**2. Preliminaries**

This section states some requisites concerned with the DHFSs and the correlation coefficients while applying in the real-life application during DM process.

**Definition 2.1 [31]** A set <sup>A</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> ð Þ <sup>x</sup> <sup>j</sup>x∈X, 0<sup>≤</sup> <sup>μ</sup>A<sup>~</sup> ð Þ <sup>x</sup> <sup>≤</sup> <sup>1</sup> , defined on the universal set XX, is said to be an fuzzy set (FS), where μA<sup>~</sup> ð Þ x represents the degree of membership of the element x xin A. ~

**Definition 2.2 [1]** A set <sup>A</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> ð Þ <sup>x</sup> , <sup>ν</sup>A<sup>~</sup> ð Þ <sup>x</sup> <sup>j</sup>x∈X, 0 <sup>≤</sup>μA<sup>~</sup> ð Þ <sup>x</sup> <sup>≤</sup> 1, 0≤νA<sup>~</sup> ð Þ x ≤1, μA<sup>~</sup> ð Þþ x νA<sup>~</sup> ð Þ x ≤1g, defined on the universal set X, is said to be an intuitionistic fuzzy set (IFS), where, μA<sup>~</sup> ð Þ x and νA<sup>~</sup> ð Þ x represents the degree of membership and degree of non-membership respectively of the element x in A. The ~ pair μA<sup>~</sup> , νA<sup>~</sup> is called an intuitionistic fuzzy number (IFN) or an intuitionistic fuzzy (IFV), where, μA<sup>~</sup> ∈½ � 0, 1 , νA<sup>~</sup> ∈½ � 0, 1 , μA<sup>~</sup> þ νA<sup>~</sup> ≤1.

**Definition 2.3 [29]** Let X be an initial universe of objects. A set A on X defined ~ as <sup>A</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> <sup>j</sup>x∈<sup>X</sup> is called a hesitant fuzzy set (HFS), where <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> is a mapping defined by <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> : <sup>X</sup> ! ½ � 0, 1 here, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> is a set of some different values in [0,1] and <sup>0</sup> s<sup>0</sup> represent the number of possible membership degrees of the element x∈X to A. For convenience, we call <sup>~</sup> <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> as a hesitant fuzzy element (HFE).

*Modified Expression to Evaluate the Correlation Coefficient of Dual Hesitant Fuzzy Sets… DOI: http://dx.doi.org/10.5772/intechopen.96474*

**Definition 2.4 [29]** Let X be an initial universe of objects. A set A on X then for ~ a given HFE <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � �, its lower and upper bounds are defined as <sup>μ</sup>� <sup>A</sup><sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � <sup>¼</sup> min <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � and <sup>μ</sup><sup>þ</sup> <sup>A</sup><sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � <sup>¼</sup> max <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � �, respectively, where <sup>0</sup> s0 represent the number of possible membership degrees of the element x∈X to A. ~

**Definition 2.5 [29]** Let X be an initial universe of objects. A set A on X then for a ~ given HFE <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � �, Aenv <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � � � is called the envelope of <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � which is denoted as μ� <sup>A</sup><sup>~</sup> , 1 � μ<sup>þ</sup> A~ � �, with the lower bound <sup>μ</sup>� <sup>A</sup><sup>~</sup> and upper bound μ<sup>þ</sup> <sup>A</sup><sup>~</sup> . Also, Aenv <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � � � establishes the relation between HFS and IFS i.e., Aenv <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � � � <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � �, <sup>ν</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � � � � � , where <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � <sup>¼</sup> <sup>μ</sup>� <sup>A</sup><sup>~</sup> and <sup>ν</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � <sup>¼</sup> <sup>1</sup> � <sup>μ</sup><sup>þ</sup> A~ .

**Definition 2.6 [29]** For a HFE μA<sup>~</sup> , s μA<sup>~</sup> � � <sup>¼</sup> <sup>1</sup> <sup>l</sup>ð Þ <sup>μ</sup>A<sup>~</sup> � �P <sup>γ</sup>∈μA<sup>~</sup> γ is called the score function of μA<sup>~</sup> , where l μA<sup>~</sup> � � is the number of the values in <sup>μ</sup>A<sup>~</sup> . For any two HFEs <sup>μ</sup>A~<sup>1</sup> and μA~<sup>2</sup> , the comparison between two HFEs is done as follows:

$$\begin{aligned} \text{i. } \text{If } \mathbf{s}\left(\mu\_{\tilde{\mathbf{A}}\_{\tilde{\mathbf{A}}}}\right) &> \mathbf{s}\left(\mu\_{\tilde{\mathbf{A}}\_{\tilde{\mathbf{A}}}}\right), \text{ then } \mu\_{\tilde{\mathbf{A}}\_{1}} > \mu\_{\tilde{\mathbf{A}}\_{2}}. \\\\ \text{ii. } \text{If } \mathbf{s}\left(\mu\_{\tilde{\mathbf{A}}\_{1}}\right) &= \mathbf{s}\left(\mu\_{\tilde{\mathbf{A}}\_{2}}\right), \text{ then } \mu\_{\tilde{\mathbf{A}}\_{1}} = \mu\_{\tilde{\mathbf{A}}\_{2}}. \end{aligned}$$

Let μA~<sup>1</sup> and μA~<sup>2</sup> be two HFEs such that l μA~<sup>1</sup> � � 6¼ <sup>l</sup> <sup>μ</sup>A~<sup>2</sup> � �. For convenience, let l <sup>¼</sup> max l μA~<sup>1</sup> � �, l <sup>μ</sup>A~<sup>2</sup> n o � � , then while comparing them, the shorter one is extended by adding the same value till both are of same length. The selection of the value to be added is dependent on the decision makers risk preferences. For example (adopted from 28), let μA~<sup>1</sup> ¼ f g 0*:*1, 0*:*2, 0*:*3 , μA~<sup>2</sup> ¼ f g 0*:*4, 0*:*5 and l μA~<sup>1</sup> � � <sup>&</sup>gt;<sup>l</sup> <sup>μ</sup>A~<sup>2</sup> � �, then for the correct arithmetic operations μA~<sup>2</sup> must be extended to μA~<sup>2</sup> 0 , i.e. either μA~<sup>2</sup> <sup>0</sup> ¼ f g 0*:*4, 0*:*5, 0*:*5 as an optimist or μA~<sup>2</sup> <sup>0</sup> ¼ f g 0*:*4, 0*:*4, 0*:*5 as a pessimist depending on the risk taking factor of the decision-maker though their results would vary definitely.

**Definition 2.7 [29]** A set A on X defined as <sup>~</sup> <sup>A</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � �, <sup>ν</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>t</sup> � � � � <sup>j</sup>x∈<sup>X</sup> � � is called a DHFS, where, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � �, <sup>ν</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>t</sup> � � is a mapping defined by <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � �, <sup>ν</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>t</sup> � � : <sup>X</sup> ! ½ � 0, 1 , here <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � �, <sup>ν</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>t</sup> � � is a set of some different values in [0,1], <sup>0</sup> s0 represent the number of possible membership degrees and <sup>0</sup> t <sup>0</sup> represent the number of possible non membership degrees of the element x∈X to A. For convenience, we ~ call d <sup>¼</sup> <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � �, <sup>ν</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>t</sup> � � � � as a dual hesitant fuzzy element (DHFE).

**Definition 2.8 [29]** Let d1 ¼ μA~<sup>1</sup> , νA~<sup>1</sup> n o and d2 <sup>¼</sup> <sup>μ</sup>A~<sup>2</sup> , νA~<sup>2</sup> n o be any two DHFEs, then the score function for DHFSs dið Þ i ¼ 1, 2 is defined as

$$\sigma(d\_i) = \left(\frac{1}{l\binom{\mu\_{\hat{\lambda}\_i}}{\mu\_{\hat{\lambda}\_i}}}\right)\_{\boldsymbol{\eta}} \sum\_{\boldsymbol{\pi} \in \mu\_{\hat{\lambda}\_i}} \boldsymbol{\eta} - \left(\frac{1}{m\binom{\boldsymbol{\nu}}{\nu\_{\hat{\lambda}\_i}}}\right)\_{\boldsymbol{\eta}} \sum\_{\boldsymbol{\pi} \in \nu\_{\hat{\lambda}\_i}} \boldsymbol{\eta} \ (i = 1, 2) \text{ and the accuracy function for } l$$

$$\text{DHFSs } \text{d}\_{i}(\text{i}=\text{1,2}) \text{ is defined as } p(d\_{i}) = \left(\frac{1}{l\binom{\mu\_{\hat{A}\_{i}}}{\mu\_{\hat{A}\_{i}}}}\right) \sum\_{\boldsymbol{\eta} \in \mu\_{\hat{A}\_{i}}} \boldsymbol{\eta} + \left(\frac{1}{m\binom{\boldsymbol{\eta}}{\nu\_{\hat{A}\_{i}}}}\right) \sum\_{\boldsymbol{\eta} \in \nu\_{\hat{A}\_{i}}} \boldsymbol{\eta} \ (\boldsymbol{i}=\textbf{1},\textbf{2})$$

where *l μA*~*<sup>i</sup>* � � and *<sup>m</sup> <sup>ν</sup>A*~*<sup>i</sup>* � � are the number of the values in *<sup>μ</sup>A*~*<sup>i</sup>* and *<sup>ν</sup>A*~*<sup>i</sup>* respectively. For any two DHFEs d1 and d2, the comparison between two DHFEs is done as follows:

i. If s dð Þ<sup>1</sup> >s dð Þ<sup>2</sup> , then d1 >d2.

ii. If s dð Þ¼ <sup>1</sup> s dð Þ<sup>2</sup> , then check the accuracy function of DHFSs

$$\mathbf{1}. \text{If } \mathbf{s}(\mathbf{d}\_1) > \mathbf{s}(\mathbf{d}\_2), \text{ then } \mathbf{d}\_1 > \mathbf{d}\_2.$$

2. If s dð Þ¼ <sup>1</sup> s dð Þ<sup>2</sup> , then d1 ¼ d2.

**Definition 2.9 [29]** Correlation coefficient of HFSs.

The values in HFEs are generally not in order, so they are arranged in descending order i.e., for HFE <sup>μ</sup>A<sup>~</sup> , let <sup>σ</sup> : ð Þ! 1, 2, … , n ð Þ 1, 2, … , n be such that <sup>μ</sup>A~σð Þ<sup>j</sup> <sup>≥</sup>μA~σð Þ <sup>j</sup>þ<sup>1</sup> for j <sup>¼</sup> 1, 2, … , n � 1 and <sup>μ</sup>A~σð Þ<sup>j</sup> be the jth largest value in <sup>μ</sup>A<sup>~</sup> .

**Definition 2.9.1** Let X ¼ f g x1, x2, … , xn be an initial universe of objects and a set A on X defined as <sup>~</sup> <sup>A</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � � � <sup>j</sup>x∈<sup>X</sup> � � be a HFS, then the information energy of A is defined as E <sup>~</sup> HFS <sup>A</sup>~� � <sup>¼</sup> <sup>P</sup><sup>n</sup> i¼1 1 li Pli <sup>j</sup>¼<sup>1</sup>μ<sup>2</sup> <sup>A</sup>~σð Þ<sup>j</sup> ð Þ xi � �, where li <sup>¼</sup> <sup>l</sup> <sup>μ</sup>A<sup>~</sup> ð Þ xi � � denotes the total number of membership values in μA<sup>~</sup> ð Þ xi , xi ∈X.

**Definition 2.9.2** Let X <sup>¼</sup> f g x1, x2, … , xn be a universal set and a set <sup>A</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � � � <sup>j</sup>x∈<sup>X</sup> � � and <sup>B</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>B<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � � � <sup>j</sup>x∈<sup>X</sup> � � be any two HFSs on X, then the correlation between A and <sup>~</sup> B is defined by C <sup>~</sup> HFS A, <sup>~</sup> <sup>B</sup><sup>~</sup> � � <sup>¼</sup>

P<sup>n</sup> i¼1 1 li Pli <sup>j</sup>¼<sup>1</sup>μA~σð Þ<sup>j</sup> ð Þ xi <sup>μ</sup>B~σð Þ<sup>j</sup> ð Þ xi � � where li <sup>¼</sup> max l <sup>μ</sup>A<sup>~</sup> ð Þ xi � �, l <sup>μ</sup>B<sup>~</sup> ð Þ xi � � � � for each xi ∈X. Also, when l μA<sup>~</sup> ð Þ xi � � 6¼ <sup>l</sup> <sup>μ</sup>B<sup>~</sup> ð Þ xi � �, then they can be made equal by adding number of membership values in HFE which has least number of membership values in it. This can be done by adding the smallest membership values to make the lengths of both HFE A and <sup>~</sup> B equal i.e. l <sup>~</sup> <sup>μ</sup>A<sup>~</sup> ð Þ xi � � <sup>¼</sup> <sup>l</sup> <sup>μ</sup>B<sup>~</sup> ð Þ xi � �. For example, <sup>A</sup><sup>~</sup> <sup>¼</sup> f g h i <sup>0</sup>*:*3, 0*:*6, 0*:*<sup>8</sup> and <sup>B</sup><sup>~</sup> <sup>¼</sup> f g h i <sup>0</sup>*:*5, 0*:*<sup>4</sup> , be any two HFSs and their lengths are not equal therefore it can be made equal as <sup>A</sup><sup>~</sup> <sup>¼</sup> f g h i <sup>0</sup>*:*3, 0*:*6, 0*:*<sup>8</sup> and <sup>B</sup><sup>~</sup> <sup>¼</sup> f g h i 0*:*5, 0*:*4, 0*:*4 respectively.

**Definition 2.9.3** Let X <sup>¼</sup> f g x1, x2, … , xn be a universal set and a set <sup>A</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � � � <sup>j</sup>x∈<sup>X</sup> � � and <sup>B</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>B<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> � � � � <sup>j</sup>x∈<sup>X</sup> � � be any two HFSs on X, then the correlation coefficient between A and ~ B is defined by ~

$$\rho\_{\text{HFS}}\left(\tilde{\mathbf{A}},\tilde{\mathbf{B}}\right) = \frac{\mathbf{c}\_{\text{HFS}}(\mathbf{A},\mathbf{\tilde{\mathbf{\mathbf{\tilde{A}}}}})}{\sqrt{\mathbf{E\_{\text{HFS}}}(\tilde{\mathbf{A}})\sqrt{\mathbf{E\_{\text{HFS}}}(\mathbf{\tilde{\mathbf{\tilde{B}}}})}}} = \frac{\sum\_{i=1}^{n}\left(\frac{1}{2}\sum\_{j=1}^{l\_{i}}\mu\_{\text{A\sigma(j)}}(\mathbf{x}\_{i})\,\mu\_{\text{B\sigma(j)}}(\mathbf{x}\_{i})\right)}{\sqrt{\sum\_{i=1}^{n}\left(\frac{1}{l\_{i}}\sum\_{j=1}^{l\_{i}}\mu\_{\text{A\sigma(j)}}^{2}(\mathbf{x}\_{i})\right)}\sqrt{\sum\_{i=1}^{n}\left(\frac{1}{l\_{i}}\sum\_{j=1}^{l\_{i}}\mu\_{\text{B\sigma(j)}}^{2}(\mathbf{x}\_{i})\right)}}.$$

#### **Definition 2.10 [29] Correlation coefficient of DHFSs.**

The values in DHFEs are generally not in order, so they are arranged in descending order i.e., for DHFE d ¼ μA<sup>~</sup> , νA<sup>~</sup> � �, let <sup>σ</sup> : ð Þ! 1, 2, … , n ð Þ 1, 2, … , n be such that <sup>μ</sup>A~σð Þ<sup>s</sup> <sup>≥</sup>μA~σð Þ <sup>s</sup>þ<sup>1</sup> for s <sup>¼</sup> 1, 2, … , n � 1, and <sup>μ</sup>A~σð Þ<sup>s</sup> be the sth largest value in <sup>μ</sup>A<sup>~</sup> ; let <sup>δ</sup> : ð Þ! 1, 2, … , m ð Þ 1, 2, … , m be such that <sup>ν</sup>A~δð Þ<sup>t</sup> <sup>≥</sup>νA~δð Þ <sup>t</sup>þ<sup>1</sup> for t <sup>¼</sup> 1, 2, … , m � 1, and <sup>ν</sup>A~δð Þ<sup>t</sup> be the tth largest value in <sup>ν</sup>A<sup>~</sup> .

**Definition 2.10.1** Let X ¼ f g x1, x2, … , xn be an initial universe of objects and a set A on X defined as <sup>~</sup> <sup>A</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> , <sup>ν</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>t</sup> , � � � � � � <sup>j</sup>x∈<sup>X</sup> � � be a DHFS, then the information energy of A is defined as E ~ DHFS A~ � � <sup>¼</sup>

P<sup>n</sup> i¼1 1 ki Pki <sup>s</sup>¼<sup>1</sup>μ<sup>2</sup> <sup>A</sup>~σð Þ<sup>s</sup> ð Þþ xi 1 li Pli <sup>t</sup>¼<sup>1</sup>ν<sup>2</sup> <sup>A</sup>~σð Þ<sup>t</sup> ð Þ xi � �, where ki <sup>¼</sup> <sup>k</sup> <sup>μ</sup>A<sup>~</sup> ð Þ xi � � denotes the total number of membership values in μA<sup>~</sup> ð Þ xi and li ¼ l νA<sup>~</sup> ð Þ xi � � denotes the total number of non-membership values in νA<sup>~</sup> ð Þ xi respectively.

*Modified Expression to Evaluate the Correlation Coefficient of Dual Hesitant Fuzzy Sets… DOI: http://dx.doi.org/10.5772/intechopen.96474*

**Definition 2.10.2** Let X <sup>¼</sup> f g x1, x2, … , xn be a universal set and a set <sup>A</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> , <sup>ν</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>t</sup> , � � � � � � <sup>j</sup>x∈<sup>X</sup> � � and <sup>B</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>B<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> , <sup>ν</sup>B<sup>~</sup> <sup>x</sup>ð Þ<sup>t</sup> , � � � � � � <sup>j</sup>x∈<sup>X</sup> � � be any two

DHFSs on X, then the correlation between A and <sup>~</sup> B is defined by C <sup>~</sup> DHFS A, <sup>~</sup> <sup>B</sup><sup>~</sup> � � <sup>¼</sup> P<sup>n</sup> i¼1 1 ki Pki <sup>s</sup>¼1μA~σð Þ<sup>s</sup> ð Þ xi <sup>μ</sup>B~σð Þ<sup>s</sup> ð Þþ xi 1 li Pli <sup>t</sup>¼1νA~δð Þ<sup>t</sup> ð Þ xi <sup>ν</sup>B~δð Þ<sup>t</sup> ð Þ xi � � where ki <sup>¼</sup> max k μA<sup>~</sup> ð Þ xi � �, k <sup>μ</sup>B<sup>~</sup> ð Þ xi � � � � li <sup>¼</sup> max l <sup>ν</sup>A<sup>~</sup> ð Þ xi � �, l <sup>ν</sup>B<sup>~</sup> ð Þ xi � � � � for each xi ∈X. Also, when k μA<sup>~</sup> ð Þ xi � � 6¼ <sup>k</sup> <sup>μ</sup>B<sup>~</sup> ð Þ xi � � or l <sup>ν</sup>A<sup>~</sup> ð Þ xi � � 6¼ <sup>l</sup> <sup>ν</sup>B<sup>~</sup> ð Þ xi � �, then they can be made equal by adding some elements in DHFE which has least number of elements in it. This can be done by adding the smallest membership values or smallest non-membership values to make the lengths of both DHFE A and <sup>~</sup> B equal i.e. k <sup>~</sup> <sup>μ</sup>A<sup>~</sup> ð Þ xi � � <sup>¼</sup> <sup>k</sup> <sup>μ</sup>B<sup>~</sup> ð Þ xi � � or l νA<sup>~</sup> ð Þ xi � � <sup>¼</sup> <sup>l</sup> <sup>ν</sup>B<sup>~</sup> ð Þ xi � �. For example, <sup>A</sup><sup>~</sup> <sup>¼</sup> f g h i f g <sup>0</sup>*:*3, 0*:*<sup>8</sup> , 0f g *:*2, 0*:*<sup>5</sup> and <sup>B</sup><sup>~</sup> <sup>¼</sup> f g h i f g 0*:*1, 0*:*7 , 0f g *:*8, 0*:*9, 0*:*4 , be any two DHFSs and their lengths are not equal therefore it can be made equal as <sup>A</sup><sup>~</sup> <sup>¼</sup> f g h i f g <sup>0</sup>*:*3, 0*:*<sup>8</sup> , 0f g *:*2, 0, 5, 0*:*<sup>2</sup> and <sup>B</sup><sup>~</sup> <sup>¼</sup> f g h i f g 0*:*1, 0*:*7 , 0f g *:*8, 0*:*9, 0*:*4 respectively.

**Definition 2.10.3** Let X <sup>¼</sup> f g x1, x2, … , xn be a universal set and a set <sup>A</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> , <sup>ν</sup>A<sup>~</sup> <sup>x</sup>ð Þ<sup>t</sup> , � � � � � � <sup>j</sup>x∈<sup>X</sup> � � and <sup>B</sup><sup>~</sup> <sup>¼</sup> x, <sup>μ</sup>B<sup>~</sup> <sup>x</sup>ð Þ<sup>s</sup> , <sup>ν</sup>B<sup>~</sup> <sup>x</sup>ð Þ<sup>t</sup> , � � � � � � <sup>j</sup>x∈<sup>X</sup> � � be any two DHFSs on X, then the correlation coefficient between A and ~ B is defined by ~

$$\rho\_{\rm DHFS}(\tilde{\mathbf{A}}, \tilde{\mathbf{B}}) = \frac{\mathbf{C}\_{\rm DHES}(\tilde{\mathbf{A}}, \tilde{\mathbf{B}})}{\sqrt{\mathbf{E}\_{\rm DHES}(\tilde{\mathbf{A}})} \sqrt{\mathbf{E}\_{\rm DHFS}(\tilde{\mathbf{B}})}}$$

$$=\frac{\sum\_{i=1}^{n}\left(\frac{1}{\mathbf{k}\_{\mathrm{i}}}\sum\_{\mathbf{s}=1}^{\mathbf{k}\_{\mathrm{i}}}\mu\_{\mathrm{A\sigma(\mathbf{s})}}^{\cdot}(\mathbf{x}\_{\mathrm{i}})\,\mu\_{\mathrm{B\sigma(\mathbf{s})}}^{\cdot}(\mathbf{x}\_{\mathrm{i}})+\frac{1}{\mathbf{k}\_{\mathrm{i}}}\sum\_{\mathbf{s}=1}^{\mathbf{k}\_{\mathrm{i}}}\nu\_{\mathrm{A\bar{\sigma}(\mathbf{t})}}^{\cdot}(\mathbf{x}\_{\mathrm{i}})\,\nu\_{\mathrm{B\bar{\sigma}(\mathbf{t})}}^{\cdot}(\mathbf{x}\_{\mathrm{i}})\right)}{\sqrt{\sum\_{i=1}^{n}\left(\frac{1}{\mathbf{k}\_{\mathrm{i}}}\sum\_{\mathbf{s}=1}^{\mathbf{k}\_{\mathrm{i}}}\mu\_{\mathrm{A\sigma(\mathbf{s})}}^{\cdot}(\mathbf{x}\_{\mathrm{i}})+\frac{1}{\mathbf{k}\_{\mathrm{i}}}\sum\_{\mathbf{s}=1}^{\mathbf{k}\_{\mathrm{i}}}\nu\_{\mathrm{A\sigma(\mathbf{t})}}^{\cdot}(\mathbf{x}\_{\mathrm{i}})\right)}\sqrt{\sum\_{i=1}^{n}\left(\frac{1}{\mathbf{k}\_{\mathrm{i}}}\sum\_{\mathbf{s}=1}^{\mathbf{k}\_{\mathrm{i}}}\mu\_{\mathrm{B\sigma(\mathbf{s})}}^{\cdot}(\mathbf{x}\_{\mathrm{i}})+\frac{1}{\mathbf{k}\_{\mathrm{i}}}\sum\_{\mathbf{s}=1}^{\mathbf{k}\_{\mathrm{i}}}\nu\_{\mathrm{B\sigma(\mathbf{t})}}^{\cdot}(\mathbf{x}\_{\mathrm{i}})\right)}}\,\mathrm{I}$$

#### **3. Brief review of the existing CoCf between two DHFSs**

In the existing literature [29] it is claimed that, there does not exist any expression to evaluate the CoCf between two DHFSs, so to fill this gap, the expression 1ð Þ is proposed to evaluate the weighted CoCf between two DHFSs A ¼ hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi , gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi n o D E and B <sup>¼</sup> hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi , gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi n o D E , where i <sup>¼</sup> 1, 2, … , n, and s, t represents the number of values in hA<sup>σ</sup>ð Þ<sup>s</sup> and gA<sup>σ</sup>ð Þ<sup>t</sup> respectively.

$$\rho\_{\text{WDHFS}}(\mathbf{A},\mathbf{B}) = \frac{\sum\_{i=1}^{n} \text{w}\_{i} \Big( \frac{1}{\mathbf{h}} \sum\_{s=1}^{\mathbf{h}\_{i}} \Big( \mathbf{h}\_{\text{Av}(\mathbf{z})}(\mathbf{x}) \, \mathbf{h}\_{\text{Bw}(\mathbf{z})}(\mathbf{x}) \Big) + \frac{1}{\mathbf{h}} \sum\_{s=1}^{\mathbf{h}\_{i}} \Big( \mathbf{g}\_{\text{Av}(\mathbf{z})}(\mathbf{x}) \, \mathbf{g}\_{\text{Bw}(\mathbf{z})}(\mathbf{x}) \Big) \Big)}{\sqrt{\sum\_{s=1}^{\mathbf{n}} \text{w}\_{i} \Big( \frac{1}{\mathbf{h}} \sum\_{s=1}^{\mathbf{h}\_{i}} \Big( \mathbf{h}\_{\text{Av}(\mathbf{z})}^{2}(\mathbf{x}) \Big) + \frac{1}{\mathbf{h}} \sum\_{s=1}^{\mathbf{h}\_{i}} \text{w}\_{i} \Big( \frac{1}{\mathbf{h}} \sum\_{s=1}^{\mathbf{h}\_{i}} \Big( \mathbf{h}\_{\text{Bw}(\mathbf{z})}^{2}(\mathbf{x}) \Big) + \frac{1}{\mathbf{h}} \sum\_{s=1}^{\mathbf{h}\_{i}} \Big( \mathbf{g}\_{\text{Bw}(\mathbf{z})}^{2}(\mathbf{x}) \Big)}} \tag{1}$$

where,


iv. ki represents the number of values in hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi .

v. li represents the number of values in gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi .

Its claimed that if wi <sup>¼</sup> <sup>1</sup> <sup>n</sup> for all i then the expression 1ð Þ will be transformed into expression 2ð Þ.

$$\rho\_{\text{DHFS}}(\mathbf{A}, \mathbf{B}) = \frac{\sum\_{i=1}^{n} \left( \frac{1}{\hbar} \sum\_{s=1}^{\mathbf{k}\_{i}} \left( \mathbf{h}\_{\text{A\sigma(s)}}(\mathbf{x}\_{i}) \mathbf{h}\_{\text{B\sigma(a)}}(\mathbf{x}\_{i}) \right) + \frac{1}{\hbar} \sum\_{i=1}^{l} \left( \mathbf{g}\_{\text{A\sigma(i)}}(\mathbf{x}\_{i}) \mathbf{g}\_{\text{B\sigma(v)}}(\mathbf{x}\_{i}) \right) \right)}{\sqrt{\sum\_{i=1}^{n} \left( \frac{1}{\hbar} \sum\_{s=1}^{\mathbf{k}\_{i}} \left( \mathbf{h}\_{\text{A\sigma(a)}}^{2}(\mathbf{x}\_{i}) \right) + \frac{1}{\hbar} \sum\_{i=1}^{l} \left( \mathbf{g}\_{\text{A\sigma(i)}}^{2}(\mathbf{x}\_{i}) \right) \right)} \sqrt{\sum\_{i=1}^{n} \left( \frac{1}{\hbar} \sum\_{s=1}^{\mathbf{k}\_{i}} \left( \mathbf{h}\_{\text{B\sigma(a)}}^{2}(\mathbf{x}\_{i}) \right) + \frac{1}{\hbar} \sum\_{i=1}^{l} \left( \mathbf{g}\_{\text{B\sigma(i)}}^{2}(\mathbf{x}\_{i}) \right) \right)}} \tag{2}$$

#### **3.1 Gaps in the existing weighted CoCf for DHFSs**

In this paper, it is claimed that the existing CoCf 1ð Þ [29] is not valid in its present form. To prove that this claim is valid, there is a need to discuss the origin of the expressions 1ð Þ. Therefore, the same is discussed in this section.

It can be easily verified that the expression 1ð Þ can be obtained mathematically in the following manner:

Xn i¼1 wi 1 ki X ki s¼1 hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þþ xi 1 li X li t¼1 gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ! <sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1 X ki s¼1 wi 1 ki hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi � � ! þX<sup>n</sup> i¼1 X li t¼1 wi 1 li gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi � � ! <sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1 X ki s¼1 ffiffiffiffiffi wi p ffiffiffiffi ki p hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ! � <sup>X</sup><sup>n</sup> i¼1 X ki s¼1 ffiffiffiffiffi wi p ffiffiffiffi ki p hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ! ! <sup>þ</sup> <sup>X</sup><sup>n</sup> i¼1 X li t¼1 ffiffiffiffiffi wi p ffiffiffi li <sup>p</sup> gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ! � <sup>X</sup><sup>n</sup> i¼1 X li t¼1 ffiffiffiffiffi wi p ffiffiffi li <sup>p</sup> gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ! !

Assuming, X1 <sup>¼</sup> <sup>P</sup><sup>n</sup> i¼1 P ki s¼1 ffiffiffiffi wi p ffiffiffi ki <sup>p</sup> hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi , Y1 <sup>¼</sup> <sup>P</sup><sup>n</sup> i¼1 P ki s¼1 ffiffiffiffi wi p ffiffiffi ki p hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi , X2 <sup>¼</sup> <sup>P</sup><sup>n</sup> i¼1 P li t¼1 ffiffiffiffi wi p ffiffi li <sup>p</sup> gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi and Y2 <sup>¼</sup> <sup>P</sup><sup>n</sup> i¼1 P li t¼1 ffiffiffiffi wi p ffiffi li <sup>p</sup> gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi . Xn i¼1 wi 1 ki X ki s¼1 hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þþ xi 1 li X li t¼1 gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ! ¼ ð Þ X1Y1 þ X2Y2 ≤ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2 <sup>1</sup> <sup>þ</sup> <sup>X</sup><sup>2</sup> 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>Y</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> Y2 2 q

$$\begin{aligned} &\leq \left(\sqrt{\left(\sum\_{i=1}^{\mathrm{n}} \sum\_{s=1}^{\mathrm{k}\_{i}} \frac{\sqrt{\mathrm{w}\_{i}}}{\sqrt{\mathrm{k}\_{i}}} \mathrm{h}\_{\mathrm{A\sigma(s)}}(\mathbf{x}\_{i})\right)^{2} + \left(\sum\_{i=1}^{\mathrm{n}} \sum\_{t=1}^{\mathrm{l}\_{i}} \frac{\sqrt{\mathrm{w}\_{i}}}{\sqrt{\mathrm{l}\_{i}}} \mathrm{g}\_{\mathrm{A\sigma(t)}}(\mathbf{x}\_{i})\right)^{2}\right) \times \\ &\leq \left(\sqrt{\left(\sum\_{i=1}^{\mathrm{n}} \sum\_{s=1}^{\mathrm{k}\_{i}} \frac{\sqrt{\mathrm{w}\_{i}}}{\sqrt{\mathrm{k}\_{i}}} \mathrm{h}\_{\mathrm{B\sigma(s)}}(\mathbf{x}\_{i})\right)^{2} + \left(\sum\_{i=1}^{\mathrm{n}} \sum\_{t=1}^{\mathrm{l}\_{i}} \frac{\sqrt{\mathrm{w}\_{i}}}{\sqrt{\mathrm{l}\_{i}}} \mathrm{g}\_{\mathrm{B\sigma(t)}}(\mathbf{x}\_{i})\right)^{2}\right) \end{aligned}$$

*Modified Expression to Evaluate the Correlation Coefficient of Dual Hesitant Fuzzy Sets… DOI: http://dx.doi.org/10.5772/intechopen.96474*

≤ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P<sup>n</sup> i¼1 ffiffiffiffiffi wi � � p <sup>2</sup> Pki s¼1 hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ffiffiffi ki p � �<sup>2</sup> <sup>þ</sup> <sup>P</sup>li t¼1 gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ffiffi li p � �<sup>2</sup> vu ! ut 0 @ 1 A 0 @ 1 A� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P<sup>n</sup> i¼1 ffiffiffiffiffi wi � � p <sup>2</sup> Pki s¼1 hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ffiffiffi ki p � �<sup>2</sup> <sup>þ</sup> <sup>P</sup>li t¼1 gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ffiffi li p � �<sup>2</sup> vu ! ut 0 @ 1 A ≤ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P<sup>n</sup> <sup>i</sup>¼<sup>1</sup>wi Pki s¼1 hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ffiffiffi ki p � �<sup>2</sup> <sup>þ</sup> <sup>P</sup>li t¼1 gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ffiffi li p � �<sup>2</sup> vu ! ut 0 @ 1 A 0 @ 1 A� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P<sup>n</sup> <sup>i</sup>¼<sup>1</sup>wi Pki s¼1 hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ffiffiffi ki p � �<sup>2</sup> <sup>þ</sup> <sup>P</sup>li t¼1 gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ffiffi li p � �<sup>2</sup> vu ! ut 0 @ 1 A ≤ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P<sup>n</sup> <sup>i</sup>¼<sup>1</sup>wi 1 ki Pki <sup>s</sup>¼<sup>1</sup>h<sup>2</sup> <sup>A</sup>σð Þ<sup>s</sup> ð Þþ xi 1 li Pli <sup>t</sup>¼<sup>1</sup>g<sup>2</sup> <sup>A</sup>σð Þ<sup>t</sup> ð Þ xi � � � � <sup>r</sup> � � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P<sup>n</sup> <sup>i</sup>¼<sup>1</sup>wi 1 ki Pki <sup>s</sup>¼<sup>1</sup>h<sup>2</sup> <sup>B</sup>σð Þ<sup>s</sup> ð Þþ xi 1 li Pli <sup>t</sup>¼<sup>1</sup>g2 <sup>B</sup>σð Þ<sup>t</sup> ð Þ xi � � r � � ) <sup>X</sup><sup>n</sup> i¼1 wi 1 ki X ki s¼1 hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þþ xi 1 li X li t¼1 gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ! ≤ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X<sup>n</sup> i¼1 wi 1 ki Xki s¼1 h2 <sup>A</sup>σð Þ<sup>s</sup> ð Þþ xi 1 li Xli t¼1 g2 <sup>A</sup>σð Þ<sup>t</sup> ð Þ xi ! ! s � � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X<sup>n</sup> i¼1 wi 1 ki Xki s¼1 h2 <sup>B</sup>σð Þ<sup>s</sup> ð Þþ xi 1 li Xli t¼1 g2 <sup>B</sup>σð Þ<sup>t</sup> ð Þ xi ! s � � 8 >>>>>>>< >>>>>>>: 9 >>>>>>>= >>>>>>>; ) Pn i¼1 wi <sup>1</sup> ki Pki s¼1 hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þþ xi <sup>1</sup> li Pli t¼1 gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P<sup>n</sup> <sup>i</sup>¼1wi <sup>1</sup> ki Pki <sup>s</sup>¼1h2 <sup>A</sup>σð Þ<sup>s</sup> ð Þþ xi <sup>1</sup> li Pli <sup>t</sup>¼1g<sup>2</sup> <sup>A</sup>σð Þ<sup>t</sup> ð Þ xi � � � � <sup>r</sup> � � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P<sup>n</sup> <sup>i</sup>¼1wi <sup>1</sup> ki Pki <sup>s</sup>¼1h2 <sup>B</sup>σð Þ<sup>s</sup> ð Þþ xi <sup>1</sup> li Pli <sup>t</sup>¼1g<sup>2</sup> <sup>B</sup>σð Þ<sup>t</sup> ð Þ xi � � � r � � � ≤1*:*

#### **3.2 Mathematical incorrect assumptions**

In this section, the mathematical incorrect assumptions, considered in existing literature [29] to obtain the expressions 1ð Þ have been discussed.

It can be easily verified from Section 3.1 that to obtain the expressions 1ð Þ it have been assumed that,

i. P<sup>n</sup> i¼1 Pki s¼1 wi ki hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi � � <sup>¼</sup> <sup>P</sup><sup>n</sup> i¼1 Pki s¼1 ffiffiffiffi wi p ffiffiffi ki p hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi � � � <sup>P</sup><sup>n</sup> i¼1 Pki s¼1 ffiffiffiffi wi p ffiffiffi ki p hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi � �, ii. P<sup>n</sup> i¼1 Pli t¼1 wi li gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi � � <sup>¼</sup> <sup>P</sup><sup>n</sup> i¼1 Pli s¼1 ffiffiffiffi wi p ffiffi li <sup>p</sup> gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi � � � <sup>P</sup><sup>n</sup> i¼1 Pli s¼1 ffiffiffiffi wi p ffiffi li <sup>p</sup> gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi � � iii. P<sup>k</sup> s¼1 1ffiffi k <sup>p</sup> hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ x1 � �<sup>2</sup> ¼ 1 k P<sup>k</sup> <sup>s</sup>¼<sup>1</sup>h<sup>2</sup> <sup>A</sup>σð Þ<sup>s</sup> ð Þ x1 iv. P <sup>l</sup> t¼1 1ffi l <sup>p</sup> gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ x1 � �<sup>2</sup> ¼ 1 l P<sup>l</sup> <sup>t</sup>¼<sup>1</sup>g<sup>2</sup> <sup>A</sup>σð Þ<sup>t</sup> ð Þ x1

$$\begin{array}{l} \text{v. } \left(\sum\_{\mathbf{s=1}}^{\mathbf{k}} \frac{1}{\sqrt{\mathbf{k}}} \mathbf{h}\_{\mathbf{B}\sigma(\mathbf{s})} \left(\mathbf{x\_{1}}\right)\right)^{2} = \frac{1}{\mathbf{k}} \sum\_{\mathbf{s=1}}^{\mathbf{k}} \mathbf{h}\_{\mathbf{B}\sigma(\mathbf{s})}^{2} \left(\mathbf{x\_{1}}\right), \\\text{vi. } \left(\sum\_{\mathbf{t=1}}^{\mathbf{l}} \frac{1}{\sqrt{\mathbf{l}}} \mathbf{g}\_{\mathbf{B}\sigma(\mathbf{t})} \left(\mathbf{x\_{1}}\right)\right)^{2} = \frac{1}{\mathbf{l}} \sum\_{\mathbf{t=1}}^{\mathbf{l}} \mathbf{g}\_{\mathbf{B}\sigma(\mathbf{t})}^{2} \left(\mathbf{x\_{1}}\right). \end{array}$$

Let us consider an example, **Example 1:** Let

$$\mathbf{A} = \left\{ \begin{array}{c} \langle \mathbf{x}\_1, \{0.1, 0.2, 0.5\}, \{0.3\} \rangle, \langle \mathbf{x}\_2, \{0.2, 0.4, 0.6\}, \{0.4, 0.5, 0.8\} \rangle, \\\\ \langle \mathbf{x}\_3, \{0.1, 0.2, 0.4\}, \{0.6, 0.8, 0.9\} \rangle, \langle \mathbf{x}\_4, \{\{0.2, 0.4, 0.1\}, \{0.8, 0.9, 0.6\} \} \rangle \end{array} \right\} \text{ and} \\\\ \mathbf{B} = \left\{ \begin{array}{c} \langle \mathbf{x}\_1, \{\{0.2, 0.3, 0.5\}, \{0.3, 0.6, 0.9\} \} \rangle, \langle \mathbf{x}\_2, \{\{0.2, 0.3, 0.7\}, \{0.1, 0.9\} \} \rangle, \\\\ \langle \mathbf{x}\_3, \{\{0.6, 0.3, 0.5\}, \{0.9, 0.2, 0.3\} \} \rangle \rangle, \langle \mathbf{x}\_4, \{\{0.5\}, \{0.9\} \} \rangle \end{array} \right\} \end{array} \tag{3}$$

be two DHFS and let w <sup>¼</sup> ð Þ <sup>0</sup>*:*3, 0*:*2, 0*:*1, 0*:*<sup>4</sup> <sup>T</sup> be the weight vector of xi. Then, it can be easily verified that

$$\sum\_{i=1}^{n} \left( \sum\_{s=1}^{\mathbf{k}\_i} \frac{\mathbf{w}\_i}{\mathbf{k}\_i} \left( \mathbf{h}\_{\mathbf{A}\sigma(s)}(\mathbf{x}\_i) \mathbf{h}\_{\mathbf{B}\sigma(s)}(\mathbf{x}\_i) \right) \right) = \mathbf{0}.1289,$$

$$\left( \sum\_{i=1}^{n} \sum\_{s=1}^{\mathbf{k}\_i} \frac{\sqrt{\mathbf{w}\_i}}{\sqrt{\mathbf{k}\_i}} \mathbf{h}\_{\mathbf{A}\sigma(s)}(\mathbf{x}\_i) \right) \times \left( \sum\_{i=1}^{n} \sum\_{s=1}^{\mathbf{k}\_i} \frac{\sqrt{\mathbf{w}\_i}}{\sqrt{\mathbf{k}\_i}} \mathbf{h}\_{\mathbf{B}\sigma(s)}(\mathbf{x}\_i) \right) = \mathbf{1.1335}.$$

It is obvious that

$$\sum\_{i=1}^{n} \left( \sum\_{\mathbf{s}=1}^{\mathbf{k}\_{\mathrm{i}}} \frac{\mathbf{w}\_{\mathbf{i}}}{\mathbf{k}\_{\mathrm{i}}} \mathbf{h}\_{\mathrm{A\sigma(\mathbf{s})}}(\mathbf{x}\_{\mathbf{i}}) \mathbf{h}\_{\mathrm{B\sigma(\mathbf{s})}}(\mathbf{x}\_{\mathbf{i}}) \right) \neq \left( \sum\_{i=1}^{n} \sum\_{\mathbf{s}=1}^{\mathbf{k}\_{\mathrm{i}}} \frac{\sqrt{\mathbf{w}\_{\mathbf{i}}}}{\sqrt{\mathbf{k}\_{\mathrm{i}}}} \mathbf{h}\_{\mathrm{A\sigma(\mathbf{s})}}(\mathbf{x}\_{\mathbf{i}}) \right) \times \left( \sum\_{i=1}^{n} \sum\_{\mathbf{s}=1}^{\mathbf{k}\_{\mathrm{i}}} \frac{\sqrt{\mathbf{w}\_{\mathbf{i}}}}{\sqrt{\mathbf{k}\_{\mathrm{i}}}} \mathbf{h}\_{\mathrm{B\sigma(\mathbf{s})}}(\mathbf{x}\_{\mathbf{i}}) \right) .$$

Also, it can be easily verified that

$$\sum\_{i=1}^{n} \left( \sum\_{t=1}^{l\_i} \frac{\mathbf{w}\_i}{\mathbf{l}\_i} \left( \mathbf{g}\_{\mathbf{A}\sigma(t)}(\mathbf{x}\_i) \mathbf{g}\_{\mathbf{B}\sigma(t)}(\mathbf{x}\_i) \right) \right) = \mathbf{0}.4003,$$

$$\left( \sum\_{i=1}^{n} \sum\_{s=1}^{l\_i} \frac{\sqrt{\mathbf{w}\_i}}{\sqrt{l\_i}} \mathbf{g}\_{\mathbf{A}\sigma(t)}(\mathbf{x}\_i) \right) \times \left( \sum\_{i=1}^{n} \sum\_{s=1}^{l\_i} \frac{\sqrt{\mathbf{w}\_i}}{\sqrt{l\_i}} \mathbf{g}\_{\mathbf{B}\sigma(t)}(\mathbf{x}\_i) \right) = \mathbf{3.1862.7}$$

It is obvious that,

$$\sum\_{i=1}^{n} \left( \sum\_{t=1}^{\mathsf{l}} \frac{\mathbf{w}\_{i}}{\mathsf{l}\_{\mathsf{i}}} \mathbf{g}\_{\mathsf{A}\sigma(\mathsf{t})}(\mathbf{x}\_{\mathsf{i}}) \mathbf{g}\_{\mathsf{B}\sigma(\mathsf{t})}(\mathbf{x}\_{\mathsf{i}}) \right) \neq \left( \sum\_{i=1}^{n} \sum\_{t=1}^{\mathsf{l}} \frac{\sqrt{\mathbf{w}\_{i}}}{\sqrt{\mathsf{l}\_{\mathsf{i}}}} \mathbf{g}\_{\mathsf{A}\sigma(\mathsf{t})}(\mathbf{x}\_{\mathsf{i}}) \right) \times \left( \sum\_{i=1}^{n} \sum\_{t=1}^{\mathsf{l}\_{\mathsf{i}}} \frac{\sqrt{\mathbf{w}\_{i}}}{\sqrt{\mathsf{l}\_{\mathsf{i}}}} \mathbf{g}\_{\mathsf{B}\sigma(\mathsf{t})}(\mathbf{x}\_{\mathsf{i}}) \right) \cdot \mathbf{g}\_{\mathsf{A}\sigma(\mathsf{t})}(\mathbf{x}\_{\mathsf{i}}) $$

Furthermore, it can be easily verified that

$$\begin{cases} \sum\_{\mathbf{s=1}}^{\mathbf{k}\_{\mathbf{i}}} \frac{1}{\sqrt{\mathbf{k}\_{\mathbf{i}}}} \mathbf{h}\_{\mathbf{A}\sigma(\mathbf{s})} \left(\mathbf{x}\_{\mathbf{1}}\right)^{2} = \mathbf{1.02}, & \frac{1}{\mathbf{k}\_{\mathbf{i}}} \sum\_{\mathbf{s=1}}^{\mathbf{k}\_{\mathbf{i}}} \mathbf{h}\_{\mathbf{A}\sigma(\mathbf{s})}^{2}(\mathbf{x}\_{\mathbf{1}}) = \mathbf{0.4267},\\ \left(\sum\_{\mathbf{t=1}}^{\mathbf{l}\_{\mathbf{i}}} \frac{1}{\sqrt{\mathbf{l}\_{\mathbf{i}}}} \mathbf{g}\_{\mathbf{A}\sigma(\mathbf{t})} \left(\mathbf{x}\_{\mathbf{1}}\right)\right)^{2} = \mathbf{4.5800}, & \frac{1}{\mathbf{l}\_{\mathbf{i}}} \sum\_{\mathbf{t=1}}^{\mathbf{l}\_{\mathbf{i}}} \mathbf{g}\_{\mathbf{A}\sigma(\mathbf{t})}^{2}(\mathbf{x}\_{\mathbf{1}}) = \mathbf{1.6467}. \end{cases}$$

*Modified Expression to Evaluate the Correlation Coefficient of Dual Hesitant Fuzzy Sets… DOI: http://dx.doi.org/10.5772/intechopen.96474*

$$\begin{pmatrix} \sum\_{\mathbf{s}=1}^{\mathbf{l}\_{\mathbf{l}}} \frac{1}{\sqrt{\mathbf{k}\_{\mathbf{l}}}} \mathbf{h}\_{\mathrm{B\sigma(s)}}\left(\mathbf{x}\_{\mathbf{l}}\right) \\ \vdots \\ \sum\_{\mathbf{t}=1}^{\mathbf{l}\_{\mathbf{l}}} \frac{1}{\sqrt{\mathbf{l}\_{\mathbf{l}}}} \mathbf{g}\_{\mathrm{B\sigma(t)}}\left(\mathbf{x}\_{\mathbf{l}}\right) \end{pmatrix}^{2} = 2.8232, \quad \frac{1}{\mathbf{s}\_{\mathbf{l}}} \sum\_{\mathbf{s}=1}^{\mathbf{k}\_{\mathbf{l}}} \mathbf{h}\_{\mathrm{B\sigma(s)}}^{2}(\mathbf{x}\_{\mathbf{l}}) = \mathbf{0.8166}.$$

$$\left(\sum\_{\mathbf{t}=1}^{\mathbf{l}\_{\mathbf{l}}} \frac{1}{\sqrt{\mathbf{l}\_{\mathbf{l}}}} \mathbf{g}\_{\mathrm{B\sigma(t)}}\left(\mathbf{x}\_{\mathbf{l}}\right)\right)^{2} = \mathbf{4.1960}, \quad \frac{1}{\mathbf{l}\_{\mathbf{l}}} \sum\_{\mathbf{t}=1}^{\mathbf{l}\_{\mathbf{l}}} \mathbf{g}\_{\mathrm{B\sigma(t)}}^{2}(\mathbf{x}\_{\mathbf{l}}) = \mathbf{1.4133}.$$

It is obvious that

$$\begin{split} & \text{i. } \left(\sum\_{s=1}^{\mathbf{k}} \frac{1}{\sqrt{\mathbf{k}}} \mathbf{h}\_{\mathbf{A}\sigma(s)} \left(\mathbf{x}\_{1}\right)\right)^{2} \neq \frac{1}{\mathbf{k}} \sum\_{s=1}^{\mathbf{k}} \mathbf{h}\_{\mathbf{A}\sigma(s)}^{2} \left(\mathbf{x}\_{1}\right) \\ & \text{ii. } \left(\sum\_{t=1}^{\mathbf{l}} \frac{1}{\sqrt{\mathbf{l}}} \mathbf{g}\_{\mathbf{A}\sigma(t)} \left(\mathbf{x}\_{1}\right)\right)^{2} \neq \frac{1}{\mathbf{l}} \sum\_{t=1}^{\mathbf{l}} \mathbf{g}\_{\mathbf{A}\sigma(t)}^{2} \left(\mathbf{x}\_{1}\right) \\ & \text{iii. } \left(\sum\_{s=1}^{\mathbf{k}} \frac{1}{\sqrt{\mathbf{k}}} \mathbf{h}\_{\mathbf{B}\sigma(s)} \left(\mathbf{x}\_{1}\right)\right)^{2} \neq \frac{1}{\mathbf{k}} \sum\_{s=1}^{\mathbf{k}} \mathbf{h}\_{\mathbf{B}\sigma(s)}^{2} \left(\mathbf{x}\_{1}\right) \\ & \text{iv. } \left(\sum\_{t=1}^{\mathbf{l}} \frac{1}{\sqrt{\mathbf{l}}} \mathbf{g}\_{\mathbf{B}\sigma(t)} \left(\mathbf{x}\_{1}\right)\right)^{2} \neq \frac{1}{\mathbf{l}} \sum\_{t=1}^{\mathbf{l}} \mathbf{g}\_{\mathbf{B}\sigma(t)}^{2} \left(\mathbf{x}\_{1}\right). \end{split}$$

Thus, Example 1 verifies that the considered mathematical assumptions in the existing literature [29] to obtain the weighted correlation coefficient expressions 1ð Þ for DHFSs are not valid.

#### **4. Proposed CoCf for the DHFSs**

Considering the above mentioned limitation in Section 3 as a motivation, an attempt has been made to modify the existing expression 1ð Þ [29], and hence the weighted CoCf for DHFSs is proposed which is represented in expression 3ð Þ.

$$\rho\_{\text{WDHFS}}(\mathbf{A},\mathbf{B}) = \frac{\sum\_{i=1}^{n} \text{w}\_{i} \left( \frac{1}{\hbar} \sum\_{s=1}^{\mathbf{k}\_{i}} \mathbf{h}\_{\text{Av}(\mathbf{s})}(\mathbf{x}) \mathbf{h}\_{\text{Bt}(\mathbf{s})}(\mathbf{x}) + \frac{1}{\hbar} \sum\_{s=1}^{\mathbf{k}\_{i}} \mathbf{g}\_{\text{Av}(\mathbf{t})}(\mathbf{x}) \mathbf{g}\_{\text{Bt}(\mathbf{t})}(\mathbf{x}) \right)}{\sum\_{i=1}^{n} \text{w}\_{i} \left( \left( \sqrt{\sum\_{s=1}^{\mathbf{k}\_{i}} \left( \frac{\mathbf{h}\_{\text{Av}(\mathbf{s})}}{\sqrt{\mathbf{k}\_{i}}} \right)^{2} \sum\_{s=1}^{\mathbf{k}\_{i}} \left( \frac{\mathbf{h}\_{\text{Bt}(\mathbf{s})}(\mathbf{x})}{\sqrt{\mathbf{k}\_{i}}} \right)^{2} \right)} + \left( \sqrt{\sum\_{s=1}^{\mathbf{k}\_{i}} \left( \frac{\mathbf{g}\_{\text{At}(\mathbf{v})}(\mathbf{x})}{\sqrt{\mathbf{k}\_{i}}} \right)^{2} \sum\_{s=1}^{\mathbf{k}\_{i}} \left( \frac{\mathbf{g}\_{\text{Bt}(\mathbf{v})}(\mathbf{x})}{\sqrt{\mathbf{k}\_{i}}} \right)^{2} \right)} \tag{4}$$

where,


### **5. Origin of the proposed CoCf for the DHFSs**

The modified expression 3ð Þ has been obtained mathematically as follows:

$$\sum\_{i=1}^{n} \mathbf{w}\_{i} \left( \frac{1}{\mathbf{k}\_{\mathrm{i}}} \sum\_{s=1}^{\mathrm{k}} \mathbf{h}\_{\mathrm{A\sigma(s)}}(\mathbf{x}\_{\mathrm{i}}) \mathbf{h}\_{\mathrm{B\sigma(s)}}(\mathbf{x}\_{\mathrm{i}}) + \frac{1}{\mathbf{l}\_{\mathrm{i}}} \sum\_{t=1}^{\mathrm{l}\_{\mathrm{i}}} \mathbf{g}\_{\mathrm{A\sigma(t)}}(\mathbf{x}\_{\mathrm{i}}) \mathbf{g}\_{\mathrm{B\sigma(t)}}(\mathbf{x}\_{\mathrm{i}}) \right) \right.$$

$$= \sum\_{i=1}^{n} \mathbf{w}\_{\mathrm{i}} \left( \left( \sum\_{s=1}^{\mathrm{k}\_{\mathrm{i}}} \frac{\mathbf{h}\_{\mathrm{A\sigma(s)}}(\mathbf{x}\_{\mathrm{i}})}{\sqrt{\mathbf{k}\_{\mathrm{i}}}} \frac{\mathbf{h}\_{\mathrm{B\sigma(s)}}(\mathbf{x}\_{\mathrm{i}})}{\sqrt{\mathbf{k}\_{\mathrm{i}}}} \right) + \left( \sum\_{t=1}^{\mathrm{l}\_{\mathrm{i}}} \frac{\mathbf{g}\_{\mathrm{A\sigma(t)}}(\mathbf{x}\_{\mathrm{i}})}{\sqrt{\mathbf{l}\_{\mathrm{i}}}} \frac{\mathbf{g}\_{\mathrm{B\sigma(t)}}(\mathbf{x}\_{\mathrm{i}})}{\sqrt{\mathbf{l}\_{\mathrm{i}}}} \right) \right).$$

Assuming,

<sup>X</sup>ð Þ<sup>s</sup> <sup>¼</sup> hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ffiffiffi ki <sup>p</sup> , Yð Þ<sup>s</sup> <sup>¼</sup> hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ffiffiffi ki p , <sup>X</sup>ð Þ<sup>t</sup> <sup>¼</sup> gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ffiffi li <sup>p</sup> and Yð Þ<sup>t</sup> <sup>¼</sup> gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ffiffi li p Xn i¼1 wi 1 ki Xki s¼1 hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þþ xi 1 li Xli t¼1 gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ! <sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1 wi Xki s¼1 <sup>X</sup>ð Þ<sup>s</sup> <sup>Y</sup>ð Þ<sup>s</sup> <sup>þ</sup>Xli t¼1 Xð Þ<sup>t</sup> Yð Þ<sup>t</sup> ! ≤ X<sup>n</sup> i¼1 wi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>X</sup>ki s¼1 <sup>X</sup>ð Þ<sup>s</sup> � �<sup>2</sup> �Xki s¼1 <sup>Y</sup>ð Þ<sup>s</sup> � �<sup>2</sup> r� � <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xli t¼1 <sup>X</sup>ð Þ<sup>t</sup> � �<sup>2</sup> �Xli t¼1 <sup>Y</sup>ð Þ<sup>t</sup> � �<sup>2</sup> ! r ≤ X<sup>n</sup> i¼1 wi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xki s¼1 hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ffiffiffiffi ki p � �<sup>2</sup> �Xki s¼1 hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ffiffiffiffi ki p � �<sup>2</sup> 0s @ 1 A þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xli t¼1 gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ffiffiffi li p !<sup>2</sup> �Xli t¼1 gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ffiffiffi li p !<sup>2</sup> vuut 0 B@ 1 CA 0 B@ 1 CA ) P<sup>n</sup> <sup>i</sup>¼<sup>1</sup>wi <sup>1</sup> ki Pki <sup>s</sup>¼<sup>1</sup>hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þþ xi <sup>1</sup> li Pli <sup>t</sup>¼<sup>1</sup>gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi � � P<sup>n</sup> <sup>i</sup>¼<sup>1</sup>wi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pki s¼1 hA<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ffiffiffi ki p � �<sup>2</sup> � <sup>P</sup>ki s¼1 hB<sup>σ</sup>ð Þ<sup>s</sup> ð Þ xi ffiffiffi ki p � �2 ! <sup>s</sup> þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pli t¼1 gA<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ffiffi li p � �<sup>2</sup> � <sup>P</sup>li t¼1 gB<sup>σ</sup>ð Þ<sup>t</sup> ð Þ xi ffiffi li p � �2 ! <sup>s</sup> ! <sup>≤</sup>1*:*

#### **6. Exact results of the existing real life problem**

There is an investment company, which intends to invest a sum of money in the best alternative [29]. There are four available alternatives, A1: a car company, A2: a food company, A3: a computer company, and A4: an arms company. The investment company considers three attributes, C1: the risk analysis, C2: the growth analysis, and C3: the environment impact analysis to consider the best alternatives. Since, there is a need to identify the best investment company among A1, A2, A3 and A4, with respect to an ideal alternative A<sup>∗</sup> on the basis of three different attributes C1, C2, and C3, it is assumed that:


*Modified Expression to Evaluate the Correlation Coefficient of Dual Hesitant Fuzzy Sets… DOI: http://dx.doi.org/10.5772/intechopen.96474*


#### **Table 1.**

*Rating values of the alternatives over the attributes.*


#### **Table 2.**

*Results of the considered real-life problem.*

Then, by applying the existing expression 1ð Þ [29] the obtained preferred company is A2 i.e. the food company is the best alternative for the investment. However it is discussed in Section 3 that the expression 1ð Þ [29] is not valid in its present form since it is scientifically incorrect. Therefore, the result of the considered real-life problem, obtained in existing literature [29], is also not exact. Thus, to obtain the exact results of the existing problem [29], the proposed CoCf represented by expression 3ð Þ is utilized and the solution is obtained successfully. Furthermore, comparison of the results of the considered real-life problem is obtained by the existing expression 1ð Þ [29] as well as by the modified expression 3ð Þ, and the results are shown below in **Table 2**.

From the above obtained results as shown in **Table 2**, it is obvious that according to existing expression 1ð Þ, A2 i.e. the food company is the most preferred company to invest the money, while, according to the proposed expression 3ð Þ, A4 i.e. arms company is the most preferred company to invest the sum of the money by the investment company.

#### **7. Advantages of the proposed measure**

The proposed correlation coefficient measure is an efficient tool which has the following advantages for solving the decision-making problems under the dual hesitant fuzzy environment.

i. Dual hesitant fuzzy set is an extension of hesitant fuzzy set (HFS), and intuitionistic fuzzy set (IFS) which contains more information i.e., it has wider range of hesitancy included both in membership and nonmembership of an object than the others fuzzy sets (HFSs, deals with only membership hesitant degrees and IFSs deals with both membership degree and non-membership degree).

