Correlation coefficients on normal wiggly dual hesitant fuzzy sets: an application in the selection of real estate agents

Introduction

A correlation coefficient (CC) is a statistical measure that demonstrate the strength of the association between two variables as well as the direction in which the relationship is changing. It assesses the level to which changes in one variable are dependent with changes in another variable. The fuzzy CCs are extensions of traditional CCs, such as Pearson’s CC, it designed to handle data that is imprecise, uncertain, or ambiguous. In traditional statistics, CCs identify the strength and direction of a linear relationship between two variables. However, when information of an object contains fuzziness (e.g., when resultant values are vague or imprecise), traditional correlation frameworks may not be suitable or applicable. The fuzzy CCs are designated to quantify the magnitude of similarity or relationship between fuzzy sets (FSs) (Zadeh, 1965). Initially, Yu (1993) proposed the concept of CCs on fuzzy numbers. Chiang & Lin (1999) researched an alternative approach for CCs on FSs.

The MCDM approaches that are used in the modern world make extensive use of CCs to define the relationship between objects, while FSs are utilized to label an enormous parameterized fuzziness in systems. The FSs are insufficient to classify information when multiple credible grades occur in complex systems. Torra & Narukawa (2009) expanded conventional FSs by introducing the concept Hesitant Fuzzy Set (HFS) (Torra, 2010), which permit the assignment of several believable degrees of uncertain information. Researchers concur that HFSs outshine FSs and are utilized in various domains, including image segmentation, distance analysis (Xu & Xia, 2011a), outranking approaches (Wang et al., 2014), linguistic decision making (Lee & Chen, 2015; Rodriguez, Martinez & Herrera, 2011; Wu, Wang & Wangzhu, 2025), information measures (Ajaib et al., 2025; Farhadinia, 2013), aggregation operators (Mehmood et al., 2019; Mehmood et al., 2018) and pattern recognition (Luo et al., 2025; Qahtan et al., 2025).

Several features of HFS have been investigated that can be used to build on new ideas in hesitant fuzzy set theory. These include aggregation operators on HFS by Xia, Xu & Chen (2013), entropy and cross-entropy on HFSs by Xu & Xia (2012), distance measures (DMs) and similarity measures (SMs) on HFSs by Xu & Xia (2011a), scores on HF elements (HFEs) (Alcantud et al., 2023) and correlation measures on HFSs by Xu & Xia (2011b).

As in elements of HFSs, the information is oscillating between values in the interval $[0,1]$, it is important to make a better decision on such information; it makes sense to define correlation among HFSs. Various sort of CCs exist concerning HFSs and their utilization throughout multiple disciplines. The essential CCs of HFSs were presented by Chen, Xu & Xia (2013), Meng & Chen (2015) and Liao, Xu & Zeng (2015), who utilized them in decision-making and clustering analysis. Probabilistic HFS (PHFS) is another significant type of HFS that takes into account the probability that each membership in HFEs will occur (Zhang, Xu & He, 2017; Zhou & Xu, 2017).

Given that HFEs in HFSs possess several grades assigned by DMs, a suitable technique was necessary to provide a range of hesitation factors to address the reason of hesitation. To support this motive, Ren, Xu & Wang (2018) established the concept of normal wiggly HFSs (NWHFSs) and exploited it for assessments of environmental quality. The NWHFSs consist of normal wiggly elements (NWEs), identified using the statistical concept of standard deviation and the real preference degree proposed by Yager (1988). Liu et al. (2019), Liu & Zhang (2021) proposed two categories of aggregation operators for NWHFSs: Muirhead mean operators and Maclaurian mean operators. Subsequently, these operators were employed in MCDM methodologies. Xia, Chen & Fang (2022) introduced probabilistic NWHFSs and demonstrated their use in battle threat assessments. Narayanamoorthy, Ramya & Kang (2020) and Wang et al. (2024) investigated the application of NWHFSs in several aspects of MCDM.

The Atanassov’s idea of intuitionistic fuzzy sets (IFSs) (Atanassov, 1986) is beneficial when considering NMGs in conjunction with MGs. Inspired by Atanassov’s concept, Zhu & Xu (2014) and Zhu, Xu & Xia (2012) proposed dual HFS (DHFS), wherein both MGs and NMGs encompass value sets from the interval $[0,1]$. The DHFSs have extended in different version such as, Alcantud et al. (2019) consider an additional form of duality factor by proposing dual extended fuzzy sets. Ye (2013) has examined the fundamental CCs on DHFSs and utilized them in MCDM. Following Ye (2013), various foundational aspects of CCs have been demonstrated by researchers (Tyagi, 2015). Meng, Xu & Wang (2019) defined CCs of DHFSs utilizing DMs for applications in engineering management. Garg, Sun & Liu (2023) proposed an algorithm employing CCs on DHFSs for engineering cost management. Boulaaras et al. (2024) investigated DMs on DHFSs and application in medical diagnoses. Ning, Wei & Wei (2024) introduced probabilistic DHFSs and CCs, while Karaaslan & Özlü (2020) explored CCs on type-2 DHFSs and their application in clustering analysis. Furthermore, a robust CCs for probabilistic DHFSs have been investigated by Garg & Kaur (2020).

The concept of NWDHFS is investigated by Narayanamoorthy et al. (2019), it is a contemporary mathematical instrument utilized to describe MGs and NMGs of the uncertain information inherent in the cognitive processes of DMs. Subsequently, Ali & Naeem (2022) established DMs and SMs for NWDHFSs and explored its applicability in medical diagnostics. The NWDHFSs consist of duality preferences, hesitancy, and different levels of preference associated with dual hesitant fuzzy elements (DHFE). On the other hand, dual hesitant fuzzy sets have been applied in multi-expert decision-making, Risk assessments and social choice theory. Data’s relationship to the best or worse case scenario is crucial when it comes to deep accounting.

Therefore, many scenarios addressing uncertainty in NWDHFSs necessitated a correlation among NWDHFSs to enhance outcomes in MCDM. In light of this purpose, we present CCs on NWDHFSs and weighted CCs on NWDHFSs. This study examines the essential axioms of CCs about NWDHFSs and the interrelations among CCs. We present a multi-criteria decision-making technique and associated algorithms based on these CCs. Through the examination of a real estate case study, we derive appropriate assessments of real estate agents for real estate firms utilizing NWDHFSs. We will analyze the techniques and results of our approach in comparison to several existing techniques.

Preliminaries

This part will look at fundamental ideas of HFSs and DHFSs, along with their correlation coefficients developed by Ye (2013) and Garg, Sun & Liu (2023). Through the assessments of specific instances and properties of DHFSs, we assess the limitations of current computational methods comprehensively. We will review particular concepts of NWDHFSs and identify the distinctions between NWDHFS and DHFS.

The concept of HFS enhanced the notion of FSs, as it encompasses a range of values from the internal interval $[0,1]$. It is beneficial when a group of prospectors offers evaluations of an object in an uncertain environment. The notion of HFS, proposed by Torra (2010), provides profound insights about its relevance in real-world uncertain circumstances.

Definition 1 (Torra, 2010). Let $\mathcal{Z}=\left\{z_1,z_2,\cdots ,z_n\right\}$ indicates an original fixed universe of discourse. The HFS $\tilde{\mathcal{S}}$ on $\mathcal{Z}$ is expressed as;

$$\tilde{\mathcal{S}}=\left\{⟨z_i,{\mathrm{\aleph }}_{\tilde{\mathcal{S}}}(z_i)⟩\mid z_i\in \mathcal{Z}\right\},$$where ${\mathrm{\aleph }}_{\tilde{\mathcal{S}}}(z_i)$ represents all possible membership grades from $[0,1]$, it showing hesitancy of certain prospectors toward the object $z_i\in \mathcal{Z}$, that is ${\mathrm{\aleph }}_{\tilde{\mathcal{S}}}(z_i)=\underset{\delta \in [0,1]}{\cup }\left\{\delta \right\}$. For any $i\in (1,2,\cdots ,n)$, ${\mathrm{\aleph }}_{\tilde{\mathcal{S}}}(z_i)$ is known as HFE of HFS $\tilde{\mathcal{S}}$.

Zhu, Xu & Xia (2012) provides the definition of DHFSs from the perspective of Atanassov’s IFSs (Atanassov, 1986). This definition states that both MGs and NMGs are HFEs. It appears that DHFS is a more flexible tool that may be applied in multiple ways DHFS seems to be a more adaptable instrument that may be utilized in many approaches based on current requirements, in contrast to the present FSs and HFSs, while including significantly more information supplied by DMs. DHFS has specific and advantageous characteristics (Narayanamoorthy et al., 2019).

Definition 2 (Narayanamoorthy et al., 2019; Zhu, Xu & Xia, 2012). Let $\mathcal{Z}=\left\{z_1,z_2,\cdots ,z_n\right\}$ indicates an origional fixed universe of discourse. The DHFS $\tilde{\mathcal{D}}$ on $\mathcal{Z}$ is expressed as;

$$\tilde{\mathcal{D}}=\left\{⟨z_i,{\mathrm{\aleph }}_{\tilde{\mathcal{D}}}(z_i),{\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}(z_i)⟩\mid z_i\in \mathcal{Z}\right\},$$where ${\mathrm{\aleph }}_{\tilde{\mathcal{D}}}(z_i)=\underset{\delta \in [0,1]}{\cup }\left\{\delta \right\}$ and ${\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}(z_i)=\underset{\nu \in [0,1]}{\cup }\left\{\nu \right\}$ represents the membership and non membership grades of $z_i\in \mathcal{Z}$ for $\tilde{\mathcal{D}}$, respectively. It holds following axioms; $0\le {\delta }^{+}+{\nu }^{-}\le 1$ and $0\le {\delta }^{-}+{\nu }^{+}\le 1$, where, ${\delta }^{+}=\underset{\delta \in {\mathrm{\aleph }}_{\tilde{\mathcal{D}}}}{\mathrm{m}\mathrm{a}\mathrm{x}\{\delta \}}$, ${\delta }^{-}=\underset{\delta \in {\mathrm{\aleph }}_{\tilde{\mathcal{D}}}}{min\{\delta \}}$, ${\nu }^{+}=\underset{\nu \in {\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}}{max\{\nu \}}$, and ${\nu }^{-}=\underset{\nu \in {\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}}{min\{\nu \}}$. For any $z\in \mathcal{Z}$, $⟨{\mathrm{\aleph }}_{\tilde{\mathcal{D}}}(z),{\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}(z)⟩$ is called duel hesitant fuzzy element (DHFE).

In order to investigate the joint relationship and interconnection between two DHFSs sets with the assistance of an interdependence measure, the CC is an efficient instrument to use. Ye (2013) is the one who presented the fundamental principle of CC on DHFSs.

Definition 3 (Ye, 2013). Let $\mathcal{Z}=\left\{z_1,z_2,\cdots ,z_n\right\}$ indicates an original fixed universe of discourse. Let $\tilde{\mathcal{D}}=\left\{⟨z_i,{\mathrm{\aleph }}_{\tilde{\mathcal{D}}}(z_i),{\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}(z_i)⟩\mid z_i\in \mathcal{Z}\right\}$ and ${\tilde{\mathcal{D}}}^{\prime }=\left\{⟨z_i,{\mathrm{\aleph }}_{\tilde{\mathcal{D}}}^{\prime }(z_i),{\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}^{\prime }(z_i)⟩\mid z_i\in \mathcal{Z}\right\}$. Then CC on $\tilde{\mathcal{D}}$ and ${\tilde{\mathcal{D}}}^{\prime }$ is indicted by

(1) $${\rho }_{DHFS}(\tilde{\mathcal{D}},{\tilde{\mathcal{D}}}^{\prime })=\frac{\sum _{i=1}^n\left(\begin{array}{c}\frac{1}{{\theta }_i}\sum _{k=1}^{{\theta }_i}{\left({\mathrm{\aleph }}_{\tilde{\mathcal{D}}}\right)}_{\varpi (k)}(z_i){\left({\mathrm{\aleph }}_{\tilde{\mathcal{D}}}^{\prime }\right)}_{\varpi (k)}(z_i)\\ +\frac{1}{{\varkappa }_i}\sum _{k^{\prime }=1}^{{\varkappa }_i}{\left({\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}\right)}_{\varpi (k^{\prime })}(z_i){\left({\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}^{\prime }\right)}_{\varpi (k^{\prime })}(z_i)\end{array}\right)}{\sqrt{\sum _{i=1}^n\left(\begin{array}{c}\frac{1}{{\theta }_i}\sum _{k=1}^{{\theta }_i}{\left({\mathrm{\aleph }}_{\tilde{\mathcal{D}}}\right)}_{\varpi (k)}^2(z_i)\\ +\frac{1}{{\varkappa }_i}\sum _{k^{\prime }=1}^{{\varkappa }_i}{\left({\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}\right)}_{\varpi (k^{\prime })}^2(z_i)\end{array}\right)}\sqrt{\sum _{i=1}^n\left(\begin{array}{c}\frac{1}{{\theta }_i}\sum _{k=1}^{{\theta }_i}{\left({\mathrm{\aleph }}_{\tilde{\mathcal{D}}}^{\prime }\right)}_{\varpi (k)}^2(z_i)\\ +\frac{1}{{\varkappa }_i}\sum _{k^{\prime }=1}^{{\varkappa }_i}{\left({\mathrm{{\rm Y}}}_{\tilde{\mathcal{D}}}^{\prime }\right)}_{\varpi (k^{\prime })}^2(z_i)\end{array}\right)}}$$where ${\theta }_i$ and ${\varkappa }_i$ are number values in MGs and NMGs of an element $z_i\in \mathcal{Z}$ respectively and $\varpi (k)\ge \varpi (k+1)$ and $\varpi (k^{\prime })\ge \varpi (k^{\prime }+1)$.

Example 1 Let ${\tilde{\mathcal{D}}}_1,{\tilde{\mathcal{D}}}_2,{\tilde{\mathcal{D}}}_3$ and ${\tilde{\mathcal{D}}}_4$ be four DHFSs;

$$\begin{array}{rl}{\tilde{\mathcal{D}}}_1\hfill & =\left\{⟨z_1,(0.3,0.19),(0.25,0.21)⟩,⟨z_2,(0.4,0.2),(0.24,0.22)⟩\right\}\\ {\tilde{\mathcal{D}}}_2\hfill & =\left\{⟨z_1,(0.45,0.285),(0.375,0.315)⟩,⟨z_2,(0.6,0.3),(0.36,0.33)⟩\right\}\\ {\tilde{\mathcal{D}}}_3\hfill & =\left\{⟨z_1,(0.4,0.3),(0.26,0.20)⟩,⟨z_2,(0.35,0.22),(0.38,0.26)⟩\right\}\\ {\tilde{\mathcal{D}}}_4\hfill & =\left\{⟨z_1,(0.48,0.36),(0.312,0.24)⟩,⟨z_2,(0.42,0.264),(0.456,0.31)⟩\right\}.\end{array}$$

According to the Definition (3), we concluded, ${\rho }_{DHFS}({\tilde{\mathcal{D}}}_1,{\tilde{\mathcal{D}}}_3)=0.9759={\rho }_{DHFS}({\tilde{\mathcal{D}}}_2,{\tilde{\mathcal{D}}}_4)$. The CC ${\rho }_{DHFS}$ (Ye, 2013), have exactly same result for DHFS ${\tilde{\mathcal{D}}}_1,{\tilde{\mathcal{D}}}_3$ and ${\tilde{\mathcal{D}}}_2,{\tilde{\mathcal{D}}}_4$.

One can check that ${\tilde{\mathcal{D}}}_1$ and ${\tilde{\mathcal{D}}}_3$ are proportional to ${\tilde{\mathcal{D}}}_2$ and ${\tilde{\mathcal{D}}}_4$ respectively. As a result, there is a gap in research that has to be filled in order to design more effective CCs that are able to deal with proportional numbers of DHFSs. A more concise description of CC is required in order to solve such a challenge, hence in this article we will utilize the concept of NWDHFSs for some new CCs.

In the MCDM process, the degree of hesitation is influenced not just by the format or magnitude of the DM’s weighted input but also by the vague and subjective feelings of the DMs. According to constrained logic, an inherent characteristic of humanity, individuals’ perceptions of any certain numerical value are likely to exist within a spectrum. The actual configuration of the spectrum may be interval, triangle, trapezoidal, or other forms, which corresponds to the initial assessment data provided by the DMs. To explore the enormous uncertainty of hesitant fuzzy information, Ren, Xu & Wang (2018) developed a visualization that identifies the standard oscillatory spectrum for each value in a HFE.

Definition 4 (Ren, Xu & Wang, 2018). A NWHFS on a given universal set $\mathcal{Z}=\left\{z_1,z_2,\cdots ,z_n\right\}$ is expressed mathematically by;

$$NW=\left\{⟨z,\mathrm{\aleph }(z),\mathrm{\Gamma }\left(\mathrm{\aleph }(z)\right)⟩|z\in \mathcal{Z}\right\}$$where $\mathrm{\aleph }(z)=\{{\tau }_1,{\tau }_2,\cdots ,{\tau }_{\mathrm{\#}\mathrm{\aleph }}\}$ is know as HFE, which can be symbolized as $\mathrm{\aleph }$ and $\mathrm{\#}\mathrm{\aleph }$ denote the number of values in HFE. The function $\mathrm{\Gamma }\left(\mathrm{\aleph }(z)\right)$ is designated as the NWE, enveloping the particulars of DM’s preferences. The NWE indicated as $\mathrm{\Gamma }\left(\mathrm{\aleph }(z)\right)=\{{\tilde{\tau }}_1,{\tilde{\tau }}_2,\cdots ,{\tilde{\tau }}_{\mathrm{\#}\mathrm{\aleph }}\}$ where ${\tilde{\tau }}_k=\{{\alpha }_k^L,{\alpha }_k^M,{\alpha }_k^U\}$ for all $k=\{1,2,\cdots \mathrm{\#}\mathrm{\aleph }\}$, such that ${\alpha }_k^L,{\alpha }_k^M$ and ${\alpha }_k^U$ indicate the lower, middle and upper bounds of the preference degree respectively. The range ${\tilde{\tau }}_k=\{{\alpha }_k^L,{\alpha }_k^M,{\alpha }_k^U\}$ of HFE in NWE can be considered as:

(2) $$\{{\alpha }_k^L,{\alpha }_k^M,{\alpha }_k^U\}=\left\{\begin{array}{c}max\left\{({\tau }_k-g({\tau }_k)),0\right\}\\ (-1+2rpd(\mathrm{\aleph }(z_k)))g({\tau }_k)+{\tau }_k,\\ min\left\{(g({\tau }_k)+{\tau }_k),1\right\}\end{array}\right\}$$where the oscillatory function $g({\tau }_k)$ is defined for regulating the membership degrees,

(3) $$g({\tau }_k)=\sigma .e^{-\frac{{({\tau }_k-\overline{\mathrm{\aleph }})}^2}{2{\sigma }^2}},$$it illustrates the fluctuating nature of preferences. The mean $\overline{\mathrm{\aleph }}$ and the variance $\sigma$ of a HFE are expressed by

(4) $$\overline{\mathrm{\aleph }}=\frac{1}{\mathrm{\#}\mathrm{\aleph }}\stackrel{\mathrm{\#}\mathrm{\aleph }}{\sum _{k=1}}\tau ,$$

(5) $$\sigma =\sqrt{\frac{1}{\mathrm{\#}\mathrm{\aleph }}\stackrel{\mathrm{\#}\mathrm{\aleph }}{\sum _{k=1}}{({\tau }_k-\overline{\mathrm{\aleph }})}^2}.$$

The real preference degree $(rpd)\mathrm{\aleph }$ (Ren, Xu & Wang, 2018; Yager, 1988) of NWHFS is a key component which measures the pattern of DMs towards specific membership degrees. It is computed as follows:

(6) $$rpd(\mathrm{\aleph })=\{\begin{array}{cc}\stackrel{\mathrm{\#}\mathrm{\aleph }}{\sum _{i=1}}{\overline{\tau }}_i\left(\frac{\mathrm{\#}\mathrm{\aleph }-i}{\mathrm{\#}\mathrm{\aleph }-1}\right)if\overline{\mathrm{\aleph }}<0.5& \\ 1-\stackrel{\mathrm{\#}\mathrm{\aleph }}{\sum _{i=1}}{\overline{\tau }}_i\left(\frac{\mathrm{\#}\mathrm{\aleph }-i}{\mathrm{\#}\mathrm{\aleph }-1}\right)if\overline{\mathrm{\aleph }}>0.5\\ 0.5if\overline{\mathrm{\aleph }}=0.5& \end{array}$$where the values ${\tau }_i(i=1,2,\cdots \mathrm{\#}\mathrm{\aleph })$ are normalized as, $\overline{{\tau }_i}={\tau }_i/\left(\stackrel{\mathrm{\#}\mathrm{\aleph }}{\sum _{i=1}}{\tau }_i\right)$. Therefore, the pair $⟨\mathrm{\aleph }(z),\mathrm{\Gamma }\left(\mathrm{\aleph }(z)\right)⟩$ for $z\in \mathcal{Z}$ is known as the normal wiggly hesitant fuzzy element (NWHFE).

The concept of NWDHFS is investigated by Narayanamoorthy et al. (2019), it is a contemporary mathematical instrument utilized to describe MGs and NMGs of the uncertain information inherent in the cognitive processes of DMs. It is expressed as follows;

Definition 5 (Narayanamoorthy et al., 2019). Let $\tilde{\mathcal{D}}=\left\{⟨z,\mathrm{\aleph }(z),\mathrm{{\rm Y}}(z)⟩\mid z\in \mathcal{Z}\right\}$ be a DHFS over $\mathcal{Z}$. A NWDHFS on a given universal set $\mathcal{Z}$ is given mathematically by;

$$A=\left\{⟨z,\mathrm{\aleph }(z),\mathrm{{\rm Y}}(z),\mathrm{\Gamma }\left(\mathrm{\aleph }(z)\right),\mathrm{\Psi }\left(\mathrm{{\rm Y}}(z)\right)⟩|z\in \mathcal{Z}\right\}$$where $\mathrm{\aleph }(z)=\underset{\delta \in [0,1]}{\cup }\left\{\delta \right\}$ and $\mathrm{{\rm Y}}(z)=\underset{\gamma \in [0,1]}{\cup }\left\{\gamma \right\}$ are called membership and non-memberhsip of $z\in \mathcal{Z}$. The functions $\mathrm{\Gamma }\left(\mathrm{\aleph }(z)\right)$ and $\mathrm{\Psi }\left(\mathrm{{\rm Y}}(z)\right)$ are expressed as NWE for $\mathrm{\aleph }(z)$ and $\mathrm{{\rm Y}}(z)$ respectively. Let $\mathrm{\#}\mathrm{\aleph }(z)$ denotes number of values in $\mathrm{\aleph }(z)$. Then $\mathrm{\Gamma }\left(\mathrm{\aleph }(z)\right)=\{{\tilde{\tau }}_1,{\tilde{\tau }}_2,\cdots ,{\tilde{\tau }}_{\mathrm{\#}\mathrm{\aleph }(z)}\}$, ${\tilde{\tau }}_k=\{{\alpha }_k^L,{\alpha }_k^M,{\alpha }_k^U\}$ $(k=1,2,\cdots ,\mathrm{\#}\mathrm{\aleph }(z))$ such that

$$\{{\alpha }_k^L,{\alpha }_k^M,{\alpha }_k^U\}=\left\{\begin{array}{c}max\{({\tau }_k-g({\tau }_k)),0\}\hfill \\ \{-1+2rpd(\mathrm{\aleph }(z_k)\}g({\tau }_k)+{\tau }_k,\hfill \\ min\{(g({\tau }_k)+{\tau }_k),1\}\hfill \end{array}\right\}.$$Furthermore, $\mathrm{\Psi }\left(\mathrm{{\rm Y}}(z)\right)=\{{\tilde{\delta }}_1,{\tilde{\delta }}_2,\cdots ,{\tilde{\delta }}_{\mathrm{\#}\mathrm{{\rm Y}}(z)}\}$, ${\tilde{\delta }}_k=\{{\beta }_j^L,{\beta }_j^M,{\beta }_j^U\}$ $(j=1,2,\cdots \mathrm{\#}\mathrm{{\rm Y}}(z))$ such that

$$\{{\beta }_j^L,{\beta }_j^M,{\beta }_j^U\}=\left\{\begin{array}{c}max\{({\delta }_j-g({\delta }_j)),0\}\hfill \\ \{-1+2rpd(\mathrm{{\rm Y}}(z_j)\}g({\delta }_j)+{\delta }_j,\hfill \\ min\{(g({\delta }_j)+{\delta }_j),1\}\hfill \end{array}\right\}$$where the functions $g({\tau }_j)$, $g({\delta }_j)$ are calculated using the Eqs. (3)–(5) and the reference grades $rpd(\mathrm{\aleph })$, $rpd(\mathrm{{\rm Y}})$ are obtained using the Eq. (6).

To quantitatively demonstrate the notion of NWDHFS, we use the following example.

Example 2 Let $\mathcal{Z}=\left\{z_1,z_2,z_3\right\}$ be a fix set and $\tilde{D}$ be a DHFS in it, then

$$\tilde{D}=\left\{\begin{array}{c}⟨z_1,(0.1,0.2,0.4),(0.2,0.6)⟩,\\ ⟨z_2,(0.4,0.5),(0.3,0.4,0.5)⟩,\\ ⟨z_3,(0.1,0.4),(0.1,0.2,0.4)⟩\end{array}\right\}.$$Utilizing Eqs. (3)–(6), we computed means, standard deviations, and preference degrees, which are presented in Table 1.

Based on the calculation computed in Table 1 and using Definition (4), we formulated NWDHFS $A_{\tilde{D}}$ as follows;

$$A_{\tilde{D}}=\left\{\begin{array}{c}\left(\begin{array}{c}{z}_1,(0.1,0.2,0.4),(0.2,0.6),\hfill \\ \left\{\begin{array}{c}(0.0296,0.0698,0.1704),(0.0797,0.1484,0.3203),\\ (0.349,0.3781,0.451)\end{array}\right\},\hfill \\ \left\{(0.0787,0.1394,0.3213),(0.4787,0.5394,0.7213)\right\}\hfill \end{array}\right),\\ \left(\begin{array}{c}{z}_2,(0.4,0.5),(0.3,0.4,0.5),\hfill \\ \left\{(0.3697,0.3966,0.4303),(0.4697,0.4966,0.5303)\right\},\hfill \\ \left\{\begin{array}{c}(0.2615,0.2936,0.3385),(0.3184,0.3864,0.4816),\\ (0.4615,0.4936,0.5385)\end{array}\right\}\hfill \end{array}\right),\\ \left(\begin{array}{c}{z}_3,(0.1,0.4),(0.1,0.2,0.4),\hfill \\ \left\{(0.009,0.1,0.191),(0.309,0.4,0.491)\right\},\hfill \\ \left\{\begin{array}{c}(0.0296,0.0698,0.1704),(0.0797,0.1484,0.3203),\\ (0.349,0.3781,0.451)\end{array}\right\}\hfill \end{array}\right)\end{array}\right\}.$$

Correlation coefficients on normal wiggly dual hesitant fuzzy sets

The NWDHFS has enhanced preference ranges and resolved ambiguities among grades within DHFS. The establishment of CCs between two NWDHFSs is therefore required in order to establish a relationship between them. This section is dedicated to defining various characteristics of CCs, namely CCs and weighted CCs on NWDHFSs.

Definition 6 Let a reference set $\mathcal{Z}=\left\{z_1,z_2,\cdots ,z_n\right\}$. Consider two NWDHFSs on $\mathcal{Z}$;

$$\begin{array}{rl}A& =\left\{⟨z,{\mathrm{\aleph }}_A(z),{\mathrm{{\rm Y}}}_A(z),{\mathrm{\Gamma }}_A\left({\mathrm{\aleph }}_A(z)\right),{\mathrm{\Psi }}_A\left({\mathrm{{\rm Y}}}_A(z)\right)⟩|z\in \mathcal{Z}\right\},\\ B& =\left\{⟨z,{\mathrm{\aleph }}_B(z),{\mathrm{{\rm Y}}}_B(z),{\mathrm{\Gamma }}_B\left({\mathrm{\aleph }}_B(z)\right),{\mathrm{\Psi }}_B\left({\mathrm{{\rm Y}}}_B(z)\right)⟩|z\in \mathcal{Z}\right\}\end{array}$$where ${\mathrm{\aleph }}_A(z_i)=\{{\tau }_{i1},{\tau }_{i2},\cdots ,{\tau }_{i\mathrm{\#}{\mathrm{\aleph }}_A}\}$, ${\mathrm{{\rm Y}}}_A(z_i)=\{{\delta }_{i1},{\delta }_{i2},\cdots ,{\delta }_{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\}$, ${\mathrm{\aleph }}_B(z_i)=\{{\eta }_{i1},{\eta }_{i2},\cdots ,{\eta }_{i\mathrm{\#}{\mathrm{\aleph }}_B}\}$, ${\mathrm{{\rm Y}}}_B(z_i)=\{{\zeta }_{i1},{\zeta }_{i2},\cdots ,{\zeta }_{i\mathrm{\#}{\mathrm{{\rm Y}}}_B}\}$, and ${\mathrm{\Gamma }}_A\left({\mathrm{\aleph }}_A(z_i)\right)=\{{\tilde{\tau }}_{i1},{\tilde{\tau }}_{i2},\cdots ,{\tilde{\tau }}_{i\mathrm{\#}{\mathrm{\aleph }}_A}\}$, ${\mathrm{\Psi }}_A\left({\mathrm{{\rm Y}}}_A(z_i)\right)=\{{\tilde{\delta }}_{i1},{\tilde{\delta }}_{i2},\cdots ,{\tilde{\delta }}_{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\}$ with ${\tilde{\tau }}_{ik}=\{{\alpha }_{ik}^L,{\alpha }_{ik}^M,{\alpha }_{ik}^U\}$ $(k=1,2,\cdots ,\mathrm{\#}{\mathrm{\aleph }}_A(z_i))$, ${\tilde{\delta }}_{i}j=\{{\beta }_{ij}^L,{\beta }_{ij}^M,{\beta }_{ij}^U\}$ $(j=1,2,\cdots ,\mathrm{\#}{\mathrm{{\rm Y}}}_A(z_i))$ and ${\mathrm{\Gamma }}_B\left({\mathrm{\aleph }}_B(z_i)\right)=\{{\stackrel{⌣}{\eta }}_1,{\stackrel{⌣}{\eta }}_2,\cdots ,{\stackrel{⌣}{\eta }}_{i\mathrm{\#}{\mathrm{\aleph }}_B(z_i)}\}$, ${\mathrm{\Psi }}_B\left({\mathrm{{\rm Y}}}_B(z_i)\right)=\{{\stackrel{⌣}{\zeta }}_1,{\stackrel{⌣}{\zeta }}_2,\cdots ,{\stackrel{⌣}{\zeta }}_{\mathrm{\#}{\mathrm{{\rm Y}}}_B(z_i)}\}$ with ${\stackrel{⌣}{\eta }}_{i{k}^{\prime }}=\{{\mu }_{i{k}^{\prime }}^L,{\mu }_{i{k}^{\prime }}^M,{\mu }_{i{k}^{\prime }}^U\}$ $(k^{\prime }=1,2,\cdots ,\mathrm{\#}{\mathrm{\aleph }}_B(z_i))$, ${\stackrel{⌣}{\zeta }}_{i{j}^{\prime }}=\{{\nu }_{i{j}^{\prime }}^L,{\nu }_{i{j}^{\prime }}^M,{\nu }_{i{j}^{\prime }}^U\}$ $(j^{\prime }=1,2,\cdots ,\mathrm{\#}{\mathrm{{\rm Y}}}_B(z_i))$.

Let $i\mathrm{\#}{\mathrm{\aleph }}_A,i\mathrm{\#}{\mathrm{{\rm Y}}}_A,i\mathrm{\#}{\mathrm{\aleph }}_B$, and $i\mathrm{\#}{\mathrm{{\rm Y}}}_B$ denote $\mathrm{\#}{\mathrm{\aleph }}_A(z_i)$, $\mathrm{\#}{\mathrm{{\rm Y}}}_A(z_i),\mathrm{\#}{\mathrm{\aleph }}_B(z_i)$ and $\mathrm{\#}{\mathrm{{\rm Y}}}_B(z_i)$ for all $i\in (1,2,\cdots ,n)$. Then, the correlation between NWDHFSs A and B is expressed by;

(7) $$\mathfrak{C}(A,B)=\frac{1}{n}\stackrel{n}{\sum _{i=1}}\left[\begin{array}{c}\left({\overline{\mathrm{\aleph }}}_A(z_i)-{\overline{\mathrm{\Gamma }}}_A\left(\mathrm{\aleph }(z_i)\right)\right).\left({\overline{\mathrm{\aleph }}}_B(z_i)-{\overline{\mathrm{\Gamma }}}_B\left(\mathrm{\aleph }(z_i)\right)\right)+\\ \left({\overline{\mathrm{{\rm Y}}}}_A(z_i)-{\overline{\mathrm{\Psi }}}_A\left(\mathrm{{\rm Y}}(z_i)\right)\right).\left({\overline{\mathrm{{\rm Y}}}}_B(z_i)-{\overline{\mathrm{\Psi }}}_B\left(\mathrm{{\rm Y}}(z_i)\right)\right)\end{array}\right]$$where the score functions are given in the following equations,

(8) $${\overline{\mathrm{\aleph }}}_A(z_i)=\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_A}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_A}{\sum _{k=1}}{\tau }_{Aik}$$

(9) $${\overline{\mathrm{\aleph }}}_B(z_i)=\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_B}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_B}{\sum _{k^{\prime }=1}}{\eta }_{Bi{k}^{\prime }}$$

(10) $${\overline{\mathrm{{\rm Y}}}}_A(z_i)=\frac{1}{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\stackrel{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}{\sum _{j=1}}{\delta }_{Aij}$$

(11) $${\overline{\mathrm{{\rm Y}}}}_B(z_i)=\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_B}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_B}{\sum _{j^{\prime }=1}}{\zeta }_{Bi{j}^{\prime }}$$

(12) $${\overline{\mathrm{\Gamma }}}_A\left({\mathrm{\aleph }}_A(z_i)\right)=\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_A}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_A}{\sum _{k=1}}\overline{{\tilde{\tau }}_{Aik}}=\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_A}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_A}{\sum _{k=1}}\frac{{\alpha }_{Aik}^L+{\alpha }_{Aik}^M+{\alpha }_{Aik}^U}{3}$$

(13) $${\overline{\mathrm{\Psi }}}_A\left({\mathrm{{\rm Y}}}_A(z_i)\right)=\frac{1}{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\stackrel{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}{\sum _{j=1}}\overline{{\tilde{\delta }}_{Aij}}=\frac{1}{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\stackrel{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}{\sum _{j=1}}\frac{{\beta }_{Aij}^L+{\beta }_{Aij}^M+{\beta }_{Aij}^U}{3}$$

(14) $${\overline{\mathrm{\Gamma }}}_B\left({\mathrm{\aleph }}_B(z_i)\right)=\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_B}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_B}{\sum _{k^{\prime }=1}}\overline{{\tilde{\eta }}_{Bi{k}^{\prime }}}=\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_B}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_B}{\sum _{k^{\prime }=1}}\frac{{\mu }_{Bi{k}^{\prime }}^L+{\mu }_{Bi{k}^{\prime }}^M+{\mu }_{Bi{k}^{\prime }}^U}{3}$$

(15) $${\overline{\mathrm{\Psi }}}_B\left({\mathrm{{\rm Y}}}_B(z_i)\right)=\frac{1}{i\mathrm{\#}{\mathrm{{\rm Y}}}_B}\stackrel{i\mathrm{\#}{\mathrm{{\rm Y}}}_B}{\sum _{j^{\prime }=1}}\overline{{\stackrel{⌣}{\zeta }}_{Bi{j}^{\prime }}}=\frac{1}{i\mathrm{\#}{\mathrm{{\rm Y}}}_B}\stackrel{i\mathrm{\#}{\mathrm{{\rm Y}}}_B}{\sum _{j^{\prime }=1}}\frac{{\nu }_{Bi{j}^{\prime }}^L+{\nu }_{Bi{j}^{\prime }}^M+{\nu }_{Bi{j}^{\prime }}^U}{3}.$$Therefore, we can easily get correlation $\mathfrak{C}(A,A)$ as given below:

(16) $$\mathfrak{C}(A,A)=\frac{1}{n}\stackrel{n}{\sum _{i=1}}\left[{\left({\overline{\mathrm{\aleph }}}_A(z_i)-{\overline{\mathrm{\Gamma }}}_A\left(\mathrm{\aleph }(z_i)\right)\right)}^2+{\left({\overline{\mathrm{{\rm Y}}}}_A(z_i)-{\overline{\mathrm{\Psi }}}_A\left(\mathrm{{\rm Y}}(z_i)\right)\right)}^2\right]$$

(17) $$=\frac{1}{n}\stackrel{n}{\sum _{i=1}}\left[\begin{array}{c}{\left(\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_A}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_A}{\sum _{k=1}}{\tau }_{Aik}-\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_A}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_A}{\sum _{k=1}}{\tilde{\tau }}_{Aik}\right)}^2+\\ {\left(\frac{1}{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\stackrel{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}{\sum _{j=1}}{\delta }_{Aij}-\frac{1}{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\stackrel{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}{\sum _{j=1}}\overline{{\stackrel{⌣}{\delta }}_{Aij}}\right)}^2\end{array}\right]$$

(18) $$=\frac{1}{n}\stackrel{n}{\sum _{i=1}}\left[\begin{array}{c}{\left(\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_A}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_A}{\sum _{k=1}}{\tau }_{Aik}-\frac{1}{i\mathrm{\#}{\mathrm{\aleph }}_A}\stackrel{i\mathrm{\#}{\mathrm{\aleph }}_A}{\sum _{k=1}}\frac{{\alpha }_{Aik}^L+{\alpha }_{Aik}^M+{\alpha }_{Aik}^U}{3}\right)}^2+\\ {\left(\frac{1}{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\stackrel{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}{\sum _{j=1}}{\delta }_{Aij}-\frac{1}{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\stackrel{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}{\sum _{j=1}}\frac{{\beta }_{Bij}^L+{\beta }_{Bij}^M+{\beta }_{Bij}^U}{3}\right)}^2\end{array}\right].$$Subsequently, utilizing the concepts of $\mathfrak{C}(A,B)$ and $\mathfrak{C}(A,A)$, we defined CC on NWDHFSs in the following approach.

Definition 7 Let a reference set $\mathcal{Z}=\left\{z_1,z_2,\cdots ,z_n\right\}$. Consider two NWDHFSs on $\mathcal{Z}$;

$$\begin{array}{rl}A& =\left\{⟨z,{\mathrm{\aleph }}_A(z),{\mathrm{{\rm Y}}}_A(z),{\mathrm{\Gamma }}_A\left({\mathrm{\aleph }}_A(z)\right),{\mathrm{\Psi }}_A\left({\mathrm{{\rm Y}}}_A(z)\right)⟩|z\in \mathcal{Z}\right\},\\ B& =\left\{⟨z,{\mathrm{\aleph }}_B(z),{\mathrm{{\rm Y}}}_B(z),{\mathrm{\Gamma }}_B\left({\mathrm{\aleph }}_B(z)\right),{\mathrm{\Psi }}_B\left({\mathrm{{\rm Y}}}_B(z)\right)⟩|z\in \mathcal{Z}\right\}\end{array}$$where ${\mathrm{\aleph }}_A(z_i)=\{{\tau }_{i1},{\tau }_{i2},\cdots ,{\tau }_{i\mathrm{\#}{\mathrm{\aleph }}_A}\}$, ${\mathrm{{\rm Y}}}_A(z_i)=\{{\delta }_{i1},{\delta }_{i2},\cdots ,{\delta }_{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\}$, ${\mathrm{\aleph }}_B(z_i)=\{{\eta }_{i1},{\eta }_{i2},\cdots ,{\eta }_{i\mathrm{\#}{\mathrm{\aleph }}_B}\}$, ${\mathrm{{\rm Y}}}_B(z_i)=\{{\zeta }_{i1},{\zeta }_{i2},\cdots ,{\zeta }_{i\mathrm{\#}{\mathrm{{\rm Y}}}_B}\}$, and ${\mathrm{\Gamma }}_A\left({\mathrm{\aleph }}_A(z_i)\right)=\{{\tilde{\tau }}_{i1},{\tilde{\tau }}_{i2},\cdots ,{\tilde{\tau }}_{i\mathrm{\#}{\mathrm{\aleph }}_A}\}$, ${\mathrm{\Psi }}_A\left({\mathrm{{\rm Y}}}_A(z_i)\right)=\{{\tilde{\delta }}_{i1},{\tilde{\delta }}_{i2},\cdots ,{\tilde{\delta }}_{i\mathrm{\#}{\mathrm{{\rm Y}}}_A}\}$ with ${\tilde{\tau }}_k=\{{\alpha }_k^L,{\alpha }_k^M,{\alpha }_k^U\}$ $(k=1,2,\cdots \mathrm{\#}{\mathrm{\aleph }}_A(z_i))$, ${\tilde{\delta }}_j=\{{\delta }_j^L,{\delta }_j^M,{\delta }_j^U\}$ $(j=1,2,\cdots \mathrm{\#}{\mathrm{{\rm Y}}}_A(z_i))$ and ${\mathrm{\Gamma }}_B\left({\mathrm{\aleph }}_B(z_i)\right)=\{{\stackrel{⌣}{\eta }}_1,{\stackrel{⌣}{\eta }}_2,\cdots ,{\stackrel{⌣}{\eta }}_{\mathrm{\#}{\mathrm{\aleph }}_B(z_i)}\}$, ${\mathrm{\Psi }}_B\left({\mathrm{{\rm Y}}}_B(z_i)\right)=\{{\stackrel{⌣}{\zeta }}_1,{\stackrel{⌣}{\zeta }}_2,\cdots ,{\stackrel{⌣}{\zeta }}_{\mathrm{\#}{\mathrm{{\rm Y}}}_B(z_i)}\}$ with ${\stackrel{⌣}{\eta }}_{k^{\prime }}=\{{\eta }_{k^{\prime }}^L,{\eta }_{k^{\prime }}^M,{\eta }_{k^{\prime }}^U\}$ $(k^{\prime }=1,2,\cdots \mathrm{\#}{\mathrm{\aleph }}_B(z_i))$, ${\stackrel{⌣}{\zeta }}_{j^{\prime }}=\{{\zeta }_{j^{\prime }}^L,{\zeta }_{j^{\prime }}^M,{\zeta }_{j^{\prime }}^U\}$ $(j^{\prime }=1,2,\cdots \mathrm{\#}{\mathrm{{\rm Y}}}_B(z_i))$. The correlation coefficient between two NWDHFSs A and B is given by;

(19) $${\mathrm{\Delta }}_{NWD{H}_1}(A,B)=\frac{\mathfrak{C}(A,B)}{\sqrt{\mathfrak{C}(A,A)}\cdot \sqrt{\mathfrak{C}}(B,B)}$$

(20) $$=\frac{\frac{1}{n}\stackrel{n}{\sum _{i=1}}\left[\begin{array}{c}\left({\overline{\mathrm{\aleph }}}_A(z_i)-{\overline{\mathrm{\Gamma }}}_A\left(\mathrm{\aleph }(z_i)\right)\right)\cdot \left({\overline{\mathrm{\aleph }}}_B(z_i)-{\overline{\mathrm{\Gamma }}}_B\left(\mathrm{\aleph }(z_i)\right)\right)\hfill \\ +\left({\overline{\mathrm{{\rm Y}}}}_A(z_i)-{\overline{\mathrm{\Psi }}}_A\left(\mathrm{{\rm Y}}(z_i)\right)\right)\cdot \left({\overline{\mathrm{{\rm Y}}}}_B(z_i)-{\overline{\mathrm{\Psi }}}_B\left(\mathrm{{\rm Y}}(z_i)\right)\right)\hfill \end{array}\right]}{\sqrt{\frac{1}{n}\stackrel{n}{\sum _{i=1}}\left[\begin{array}{c}{\left({\overline{\mathrm{\aleph }}}_A(z_i)-{\overline{\mathrm{\Gamma }}}_A\left(\mathrm{\aleph }(z_i)\right)\right)}^2\hfill \\ +{\left({\overline{\mathrm{{\rm Y}}}}_A(z_i)-{\overline{\mathrm{\Psi }}}_A\left(\mathrm{{\rm Y}}(z_i)\right)\right)}^2\hfill \end{array}\right]}\cdot \sqrt{\frac{1}{n}\stackrel{n}{\sum _{i=1}}\left[\begin{array}{c}{\left({\overline{\mathrm{\aleph }}}_B(z_i)-{\overline{\mathrm{\Gamma }}}_B\left(\mathrm{\aleph }(z_i)\right)\right)}^2\hfill \\ +{\left({\overline{\mathrm{{\rm Y}}}}_B(z_i)-{\overline{\mathrm{\Psi }}}_B\left(\mathrm{{\rm Y}}(z_i)\right)\right)}^2\hfill \end{array}\right]}}.$$

Theorem 1 The formula of CC in Eq. (19) between two NWDHFSs verifies the given properties:

• (I) ${\mathrm{\Delta }}_{NWD{H}_1}(A,B)={\mathrm{\Delta }}_{NWD{H}_1}(B,A)$

• (II) ${\mathrm{\Delta }}_{NWD{H}_1}(A,A)=1$

• (III) $|{\mathrm{\Delta }}_{NWD{H}_1}(A,B)|\le 1$