ,\left[ {\eta_{2}^{l} ,\eta_{2}^{u} } \right] \in g_{2} }} \left\{ {\left\{ {\left[ {r_{1}^{l} r_{2}^{l} ,r_{1}^{u} r_{2}^{u} } \right]} \right\}} \right., \\ & \quad \left. {\left\{ {\left[ {\left( {\left( {\eta_{1}^{l} } \right)^{q} + \left( {\eta_{2}^{l} } \right)^{q} - \left( {\eta_{1}^{l} \eta_{2}^{l} } \right)^{q} } \right)^{1/q} ,\left( {\left( {\eta_{1}^{u} } \right)^{q} + \left( {\eta_{2}^{u} } \right)^{q} - \left( {\eta_{1}^{u} \eta_{2}^{u} } \right)^{q} } \right)^{1/q} } \right]} \right\}} \right\}; \\ \end{aligned} $$

  (3)
• 3.
  $$ \lambda d_{1} = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} }} \left\{ {\left\{ {\left[ {\left( {1 - \left( {1 - \left( {r_{1}^{l} } \right)^{q} } \right)^{\lambda } } \right)^{1/q} ,\left( {1 - \left( {1 - \left( {r_{1}^{u} } \right)^{q} } \right)^{\lambda } } \right)^{1/q} } \right]} \right\},\left\{ {\left[ {\left( {\eta_{1}^{l} } \right)^{\lambda } ,\left( {\eta_{1}^{u} } \right)^{\lambda } } \right]} \right\}} \right\}; $$

  (4)
• 4.
  $$ d_{1}^{\lambda } = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} }} \left\{ {\left\{ {\left[ {\left( {r_{1}^{l} } \right)^{\lambda } ,\left( {r_{1}^{u} } \right)^{\lambda } } \right]} \right\},\left\{ {\left[ {\left( {1 - \left( {1 - \left( {\eta_{1}^{l} } \right)^{q} } \right)^{\lambda } } \right)^{1/q} ,\left( {1 - \left( {1 - \left( {\eta_{1}^{u} } \right)^{q} } \right)^{\lambda } } \right)^{1/q} } \right]} \right\}} \right\}. $$

  (5)

Definition 3

(Xu et al. 2019). Let $d=\left\{h,g\right\}$ be an IVq-RDHFE, the score function of d is defined as

$$S\left(d\right)={\left(\frac{1}{\#h}{\sum }_{\left[{r}^{l},{r}^{u}\right]\in h}{\gamma }^{l}\right)}^{q}+{\left(\frac{1}{\#h}{\sum }_{\left[{r}^{l},{r}^{u}\right]\in h}{\gamma }^{u}\right)}^{q}-{\left(\frac{1}{\#g}{\sum }_{\left[{\eta }^{l},{\eta }^{u}\right]\in g}{\eta }^{l}\right)}^{q}-{\left(\frac{1}{\#g}{\sum }_{\left[{\eta }^{l},{\eta }^{u}\right]\in g}{\eta }^{u}\right)}^{q},$$

(6)

and the accuracy function

$$H\left(d\right)={\left(\frac{1}{\#h}{\sum }_{\left[{r}^{l},{r}^{u}\right]\in h}{\gamma }^{l}\right)}^{q}+{\left(\frac{1}{\#h}{\sum }_{\left[{r}^{l},{r}^{u}\right]\in h}{\gamma }^{u}\right)}^{q}+{\left(\frac{1}{\#g}{\sum }_{\left[{\eta }^{l},{\eta }^{u}\right]\in g}{\eta }^{l}\right)}^{q}+{\left(\frac{1}{\#g}{\sum }_{\left[{\eta }^{l},{\eta }^{u}\right]\in g}{\eta }^{u}\right)}^{q},$$

(7)

where #h and #g are the numbers of the elements in h and g, respectively. Let ${d}_{i}=\left\{{h}_{i},{g}_{i}\right\}\left(i={1,2}\right)$ be any two IVq-RDHFEs, then

• (1) If $S\left({d}_{1}\right)>S\left({d}_{2}\right)$, then ${d}_{1}\succ {d}_{2}$;

• (2) If $S\left({d}_{1}\right)=S\left({d}_{2}\right)$, then

  • if $H\left({d}_{1}\right)>H\left({d}_{2}\right)$, then ${d}_{1}\succ {d}_{2}$;

  • if $H\left({d}_{1}\right)=H\left({d}_{2}\right)$, then ${d}_{1}={d}_{2}$.

Remark 1

If in an IVq-RDHFE, the upper bound and lower bound of each interval value in MD and NMD sets are equal, the IVq-RDHFE reduces to the q-RDHFE denoted as $\widetilde{d}$, and then the score function of $\widetilde{d}$ can be calculated by the formula $S\left(\widetilde{d}\right)=\frac{1}{2}S\left(d\right)$. For more information about q-RDHFE, please see Xu et al.’ study (2019).

In this paper, we provide a new distance measure between any two IVq-RDHFEs.

Definition 4

Let ${d}_{i}=\left\{{h}_{i},{g}_{i}\right\}$ and ${d}_{j}=\left\{{h}_{j},{g}_{j}\right\}$ be any two IVq-RDHFEs, then the interval-valued q-rung dual hesitant fuzzy distance between ${d}_{i}$ and ${d}_{j}$ is defined as

$$ \begin{aligned} dis\left( {d_{i} ,d_{j} } \right) & = \frac{1}{4}\left( {\frac{1}{{\# h_{i} \# h_{j} }}\mathop \sum \limits_{m = 1}^{{\# h_{i} }} \mathop \sum \limits_{n = 1}^{{\# h_{j} }} \left( {\left| {\left( {\gamma_{m}^{l} } \right)^{q} - \left( {\gamma_{n}^{l} } \right)^{q} } \right| + \left| {\left( {\gamma_{m}^{u} } \right)^{q} - \left( {\gamma_{n}^{u} } \right)^{q} } \right|} \right)} \right. \\ & + \left. {\frac{1}{{\# g_{i} \# g_{j} }}\mathop \sum \limits_{m = 1}^{{\# g_{i} }} \mathop \sum \limits_{n = 1}^{{\# g_{j} }} \left( {\left| {\left( {\eta_{m}^{l} } \right)^{q} - \left( {\eta_{n}^{l} } \right)^{q} } \right| + \left| {\left( {\eta_{m}^{u} } \right)^{q} - \left( {\eta_{n}^{u} } \right)^{q} } \right|} \right)} \right), \\ \end{aligned} $$

(8)

where $\#{h}_{i}$, $\#{h}_{j}$ denotes the number of interval values in ${h}_{i}$ and ${h}_{j}$, and $\#{g}_{i}$, $\#{g}_{j}$ denotes the number of interval values in ${g}_{i}$ and ${g}_{j}$, respectively. In addition, ${\gamma }_{m}^{l},{\gamma }_{m}^{u}\in {h}_{i}$, ${\gamma }_{n}^{l},{\gamma }_{n}^{u}\in {h}_{j}$, ${\eta }_{m}^{l},{\eta }_{m}^{u}\in {g}_{i}$, and ${\eta }_{n}^{l},{\eta }_{n}^{u}\in {g}_{j}$.

It is easy to prove that the distance measure $dis\left({d}_{i},{d}_{j}\right) \left(i,j={1,2},\ldots ,n\right)$ between any two IVq-RDHFEs have the following properties:

• (1) (Non-negativity) $dis\left({d}_{i},{d}_{j}\right)\epsilon \left[{0,1}\right]$.

• (2) (Identity of indiscernibles) $dis\left({d}_{i},{d}_{i}\right)=0$.

• (3) (Symmetry) $dis\left({d}_{i},{d}_{j}\right)=dis\left({{d}_{j},d}_{i}\right)$.

Proof

(1) According to Definition 1, we can know that, in Eq. (8)

$$\left[{r}_{m}^{l},{r}_{m}^{u}\right],\left[{\eta }_{m}^{l},{\eta }_{m}^{u}\right],\left[{r}_{n}^{l},{r}_{n}^{u}\right],\left[{\eta }_{n}^{l},{\eta }_{n}^{u}\right]\subset \left[{0,1}\right], q\ge 1$$

and

$${\left({\left({\gamma }_{m}^{u}\right)}^{+}\right)}^{q}+{\left({\left({\eta }_{m}^{u}\right)}^{+}\right)}^{q}\le 1 \left(q\ge 1\right), {\left({\left({\gamma }_{n}^{u}\right)}^{+}\right)}^{q}+{\left({\left({\eta }_{n}^{u}\right)}^{+}\right)}^{q}\le 1 \left(q\ge 1\right).$$

Therefore, we can get

$$\left|{\left({\gamma }_{m}^{l}\right)}^{q}-{\left({\gamma }_{n}^{l}\right)}^{q}\right|,\left|{\left({\gamma }_{m}^{u}\right)}^{q}-{\left({\gamma }_{n}^{u}\right)}^{q}\right|,\left|{\left({\eta }_{m}^{l}\right)}^{q}-{\left({\eta }_{n}^{l}\right)}^{q}\right|,\left|{\left({\eta }_{m}^{u}\right)}^{q}-{\left({\eta }_{n}^{u}\right)}^{q}\right|\epsilon \left[{0,1}\right],$$

then,

$$\left(\left|{\left({\gamma }_{m}^{l}\right)}^{q}-{\left({\gamma }_{n}^{l}\right)}^{q}\right|+\left|{\left({\gamma }_{m}^{u}\right)}^{q}-{\left({\gamma }_{n}^{u}\right)}^{q}\right|\right),\left(\left|{\left({\eta }_{m}^{l}\right)}^{q}-{\left({\eta }_{n}^{l}\right)}^{q}\right|+\left|{\left({\eta }_{m}^{u}\right)}^{q}-{\left({\eta }_{n}^{u}\right)}^{q}\right|\right)\epsilon \left[{0,2}\right].$$

Besides,

$$\sum_{m=1}^{\#{h}_{i}}\sum_{n=1}^{\#{h}_{j}}\left(\left|{\left({\gamma }_{m}^{l}\right)}^{q}-{\left({\gamma }_{n}^{l}\right)}^{q}\right|+\left|{\left({\gamma }_{m}^{u}\right)}^{q}-{\left({\gamma }_{n}^{u}\right)}^{q}\right|\right)\epsilon \left[{0,2}\#{h}_{i}\#{h}_{j}\right],$$

and

$$\sum_{m=1}^{\#{g}_{i}}\sum_{n=1}^{\#{g}_{j}}\left(\left|{\left({\eta }_{m}^{l}\right)}^{q}-{\left({\eta }_{n}^{l}\right)}^{q}\right|+\left|{\left({\eta }_{m}^{u}\right)}^{q}-{\left({\eta }_{n}^{u}\right)}^{q}\right|\right)\epsilon \left[{0,2}\#{g}_{i}\#{g}_{j}\right].$$

Then, we can obtain

$$\frac{1}{\#{h}_{i}\#{h}_{j}}\sum_{m=1}^{\#{h}_{i}}\sum_{n=1}^{\#{h}_{j}}\left(\left|{\left({\gamma }_{m}^{l}\right)}^{q}-{\left({\gamma }_{n}^{l}\right)}^{q}\right|+\left|{\left({\gamma }_{m}^{u}\right)}^{q}-{\left({\gamma }_{n}^{u}\right)}^{q}\right|\right)\epsilon \left[{0,2}\right],$$

and,

$$\frac{1}{\#{g}_{i}\#{g}_{j}}\sum_{m=1}^{\#{g}_{i}}\sum_{n=1}^{\#{g}_{j}}\left(\left|{\left({\eta }_{m}^{l}\right)}^{q}-{\left({\eta }_{n}^{l}\right)}^{q}\right|+\left|{\left({\eta }_{m}^{u}\right)}^{q}-{\left({\eta }_{n}^{u}\right)}^{q}\right|\right)\epsilon \left[{0,2}\right].$$

So that

$$ \begin{aligned} & \frac{1}{{\# h_{i} \# h_{j} }}\mathop \sum \limits_{m = 1}^{{\# h_{i} }} \mathop \sum \limits_{n = 1}^{{\# h_{j} }} \left( {\left| {\left( {\gamma_{m}^{l} } \right)^{q} - \left( {\gamma_{n}^{l} } \right)^{q} } \right| + \left| {\left( {\gamma_{m}^{u} } \right)^{q} - \left( {\gamma_{n}^{u} } \right)^{q} } \right|} \right) \\ & \quad + \frac{1}{{\# g_{i} \# g_{j} }}\mathop \sum \limits_{m = 1}^{{\# g_{i} }} \mathop \sum \limits_{n = 1}^{{\# g_{j} }} \left( {\left| {\left( {\eta_{m}^{l} } \right)^{q} - \left( {\eta_{n}^{l} } \right)^{q} } \right| + \left| {\left( {\eta_{m}^{u} } \right)^{q} - \left( {\eta_{n}^{u} } \right)^{q} } \right|} \right)\epsilon \left[ {0,4} \right]. \\ \end{aligned} $$

Finally, we can get

$$ \begin{aligned} & \frac{1}{4}\left( {\frac{1}{{\# h_{i} \# h_{j} }}\mathop \sum \limits_{m = 1}^{{\# h_{i} }} \mathop \sum \limits_{n = 1}^{{\# h_{j} }} \left( {\left| {\left( {\gamma_{m}^{l} } \right)^{q} - \left( {\gamma_{n}^{l} } \right)^{q} } \right| + \left| {\left( {\gamma_{m}^{u} } \right)^{q} - \left( {\gamma_{n}^{u} } \right)^{q} } \right|} \right)} \right. \\ & \quad + \left. {\frac{1}{{\# g_{i} \# g_{j} }}\mathop \sum \limits_{m = 1}^{{\# g_{i} }} \mathop \sum \limits_{n = 1}^{{\# g_{j} }} \left( {\left| {\left( {\eta_{m}^{l} } \right)^{q} - \left( {\eta_{n}^{l} } \right)^{q} } \right| + \left| {\left( {\eta_{m}^{u} } \right)^{q} - \left( {\eta_{n}^{u} } \right)^{q} } \right|} \right)} \right)\epsilon \left[ {0,1} \right]. \\ \end{aligned} $$

That is, $dis\left({d}_{i},{d}_{j}\right)\epsilon \left[{0,1}\right]$. The Non-negative property is proven.

(2)

$$ \begin{aligned} & dis\left( {d_{i} ,d_{i} } \right) = \frac{1}{4}\left( {\frac{1}{{\# h_{i} \# h_{i} }}\mathop \sum \limits_{m = 1}^{{\# h_{i} }} \mathop \sum \limits_{m = 1}^{{\# h_{i} }} \left( {\left| {\left( {\gamma_{m}^{l} } \right)^{q} - \left( {\gamma_{m}^{l} } \right)^{q} } \right| + \left| {\left( {\gamma_{m}^{u} } \right)^{q} - \left( {\gamma_{m}^{u} } \right)^{q} } \right|} \right)} \right. \\ & \quad + \left. {\frac{1}{{\# g_{i} \# g_{i} }}\mathop \sum \limits_{m = 1}^{{\# g_{i} }} \mathop \sum \limits_{m = 1}^{{\# g_{i} }} \left( {\left| {\left( {\eta_{m}^{l} } \right)^{q} - \left( {\eta_{m}^{l} } \right)^{q} } \right| + \left| {\left( {\eta_{m}^{u} } \right)^{q} - \left( {\eta_{m}^{u} } \right)^{q} } \right|} \right)} \right) \\ \end{aligned} $$

It is obvious that $\left|{\left({\gamma }_{m}^{l}\right)}^{q}-{\left({\gamma }_{m}^{l}\right)}^{q}\right|, \left|{\left({\gamma }_{m}^{u}\right)}^{q}-{\left({\gamma }_{m}^{u}\right)}^{q}\right|, \left|{\left({\eta }_{m}^{l}\right)}^{q}-{\left({\eta }_{m}^{l}\right)}^{q}\right|, \left|{\left({\eta }_{m}^{u}\right)}^{q}-{\left({\eta }_{m}^{u}\right)}^{q}\right|=0$. Therefore, $dis\left({d}_{i},{d}_{i}\right)=0.$

(3) It is obvious that $dis\left({{d}_{j},d}_{i}\right)=\frac{1}{4}\left(\frac{1}{\#{h}_{j}\#{h}_{i}}\sum_{m=1}^{\#{h}_{j}}\sum_{n=1}^{\#{h}_{i}}\left(\left|{\left({\gamma }_{m}^{l}\right)}^{q}-{\left({\gamma }_{n}^{l}\right)}^{q}\right|+\left|{\left({\gamma }_{m}^{u}\right)}^{q}-{\left({\gamma }_{n}^{u}\right)}^{q}\right|\right)\right.$ $+\left.\frac{1}{\#{g}_{j}\#{g}_{i}}\sum_{m=1}^{\#{g}_{j}}\sum_{n=1}^{\#{g}_{i}}\left(\left|{\left({\eta }_{m}^{l}\right)}^{q}-{\left({\eta }_{n}^{l}\right)}^{q}\right|+\left|{\left({\eta }_{m}^{u}\right)}^{q}-{\left({\eta }_{n}^{u}\right)}^{q}\right|\right)\right)=dis\left({{d}_{j},d}_{i}\right)$. The third property has also been proved.

Remark 2

The existing distance measurement proposed by Feng et al. (2021) requires that MDs between input IVq-RDHFEs have the identical number of interval values, as well as the requirement for NMDs. Otherwise, the maximum value or minimum value in the shorter should be repeated several times to align the lengths. However, it is difficult to compare the size of interval values, and length alignment operation may incur distortion or redundancy of the original information. For this, the distance measurement provided in Definition 4 is more concise and useable, and the following Example 1 offers further demonstration.

Example 1

Given two IVq-RDHFEs ${d}_{1}=\left\{\left\{\left[{0.2,0.3}\right]\right\},\left\{[{0.5,0.7}]\right\}\right\}$ and ${d}_{2}=\left\{\left\{\left[{0.2,0.3}\right],[{0.4,0.7}]\right\},\left\{[{0.6,0.7}]\right\}\right\}$, then their distance can be obtained according to Definition 4 (let $q=3$), as follows:

$$ \begin{aligned} dis\left( {d_{1} ,d_{2} } \right) & = \frac{1}{4} \times \left( {\frac{1}{1 \times 2}\left( {\left( {\left| {\left( {0.2} \right)^{3} - \left( {0.2} \right)^{3} } \right| + \left| {\left( {0.3} \right)^{3} - \left( {0.3} \right)^{3} } \right|} \right) + \left( {\left| {\left( {0.2} \right)^{3} - \left( {0.4} \right)^{3} } \right| + \left| {\left( {0.3} \right)^{3} - \left( {0.7} \right)^{3} } \right|} \right)} \right)} \right. \\ & \quad + \left. {\frac{1}{1 \times 1}\left( {\left| {\left( {0.5} \right)^{3} - \left( {0.6} \right)^{3} } \right| + \left| {\left( {0.7} \right)^{3} - \left( {0.7} \right)^{3} } \right|} \right)} \right). \\ \end{aligned} $$

2.2 Extended power average operator

Definition 5

(** EPA: ${R}^{n}\to R$, defined by a parameter $\rho \in \left(-\infty \right.,\left.-\left(n-1\right)\right]\cup \left[0\right.,\left.+\infty \right)$, according to the following formula

$${EPA}^{\left(\rho \right)}\left({a}_{1},{a}_{2},\ldots ,{a}_{n}\right)={\sum }_{i=1}^{n}\frac{\left(\rho +T\left({a}_{i}\right)\right){a}_{i}}{{\sum }_{j=1}^{n}\left(\rho +T\left({a}_{j}\right)\right)},$$

(9)

where $T\left({a}_{i}\right)={\sum }_{j=1;j\ne i}^{n}Sup\left({a}_{i},{a}_{j}\right)$, and $Sup\left({a}_{i},{a}_{j}\right)$ is the support for ${a}_{i}$ from ${a}_{j}$, such that

• (1) $0\le Sup\left({a}_{i},{a}_{j}\right)\le 1$;

• (2) $Sup\left({a}_{i},{a}_{j}\right)=Sup\left({a}_{j},{a}_{i}\right)$;

• (3) $Sup\left(a,b\right)\le Sup\left(c,d\right)$, if $\left|a-b\right|\ge \left|c-d\right|$.

Remark 3

**ong et al. (2019) has discussed several support functions, and $Sup\left({a}_{i},{a}_{j}\right)=1-\left|{a}_{i}-{a}_{j}\right|$ is assigned in this paper for simplicity. Additionally, it is obvious from Eq. (9) that the parameter $\rho$ plays a great role in EPA. Li et al. (2021a) has pointed out that, when $\rho =1$, the EPA reduces to the PA operator; when $\rho \to \infty$ or $T\left({a}_{i}\right)=t\left(0\le t\le n-1\right)$, the EPA reduces to the algebraic average operator.

3 Interval-valued q-rung dual hesitant fuzzy Frank operations

In this section, we present some new operational rules for IVq-RDHFEs based on FTT. We first review the concept of FTT.

3.1 The definition of Frank t-norm and t-conorm

Definition 7

(Frank 1978). The mathematical expression of FTT is presented as follows

$${T}_{F}\left(x,y\right)={{\text{log}}}_{\theta }\left(1+\frac{\left({\theta }^{x}-1\right)\left({\theta }^{y}-1\right)}{\theta -1}\right),$$

(10)

$${T}_{F}^{*}\left(x,y\right)=1-{{\text{log}}}_{\theta }\left(1+\frac{\left({\theta }^{1-x}-1\right)\left({\theta }^{1-y}-1\right)}{\theta -1}\right),$$

(11)

where $0\le x,y\le 1$ and $\theta \in \left(1,+\infty \right)$. Especially, if $\theta \to 1$, ${T}_{F}\left(x,y\right)$ and ${T}_{F}^{*}\left(x,y\right)$ degenerate into the algebraic t-norm and t-conorms.

3.2 New operations for q-rung dual hesitant fuzzy elements

Definition 2 is the original algebraic operations of IVq-RDHFEs. Then, we extend them to obtain more general operations in terms of FTT, as defined in Definition 8.

Definition 8

Let ${d}_{i}=\left\{{h}_{i},{g}_{i}\right\}\left(i={1,2}\right)$ be any two IVq-RDHFEs and $\lambda$ be a positive real number, then

• (1)
  $$ \begin{aligned} d_{1} \oplus d_{2} & = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {r_{2}^{l} ,r_{2}^{u} } \right] \in h_{2} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} ,\left[ {\eta_{2}^{l} ,\eta_{2}^{u} } \right] \in g_{2} }} \\ & \quad \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{l} } \right)^{q} }} - 1} \right)\left( {\theta^{{1 - \left( {r_{2}^{l} } \right)^{q} }} - 1} \right)}}{\theta - 1}} \right)} \right)^{1/q} ,} \right.} \right.} \right. \\ & \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{u} } \right)^{q} }} - 1} \right)\left( {\theta^{{1 - \left( {r_{2}^{u} } \right)^{q} }} - 1} \right)}}{\theta - 1}} \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left. {\left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{l} } \right)^{q} }} - 1} \right)\left( {\theta^{{\left( {\eta_{2}^{l} } \right)^{q} }} - 1} \right)}}{\theta - 1}} \right)} \right)^{1/q} ,\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{u} } \right)^{q} }} - 1} \right)\left( {\theta^{{\left( {\eta_{2}^{u} } \right)^{q} }} - 1} \right)}}{\theta - 1}} \right)} \right)^{1/q} } \right]} \right\}} \right\}; \\ \end{aligned} $$

  (12)
• (2)
  $$ \begin{aligned} d_{1} \otimes d_{2} & = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {r_{2}^{l} ,r_{2}^{u} } \right] \in h_{2} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} ,\left[ {\eta_{2}^{l} ,\eta_{2}^{u} } \right] \in g_{2} }} \\ & \quad \left\{ {\left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {r_{1}^{l} } \right)^{q} }} - 1} \right)\left( {\theta^{{\left( {r_{2}^{l} } \right)^{q} }} - 1} \right)}}{\theta - 1}} \right)} \right)^{1/q} ,\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {r_{1}^{u} } \right)^{q} }} - 1} \right)\left( {\theta^{{\left( {r_{2}^{u} } \right)^{q} }} - 1} \right)}}{\theta - 1}} \right)} \right)^{1/q} } \right]} \right\},} \right. \\ & \quad \left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {\eta_{1}^{l} } \right)^{q} }} - 1} \right)\left( {\theta^{{1 - \left( {\eta_{2}^{l} } \right)^{q} }} - 1} \right)}}{\theta - 1}} \right)} \right)^{1/q} ,} \right.} \right. \\ & \quad \left. {\left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {\eta_{1}^{u} } \right)^{q} }} - 1} \right)\left( {\theta^{{1 - \left( {\eta_{2}^{u} } \right)^{q} }} - 1} \right)}}{\theta - 1}} \right)} \right)^{1/q} } \right]} \right\}} \right\}; \\ \end{aligned} $$

  (13)
• (3)
  $$ \begin{gathered} \lambda d_{1} = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} }} \hfill \\ \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{l} } \right)^{q} }} - 1} \right)^{\lambda } }}{{\left( {\theta - 1} \right)^{\lambda - 1} }}} \right)} \right)^{1/q} } \right.} \right.} \right.,\left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{u} } \right)^{q} }} - 1} \right)^{\lambda } }}{{\left( {\theta - 1} \right)^{\lambda - 1} }}} \right)} \right)^{1/q} } \right]} \right\}, \hfill \\ \left. {\left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{l} } \right)^{q} }} - 1} \right)^{\lambda } }}{{\left( {\theta - 1} \right)^{\lambda - 1} }}} \right)} \right)^{1/q} ,\left. {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{u} } \right)^{q} }} - 1} \right)^{\lambda } }}{{\left( {\theta - 1} \right)^{\lambda - 1} }}} \right)} \right)^{1/q} } \right]} \right.} \right\}} \right\}; \hfill \\ \end{gathered} $$

  (14)
• (4)
  $$ \begin{aligned} d_{1}^{\lambda } = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} }} \left\{ {\left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {r_{1}^{l} } \right)^{q} }} - 1} \right)^{\lambda } }}{{\left( {\theta - 1} \right)^{\lambda - 1} }}} \right)} \right)^{1/q} ,\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {r_{1}^{u} } \right)^{q} }} - 1} \right)^{\lambda } }}{{\left( {\theta - 1} \right)^{\lambda - 1} }}} \right)} \right)^{1/q} } \right]} \right\},} \right. \\ & \quad \left. {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {\eta_{1}^{l} } \right)^{q} }} - 1} \right)^{\lambda } }}{{\left( {\theta - 1} \right)^{\lambda - 1} }}} \right)} \right)^{1/q} } \right.} \right.,\left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {\eta_{1}^{u} } \right)^{q} }} - 1} \right)^{\lambda } }}{{\left( {\theta - 1} \right)^{\lambda - 1} }}} \right)} \right)^{1/q} } \right]} \right\}} \right\}. \\ \end{aligned} $$

  (15)

Example 2

There are three IVq-RDHFEs ${d}_{1}=\left\{\left\{\left[{0.2,0.3}\right]\right\},\left\{[{0.5,0.7}]\right\}\right\}$ and ${d}_{2}=\left\{\left\{\left[{0.2,0.3}\right],[{0.4,0.7}]\right\},\left\{[{0.6,0.7}]\right\}\right\}$ and $d=\left\{\left\{\left[{0.3,0.4}\right],[{0.5,0.6}]\right\},\left\{[{0.4,0.8}]\right\}\right\}$. Let $\lambda =2$ and suppose $\theta =2$ and $q=3$, then

$$ \begin{aligned} & d_{1} \oplus d_{2} = \left\{ {\left\{ {\left[ {0.2517,0.3768} \right],\left[ {0.4153,0.7132} \right]} \right\},\left\{ {\left[ {0.2756,0.4647} \right]} \right\}} \right\}; \\ & d_{1} \otimes d_{2} = \left\{ {\left\{ {\left[ {0.0355,0.0801} \right],\left[ {0.0714,0.1940} \right]} \right\},\left\{ {\left[ {0.6840,0.8367} \right]} \right\}} \right\}; \\ & \lambda d = \left\{ {\left\{ {\left[ {0.3768,0.5000} \right],\left[ {0.6199,0.7336} \right]} \right\},\left\{ {\left[ {0.1437,0.6220} \right]} \right\}} \right\}; \\ & d^{\lambda } = \left\{ {\left\{ {\left[ {0.0801,0.1437} \right],\left[ {0.2275,0.3337} \right]} \right\},\left\{ {\left[ {0.5000,0.9218} \right]} \right\}} \right\}. \\ \end{aligned} $$

Theorem 1

Let ${d}_{i}=\left\{{h}_{i},{g}_{i}\right\}\left(i={1,2}\right)$ be any two IVq-RDHFEs, and ${\lambda }_{i}\left(i={1,2}\right)$ be any three positive real numbers, then

• (1)
  $$ d_{1} \oplus d_{2} = d_{2} \oplus d_{1} ; $$

  (16)
• (2)
  $$ d_{1} \otimes d_{2} = d_{2} \otimes d_{1} ; $$

  (17)
• (3)
  $$ \lambda_{1} \left( {d_{1} \oplus d_{2} } \right) = \lambda_{1} d_{1} \oplus \lambda_{1} d_{2} ; $$

  (18)
• (4)
  $$ \lambda_{1} d_{1} \oplus \lambda_{2} d_{1} = \left( {\lambda_{1} + \lambda_{2} } \right)d_{1} ; $$

  (19)
• (5)
  $$ d_{1}^{{\lambda_{1} }} \otimes d_{1}^{{\lambda_{2} }} = d_{1}^{{\lambda_{1} + \lambda_{2} }} ; $$

  (20)
• (6)
  $$ d_{1}^{{\lambda_{1} }} \otimes d_{2}^{{\lambda_{1} }} = \left( {d_{1} \otimes d_{2} } \right)^{{\lambda_{1} }} . $$

  (21)

Proof

Equations (17) and (18) can be easily derived from Definition 8.

For Eq. (19), according to the Definition 8, it can be deduced that

$$ \begin{aligned} \lambda_{1} \left( {d_{1} \oplus d_{2} } \right) & = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {r_{2}^{l} ,r_{2}^{u} } \right] \in h_{2} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} ,\left[ {\eta_{2}^{l} ,\eta_{2}^{u} } \right] \in g_{2} }} \\ & \quad \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} \left( {\theta^{{1 - \left( {r_{2}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} }}{{\left( {\theta - 1} \right)^{{2\lambda_{1} - 1}} }}} \right)} \right)^{1/q} } \right.} \right.} \right., \\ & \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} \left( {\theta^{{1 - \left( {r_{2}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} }}{{\left( {\theta - 1} \right)^{{2\lambda_{1} - 1}} }}} \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left. {\left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} \left( {\theta^{{\left( {\eta_{2}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} }}{{\left( {\theta - 1} \right)^{{2\lambda_{1} - 1}} }}} \right)} \right)^{1/q} ,\left. {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} \left( {\theta^{{\left( {\eta_{2}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} }}{{\left( {\theta - 1} \right)^{{2\lambda_{1} - 1}} }}} \right)} \right)^{1/q} } \right]} \right.} \right\}} \right\}; \\ \end{aligned} $$

$$ \begin{aligned} \lambda_{1} d_{1} \oplus \lambda_{1} d_{2} & = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {r_{2}^{l} ,r_{2}^{u} } \right] \in h_{2} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} ,\left[ {\eta_{2}^{l} ,\eta_{2}^{u} } \right] \in g_{2} }} \\ & \quad \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} \left( {\theta^{{1 - \left( {r_{2}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} }}{{\left( {\theta - 1} \right)^{{2\lambda_{1} - 1}} }}} \right)} \right)^{1/q} ,} \right.} \right.} \right. \\ & \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} \left( {\theta^{{1 - \left( {r_{2}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} }}{{\left( {\theta - 1} \right)^{{2\lambda_{1} - 1}} }}} \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left. {\left\{ {\left[ {\begin{array}{*{20}c} {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} \left( {\theta^{{\left( {\eta_{2}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} }}{{\left( {\theta - 1} \right)^{2\lambda - 1} }}} \right)} \right)^{\frac{1}{q}} ,} \\ {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} \left( {\theta^{{\left( {\eta_{2}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} }} }}{{\left( {\theta - 1} \right)^{2\lambda - 1} }}} \right)} \right)^{\frac{1}{q}} } \\ \end{array} } \right]} \right\}} \right\}. \\ \end{aligned} $$

Therefore, the formula (19), i.e., ${\lambda }_{1}\left({d}_{1}\oplus {d}_{2}\right)={\lambda }_{1}{d}_{1}\oplus {\lambda }_{1}{d}_{2}$ holds. In a similar vein, the formula (20) can be inferred from Definition 8.

$$ \begin{aligned} \lambda_{1} d_{1} \oplus \lambda_{2} d_{1} & = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} }} \\ & \quad \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} + \lambda_{2} }} }}{{\left( {\theta - 1} \right)^{{\lambda_{1} + \lambda_{2} - 1}} }}} \right)} \right)^{1/q} ,} \right.} \right.} \right.\left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} + \lambda_{2} }} }}{{\left( {\theta - 1} \right)^{{\lambda_{1} + \lambda_{2} - 1}} }}} \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} + \lambda_{2} }} }}{{\left( {\theta - 1} \right)^{{\lambda_{1} + \lambda_{2} - 1}} }}} \right)} \right)^{\frac{1}{q}} ,} \right.} \right.\left. {\left. {\left. {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} + \lambda_{2} }} }}{{\left( {\theta - 1} \right)^{{\lambda_{1} + \lambda_{2} - 1}} }}} \right)} \right)^{\frac{1}{q}} } \right]} \right\}} \right\}; \\ \end{aligned} $$

$$ \begin{aligned} \left( {\lambda_{1} + \lambda_{2} } \right)d_{1} & = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} }} \\ & \quad \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} + \lambda_{2} }} }}{{\left( {\theta - 1} \right)^{{\lambda_{1} + \lambda_{2} - 1}} }}} \right)} \right)^{1/q} } \right.} \right.} \right.,\left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} + \lambda_{2} }} }}{{\left( {\theta - 1} \right)^{{\lambda_{1} + \lambda_{2} - 1}} }}} \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left. {\left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{l} } \right)^{q} }} - 1} \right)^{{\lambda_{1} + \lambda_{2} }} }}{{\left( {\theta - 1} \right)^{{\lambda_{1} + \lambda_{2} - 1}} }}} \right)} \right)^{1/q} ,\left. {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{u} } \right)^{q} }} - 1} \right)^{{\lambda_{1} + \lambda_{2} }} }}{{\left( {\theta - 1} \right)^{{\lambda_{1} + \lambda_{2} - 1}} }}} \right)} \right)^{1/q} } \right]} \right.} \right\}} \right\}. \\ \end{aligned} $$

The Eq. (20) and (21) are validated in the same manner. When the IVq-RDHFEs reduce to the DHFEs, Theorem 1 are also verified by Tang et al. (2018).

Based on the above notions and definitions, we develop several interval-valued q-rung dual hesitant fuzzy aggregate operators in the following.

4 Some extended power average operators for interval valued q-rung dual hesitant fuzzy information

In this section, we propose some novel AOs of IVq-RDHFEs under the new Frank operation law and investigate their important properties.

4.1 The interval-valued q-rung dual hesitant fuzzy Frank extended power average operator

Definition 9

Let ${d}_{i}\left(i={1,2},\ldots ,n\right)$ be a collection of IVq-RDHFEs and the parameter $\rho \in \left(-\infty ,\left.-\left(n-1\right)\right]\cup \left[0,\left.+\infty \right)\right.\right.$, then the interval-valued q-rung dual hesitant fuzzy Frank power average (IVq-RDHFFEPA) operator is defined as

$${IVq-RDHFFEPA}^{\left(\rho \right)}\left({d}_{1},{d}_{2},\ldots ,{d}_{n}\right)={\oplus }_{i=1}^{n}\frac{\left(\rho +T\left({d}_{i}\right)\right){d}_{i}}{{\sum }_{j=1}^{n}\left(\rho +T\left({d}_{j}\right)\right)},$$

(22)

where $T\left({d}_{i}\right)=\sum_{j=1;i\ne j}^{n}Sup\left({d}_{i},{d}_{j}\right)$, and $Sup\left({d}_{i},{d}_{j}\right)$ denotes the support ${d}_{i}$ from ${d}_{j}$, calculating by $Sup\left({d}_{i},{d}_{j}\right)=1-dis\left({d}_{i},{d}_{j}\right)$ satisfying the conditions

• (1) $0\le Sup\left({d}_{i},{d}_{j}\right)\le 1$

• (2) $Sup\left({d}_{i},{d}_{j}\right)=Sup\left({d}_{j},{d}_{i}\right)$

• (3) $Sup\left({d}_{i},{d}_{j}\right)\le Sup\left({d}_{s},{d}_{t}\right)$, if $dis\left({d}_{i},{d}_{j}\right)\ge dis\left({d}_{s},{d}_{t}\right).$

For convenient description, we assume

$${\varpi }_{i}^{\left(\rho \right)}=\frac{\rho +T\left({d}_{i}\right)}{{\sum }_{j=1}^{n}\left(\rho +T\left({d}_{j}\right)\right)},$$

(23)

then Eq. (23) can be rewritten as

$${IVq-RDHFFEPA}^{\left(\rho \right)}\left({d}_{1},{d}_{2},\ldots ,{d}_{n}\right)={\oplus }_{i=1}^{n}{\varpi }_{i}^{\left(\rho \right)}{d}_{i},$$

(24)

where $\varpi ={\left({\varpi }_{1}^{\left(\rho \right)},{\varpi }_{2}^{\left(\rho \right)},\ldots ,{\varpi }_{n}^{\left(\rho \right)}\right)}^{T}$ is the called power weight vector, such that $0\le {\varpi }_{i}^{\left(\rho \right)}\le 1$ and ${\sum }_{i=1}^{n}{\varpi }_{i}^{\left(\rho \right)}=1$.

Theorem 2

Let ${d}_{i}=\left({h}_{i},{g}_{i}\right)\left(i={1,2},\ldots ,n\right)$ be a collection of IVq-RDHFEs and parameter $\rho \in \left(-\infty ,\left.-\left(n-1\right)\right]\cup \left[0,\left.+\infty \right)\right.\right.$, then the aggregated result by the IVq-RDHFFEPA operator is still a IVq-RDHFE, and

$$ \begin{aligned} & IVq - RDHFFEPA^{\left( \rho \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right) = \oplus_{i = 1}^{n} \varpi_{i}^{\left( \rho \right)} d_{i} \\ & \quad = \cup_{{\left[ {r_{i}^{l} ,r_{i}^{u} } \right] \in h_{i} ,\left[ {\eta_{i}^{l} ,\eta_{i}^{u} } \right] \in g_{i} }} \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - \left( {r_{i}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} ,} \right.} \right.} \right. \\ & \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - \left( {r_{i}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\left( {\eta_{i}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right.} \right.,\left. {\left. {\left. {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\left( {\eta_{i}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right]} \right\}} \right\}, \\ \end{aligned} $$

(25)

where ${\varpi }_{i}^{\left(\rho \right)}$ is the power weight pertaining to the argument ${d}_{i}$ and $0\le {\varpi }_{i}^{\left(\rho \right)}\le 1$ and ${\sum }_{i=1}^{n}{\varpi }_{i}^{\left(\rho \right)}=1$.

Proof

We will deduce the Eq. (22) by utilizing the Frank operation laws of IVq-RDHFEs given in Definition 8.

When $n=2$, we know ${\varpi }_{1}^{\left(\rho \right)}+{\varpi }_{2}^{\left(\rho \right)}=1$,

$$ \begin{aligned} \varpi_{1}^{\left( \rho \right)} d_{1} & = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} }} \\ & \quad \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} }} }}{{\left( {\theta - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} - 1}} }}} \right)} \right)^{1/q} } \right.} \right.} \right.,\left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{1}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} }} }}{{\left( {\theta - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} - 1}} }}} \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left. {\left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} }} }}{{\left( {\theta - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} - 1}} }}} \right)} \right)^{1/q} ,\left. {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{1}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} }} }}{{\left( {\theta - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} - 1}} }}} \right)} \right)^{1/q} } \right]} \right.} \right\}} \right\}; \\ \end{aligned} $$

and

$$ \begin{aligned} \varpi_{2}^{\left( \rho \right)} d_{2} & = \cup_{{\left[ {r_{2}^{l} ,r_{2}^{u} } \right] \in h_{2} ,\left[ {\eta_{2}^{l} ,\eta_{2}^{u} } \right] \in g_{2} }} \\ & \quad \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{2}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} }} }}{{\left( {\theta - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} - 1}} }}} \right)} \right)^{1/q} } \right.} \right.} \right.,\left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \frac{{\left( {\theta^{{1 - \left( {r_{2}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} }} }}{{\left( {\theta - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} - 1}} }}} \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left. {\left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{2}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} }} }}{{\left( {\theta - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} - 1}} }}} \right)} \right)^{1/q} ,\left. {\left( {\log_{\theta } \left( {1 + \frac{{\left( {\theta^{{\left( {\eta_{2}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} }} }}{{\left( {\theta - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} - 1}} }}} \right)} \right)^{1/q} } \right]} \right.} \right\}} \right\}; \\ \end{aligned} $$

Then,

$$ \begin{aligned} & IVq - RDHFFEPA^{\left( \rho \right)} \left( {d_{1} ,d_{2} } \right) = \varpi_{1}^{\left( \rho \right)} d_{1} \oplus \varpi_{2}^{\left( \rho \right)} d_{2} = \cup_{{\left[ {r_{1}^{l} ,r_{1}^{u} } \right] \in h_{1} ,\left[ {r_{2}^{l} ,r_{2}^{u} } \right] \in h_{2} ,\left[ {\eta_{1}^{l} ,\eta_{1}^{u} } \right] \in g_{1} ,\left[ {\eta_{2}^{l} ,\eta_{2}^{u} } \right] \in g_{2} }} \\ & \quad \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \left( {\theta^{{1 - \left( {r_{1}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} }} \left( {\theta^{{1 - \left( {r_{2}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} }} } \right)} \right)^{1/q} ,} \right.} \right.} \right. \\ & \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \left( {\theta^{{1 - \left( {r_{1}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} }} \left( {\theta^{{1 - \left( {r_{2}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \left( {\theta^{{\left( {\eta_{1}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} }} \left( {\theta^{{\left( {\eta_{2}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right.} \right., \\ & \quad \left. {\left. {\left. {\left( {\log_{\theta } \left( {1 + \left( {\theta^{{\left( {\eta_{1}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{1}^{\left( \rho \right)} }} \left( {\theta^{{\left( {\eta_{2}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{2}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right]} \right\}} \right\}, \\ \end{aligned} $$

which means that Eq. (26) holds for $n=2$.

Suppose that Eq. (26) is true for $n=k$, where ${\sum }_{i=1}^{k}{\varpi }_{i}^{\left(\rho \right)}=1$, then we can get

$$ \begin{aligned} & IVq - RDHFFEPA^{\left( \rho \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{k} } \right) = \oplus_{i = 1}^{k} \varpi_{i}^{\left( \rho \right)} d_{i} \\ & \quad = \cup_{{\left[ {r_{i}^{l} ,r_{i}^{u} } \right] \in h_{i} ,\left[ {\eta_{i}^{l} ,\eta_{i}^{u} } \right] \in g_{i} }} \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{k} \left( {\theta^{{1 - \left( {r_{i}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} ,} \right.} \right.} \right. \\ & \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{k} \left( {\theta^{{1 - \left( {r_{i}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{k} \left( {\theta^{{\left( {\eta_{i}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} ,\left. {\left. {\left. {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{k} \left( {\theta^{{\left( {\eta_{i}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right]} \right\}} \right\}} \right.} \right.; \\ \end{aligned} $$

Now let $n=k+1$, and we have

$$ \begin{aligned} & IVq - RDHFFEPA^{\left( \rho \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{k + 1} } \right) = \oplus_{i = 1}^{k} \varpi_{i}^{\left( \rho \right)} d_{i} \oplus \varpi_{k + 1}^{\left( \rho \right)} d_{k + 1} \\ & \quad = \cup_{{\left[ {r_{i}^{l} ,r_{i}^{u} } \right] \in h_{i} ,\left[ {\eta_{i}^{l} ,\eta_{i}^{u} } \right] \in g_{i} }} \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{k + 1} \left( {\theta^{{1 - \left( {r_{i}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} ,} \right.} \right.} \right. \\ \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{k + 1} \left( {\theta^{{1 - \left( {r_{i}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left. {\left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{k + 1} \left( {\theta^{{\left( {\eta_{i}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} ,\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{k + 1} \left( {\theta^{{\left( {\eta_{i}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right]} \right\}} \right\}, \\ \end{aligned} $$

which shows that the Eq. (26) also holds for $n=k+1$. Consequently, it is holds for $n$.

Example 3

There are two IVq-RDHFEs ${d}_{1}=\left\{\left\{\left[{0.2,0.3}\right]\right\},\left\{\left[{0.5,0.7}\right]\right\}\right\}$ and ${d}_{2}=\left\{\left\{\left[{0.2,0.3}\right],\left[{0.4,0.7}\right]\right\},\left\{\left[{0.6,0.7}\right]\right\}\right\}$, then we utilize IVq-RDHFFEPA to get their aggregated result (suppose $\rho =2$, $\theta =2$ and $q=3$), as follows:

• Firstly, $Sup\left({d}_{1},{d}_{2}\right)=Sup\left({d}_{2},{d}_{1}\right)=1-dis\left({d}_{1},{d}_{2}\right)=0.9307$.

• Then, $T\left({d}_{1}\right)=T\left({d}_{2}\right)=0.9307$. So, it is easily to get ${\varpi }_{1}^{\left(2\right)}={\varpi }_{2}^{\left(2\right)}=0.5$.

• Further, according to Eq. (26), we can get

  $$ \begin{aligned} & IVq - RDHFFEPA^{\left( 2 \right)} \left( {d_{1} ,d_{2} } \right) = \varpi_{1}^{\left( 2 \right)} d_{1} \oplus \varpi_{2}^{\left( 2 \right)} d_{2} \\ & \quad = \left\{ {\left\{ {\left[ {0.2,0.3} \right],\left[ {0.3311,0.5814} \right]} \right\},\left\{ {0.5481,0.7000} \right\}} \right\}. \\ \end{aligned} $$

• It is easy to show that the $IVq-RDHFFEPA$ operator has the property of commutativity.

Theorem 3

(Commutativity). Let $\left({d}_{1},{d}_{2},\ldots ,{d}_{n}\right)$ be a collection of IVq-RDHFEs, and parameter $\rho \in \left(-\infty ,\left.-\left(n-1\right)\right]\cup \left[0,\left.+\infty \right)\right.\right.$, if $\left({\overline{d} }_{1},{\overline{d} }_{2},\ldots ,{\overline{d} }_{n}\right)$ is any premutation of $\left({d}_{1},{d}_{2},\ldots ,{d}_{n}\right)$, then

$${IVq-RDHFFEPA}^{\left(\rho \right)}\left({d}_{1},{d}_{2},\ldots ,{d}_{n}\right)={IVq-RDHFFEPA}^{\left(\rho \right)}\left({\overline{d} }_{1},{\overline{d} }_{2},\ldots ,{\overline{d} }_{n}\right).$$

(26)

Special case 1

When $\rho =1$, the $IVq-RDHFFEPA$ reduces to the interval-valued q-rung dual hesitant fuzzy Frank power average ($IVq-RDHFFPA)$ operator, i.e.,

$$ \begin{aligned} & IVq - RDHFFEPA^{\left( 1 \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right) = \oplus_{i = 1}^{n} \varpi_{i}^{\left( 1 \right)} d_{i} \\ & \quad = \cup_{{\left[ {r_{i}^{l} ,r_{i}^{u} } \right] \in h_{i} ,\left[ {\eta_{i}^{l} ,\eta_{i}^{u} } \right] \in g_{i} }} \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - \left( {r_{i}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( 1 \right)} }} } \right)} \right)^{1/q} ,} \right.} \right.} \right. \\ & \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - \left( {r_{i}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( 1 \right)} }} } \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\left( {\eta_{i}^{l} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( 1 \right)} }} } \right)} \right)^{1/q} } \right.} \right.,\left. {\left. {\left. {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\left( {\eta_{i}^{u} } \right)^{q} }} - 1} \right)^{{\varpi_{i}^{\left( 1 \right)} }} } \right)} \right)^{1/q} } \right]} \right\}} \right\} \\ & \quad = IVq - RDHFFPA\left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right). \\ \end{aligned} $$

(27)

Special case 2

When $\rho \to \infty$ or $T\left({a}_{i}\right)=t\left(0\le t\le n-1\right)$, the $IVq-RDHFFEPA$ reduces to the interval-valued q-rung dual hesitant fuzzy Frank average ($IVq-RDHFFA)$ operator, i.e.,

$$ \begin{aligned} & IVq - RDHFFEPA^{\left( \infty \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right) = \frac{1}{n} \oplus_{i = 1}^{n} d_{i} \\ & \quad = \cup_{{\left[ {r_{i}^{l} ,r_{i}^{u} } \right] \in h_{i} ,\left[ {\eta_{i}^{l} ,\eta_{i}^{u} } \right] \in g_{i} }} \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - \left( {r_{i}^{l} } \right)^{q} }} - 1} \right)^{\frac{1}{n}} } \right)} \right)^{1/q} ,} \right.} \right.} \right. \\ & \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - \left( {r_{i}^{u} } \right)^{q} }} - 1} \right)^{\frac{1}{n}} } \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\left( {\eta_{i}^{l} } \right)^{q} }} - 1} \right)^{\frac{1}{n}} } \right)} \right)^{1/q} } \right.} \right.,\left. {\left. {\left. {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\left( {\eta_{i}^{u} } \right)^{q} }} - 1} \right)^{\frac{1}{n}} } \right)} \right)^{1/q} } \right]} \right\}} \right\} \\ & \quad = IVq - RDHFFA^{\left( \infty \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right). \\ \end{aligned} $$

(28)

Special case 3

When $q=1$, the $IVq-RDHFFEPA$ reduces to the interval-valued dual hesitant fuzzy Frank extended power average (IVDHFFEPA) operator, i.e.,

$$ \begin{aligned} & IVq - RDHFFEPA^{\left( \rho \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right) \\ & \quad = \cup_{{\left[ {r_{i}^{l} ,r_{i}^{u} } \right] \in h_{i} ,\left[ {\eta_{i}^{l} ,\eta_{i}^{u} } \right] \in g_{i} }} \left\{ {\left\{ {\left[ {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - r_{i}^{l} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right),} \right.} \right.} \right. \\ & \quad \left. {\left. {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - r_{i}^{u} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right]} \right\}, \\ & \quad \left\{ {\left[ {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\eta_{i}^{l} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right.} \right.,\left. {\left. {\left. {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\eta_{i}^{u} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right]} \right\}} \right\} \\ & \quad = {\text{IVDHFFEPA}}^{\left( \rho \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right). \\ \end{aligned} $$

(29)

Special case 4

When $q=2$, the $IVq-RDHFFEPA$ reduces to the interval-valued Pythagorean dual hesitant fuzzy Frank extended power average (IVPDHFFEPA) operator, i.e.,

$$ \begin{aligned} & IVq - RDHFFEPA^{\left( \rho \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right) \\ & \quad = \cup_{{\left[ {r_{i}^{l} ,r_{i}^{u} } \right] \in h_{i} ,\left[ {\eta_{i}^{l} ,\eta_{i}^{u} } \right] \in g_{i} }} \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - \left( {r_{i}^{l} } \right)^{2} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/2} ,} \right.} \right.} \right. \\ & \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - \left( {r_{i}^{u} } \right)^{2} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/2} } \right]} \right\}, \\ & \quad \left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\left( {\eta_{i}^{l} } \right)^{2} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/2} } \right.} \right.,\left. {\left. {\left. {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\left( {\eta_{i}^{u} } \right)^{2} }} - 1} \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/2} } \right]} \right\}} \right\} \\ & \quad = {\text{IVPDHFFEPA}}^{\left( \rho \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right). \\ \end{aligned} $$

(30)

Special case 5

When $\theta \to 1$, the $IVq-RDHFFEPA$ reduces to the interval-valued q-rung dual hesitant fuzzy extended power average ($IVq-RDHFEPA)$ operator, i.e.,

$$ \begin{aligned} & IVq - RDHFFEPA^{\left( \rho \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right) = \oplus_{i = 1}^{n} \varpi_{i}^{\left( \rho \right)} d_{i} \\ & \quad = \cup_{{\left[ {r_{i}^{l} ,r_{i}^{u} } \right] \in h_{i} ,\left[ {\eta_{i}^{l} ,\eta_{i}^{u} } \right] \in g_{i} }} \left\{ {\left\{ {\left[ {\left( {1 - \mathop \prod \limits_{i = 1}^{n} \left( {1 - \left( {r_{i}^{l} } \right)^{q} } \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)^{1/q} ,\left. {\left. {\left( {1 - \mathop \prod \limits_{i = 1}^{n} \left( {1 - \left( {r_{i}^{u} } \right)^{q} } \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right)^{1/q} } \right]} \right\}} \right.} \right.} \right., \\ & \quad \left\{ {\left[ {\mathop \prod \limits_{i = 1}^{n} \left( {\eta_{i}^{l} } \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right.} \right.,\left. {\left. {\left. {\mathop \prod \limits_{i = 1}^{n} \left( {\eta_{i}^{u} } \right)^{{\varpi_{i}^{\left( \rho \right)} }} } \right]} \right\}} \right\} = IVq - RDHFEPA\left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right). \\ \end{aligned} $$

(31)

In the case of $\theta \to 1$, the Frank operations of IVq-RDHFEs are transformed into the algebraic operations as described in Definition 2.

4.2 The interval-valued q-rung dual hesitant fuzzy Frank weighted extended power average operator

Definition 12

Let ${d}_{i}\left(i={1,2},\ldots ,n\right)$ be a collection of IVq-RDHFEs, and the parameter $\rho \in \left(-\infty ,\left.-\left(n-1\right)\right]\cup \left[0,\left.+\infty \right)\right.\right.$. Let $w={\left({w}_{1},{w}_{2},\ldots ,{w}_{n}\right)}^{T}$ be the corresponding weight vector of ${d}_{i}\left(i={1,2},\ldots ,n\right)$, such that $0\le {w}_{i}\le 1$ and ${\sum }_{i=1}^{n}{w}_{i}=1$. The interval-valued q-rung dual hesitant fuzzy Frank weighted extended power average (IVq-RDHFFWEPA) operator is expressed as

$$IVq-RDHFFWEPA\left({d}_{1},{d}_{2},\ldots ,{d}_{n}\right)=\frac{{\oplus }_{i=1}^{n}{w}_{i}\left(\rho +T\left({d}_{i}\right)\right){d}_{i}}{{\sum }_{j=1}^{n}{w}_{j}\left(\rho +T\left({d}_{j}\right)\right)},$$

(32)

where $T\left({d}_{i}\right)=\sum_{j=1;i\ne j}^{n}Sup\left({d}_{i},{d}_{j}\right)$, $Sup\left({d}_{i},{d}_{j}\right)$ denotes the support for ${d}_{i}$ from ${d}_{j}$, satisfying the properties presented in Definition 9. Similarly, let

$${{\tau }_{i}}^{\left(\rho \right)}=\frac{{w}_{i}\left(\rho +T\left({d}_{i}\right)\right)}{{\sum }_{j=1}^{n}{w}_{j}\left(\rho +T\left({d}_{j}\right)\right)},$$

(33)

then we can rewrite Eq. (36) as

$${IVq-RDHFFWEPA}^{\left(\rho \right)}\left({d}_{1},{d}_{2},\ldots ,{d}_{n}\right)={\oplus }_{i=1}^{n}{{\tau }_{i}}^{\left(\rho \right)}{d}_{i},$$

(34)

where $\tau ={\left({{\tau }_{1}}^{\left(\rho \right)},{{\tau }_{2}}^{\left(\rho \right)},\ldots ,{{\tau }_{n}}^{\left(\rho \right)}\right)}^{T}$ is called the power weight vector, such that $0\le {{\tau }_{i}}^{\left(\rho \right)}\le 1$ and ${\sum }_{i=1}^{n}{{\tau }_{i}}^{\left(\rho \right)}=1$.

Theorem 5

Let ${d}_{i}\left(i={1,2},\ldots ,n\right)$ be a collection of IVq-RDHFEs and the parameter $\rho \in \left(-\infty ,\left.-\left(n-1\right)\right]\cup \left[0,\left.+\infty \right)\right.\right.$. Then the aggregated value by the $IVq-RDHFFWEPA$ operator is still an IVq-RDHFE, and

$$ \begin{aligned} & IVq - RDHFFWEPA^{\left( \rho \right)} \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right) = \oplus_{i = 1}^{n} \tau_{i}^{\left( \rho \right)} d_{i} \\ & = \cup_{{\left[ {r_{i}^{l} ,r_{i}^{u} } \right] \in h_{i} ,\left[ {\eta_{i}^{l} ,\eta_{i}^{u} } \right] \in g_{i} }} \left\{ {\left\{ {\left[ {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - \left( {r_{i}^{l} } \right)^{q} }} - 1} \right)^{{\tau_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} ,} \right.} \right.} \right. \\ & \quad \left. {\left. {\left( {1 - \log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{1 - \left( {r_{i}^{u} } \right)^{q} }} - 1} \right)^{{\tau_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right]} \right\}, \\ & \quad \left\{ {\left[ {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\left( {\eta_{i}^{l} } \right)^{q} }} - 1} \right)^{{\tau_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right.} \right.,\left. {\left. {\left. {\left( {\log_{\theta } \left( {1 + \mathop \prod \limits_{i = 1}^{n} \left( {\theta^{{\left( {\eta_{i}^{u} } \right)^{q} }} - 1} \right)^{{\tau_{i}^{\left( \rho \right)} }} } \right)} \right)^{1/q} } \right]} \right\}} \right\}. \\ \end{aligned} $$

(35)

where ${{\tau }_{i}}^{\left(\rho \right)}$ is the power weight with respect to the input argument ${d}_{i}$ and $0\le {{\tau }_{i}}^{\left(\rho \right)}\le 1$ and ${\sum }_{i=1}^{n}{{\tau }_{i}}^{\left(\rho \right)}=1$.

Theorem 5 can be proved in the similar manner with Theorem 2.

It is obvious that the $IVq-RDHFFWEPA$ also has the property of commutativity.

5 Methods for determining DM and attribute weights

In MAGDM, the weight information of DMs and attributes plays a significant role in the results (Li et al. 2022; Rabiee et al. 2021). In fact, many times it is difficult to directly specify the DMs and attributes weights in advance. Therefore, it is necessary and important to determine appropriate methods for calculating weight values.

5.1 The method to determine the DM weights based on BWM

It is noted that in most MAGDM studies, the initial DM weights are usually set by experience or is equal by default (Zhang et al. 2020). To tackle possible situations more comprehensively and reasonably, here, we add an organizer (team) to provide preferences for DMs involved in decision-making. Best–worst method (BWM) is a novel technology to determine the weights of criteria proposed by Rezaei in 2015. Because of its excellent performance, BWM is quickly accepted and adopted in many decision-making problems (Amiri et al. 2020). In view of the potential of BWM to deal with DM weights in some cases (Li et al. 2021b), we expand BWM in this study to obtain DM weights, shown as Fig. 1. The main steps are as follows:

Fig. 1

DM comparisons based on BWM

Step 1. Establish a decision-making group. The DMs in the group may be experts with varied knowledge backgrounds in the field, and they are denoted as $\left\{{D}_{1},{D}_{2},\ldots ,{D}_{k}\right\}$.

Step 2. Designate the most authoritative DM and the least authoritative DM. In align with most studies adopting BWM (Amiri et al. 2020), it is suggested to identify the best and worst according to the professional judgment of the organizer.

Step 3. Determine the priority of the most authoritative DM over each of others using a number between 1 and 9 and denote it as ${P}_{Bz}=\left({p}_{M1},{p}_{M2},\ldots ,{p}_{Mk}\right)$. Clearly, ${p}_{MM}=1$.

Step 4. Determine the priority of each DM over the least authoritative DM using a number between 1 and 9 and denote it as ${P}_{zL}=\left({p}_{1L},{p}_{2L},\ldots ,{p}_{kL}\right)$. Clearly, ${p}_{LL}=1$.

Step 5. Calculate the optimal weights of the DMs according to the optimal weight model, as shown in Eq. (36):

$$ \begin{aligned} & \min \zeta \\ & \quad \begin{array}{*{20}c} {s.t.} & {\left\{ {\left\{ {\begin{array}{*{20}l} {\left| {\frac{{\gamma_{M} }}{{\gamma_{z} }} - p_{Mz} } \right| \le \zeta , } \hfill & {for all z} \hfill \\ {\left| {\frac{{\gamma_{z} }}{{\gamma_{L} }} - p_{zL} } \right| \le \zeta ,} \hfill & {for all z} \hfill \\ {\mathop \sum \limits_{z} \gamma_{z} = 1\,\gamma_{z} \ge 0, } \hfill & {for all z} \hfill \\ \end{array} } \right.} \right.} \\ \end{array} \\ \end{aligned} $$

(36)

Solving problem (36), the optimal solution $\left({\gamma }_{1}^{*},{\gamma }_{2}^{*},\ldots ,{\gamma }_{k}^{*}\right)$ and ${\zeta }^{*}$ are obtained, where ${\gamma }_{z}^{*}$ represents the weight value of DM ${D}_{z}$.

Step 6. Calculate the consistency ratio (CR) according to Eq. (37):

$${CR}_{P}=\frac{{\zeta }^{*}}{Consistency Index}.$$

(37)

It is clear that the higher the CR, the less reliable the comparisons. Remarkably, $Consistency Index$ (CI) can refer to the Table 1 in Rezaei (2015).

Table 1 The pair-wise comparison vector for the most authoritative DM

Example 4

A company aims to select a new area to launch their products. Three DMs ${D}_{1},$ ${D}_{2},$ and ${D}_{3}$ are invited to participate in the evaluation (Step 1). The organizer team identifies ${D}_{3}$ and ${D}_{1}$ as the most authoritative and the least respectively (Step 2). Then, the comparison vectors are provided in Table 1 and 2 (Steps 3 and 4).

Table 2 The pair-wise comparison vector for the least authoritative DM

Considering the data in Table 1 and 2, the model is built for solving this problem according to Eq. (36) (Step 5). As a result, we have ${\gamma }_{1}^{*}=0.1429$, ${\gamma }_{2}^{*}=0.2857$, ${\gamma }_{3}^{*}=0.5714$, and ${\zeta }^{*}=0.00$. For the CR, as ${\gamma }_{ML}={\gamma }_{31}=4$, CI for the problem is 1.63, and thus, CR is $\frac{0.00}{1.63}=0$, which implies a perfect consistency (Step 6).

Remark 4

Rezaei (2015) suggested that CR should not be greater than 0.1, meanwhile some scholars suggested the threshold can be adjusted properly according to the practical problems (Amiri et al. 2020). When the result of CR is not desirable, organizers can modify their comparison values to improve consistency, and some modification algorithms have been investigated (Zhou et al. 2022).

5.2 The method to determine the attribute weights based on entropy

When attribute weights are completely unknown in advance, entropy measure is a widely used tool to determine the weight values of fuzzy value information. Motivated by the entropy measures for DHFSs (Zhang 2020), q-RDHFSs (Kou et al. 2021), we develop an entropy measure for IVq-RDHFSs in this part, and a weight calculation of attributes method based on it. The axiom is as follows.

Definition 13

Let ${d}_{i}=\left({h}_{i},{g}_{i}\right)\left(i={1,2}\right)$ be any two IVq-RDHFEs. A real function ${\rm E}$: ${\text{IVq}}-{\text{RDHFE}}\to \left[{0,1}\right]$ is the entropy on IVq-RDHFE, if and only if the following properties are satisfied.

• (1) ${\rm E}\left({d}_{1}\right)=0$, if and only if ${d}_{1}=\left\{\left\{0\right\},\left\{1\right\}\right\}$ or ${d}_{1}=\left\{\left\{1\right\},\left\{0\right\}\right\}$;

• (2) ${\rm E}\left({d}_{1}\right)=1$, if and only if #${h}_{1}=\#{g}_{1}$ and ${\left[{\gamma }_{1}^{l},{\gamma }_{1}^{u}\right]}_{\sigma (i)}={\left[{\eta }_{1}^{l},{\eta }_{1}^{u}\right]}_{\sigma (i)}\left(i={1,2},\ldots ,m\right)$, where $\left[{\gamma }_{1}^{l},{\gamma }_{1}^{u}\right]$ and $\left[{\eta }_{1}^{l},{\eta }_{1}^{u}\right]$ are the ith smallest interval values of ${h}_{1}$ and ${g}_{1}$, respectively;

• (3) ${\rm E}\left({d}_{1}\right)\le {\rm E}\left({d}_{2}\right)$ if $dis\left({d}_{1},{d}_{1}^{C}\right)\ge dis\left({d}_{2},{d}_{2}^{C}\right)$;

• (4) ${\rm E}\left({d}_{1}\right)={\rm E}\left({d}_{1}^{C}\right)$.

Then, the entropy measure of any IVq-RDHFE $d$ is defined as:

$${\rm E}\left(d\right)=1-dis\left(d,{d}^{C}\right)$$

(38)

where $dis\left(d,{d}^{C}\right)$ is the distance between $d$ and its complement ${d}^{C}$.

Example 5

If $d=\left\{\left\{\left[{0.2,0.3}\right],[{0.4,0.7}]\right\},\left\{[{0.6,0.7}]\right\}\right\}$, then ${d}^{C}=\left\{\left\{[{0.6,0.7}]\right\},\left\{\left[{0.2,0.3}\right],[{0.4,0.7}]\right\}\right\}$. Supposing $q=3$, the distance is calculated according to Definition 4. Then, we can get

$$ \begin{aligned} & {\rm E}\left( d \right) = 1 - dis\left( {d,d^{C} } \right) = 1 - \frac{1}{4} \times \left( {\frac{1}{2 \times 1}\left( {\left( {\left| {0.2^{3} - 0.6^{3} } \right| + \left| {0.3^{3} - 0.7^{3} } \right|} \right) + \left( {\left| {0.4^{3} - 0.6^{3} } \right|} \right.} \right.} \right. \\ & \quad + \left. {\left. {\left. {\left| {0.7^{3} - 0.7^{3} } \right|} \right)} \right) + \frac{1}{2 \times 1}\left( {\left( {\left| {0.6^{3} - 0.2^{3} } \right| + \left| {0.7^{3} - 0.3^{3} } \right|} \right) + \left( {\left| {0.6^{3} - 0.4^{3} } \right| + } \right.} \right.\left. {\left. {\left| {0.7^{3} - 0.7^{3} } \right|} \right)} \right)} \right) \\ & \quad = 1 - 0.{169} = 0.{831}. \\ \end{aligned} $$

Based on the entropy measure in Eq. (38), we propose a novel method to determine the weights of aggregated IVq-RDHFEs. Let ${d}_{i}\left(i={1,2},\ldots ,n\right)$ be a collection of IVq-RDHFEs, then weights of ${d}_{i}$ is as follows:

$${w}_{i}=\frac{1-{\rm E}\left({d}_{i}\right)}{n-{\sum }_{i=1}^{n}{\rm E}\left({d}_{i}\right)}.$$

(39)

6 A novel MAGDM method under interval-valued q-rung dual hesitant fuzzy sets

In this section, based on the proposed AOs, we introduce a new MAGDM method under IVq-RDHFSs. Firstly, the structure of MAGDM under IVq-RDHFSs is introduced. The framework of the new method is shown in Fig. 2, and it will be introduced in detail.

Fig. 2

The framework of the proposed method

6.1 The structure of interval-valued q-rung dual hesitant fuzzy MAGDM problem

We specify that $A=\left\{{A}_{1},{A}_{2},\ldots ,{A}_{m}\right\}$ represents a set of alternatives, $C=\left\{{C}_{1},{C}_{2},\ldots ,{C}_{n}\right\}$ is a common attribute collection pertaining to all alternatives, and $D=\left\{{D}_{1},{D}_{2},\ldots ,{D}_{k}\right\}$ stands for a group of DMs. DMs evaluate alternative ${A}_{i}\left(i={1,2},\ldots ,m\right)$ pertinent to each attribute ${C}_{j}\left(j={1,2},\ldots ,n\right)$, and each evaluation value is expressed in the form of IVq-RDHFE, i.e., ${d}_{ij}^{l}=\left\{{h}_{ij}^{l},{g}_{ij}^{l}\right\}$. Hence, for each DM, an individual interval-valued q-rung dual hesitant fuzzy decision matrix denoted as ${M}^{l}={\left({d}_{ij}^{l}\right)}_{m\times n}$ can be derived. Let $\gamma =\left({\gamma }_{1},{\gamma }_{2},\ldots ,{\gamma }_{k}\right)$ the weight vector of DMs, such that ${\gamma }_{l}\in \left[{0,1}\right]$ and $\sum_{l}^{k}{\gamma }_{l}=1$. And let $w={\left({w}_{1},{w}_{2},\ldots ,{w}_{n}\right)}^{T}$ be the weight vector of attributes, satisfying ${w}_{j}\in \left[{0,1}\right]$ and $\sum_{j}^{n}{w}_{l}=1$. Next, a new method is introduced to solve the interval-valued q-rung dual hesitant fuzzy MAGDM problems by utilizing IVq-RDHFFWEPA operators.

6.2 The main steps of the new MAGDM method

• Step 1. Collect DMs’ opinions and specify algorithm parameters.

  • Step 1.1 Construct decision matrices ${M}^{l}={\left({d}_{ij}^{l}\right)}_{m\times n}$ based on the evaluation values provided by $k$ DMs.

  • Step 1.2 Normalize the original decision matrices. There are two kinds of attributes in the real decision-making problems, namely benefit attributes (${I}_{1}$) and cost attributes (${I}_{2}$), so the original decision matrices should be normalized by the formula:

    $${d}_{ij}^{l}=\left\{\begin{array}{c}\left\{{h}_{ij}^{l},{g}_{ij}^{l}\right\},{C}_{j}\in {I}_{1}\\ \left\{{g}_{ij}^{l},{h}_{ij}^{l}\right\},{C}_{j}\in {I}_{2}\end{array}\right..$$

    (40)

  • Step 1.3 Specify appropriate values of parameters $q$, $\theta$, $\rho$, complying to actual needs.

• Step 2. If attribute weights are not assigned in advance, utilize the entropy-based method in Sect. 5.2 to calculate them.

• Step 3. If DMs weights are not assigned in advance, invite an organizer, and then use the BWM-based method in Sect. 5.1 to obtain them.

• Step 4. If there are multiple decision matrices, i.e., $k>1$, utilize the proposed IVq-RDHFFWEPA operator to aggregate the evaluations of multiple DMs to a collective matrix.

  • Step 4.1 Compute $Sup\left({d}_{ij}^{l},{d}_{ij}^{z}\right)$, i.e., the support for ${d}_{ij}^{l}$ from ${d}_{ij}^{z}\left(l,z={1,2},\ldots ,k;l\ne z\right)$, according to the following formula:

    $$Sup\left({d}_{ij}^{l},{d}_{ij}^{z}\right)=1-d\left({d}_{ij}^{l},{d}_{ij}^{z}\right),$$

    (41)

    where $d\left({d}_{ij}^{l},{d}_{ij}^{z}\right)$ is the distance between ${d}_{ij}^{l}$ and ${d}_{ij}^{z}$.

  • Step 4.2 Compute the overall supports $T\left({d}_{ij}^{l}\right)$ for ${d}_{ij}^{l}\left(l={1,2},\ldots ,k\right)$ by

    $$T\left({d}_{ij}^{l}\right)=\sum_{z=1,z\ne l}^{k}Sup\left({d}_{ij}^{l},{d}_{ij}^{z}\right)$$

    (42)

  • Step 4.3 Compute the power weight of the evaluation value ${d}_{ij}^{l}$ given by DM ${D}_{l}$ according to the following formula:

    $${{\tau }_{l}}^{\left(\rho \right)}=\frac{{\gamma }_{l}\left(\rho +T\left({d}_{ij}^{l}\right)\right)}{{\sum }_{l=1}^{k}{\gamma }_{l}\left(\rho +T\left({d}_{ij}^{l}\right)\right)}.$$

    (43)

  • Step 4.4 Use the IVq-RDHFFWEPA operator to aggregate multiple individual decision matrices ${M}^{l}={\left({d}_{ij}^{l}\right)}_{m\times n}\left(l={1,2},\dots k\right)$ to a comprehensive decision matrix $M={\left({d}_{ij}\right)}_{m\times n}$. Here,

    $${d}_{ij}={IVq-RDHFFWEPA}^{\left(\rho \right)}\left({d}_{ij}^{1},{d}_{ij}^{2},\ldots ,{d}_{ij}^{k}\right){=\oplus }_{l=1}^{k}{{\tau }_{l}}^{\left(\rho \right)}{d}_{ij}^{l}.$$

    (44)

• Step 5. Apply the proposed IVq-RDHFFWEPA operator to fuse the attribute values of each alternative into one.

  • Step 5.1 Compute the support $Sup\left({d}_{ij},{d}_{is}\right)\left(i={1,2},\ldots ,m;j,s={1,2},\ldots ,n;j\ne s\right)$ by

    $$Sup\left({d}_{ij},{d}_{is}\right)=1-d\left({d}_{ij},{d}_{is}\right),$$

    (45)

    where $d\left({d}_{ij},{d}_{is}\right)$ is the distance between ${d}_{ij}$ and ${d}_{is}$.

  • Step 5.2 Compute the overall supports $T\left({d}_{ij}\right)$ for ${d}_{ij}$ by the following formula

    $$T\left({d}_{ij}\right)={\sum }_{s=1;s\ne j}^{n}Sup\left({d}_{ij},{d}_{is}\right).$$

    (46)

  • Step 5.3 Compute the power weight of ${d}_{ij}$ by

    $${{\delta }_{ij}}^{\left(\rho \right)}=\frac{{w}_{i}\left(\rho +T\left({d}_{ij}\right)\right)}{{\sum }_{j=1}^{n}{w}_{j}\left(\rho +T\left({d}_{ij}\right)\right)}.$$

    (47)

  • Step 5.4 Utilize the IVq-RDHFFWEPA operator to aggregate multiple attribute values ${d}_{ij}$ of each alternative ${A}_{i}\left(i={1,2},\ldots ,m\right)$, i.e.,

    $${d}_{i}={IVq-RDHFFWEPA}^{\left(\rho \right)}\left({d}_{i1},{d}_{i2},\ldots ,{d}_{in}\right)={\otimes }_{j=1}^{n}{d}_{ij}^{{{\delta }_{ij}}^{\left(\rho \right)}},$$

    (48)

    and thus, the overall preference value ${d}_{i}$ of each alternative can be obtained.

• Step 6. Obtain the optimal choice based on the alternative scores.

  • Step 6.1 Calculate the scores of ${d}_{i}$ in the light of the score function described in Definition 3.

  • Step 6.2 Rank the corresponding alternatives according to their scores and select the best one.

7 Numerical examples

In this section, we will demonstrate and verify our method presented in Fig. 2 through two comprehensive cases. Then, the parameter analysis, validity analysis, and advantage analysis are conducted through a series of examples.

7.1 Case study 1: Desalination technology (DT) selection problem

Example 6

(Revised from Mishra et al. 2022). The assessment of DTs is a very valuable but, challenging MAGDM problem, because numerous attributes of multiple DTs need to be deliberated. In Mishra et al.’s study (2022), interval-valued hesitant Fermatean fuzzy sets (IVHFFSs) are utilized to evaluate five DT alternatives $\left\{{A}_{1},{A}_{2},{A}_{3},{A}_{4},{A}_{5}\right\}$ pertinent to nine attributes, that is {${C}_{1}$(Adaptability), ${C}_{2}$(Lifetime), ${C}_{3}$(Treated water quality), ${C}_{4}$(Capital cost), ${C}_{5}$(Maintenance cost), ${C}_{6}$(Acceptance of desalination technology), ${C}_{7}$(Degree of complexity), ${C}_{8}$(Consumption of energy), ${C}_{9}$(Degree of usage of of green energy)}. Since IVHFFSs is a special case of our IVq-RDHFSs when $q=3$, this evaluation data can be completely processed by our IVq-RDHFSs-based method. The original evaluation matrix given by DM can be found in the Table 2 of (Mishra et al. 2022).

The decision-making process of our approach will be elaborated.

• Step 1. Decision matrices are transformed, and the parameters of the algorithms are specified.

  • Step 1.1. These attributes including ${C}_{3}$, ${C}_{4}$, ${C}_{5}$, ${C}_{7}$, and ${C}_{8}$ are identified non-beneficial (or cost attributes). Thus, the original decision matrix is normalized according to Eq. (40). The normalized decision matrix is shown in Online Appendix Table S1.

  • Step 1.2. Set $q=3$, $\theta =2$, and $\rho =2$ in this example.

• Step 2. The entropy weight of ${C}_{j}\left(j={1,2},\ldots ,n\right)$ is calculated for each ${A}_{i}\left(i={1,2},\ldots ,m\right)$ according to Eqs. (38)–(39). The results are as follows:

  $$ \left( {w_{{ij}} } \right)_{{5 \times 9}}^{T} = \left[ {\begin{array}{*{20}l} {0.1718} \hfill & {0.0575} \hfill & {0.1094} \hfill & {0.1410} \hfill & {0.1682} \hfill \\ {0.1763} \hfill & {0.1034} \hfill & {0.1005} \hfill & {0.0613} \hfill & {0.0729} \hfill \\ {0.0212} \hfill & {0.0952} \hfill & {0.0366} \hfill & {0.1012} \hfill & {0.0848} \hfill \\ {0.0983} \hfill & {0.0960} \hfill & {0.2361} \hfill & {0.2261} \hfill & {0.2327} \hfill \\ {0.0561} \hfill & {0.1491} \hfill & {0.1966} \hfill & {0.1345} \hfill & {0.1280} \hfill \\ {0.0730} \hfill & {0.1619} \hfill & {0.1266} \hfill & {0.0679} \hfill & {0.0842} \hfill \\ {0.1271} \hfill & {0.0846} \hfill & {0.0638} \hfill & {0.0770} \hfill & {0.0612} \hfill \\ {0.1267} \hfill & {0.1445} \hfill & {0.0772} \hfill & {0.0963} \hfill & {0.1092} \hfill \\ {0.1495} \hfill & {0.1078} \hfill & {0.0531} \hfill & {0.0947} \hfill & {0.0588} \hfill \\ \end{array} } \right] $$

• Step 3. Utilize IVq-RDHFFWEPA operator to aggregate the attribute values of each alternative into comprehensive one.

  • Step 3.1. The support $Sup\left({d}_{ij},{d}_{is}\right)$ can be computed by Eq. (45), where $i={1,2},\ldots ,5;j,s={1,2},\ldots ,9;j\ne s$. In order to save space, the results are omitted.

  • Step 3.2. The overall supports $T\left({d}_{ij}\right)$ can be obtained by the formula (46):

    $$ \left( {T\left( {d_{{ij}} } \right)} \right)^{T} = \left[ {\begin{array}{*{20}l} {6.0134} \hfill & {6.7894} \hfill & {6.5011} \hfill & {6.5102} \hfill & {6.5293} \hfill \\ {6.3366} \hfill & {6.8261} \hfill & {6.5627} \hfill & {6.9249} \hfill & {6.8526} \hfill \\ {6.6884} \hfill & {6.9291} \hfill & {6.9509} \hfill & {6.7467} \hfill & {6.8602} \hfill \\ {6.6032} \hfill & {6.7939} \hfill & {6.4611} \hfill & {6.0713} \hfill & {6.1754} \hfill \\ {6.7234} \hfill & {6.6058} \hfill & {6.6479} \hfill & {6.5088} \hfill & {6.5661} \hfill \\ {6.7743} \hfill & {6.4727} \hfill & {6.7987} \hfill & {6.8316} \hfill & {6.8232} \hfill \\ {6.0081} \hfill & {6.8260} \hfill & {6.8217} \hfill & {6.7346} \hfill & {6.9131} \hfill \\ {6.5281} \hfill & {6.6227} \hfill & {6.8768} \hfill & {6.7430} \hfill & {6.6833} \hfill \\ {6.0088} \hfill & {6.8152} \hfill & {6.9488} \hfill & {6.7424} \hfill & {6.6494} \hfill \\ \end{array} } \right]$$

  • Step 3.3. The power weight of ${d}_{ij}$ can be calculated by Eq. (47):

    $$ \left( {\delta _{{ij}} ^{{\left( \rho \right)}} } \right)_{{5 \times 9}}^{T} = \left[ {\begin{array}{*{20}l} {0.1658} \hfill & {0.0580} \hfill & {0.1075} \hfill & {0.1404} \hfill & {0.1673} \hfill \\ {0.1770} \hfill & {0.1047} \hfill & {0.0994} \hfill & {0.0640} \hfill & {0.0752} \hfill \\ {0.0222} \hfill & {0.0976} \hfill & {0.0379} \hfill & {0.1036} \hfill & {0.0877} \hfill \\ {0.1019} \hfill & {0.0969} \hfill & {0.2309} \hfill & {0.2136} \hfill & {0.2218} \hfill \\ {0.0589} \hfill & {0.1473} \hfill & {0.1964} \hfill & {0.1339} \hfill & {0.1279} \hfill \\ {0.0771} \hfill & {0.1575} \hfill & {0.1288} \hfill & {0.0701} \hfill & {0.0866} \hfill \\ {0.1226} \hfill & {0.0857} \hfill & {0.0651} \hfill & {0.0787} \hfill & {0.0636} \hfill \\ {0.1301} \hfill & {0.1431} \hfill & {0.0792} \hfill & {0.0985} \hfill & {0.1105} \hfill \\ {0.1442} \hfill & {0.1091} \hfill & {0.0549} \hfill & {0.0969} \hfill & {0.0593} \hfill \\ \end{array} } \right] $$

  • Step 3.4. According to Eq. (48), apply the IVq-RDHFFWEPA operator to aggregate the nine attribute values of each alternative to a comprehensive value ${d}_{i}(i={1,2},3,4,5)$. The results are omitted here because they are so complicated.

• Step 4. Calculate alternative scores and obtain the best choice.

  • Step 4.1. According to Definition 3, the score of ${d}_{i}$ can be calculated to represent the final score of each alternative, i.e.,

    $$ \begin{aligned} S\left( {A_{1} } \right) & = S\left( {d_{1} } \right) = - 0.0424,S\left( {A_{2} } \right) = S\left( {d_{2} } \right) = 0.0847,S\left( {A_{3} } \right) = S\left( {d_{3} } \right) = - 0.1507, \\ S\left( {A_{4} } \right) & = S\left( {d_{4} } \right) = 0.0493,S\left( {A_{5} } \right) = S\left( {d_{5} } \right) = 0.0435. \\ \end{aligned} $$

  • Step 4.2. According to the scores, the ranking order of alternatives is that ${A}_{2}\succ {A}_{4}\succ {A}_{5}\succ {A}_{1}\succ {A}_{3}$, which means that ${A}_{2}$ is the best DT choice.

7.2 Case study 2: Evaluation of rural green eco-tourism projects

Example 7

The COVID-19 pandemic has brought heavy losses to the tourism industry around the world. Subjectively, people are worried about the health crisis in travel, and objectively, epidemic prevention measures in different regions also restrict people's travel. Recently, with the gradual recovery of tourism economy, tourism consumption habits have changed under the new external environment. For example, it is reported that suburban weekend leisure and ecological health tourism are becoming more popular in China. To seek a breakthrough under the normalization of COVID-19, a large domestic tourism enterprise decides to develop new rural green ecotourism products. Due to the great complexity and uncertainty of the external environment, the company invites a group of professionals to comprehensively evaluate all candidate destination projects, and the method proposed in this paper is appropriate. Suppose there are three destination alternatives, denoted as ${A}_{i}\left(i={1,2},3\right)$. Three DMs ${D}_{l}(l={1,2},3)$ are required to evaluation the three projects in terms of four attributes ${C}_{j}(j={1,2},{3,4})$, i.e., tourism resources (${C}_{1}$), ecological environment quality (${C}_{2}$), economic conditions (${C}_{3}$), society conditions (${C}_{4}$). The weight vector of DMs is $\gamma =\left({0.4,0.3,0.3}\right)$, and that of attributes is $w={\left({0.3,0.25,0.25,0.2}\right)}^{T}$. DMs record their evaluation values in the form of IVq-RDHFE, and ${d}_{ij}^{l}$ represents the opinion given by DM ${D}_{l}$ on the attribute ${C}_{j}$ of alternative ${A}_{i}$. Then, the assessments provided by the three DMs is summarized into three decision matrices, denoted as ${M}^{l}={\left({d}_{ij}^{l}\right)}_{3\times 4}(l={1,2},3)$, which are listed in Tables 3, 4, 5.

Table 3 The interval-valued q-rung orthopair dual hesitant fuzzy decision matrix M¹ provided by D₁ in Example 7

Table 4 The interval-valued q-rung orthopair dual hesitant fuzzy decision matrix M² provided by D₂ in Example 7

Table 5 The interval-valued q-rung orthopair dual hesitant fuzzy decision matrix M³ provided by D₃ in Example 7

In this example, the weight vectors of DMs and attributes are given in advance based on preliminary analysis and expert knowledge. Next, we will demonstrate the detailed decision-making process of solving the problem through our method.

• Step 1. Transform decision matrices and set algorithm parameters.

  • Step 1.1. Normalize the original decision matrices according to Eq. (40). Because all attributes are benefit types, the three decision matrices remain unchanged.

  • Step 2.2. Suppose $q=3$, $\theta =2$, and $\rho =2$ to tackle this example.

• Step 2. Utilize the proposed IVq-RDHFFWEPA operator to gain the comprehensive decision matrix.

  • Step 2.1. The support $Sup\left({d}_{ij}^{l},{d}_{ij}^{z}\right)\left(l,z={1,2},3;l\ne z\right)$ can be calculated according to Eq. (41):

    $$Sup\left({d}_{ij}^{1},{d}_{ij}^{2}\right)=Sup\left({d}_{ij}^{2},{d}_{ij}^{1}\right)=\left[\begin{array}{ccc}0.8534& 0.7259& \begin{array}{cc}0.9490& 0.7389\end{array}\\ 0.8649& 0.9371& \begin{array}{cc}0.8969& 0.8271\end{array}\\ 0.9299& 0.8044& \begin{array}{cc}0.7218& 0.8620\end{array}\end{array}\right]$$
    $$Sup\left({d}_{ij}^{1},{d}_{ij}^{3}\right)=Sup\left({d}_{ij}^{3},{d}_{ij}^{1}\right)=\left[\begin{array}{ccc}0.8656& 0.8729& \begin{array}{cc}0.9555& 0.9840\end{array}\\ 0.9663& 0.8429& \begin{array}{cc}0.8499& 0.9596\end{array}\\ 0.8295& 0.7255& \begin{array}{cc}0.8579& 0.8626\end{array}\end{array}\right]$$
    $$Sup\left({d}_{ij}^{2},{d}_{ij}^{3}\right)=Sup\left({d}_{ij}^{3},{d}_{ij}^{2}\right)=\left[\begin{array}{ccc}0.8491& 0.8384& \begin{array}{cc}0.9654& 0.7321\end{array}\\ 0.8807& 0.8648& \begin{array}{cc}0.9270& 0.8271\end{array}\\ 0.8501& 0.9119& \begin{array}{cc}0.7554& 0.9587\end{array}\end{array}\right]$$

  • Step 2.2. The overall supports $T\left({d}_{ij}^{l}\right)$ can be calculated by Eq. (42):

    $$T\left({d}_{ij}^{1}\right)=\left[\begin{array}{ccc}1.7190& 1.5987& \begin{array}{cc}1.9045& 1.7229\end{array}\\ 1.8311& 1.7800& \begin{array}{cc}1.7468& 1.7868\end{array}\\ 1.7594& 1.5299& \begin{array}{cc}1.5796& 1.7246\end{array}\end{array}\right]$$
    $$T\left({d}_{ij}^{2}\right)=\left[\begin{array}{ccc}1.7025& 1.5642& \begin{array}{cc}1.9144& 1.4710\end{array}\\ 1.7456& 1.8019& \begin{array}{cc}1.8239& 1.6543\end{array}\\ 1.7800& 1.7163& \begin{array}{cc}1.4771& 1.8207\end{array}\end{array}\right]$$
    $$T\left({d}_{ij}^{3}\right)=\left[\begin{array}{ccc}1.7147& 1.7112& \begin{array}{cc}1.9209& 1.7161\end{array}\\ 1.8470& 1.7076& \begin{array}{cc}1.7769& 1.7868\end{array}\\ 1.6796& 1.6374& \begin{array}{cc}1.6133& 1.8214\end{array}\end{array}\right]$$

  • Step 2.3. The power weight of the evaluation value ${d}_{ij}^{l}$ given by DM ${D}_{l}$ can be obtained according to Eq. (43):

    $${{\tau }_{1}}^{\left(\rho \right)}=\left[\begin{array}{ccc}0.4007& 0.3974& \begin{array}{cc}0.3992& 0.4085\end{array}\\ 0.4022& 0.4016& \begin{array}{cc}0.3966& 0.4042\end{array}\\ 0.4019& 0.3903& \begin{array}{cc}0.4023& 0.3939\end{array}\end{array}\right]$$
    $${{\tau }_{2}}^{\left(\rho \right)}=\left[\begin{array}{ccc}0.2992& 0.2952& \begin{array}{cc}0.3002& 0.2857\end{array}\\ 0.2949& 0.3030& \begin{array}{cc}0.3036& 0.2926\end{array}\\ 0.3031& 0.3081& \begin{array}{cc}0.2931& 0.3030\end{array}\end{array}\right]$$
    $${{\tau }_{3}}^{\left(\rho \right)}=\left[\begin{array}{ccc}0.3002& 0.3074& \begin{array}{cc}0.3007& 0.3058\end{array}\\ 0.3029& 0.2954& \begin{array}{cc}0.2998& 0.3032\end{array}\\ 0.2950& 0.3016& \begin{array}{cc}0.3046& 0.3031\end{array}\end{array}\right]$$

  • Step 2.4. According to Eq. (44), the IVq-RDHFFWEPA operator is used to aggregate three individual decision matrices ${M}^{l}={\left({d}_{ij}^{l}\right)}_{m\times n}$ to a comprehensive decision matrix $M={\left({d}_{ij}\right)}_{m\times n}$. As the results are litte complex, we omit them here.

• Step 3. Apply the proposed IVq-RDHFFWEPA operator to obtain the comprehensive evaluation value of each alternative (suppose $q=3$, $\theta =2$, $\widetilde{\rho }=2$).

  • Step 3.1. The support $Sup\left({d}_{ij},{d}_{is}\right)$ (suppose $q=3$) can be computed by Eq. (45), where $i={1,2},3;j,s={1,2},{3,4};j\ne s$:

    $$Sup\left({d}_{i1},{d}_{i2}\right)=Sup\left({d}_{i2},{d}_{i1}\right)=[\begin{array}{ccc}0.7600& 0.9272& 0.9270\end{array}]$$
    $$Sup\left({d}_{i1},{d}_{i3}\right)=Sup\left({d}_{i3},{d}_{i1}\right)=[\begin{array}{ccc}0.8777& 0.8758& 0.8400\end{array}]$$
    $$Sup\left({d}_{i1},{d}_{i4}\right)=Sup\left({d}_{i4},{d}_{i1}\right)=[\begin{array}{ccc}0.9502& 0.9182& 0.9102\end{array}]$$
    $$Sup\left({d}_{i2},{d}_{i3}\right)=Sup\left({d}_{i3},{d}_{i2}\right)=[\begin{array}{ccc}0.8422& 0.9258& 0.8838\end{array}]$$
    $$Sup\left({d}_{i2},{d}_{i4}\right)=Sup\left({d}_{i4},{d}_{i2}\right)=[\begin{array}{ccc}0.7994& 0.9661& 0.8589\end{array}]$$
    $$Sup\left({d}_{i3},{d}_{i4}\right)=Sup\left({d}_{i4},{d}_{i3}\right)=[\begin{array}{ccc}0.9116& 0.9395& 0.7752\end{array}]$$

  • Step 3.2. The overall supports $T\left({d}_{ij}\right)$ can be obtained by the formula (46):

    $$T\left({d}_{ij}\right)=\left[\begin{array}{ccc}2.5879& 2.4016& \begin{array}{cc}2.6315& 2.6612\end{array}\\ 2.7212& 2.8191& \begin{array}{cc}2.7411& 2.8239\end{array}\\ 2.6772& 2.6697& \begin{array}{cc}2.4990& 2.5442\end{array}\end{array}\right]$$

  • Step 3.3. The power weight of ${d}_{ij}$ can be calculated by Eq. (47):

    $${{\delta }_{ij}}^{\left(\rho \right)}=\left[\begin{array}{ccc}0.3014& 0.2410& \begin{array}{cc}0.2535& 0.2041\end{array}\\ 0.2969& 0.2525& \begin{array}{cc}0.2484& 0.2022\end{array}\\ 0.3048& 0.2536& \begin{array}{cc}0.2443& 0.1974\end{array}\end{array}\right]$$

  • Step 3.4. According to Eq. (48), the IVq-RDHFFWEPA operator are applied to aggregate the four attribute values of each alternative to a comprehensive value ${d}_{i}(i={1,2},3)$. The results are omitted here because they are so complicated.

• Step 4. Obtain the optimal alternative.

• Step 4.1. According to Definition 3, the score of ${d}_{i}$ can be gathered to represent the final score of each alternative, i.e., $S\left({A}_{1}\right)=S\left({d}_{1}\right)=0.3169, S\left({A}_{2}\right)=S\left({d}_{2}\right)=0.2530, S\left({A}_{3}\right)=S\left({d}_{3}\right)=0.2093.$

• Step 4.2. According to the corresponding scores, the order of alternatives is that ${A}_{1}\succ {A}_{2}\succ {A}_{3}$, which means ${A}_{1}$ is the optimal investment project and should be given priority.

7.3 Analysis of the impact of parameters on the decision results

It is worth noting that the parameters of IVq-RDHFFWEPA operator play a significant role on the final decision result. For example, the parameter $\rho$ of the EPA operator makes the proposed method available for diverse application scenarios. And the parameter $\theta$ from FTT can reflect the risk attitudes of DMs. Likewise, the parameter $q$ inherited from IVq-RDHFSs provides DMs with great flexibility in the face of complicated decision problems. To better understand and use these parameters, we perform a set of experiments to analyze the performance of the proposed method under diverse parameter values.

7.3.1 The influence of parameter ${\varvec{\rho}}$ on decision results

Parameter $\rho$ is the core of EPA operator, which enables EPA to adapt to different types of MAGDM problems. In practical, through varying $\rho$ in ($-\infty ,\left.-\left(n-1\right)\right]\cup \left[0,\left.+\infty \right)\right.$, the weights assigned to the input arguments can be dynamically adjusted according to the specific decision issues to tackle. Specially, if $\rho \ge 0$, the unduly low or high arguments are regarded as unreasonable evaluation values and allocated lower weights; By contrast, if $\rho \le -\left(n-1\right)$($n$ is the number of input arguments), the unduly low or high inputs are esteemed to be of sufficient importance and given higher weights. Therefore, in this subsection, we would investigate how the parameter $\rho$ affects the decision outcomes. To shed light on the influence of $\rho$, we redesign Example 7 by introducing an extreme evaluation value, and the modified example is denoted as Example 8.

Example 8

(Modified Example 7 by introducing extreme evaluation values). Assuming that DM D₂ give some extreme evaluations, the modified decision matrix is listed in Table 6. In which, the value of attribute C₃ of alternative A₃ is substituted with an extremely large IVq-RDHFE $\left\{\left\{[{0.8,0.9}]\right\},\left\{[{0.1,0.2}]\right\}\right\}$. At the same time, the other conditions remain the same as Example 7.

Table 6 The interval-valued q-rung orthopair dual hesitant fuzzy decision matrix provided by D₂ in Example 8

Then, we allocate different positive and negative values to $\rho$, and apply our method to solve Example 8. For the sake of comparability, we still assume $q=3$ and $\theta =2$, then the corresponding decision results are presented in Fig. 3.

Fig. 3

The scores of alternatives of Example 8 with a variation in $\rho$. The left part (a) is for positive $\rho$ and the right part (b) is for negative $\rho$

As can been seen from the panel (a) of Fig. 3, when $\rho \ge 0$, the scores of the alternative ${A}_{3}$ obviously increase compared with Example 4 because of the added large value. However, the optimal scheme to choose is always ${A}_{1}$ although the ranking order is changed to ${A}_{1}\succ {A}_{3}\succ {A}_{2}$. The reason is that the positive $\rho$ implies less importance to the extreme value, and consequently, the influence of the added maximal value is partially eliminated. In contrast, from the panel (b) of Fig. 3 the scores of the alternative ${A}_{3}$ change dramatically and are universally higher than that when $\rho \ge 0$. This is because when $\rho$ is negative, the extremely large input is considered important and allocated a high weight, which further leads to the changes in the ranking order especially in the case of $\rho =-3$ and $-4$ in which ${A}_{3}$ exceeds ${A}_{1}$ as the optimal option. This explains the ability of our method to flexibly model different decision problems with diverse values of $\rho$.

Additionally, in both cases of the panels (a) and (b), the scores of each alternative tend to be stable as the absolute value of $\rho$ increases. More specially, it can be found that when $\rho$ tends to positive infinity or negative infinity, the score value of the same alternative converges to the same constant. This reveals a rule that the greater the absolute value of $\rho$, the smaller the weight adjustment for extreme value. This conclusion is in accord with these special cases, that is, when $\rho \to \infty$, our IVq-RDHFFWEPA operator finally become the simple weighted average operator for IVq-RDHFEs.

7.3.2 The influence of parameter ${\varvec{q}}$ on decision results

To study the influence of parameter q on the results, we choose IVq-RDHFEs-based Example 7 as a case. Then, when applying the proposed method to solve this example, we assign a series of different q values. In the context of $\rho =\theta =2$, the corresponding results of the three alternatives with a variation in q is listed in Table 7.

Table 7 The results of Example 7 with a variation in $q$

From Table 7, we can see that for each alternative, the score of different q is diverse, and the score decreases with the increase of q. Notably, when q = 3, the ranking order of the three alternatives is ${A}_{1}\succ {A}_{2}\succ {A}_{3}$, while the order changes to ${A}_{1}\succ {A}_{2}\succ {A}_{3}$ when $q={4,5},\ldots ,10$. This shows that the parameter q indeed has a key impact on the final decision, and the choice of $q$ is a significant issue in the proposed MAGDM method. Generally, the parameter q permits DMs to express their evaluation in a more flexible manner, because no matter how large their evaluation values is, q can be adjusted to meet the requirement of IVq-RDHFE. As for the selection of q in practical use, an efficient manner has been proposed in our article (Xu et al. 2020), and readers are recommended to get more details from it.

7.3.3 The influence of parameter ${\varvec{\theta}}$ on decision results

In the proposed method, the parameter $\theta$ introduced by Frank operation laws likewise have a significant effect on the decision-making results. To examine the the influence of parameter $\theta$, the proposed method is utilized to solve Example 7 and it is assumed $\theta$ varies from 1.05 to 15 with a step 1. In the situation of $q=3$ and $\rho =2$, the score curves of the three alternatives are displayed in Fig. 4.

Fig. 4

The score values of Example 7 with a variation in $\theta$

As evident in Fig. 4, the score of each alternative exhibits a monotonic decrease with the escalation of the parameter $\theta$. This phenomena is inherently connected to the practical implication of the parameter $\theta$ in the realm of MAGDM, notably reflecting the risk preference of decision makers (DMs) as highlighted by Tang et al. (2018). Hence, a larger $\theta$ suggests a lean towards pessimism by the DM, who then prefers lower aggregated estimation values. The selection of $\theta$, therefore, hinges on the DMs’ risk preference in dealing with real-world decision scenarios. To elaborate, a DM with a risk-seeking nature might opt for a smaller $\theta$, while a risk-averse individual would likely find a larger $\theta$ more suitable. In this study, we refrain from imposing any constraints on the value of $\theta$, thereby allowing DMs the freedom to select the value that best aligns with their personal preferences and the actual situation. Interestingly, the minimal possible $\theta$ is near 1, and as $\theta$ approaches 1, the Frank operation mode of AOs transitions into its algebraic mode, further demonstrating the flexibility of the Frank operation. It’s also noteworthy that despite the varied values of $\theta$, the ranking order of the three alternatives remains consistent. This consistency further attests to the robustness and stability of our proposed method, regardless of the risk attitude expressed through the parameter $\theta$.

7.4 Validity of the novel IVq-RDHFFWEPA operator

In this section, we conduct validity analysis to verify the effectiveness of the proposed new operator. Peng et al.’s (2018) Archimedean t-norm and t-conorm based interval-valued dual hesitant fuzzy weighted average (A-IVDHFWA) operator as well as our method based on IVq-RDHFFWEPA operator are respectively applied to process the same MAGDM problem under the interval-valued dual hesitant fuzzy environment, as presented in the Example 6 below.

Example 9

(Revised from Peng et al.’s 2018). An investment company plans to support the best of all possible software development projects. Specifically, a panel with three alternatives, including an e-commerce development project (${A}_{1}$), a game development project (${A}_{2}$) and a game development project (${A}_{3}$), will be evaluated with respect to four attributes, namely economic feasibility (${C}_{1}$), technological feasibility (${C}_{2}$), social feasibility (${C}_{3}$) and staff feasibility (${C}_{4}$). And the weight vector of attributes is $w={\left({0.3,0.25,0.25,0.2}\right)}^{T}$. Then, the DMs are required to evaluate the three alternatives ${A}_{i}(i={1,2},3)$ under the four attributes ${C}_{j}(j={1,2},{3,4})$, and the evaluation values are provided in the form of interval-valued dual hesitant fuzzy elements (IVDHFEs). It is obvious that the attributes ${C}_{1}$ and ${C}_{2}$ are benefit attributes, while ${C}_{3}$ and ${C}_{4}$ are cost attributes, which implies the initial evaluation needs to be normalized. For convenience, we directly quote the normalized decision matrix form Peng et al.’s study (2018), as shown in Table 8.

Table 8 Normalized interval-valued dual hesitant fuzzy decision matrix (Peng et al. 2018)

We adopt the MAGDM methods based on the above-mentioned operators to address Example 9 and compare their decision results, as displayed in Table 9. It can be seen from Table 8 that both the scores and ranking order of alternatives derived by A-IVDHFFWA-based method (Peng et al. 2018) are the same as those gained by our IVq-DHFFWEPA-based method (suppose $q=1$, $\rho \to \infty$). This is due to when setting $q=1$, $\rho \to \infty$ and using Frank operation, our IVq-DHFFWEPA operator degenerates to Peng et al.’s A-IVDHFFWA operator (2018). In short, through the degeneration of operators, our method produces the same outcome as the existing MAGDM method when encountering the same decision problem, which fully demonstrates the correctness and validity of the newly developed operator.

Table 9 The results of Example 9 obtained by different methods ($\theta =2$)

7.5 Advantage analysis of the proposed method

Our method is endowed with the merits of IVq-RDHFEs, EPA operator, as well as FTT. In this section, the superiorities of the presented approach in four aspects are demonstrated by comparing against some state-of-the-art MAGDM methods under the q-RDHFSs (Li et al. 2021a), IVDHFSs (Ju et al. 2014; Peng et al. 2018; Qu et al. 2018; Zang et al. 2018) as well as IVq-RDHFSs environments (Xu et al. 2019; Feng et al. 2021).

7.5.1 Dynamically adjusting the role of extreme evaluation values

As mentioned earlier, EPA operator is excellent because it can dynamically adjust the role of extreme evaluation values. Absorbed the merit of EPA, our MAGDM method is hence able to properly deal with extreme values according to the actual needs in the decision-making process. For instance, in some situations the extreme assessment is considered unreasonable for some reasons, such as DMs’ bias or expertise limitation, then the parameter $\rho$ of EPA should be equal to or greater than 0 to eliminate the negative impact of extreme assessment values to a certain extent; However, sometimes, the extreme assessment rather than normal assessment is more important for the final decision, such as anomaly detection problem, then the parameter $\rho$ should be equal to or less than $-\left(n-1\right)$ (n is the number of inputs) to give more weight to attribute parameters with unduly large or small performance. This characteristic makes our approach superior to some of the latest MAGDM developed for IVq-RDHFEs. In the following, Example 8 containing extreme evaluation value is chosen as a case to compare our method with Xu et al. (2019) method based on IVq-RDHFWMM operator ($R=({1,0},\ldots ,0)$). For the sake of comparability, the algebraic operation mode of our method (i.e., $\theta \to 1$) would be taken, and then the results obtained by the two methods are summarized in Table 10.

Table 10 The results of Example 8 obtained by different methods (q = 3)

It can be found from Table 10 that the adjusting effect of the EPA operators on the extreme value has significantly affected the decision results. Firstly, compare our method based on IVq-RDHFWEPA in the situation of $\rho =0$ with Xu et al.’s method (2019) based on the IVq-RDHFWMM operator (i.e., the first and second rows in Table 8). It is noted that the ranking order obtained by our method is ${A}_{1}\succ {A}_{2}\succ {A}_{3}$, whereas it is ${A}_{1}\succ {A}_{3}\succ {A}_{2}$ gained by Xu et al.’s. This significant difference is attributed to the balance effect of our method on extreme values, and specifically in the scope of $\rho \ge 0$, the smaller the $\rho$ is, the stronger the weakening effect of extreme evaluation values. Especially, when $\rho =0$, the effect of the extreme value can be eliminated to the most extent in our method. Secondly, compare the negative $\rho$ and positive $\rho$ results of the IVq-RDHFWEPA operator proposed in this paper (i.e., the second and third row in Table 8). Evidently, they have significant differences in the score values and orders. This is because when $\rho \le -\left(n-1\right)$, the effect of $\rho$ on extreme assessment is opposite, and the larger $\rho$ means the greater impact of extreme assessment. So, based on the newest EPA operator, our developed method can dynamically control the the weight adjustment degree of extreme evaluation, which gives our method more flexibility than Xu et al.’s (2019). More importantly, our method allows to flexibly set $\rho$ adhering to the requirements of different decision-making contexts, which indicates our method is a stronger tool in practical application.

7.5.2 Portraying fuzzy evaluation information more effectively

The developed MAGDM in this paper adopts the IVq-RDHFEs to portray DMs’ assessment information. In IVq-RDHFEs, a collection of interval values instead of crisp values can be used to represent the MDs and NMDs evaluation of DMs. Whereas Li et al.’s (2021a) method is based on q-RDHFEs in which several crisp numbers are utilized to denote the MDs and NMDs of DMs. Obviously, the proposed method is stronger than Li et al.’s (2021a) method. In theory, the q-RDHFE becomes a special case of IVq-RDHFE, when the upper bound and lower bound of each interval value in MD and NMD sets of IVq-RDHFE are equal. This indicates that our method can also successfully cope with the decision problems with the evaluation values in the form of q-RDHFEs. Then, to better demonstrate this advantage, our method is employed to process the example constructed by Li et al. (2021a).

Example 10

(From Li et al.’s (2021a)). For the purpose of improving the air quality in the office, a company plans to purchase a large volume of air purifiers. The DMs panel decides to evaluation the performance of four possible suppliers ${X}_{i}(i={1,2},{3,4})$ in terms of four attributes including ${A}_{1}$ (product quality), ${A}_{2}$ (speed of delivery), ${A}_{3}$ (popularity of brand) as well as ${A}_{4}$ (reputation of supplier). The weights of the four attributes are $w={\left({0.25,0.2,0.2,0.35}\right)}^{T}$. The decision panel applies the q-RDHFEs to denote their opinions, then the original decision matrix is shown in Table 11.

Table 11 The q-rung orthopair dual hesitant fuzzy decision matrix of Example 10 (Li et al. 2021a)

In Example 10, it is suggested to use q-RDHFEs to express DMs’ evaluation. As said above, q-RDHFEs is just a special case of IVq-RDHFEs, so that we can transform the form of theses evaluation values from q-RDHFE to IVq-RDHFE. For example, a q-RDHFE $\left\{\left\{{0.5,0.6}\right\},\left\{{0.3,0.9}\right\}\right\}$ can be rewritten as an IVq-RDHFE $\left\{\left\{\left[{0.5,0.5}\right],\left[{0.6,0.6}\right]\right\},\left\{\left[{0.3,0.3}\right],\left[{0.9,0.9}\right]\right\}\right\}$. Hence, the original decision matrix (i.e., Table 11) is converted into the new interval-valued q-rung orthopair dual hesitant fuzzy decision matrix, as displayed in Table 12. Next, our IVq-RDHFFWEPA-based method and Li et al.’s (2021a) q-RDHFAWEPA-based method are respectively applied to address Example 10. Considering the comparability, algebraic operation rules are selected for both methods, and their calculation outcomes are listed in Table 13.

Table 12 The interval-valued q-rung orthopair dual hesitant fuzzy decision matrix of Example 10

Table 13 The outcomes of Example 10 obtained by different methods ($q=3$, $\rho =2$)

Our method’s outcomes align perfectly with the research findings of Li et al. (2019), providing strong validation for the method we have proposed in this study. Notably, our method, grounded in the use of IVq-RDHFSs, offers superior accuracy and effectiveness when representing fuzzy information compared to Li et al.’s approach. In the face of complex decision-making problems, decision makers (DMs) often exhibit high levels of hesitation. Our method accommodates this uncertainty by allowing DMs to express their views using interval values rather than precise numbers. This approach avoids the need for DMs to exclude certain information or to record their views as exact values. Additionally, our method exhibits greater flexibility than that of Li et al. (2019) by addressing MAGDM problems not only under IVq-RDHFSs information but also q-RDHFSs information. A case in point is the inability of Li et al.'s method to process an IVq-RDHFE evaluation $\left\{\left\{\left[{0.2,0.3}\right],[{0.3,0.4}]\right\},\left\{\left[{0.1,0.2}\right]\right\}\right\}$ without any loss of information. In contrast, our method can handle these evaluations without sacrificing any details, further reinforcing its effectiveness and adaptability.

7.5.3 Providing greater freedom of information representation

Given another good characteristic of IVq-RDHFS, our method can give DMs greater freedom when expressing their evaluation information, which makes our method more suitable for MAGDM applications. Our method requires that the evaluation values should meet the constrain of IVq-RDHFS, that is, the sum of qth power of the maximum upper bound in MD and that in NMD should not be greater than 1. In practical use, the value of q can always be flexibly devised to maintain this constrain. By comparison, the IVDHFS, the predecessor of IVq-RDHFS, is inferior in this regard because it strictly limits that the sum of the maximum upper bound in MD and that in NMD should be less than or equal to 1. We enumerate the existing decision-making methods established on IVDHFSs in Table 13, including the approach proposed by Ju et al. (2014) based on interval-valued dual hesitant fuzzy weighted average (IVDHFWA) operator, the method introduced by Qu et al. (2018) based on interval-valued dual hesitant fuzzy choquet ordered average (IVDHFCOA) operator, the manner developed by Peng et al. (2018) based on A-IVDHFWA operator and that presented by Zang et al. (2018) based on interval-valued dual hesitant fuzzy weighted Heronian mean (IVDHFWHM) operator.

Table 14 exhibits the results of applying these methods and our method to handle Example 7. Clearly, owing to the rigorous constrain of IVDHFS, all IVDHFS-based approaches fail when encountering Example 7, whereas our IVq-RDHFS-based approach is perfectly capable. For example, in Example 7, the evaluation value of the attribute ${C}_{2}$ of the alternative ${A}_{1}$ given by DM ${D}_{1}$ is $\left\{\left\{\left[{0.8,0.9}\right]\right\},\left\{\left[{0.2,0.3}\right],[{0.4,0.6}]\right\}\right\}$. Then, as $0.9+0.6>1$, the above IVDHFS-based methods cannot process this situation. However, our method can still be successful because we can specify $q=3$ to achieve ${0.9}^{3}+{0.6}^{3}=0.945<1$. And hence, in terms of fuzzy information representation, our method makes more relaxed restrictions and offer greater freedom. Commonly, the requirement of IVDHFS is unable to be satisfied in many decision-making contexts, and thus, our method is more practical and can enjoy wider applications.

Table 14 The results of Example 7 obtained by several different methods

7.5.4 Owning flexible and functional operation

On the foundation of Frank operation rules, our method gains great flexibility and practicability. In principle, algebraic operation is just a special case of Frank operation when the parameter $\theta$ is infinitely close to 1. Therefore, it is easily to deduce the algebraic mode of our proposed operators, i.e., IVq-RDHFEPA and IVq-RDHFWEPA operators. This reveals the flexibility of the proposed method in this paper. More significantly, in the context of MAGDM, the monotonically decreasing aggregation results continent to $\theta$ are related to the risk preference of DMs. This permits DMs to choose the proper parameter value in accord with their risk attitude. To better illustrate this characteristic, we utilize our method and Feng et al. method (2021) based on the IVq-RDHFWPHM operator to solve Example 7, then show and compare their results in Table 15.

Table 15 The results of Example 7 obtained by two different methods ($q=3$)

From Table 15, the ranking order of alternatives obtained by both methods is the same, i.e., ${A}_{1}\succ {A}_{2}\succ {A}_{3}$, despite the difference in their scores. Basically, Feng et al. (2021) method is established on the power Hamy mean (PHM) operator and our method is on the basis of EPA operator. For the PHM, the setting of k = 1 is means that it would reduce to the PA operator which is exactly case of EPA when $\rho =1$. A special case is that when $\rho =1$, the EPA degenerates into PA operator, so if setting $\theta \to 1$ (i.e., algebraic operation rules), our IVq-RDHFWEPA is converted to interval-valued q-rung dual hesitant fuzzy weighted power average (IVq-RDHFWPA) operator, which is the same as IVq-RDHFWPHM operator (Feng et al. 2021) in the case of k = 1. And thus, the two methods can deliver the totally same score and order in this situation, which likewise exhibits the effectiveness of the proposed method in this paper. In Table 13, the calculation of Feng et al.’s method (2021) complies to algebraic operation, while Frank operation is our calculation rule. This indicates that the difference in the scores is completely caused by the different operation rules. Compared with Feng et al.’s (2021) method, because of the parameter $\theta$ of Frank operation, our method possesses the ability to model the risk preference, in other words, the results gained by our method integrate the risk cognitive of DMs. Considering the risk attitude of DMs is a common issue in decision scenes, our method is more consistent with the real decision needs and has higher credibility.

7.5.5 Summary

From four aspects, the merits of the proposed method in this paper are elaborated through a variety of comparative experiments. To exhibit these merits more clearly, we summarize the main characteristics of the above existing methods (Feng et al. 2021; Ju et al. 2014; Peng et al. 2018; Li et al. 2021a; Qu et al. 2018; Xu et al. 2019; Zang et al. 2018) as well as our method in Table 16.

Table 16 Characteristics of different decision-making methods

Firstly, the ability to denote MD and NMD by interval values. As an application and extension of IVq-RDHFSs, the developed approach in this paper allows DMs to represent their MDs and NMDs by interval values rather that crisp values. Capturing DMs’ hesitation with less fuzzy information loss, our approach is consequently more effective than some existing approaches (Ju et al. 2014; Peng et al. 2018; Li et al. 2021a; Qu et al. 2018; Zang et al. 2018).

Secondly, the ability to reflect DMs’ risk attitudes. The manner presented in this paper is worked on the foundation of FTT. Apart from flexibly reducing the basic algebraic operation rules used in (Feng et al. 2021; Ju et al. 2014; Qu et al. 2018; Xu et al. 2019; Zang et al. 2018), the parameter $\theta$ of FTT is associated with the risk preference of DM. And thus, risk seeker can determine a small value of $\theta$ while risk averter choose a large value, which makes the results produced by our manner are more reasonable and reliable than these existing methods.

Thirdly, the freedom for DMs to express their evaluation information. Based on the IVq-RDHFS, requiring the sum of qth power of the maximum upper bound in MD and that in NMD should not be greater than 1, our method is always able to make the constrain hold by adjusting the value of q. So, our method provides a larger space for the free information representation of DMs than some existing methods (Ju et al. 2014; Peng et al. 2018; Qu et al. 2018; Zang et al. 2018).

At last, the flexibility to handle extreme evaluation values. Inheriting the function of EPA operator, our approach can determine the positive or negative $\rho$ to realize the diverse role of extreme values. More than this, the the adjustment power weight of extreme assessment can change dynamically with value of $\rho$. Thus, the approach given in this paper possesses great power to deal with extreme inputs than most existing approaches (Feng et al. 2021; Ju et al. 2014; Peng et al. 2018; Qu et al. 2018; Xu et al. 2019; Zang et al. 2018).

8 Conclusions

This study introduces a novel multi-attribute group decision making (MAGDM) method that incorporates DMs’ evaluations as interval-valued q-rung dual hesitant fuzzy sets (IVq-RDHFSs). The contributions of our research include the introduction of a new distance measurement method, the development of operational rules based on Frank t-norm and t-conorm (FTT), and the extension of powerful extended power average (EPA) operators into an interval-valued q-rung dual hesitant fuzzy context.

We have developed novel aggregation operators (AOs) for merging IVq-RDHFSs and have thoroughly explored the properties of these new AOs. We also introduced a best–worst method (BWM) for determining DM weights and utilized the entropy measure of IVq-RDHFSs to calculate attribute weights. Based on these advancements, we have devised a complete MAGDM method and demonstrated its detailed implementation steps. Our method outperforms existing techniques in several ways. It dynamically adjusts the weights of extreme evaluation values using the EPA operator, meeting practical requirements. It also accurately represents DMs’ fuzzy evaluation information through interval values. Further, our approach provides more flexibility in information representation by relaxing restrictions on membership and non-membership degrees. Finally, it uses FTT for flexible transformation of multiple operations and accurate depiction of DMs’ risk attitudes.

We have provided two numerical examples to illustrate the practical application of our decision-making approach, and conducted sensitivity, validity, and comparison analyses through various examples. Furthermore, two case studies demonstrate the potential applicability of our method to practical MAGDM problems (Narang et al., 2022; Tang et al., 2022). Despite the inherent complexity of IVq-RDHFSs and the increased computational requirements, our method effectively handles diverse complex cognitive information. However, we acknowledge that our method overlooks the dynamic adjustment mechanism of decision-maker weights based on feedback information, a significant research question in decision science that warrants further exploration.

In future work, we plan to apply our methods to complex decision-making problems, combine EPA with hybrid models, as well as develop some extensions of extended power geometric averaging operators. Besides, we will explore the potential of our method in large-scale group decision-making (LSGDM) scenarios. We also have a keen interest in develo** more efficient techniques for opinion classification, consensus building, and information expression in the context of LSGDM (Liu et al., 2018). Overall, this study presents a significant advancement in the field of MAGDM, opening up new avenues for future research and practical applications.