**2. Preliminaries**

In this part, the process of traditional TODIM approach in decision making will be introduced briefly. Then, we retrospect the concepts and theories of dual hesitant Pythagorean fuzzy set (DHPFS).

#### *2.1. The TODIM Approach*

 TODIM approach

Let *A* = {*A*1, *A*2, ... , *Am*} be a discrete set with *m* alternatives, and *Ai* represents the *ith* alternative in the set. Let *C* = {*C*1,*C*2, ... ,*Cn*} be a set with *n* attributes, and *Cj* represents the *jth* attribute in the set. Let ω = {<sup>ω</sup>1, ω2, ... , <sup>ω</sup>*n*}*<sup>T</sup>* be the attributes set's weight vector, with <sup>ω</sup>*j* ∈ [0, 1] and *nj*=1 <sup>ω</sup>*j* = 1. Suppose that *X* = *xijm*×*n* be a decision matrix, where *xij* is the attribute value according to a decision maker over the alternative *Ai* with respect to attribute *Cj*. We define <sup>ω</sup>*jr* = <sup>ω</sup>*j*/<sup>ω</sup>*r*(*j*,*<sup>r</sup>* = 1, 2, ... , *n*) as the relative weight of the attribute *Cj* to *Cr*, ω*r* = max<sup>ω</sup>*j j* = 1, 2, ... , *n*, and <sup>ω</sup>*jr* ∈ [0, 1]. Then, the

 Step 1. Standardize the decision matrix *X* = *xijm*×*n* into the matrix *Y* = *yijm*×*n*. Step2. Computethedominancedegreeof *Ai* overeachalternative *Ak* forattribute *Cj*:

are as follows:

$$\delta(A\_{i\prime}A\_k) = \sum\_{j=1}^{n} \phi\_j(A\_{i\prime}A\_k)\_{\prime\prime}(i,k=1,2,\ldots,m) \tag{1}$$

where

steps of the traditional

$$\phi\_{\vec{j}}(A\_i, A\_k) = \begin{cases} \sqrt{(y\_{ij} - y\_{kj})} \omega\_{jr} / \left(\sum\_{j=1}^n \omega\_{jr}\right), & y\_{ij} > y\_{kj} \\ 0, & y\_{ij} = y\_{kj} \\ -\frac{1}{\mathcal{D}} \sqrt{(y\_{kj} - y\_{ij}) \left(\sum\_{j=1}^n \omega\_{jr}\right) / \omega\_{jr}}, & y\_{ij} < y\_{kj} \end{cases} \tag{2}$$

and the attenuation factor of the losses is expressed by the parameter value θ. Step 3. Compute the overall value of the alternative *Ai* by:

$$\xi(A\_i) = \frac{\sum\_{k=1}^{m} \delta(A\_{i\prime} A\_k) - \min\_{i \in M} \{ \sum\_{k=1}^{m} \delta(A\_{i\prime} A\_k) \}}{\max\_{i \in M} \{ \sum\_{k=1}^{m} \delta(A\_{i\prime} A\_k) \} - \min\_{i \in M} \{ \sum\_{k=1}^{m} \delta(A\_{i\prime} A\_k) \}}, (i = 1, 2, \dots, m) \tag{3}$$

Step 4. Rank all the alternatives according to the overall values ξ(*Ai*),(*<sup>i</sup>* = 1, 2, ... , *<sup>m</sup>*). Based on this result, we can ge<sup>t</sup> the conclusion that the most ideal alternative has largest overall value, and conversely, the worst alternative has the smallest value.

#### *2.2. Dual Pythagorean Hesitant Fuzzy Set*

In this part, some definitions of PFS, DHFS and DHPFS are introduced. The score function and accuracy function of DHPFN are also provided.

**Definition 1 [22]).** *A Pythagorean fuzzy set (PFS) P with an object X is defined as follows:*

$$P = \{ \langle \mathbf{x}, \mu\_P(\mathbf{x}), \upsilon\_P(\mathbf{x}) \rangle | \mathbf{x} \in X \} \tag{4}$$

*where uP*(*x*) ∈ [0, 1] *is defined as the membership degree function, vP*(*x*) ∈ [0, 1] *is defined as the non-membership degree function. For all x* ∈ *X, uP*(*x*) *and vP*(*x*) *meet the following requirements: uP*(*x*)<sup>2</sup> + *vP*(*x*)<sup>2</sup> ≤ 1*. The hesitant degree of x* ∈ *X is presented as:*

$$
\pi\_P(\mathbf{x}) = \sqrt{1 - \mu\_P(\mathbf{x})^2 - \upsilon\_P(\mathbf{x})^2}, \forall \mathbf{x} \in X \tag{5}
$$

*For convenience,* (*uP*(*x*), *vP*(*x*)) *is called a PFN, which can be denoted by P* = (*uP*, *vP*).

**Definition 2 ([23]).** *Let X be a fixed set, then a dual hesitant fuzzy set (DHFS) D with an object X is defined as follows:*

$$D = \{ \langle \mathbf{x}, \Gamma\_D(\mathbf{x}), \Psi\_D(\mathbf{x}) \rangle | \mathbf{x} \in X \} \tag{6}$$

*where* <sup>Γ</sup>*D*(*x*) ∈ [0, 1] *is defined as a set of all the possible membership degrees,* <sup>Ψ</sup>*D*(*x*) ∈ [0, 1] *is defined as a set of all the possible non-membership degrees.*

**Definition 3 ([5]).** *Let X be a fixed set, then a dual Pythagorean hesitant fuzzy set (DHPFS) A with an object X is defined as follows:*

$$A = \{ \langle \mathbf{x}, \Gamma\_A(\mathbf{x}), \Psi\_A(\mathbf{x}) \rangle | \mathbf{x} \in X \} \tag{7}$$

*where* <sup>Γ</sup>*A*(*x*) ∈ [0, 1] *is defined as a set of all the possible membership degrees,* <sup>Ψ</sup>*A*(*x*) ∈ [0, 1] *is defined as a set of all the possible non-membership degrees. Furthermore, for all uA*(*x*) ∈ <sup>Γ</sup>*A*(*x*) *and vA*(*x*) ∈ <sup>Ψ</sup>*A*(*x*) *meet the following requirements: uA*(*x*)<sup>2</sup> + *vA*(*x*)<sup>2</sup> ≤ 1*. The set of hesitant degree x* ∈ *X is presented as:*

$$\Pi\_A(\mathbf{x}) = \begin{array}{c} \cup \\ u\_A(\mathbf{x}) \in \Gamma\_A(\mathbf{x}) \end{array} \left\{ \pi\_A(\mathbf{x}) = \sqrt{1 - u\_A(\mathbf{x})^2 - v\_A(\mathbf{x})^2} \ge 0 \right\}, \forall\_x \in X \tag{8}$$

*For convenience,* (<sup>Γ</sup>*A*(*x*), <sup>Ψ</sup>*A*(*x*)) *is called a DHPFN, which can be denoted by A* = (<sup>Γ</sup>*A*, <sup>Ψ</sup>*A*).

Wei and Lu [5] have proposed some operational rules for DHPFN.

**Definition 4 ([5]).** *Let* β1 = (<sup>Γ</sup>β1, <sup>Ψ</sup>β<sup>1</sup>), β2 = (<sup>Γ</sup>β2, <sup>Ψ</sup>β2) *and* β = (<sup>Γ</sup>β, <sup>Ψ</sup>β) *be three DHPFNs, then*

*(1)* β1 ⊕ β2 =< ∪ *<sup>u</sup>*β<sup>1</sup>(*x*) ∈ <sup>Γ</sup>β<sup>1</sup>(*x*) *<sup>u</sup>*β<sup>2</sup>(*x*) ∈ <sup>Γ</sup>β<sup>2</sup>(*x*) *<sup>u</sup>*β1<sup>2</sup> + *<sup>u</sup>*β2<sup>2</sup> − *<sup>u</sup>*β12*<sup>u</sup>*β1<sup>2</sup>, ∪ *<sup>v</sup>*β<sup>1</sup>(*x*) ∈ <sup>Ψ</sup>β<sup>1</sup>(*x*) *<sup>v</sup>*β<sup>2</sup>(*x*) ∈ <sup>Ψ</sup>β<sup>2</sup>(*x*) *<sup>v</sup>*β1*<sup>v</sup>*β2 >; *(2)* β1 ⊗ β2 =< ∪ *<sup>u</sup>*β<sup>1</sup>(*x*) ∈ <sup>Γ</sup>β<sup>1</sup>(*x*) *<sup>u</sup>*β<sup>2</sup>(*x*) ∈ <sup>Γ</sup>β<sup>2</sup>(*x*) *<sup>v</sup>*β1*<sup>v</sup>*β<sup>2</sup>, ∪ *<sup>v</sup>*β<sup>1</sup>(*x*) ∈ <sup>Ψ</sup>β<sup>1</sup>(*x*) *<sup>v</sup>*β<sup>2</sup>(*x*) ∈ <sup>Ψ</sup>β<sup>2</sup>(*x*) *<sup>u</sup>*β1<sup>2</sup> + *<sup>u</sup>*β2<sup>2</sup> − *<sup>u</sup>*β12*<sup>u</sup>*β1<sup>2</sup> >; *(3)* λβ =< ∪ *u*β∈Γβ 1 − (1 − *<sup>u</sup>*β<sup>2</sup>)<sup>λ</sup>, ∪*<sup>v</sup>*β<sup>∈</sup>Ψβ*<sup>v</sup>*β<sup>λ</sup> >; *(4)* βλ =< ∪ *<sup>u</sup>*β<sup>∈</sup>Γβ*<sup>u</sup>*β<sup>λ</sup>, ∪*<sup>v</sup>*β<sup>∈</sup>Ψβ 1 − (1 − *<sup>v</sup>*β<sup>2</sup>)<sup>λ</sup> >;

*Mathematics* **2020**, *8*, 8

*(5)* β*c* =< <sup>Ψ</sup>β, Γβ >;

> Wei and Lu [5] gave a score function of DHPFNs to rank two different numbers.

**Definition 5 ([5]).** *Let* β = (<sup>Γ</sup>β, <sup>Ψ</sup>β) *be a DHPFN, then the score function of* β *is defined as:*

$$S\_{\beta} = \frac{1}{2} \left( 1 + \frac{1}{\left| \Gamma\_{\beta} \right|} \sum\_{\boldsymbol{\mu}\_{\beta} \in \Gamma\_{\beta}} \boldsymbol{u}\_{\beta}^{-2} - \frac{1}{\left| \boldsymbol{\Psi}\_{\beta} \right|} \sum\_{\boldsymbol{v}\_{\beta} \in \boldsymbol{\Psi}\_{\beta}} \boldsymbol{v}\_{\beta}^{-2} \right) \tag{9}$$

*the accuracy function of* β *is defined as:*

$$p\_{\beta} = \frac{1}{|\Gamma\_{\beta}|} \sum\_{\boldsymbol{\nu}\_{\beta} \in \Gamma\_{\beta}} \boldsymbol{u}\_{\beta}^{2} + \frac{1}{|\boldsymbol{\Psi}\_{\beta}|} \sum\_{\boldsymbol{\nu}\_{\beta} \in \boldsymbol{\Psi}\_{\beta}} \boldsymbol{v}\_{\beta}^{2} \tag{10}$$

Γβ *and* <sup>Ψ</sup>β *represents the length of set* Γβ *and* <sup>Ψ</sup>β *respectively. The higher the score, the better the number.*

#### **3. Dual Pythagorean Hesitant Fuzzy TODIM Approach**

In order to describe the dual Pythagorean hesitant fuzzy TODIM approach in detail, first of all, we put forward some fundamental concepts that are essential to this paper.

*3.1. Fundamental Knowledge*

> According to Definition 5, Wei and Lu [5] gave a solution to compare two DHPFNs.

**Definition 6 ([5]).** *Let* β1 = (<sup>Γ</sup>β1, <sup>Ψ</sup>β1) *and* β2 = (<sup>Γ</sup>β2, <sup>Ψ</sup>β2) *be two DHPFNs, <sup>S</sup>*β1 *and <sup>S</sup>*β2 *be the scores value of* β1 *and* β2 *respectively, p*β1 *and p*β2 *be the accuracy values of* β1 *and* β2 *respectively. Then, if <sup>S</sup>*β1> *<sup>S</sup>*β2*,then* β1 > β2*; if <sup>S</sup>*β1= *<sup>S</sup>*β2,then

 *(1) if p*β1= *p*β2 *, then* β1 = β2;

*(2) if p*β1 > *p*β2 *, then* β1 > β2;

**Definition 7 ([24]).** *Let* β1 = (<sup>Γ</sup>β1, <sup>Ψ</sup>β1) *and* β2 = (<sup>Γ</sup>β2, <sup>Ψ</sup>β2) *be two DHPFNs, then the Hamming distance between* β1 *and* β2 *is:*

$$d(\beta\_1, \beta\_2) = \frac{1}{2} \Big( \frac{1}{l} \sum\_{j=1}^{l} \left| \Gamma\_{\beta 1} \,^{\sigma(j)} - \Gamma\_{\beta 2} \,^{\sigma(j)} \right| + \frac{1}{m} \sum\_{j=1}^{m} \left| \Psi\_{\beta 1} \,^{\sigma(j)} - \Psi\_{\beta 2} \,^{\sigma(j)} \right| \Big) \tag{11}$$

*where l*, *m are the lengths of the set* Γ and Ψ, *respectively.* <sup>Γ</sup>β1<sup>σ</sup>(*j*), <sup>Γ</sup>β2<sup>σ</sup>(*j*) *represents the jth largest element in set* <sup>Γ</sup>β1 *and* <sup>Γ</sup>β2*, respectively.* <sup>Ψ</sup>β1<sup>σ</sup>(*j*), <sup>Ψ</sup>β2<sup>σ</sup>(*j*) *represents the jth largest element in set* <sup>Ψ</sup>β1 *and* <sup>Ψ</sup>β2*, respectively.*

In general, the number of elements in two DHPFS is different. Therefore, we use the extension principle to compare two sets, that is, adding elements to a set with fewer elements, so that the number of elements in the two sets is the same. There are two types of extension principles. The first is to add the largest element in the set, while the other is to add the smallest element in the set. In this article, we use the first principle.

An example is shown to illustrate the Hamming distance between β1 and β2: If β1 = ({0.4, 0.2}, {0.7, 0.3, 0.1} and β1 = ({0.7, 0.3, 0.1}, {0.4, 0.1} are two DHPFNs. Ar first, we extend set <sup>Γ</sup>β1(*x*) = {0.4, 0.2} to {0.4, 0.4, 0.2} and set <sup>Ψ</sup>β2(*x*) = {0.4, 0.1} to {0.4, 0.4, 0.1}. Then the distance between β1 and β2 is: *<sup>d</sup>*(β1, β2) = 0.1500.

#### *3.2. TODIM Method for Dual Hesitant Pythagorean Fuzzy MADM Problems*

Now, we give procedures to use TODIM approach to deal with MADM Problems with dual hesitant Pythagorean fuzzy set. In order to understand it easily, we describe the problems briefly as follows:

Let *A* = {*A*1, *A*2, ... , *Am*} be a discrete set of alternatives and *C* = {*C*1,*C*2, ... ,*Cn*} be the set of attributes; Let = {<sup>ω</sup>1, ω2, ... , <sup>ω</sup>*n*} be the attributes set's weight vector, with <sup>ω</sup>*j* ∈ [0, 1] and *n j*=1 <sup>ω</sup>*j* = 1.

Suppose *R* = *rijm*×*n* = <sup>Γ</sup>*ij*, <sup>Ψ</sup>*ijm*×*n* be a dual hesitant Pythagorean fuzzy decision matrix, where *rij* is a criterion value according to a decision maker over the alternative. *Ai* with respect to attribute *Cj*, where <sup>Γ</sup>*A*(*x*) ∈ [0, 1] is defined as a set of all the possible membership degrees, <sup>Ψ</sup>*A*(*x*) ∈ [0, 1] is defined as a set of all the possible non-membership degrees. Furthermore, for all *uA*(*x*) ∈ <sup>Γ</sup>*A*(*x*) and *vA*(*x*) ∈ <sup>Ψ</sup>*A*(*x*) meet the following requirements: *uA*(*x*)<sup>2</sup> + *vA*(*x*)<sup>2</sup> ≤ 1.

Then, we extend the TODIM model to deal with the MADM problems with dual hesitant Pythagorean fuzzy set. At first, it is necessary to standardize the decision matrix, because there may be some benefit attribute and cost attribute in *C*, as they are two opposite DHPFNs:

$$l\_{ij} = \begin{cases} r\_{ij\prime} forbene ftattribute \text{C}\_j\\ r\_{ij\prime}^\varepsilon for \cos\text{sharp} turbute \text{C}\_j \end{cases} \tag{12}$$

where *rcij* = (<sup>Ψ</sup>*ij*, <sup>Γ</sup>*ij*) is the complement of *rcij*. After that, we can ge<sup>t</sup> the normalized dual hesitant Pythagorean fuzzy decision matrix *L* = *lijm*×*n*.

Next, we calculate the relative weight of the attribute *Cj* to *Cr* as

$$
\omega\_{\rm jr} = \omega\_{\rm j} / \omega\_{\rm r}(\rm j, r = 1, 2, \dots, n) \tag{13}
$$

where ω*r* = max<sup>ω</sup>*j j* = 1, 2, ... , *n*, <sup>ω</sup>*jr* ∈ [0, 1].

Then we can ge<sup>t</sup> the dominance degree of *Ai* over attribute *Cj* for each alternative *Ak*:

$$\phi\_j \phi\_j(A i, Ak) = \begin{cases} \sqrt{d(l\_{ij}, l\_{kj})} \omega\_{jr} / \left(\sum\_{j=1}^n \omega\_{jr}\right) & l\_{ij} > l\_{kj} \\ 0, & l\_{ij} = l\_{kj} \\ -\frac{1}{\partial} \sqrt{d(l\_{ij}, l\_{kj})} \left(\sum\_{j=1}^n \omega\_{jr}\right) / \omega\_{jr} & l\_{ij} < l\_{kj} \end{cases} \tag{14}$$

where the parameter θ represents the attenuation factor of the losses, *<sup>d</sup>lij*.*lkj*describe the distance between the DHPFNs *lij* and *lkj*. In order to show the function φ*j*(*Ai*, *Ak*) visually, we express it in a matrix way:

$$\phi\_j(A\_i, A\_k) = \begin{array}{c} \text{A1} \\ \text{A2} \\ \phi\_j(A\_i, A\_k) = \begin{array}{c} \text{A1} \\ \text{A2} \\ \vdots \\ \text{A3} \end{array} \begin{array}{c} \text{A2} \\ \text{O}\_j(A\_1, A\_2) \\ \text{O} \end{array} \dots \begin{array}{c} \text{A3} \\ \text{O}\_j(A\_1, A\_m) \\ \vdots \\ \vdots \\ \phi\_j(A\_m, A\_1) \\ \phi\_j(A\_m, A\_1) \end{array} \begin{array}{c} \text{A4} \\ \dots \\ \text{O}\_j(A\_1, A\_m) \\ \vdots \\ \phi\_j(A\_m, A\_1) \end{array} \dots \begin{array}{c} \text{A5} \\ \text{A6} \\ \vdots \\ \vdots \\ \phi\_j(A\_1, A\_j) \end{array} \tag{15}$$

based on which, we can calculate the overall dominance degree of the *Ai* over each alternative *Ak* by:

$$\delta(A\_{i\prime}A\_k) = \sum\_{j=1}^{n} \phi\_j(A\_{i\prime}A\_k)\_{\prime}(i,k=1,2,\ldots,m) \tag{16}$$

and we also show the function <sup>δ</sup>(*Ai*, *Ak*) visually in a matrix:

A1 A2 ... *Am* δ = <sup>δ</sup>(*Ai*, *Ak*)*m*×*n* = A1 A2 . . . *Am* ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 φ*j*(*<sup>A</sup>*1, *<sup>A</sup>*2) ... φ*j*(*<sup>A</sup>*1, *Am*) φ*j*(*<sup>A</sup>*2, *<sup>A</sup>*1) 0 ... φ*j*(*<sup>A</sup>*2, *Am*) ... ... ... ... φ*j*(*Am*, *<sup>A</sup>*1) φ*j*(*Am*, *<sup>A</sup>*2) ... 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (17)

Finally, the overall value of each alternative *Ai* can be computed by the following formula:

$$\xi(A\_i) = \frac{\sum\_{k=1}^{m} \delta(A\_i, A\_k) - \min\_{i \in M} \left\{ \sum\_{k=1}^{m} \delta(A\_i, A\_k) \right\}}{\max\_{i \in M} \left\{ \sum\_{k=1}^{m} \delta(A\_i, A\_k) \right\} - \min\_{i \in M} \left\{ \sum\_{k=1}^{m} \delta(A\_i, A\_k) \right\}}, (i = 1, 2, \dots, m) \tag{18}$$

In order to rank all alternatives, we construct a principle whereby that the greater the overall value of ξ(*Ai*),(*<sup>i</sup>* = 1, 2, ... , *<sup>m</sup>*), the better the alternative *Ai*.

#### **4. Numerical Example**

#### *4.1. The Application of Dual Hesitant Pythagorean Fuzzy TODIM Approach*

In order to evaluate the borrower's default risk accurately, many models have been constructed. The first step in building the model is to select some reasonable credit evaluation indicators. Lin et al.[25] explored the factors that determine the default risk based on the demographic characteristics of borrowers. The empirical results show that the following ten factors played an important role in loan default: (1) gender, (2) age, (3) marital status, (4) educational level, (5) working years, (6) company size, (7) monthly payment, (8) loan amount, (9) debt to income ratio, and (10) delinquency history.

We denote these ten attributes as *Cj*(*j* = 1, 2, ... , 10) respectively, and their weight vector is ω = (0.0550, 0.0466, 0.2005, 0.1692, 0.0766, 0.0804, 0.0821, 0.1530, 0.0655, 0.0711)*<sup>T</sup>*, which is given by the decision maker. Suppose that there are four applicants to be ranked, which denoted as *Aj*(*j* = 1, 2, 3, <sup>4</sup>), and dual hesitant Pythagorean numbers *rij*(<sup>Γ</sup>*ij*, <sup>Ψ</sup>*ij*) appear in the MADM process. *rij*(<sup>Γ</sup>*ij*, <sup>Ψ</sup>*ij*)(<sup>i</sup> = 1,2,3,4; *j* = 1, 2, ... , 10) represents the value of alternative *Ai* relative to attribute *Cj*. All these values are contained in the decision matrix *R* = *rijm*×*n* = <sup>Γ</sup>*ij*, <sup>Ψ</sup>*ijm*×*n*, as shown in Table 1.


**Table 1.** The decision matrix *R*.

Then, we can apply dual hesitant Pythagorean fuzzy TODIM approach to rank the personal default risk. Before that, we use extension principle which is claimed in this paper to extend the decision matrix, as shown in Table 2.


**Table 2.** The extended decision matrix *R*.

Firstly, ω3 = max<sup>ω</sup>*j j* = 1, 2, ... , 10= 0.2005, so we calculate the relative weights of the attribute are:

$$\omega\_{13} = 0.2743, \omega\_{23} = 0.2324, \omega\_{33} = 1.0000, \omega\_{43} = 0.8439, \omega\_{53} = 0.3820, \omega\_{63} = 0.4010, \omega\_{73} = 0.4010, \omega\_{83} = 0.4095, \omega\_{93} = 0.3267, \omega\_{10\cdot 3} = 0.3546.$$

Then, we need to calculate the dominance degree of *Ai* over each alternative *Ak* with respect to the attribute *Cj*(*<sup>i</sup>*, *k* = 1, 2, 3, 4; *j* = 1, 2, ... , <sup>10</sup>). Let θ = 2.5:

A1 A2 A3 A4 φ1 = A1 A2 A3 A4 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 0.0981 −0.7340 −0.7340 −0.7928 0 −0.5190 −0.5190 0.0908 0.0642 0 0 0.0908 0.0642 0 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ A1 A2 A3 A4 φ2 = A1 A2 A3 A4 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 −0.7974 −0.5638 0.0483 0.0836 0 0.0591 0.0836 0.0591 −0.5638 0 0.0591 −0.4604 −0.7974 −0.5638 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ A1 A2 A3 A4 φ3 = A1 A2 A3 A4 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 0.0708 −0.3509 −0.4152 −0.1569 0 −0.3844 −0.4439 0.1583 0.1734 0 −0.3844 0.1873 0.2002 0.1734 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ A1 A2 A3 A4 φ4 = A1 A2 A3 A4 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 −0.3417 −0.1708 −0.6160 0.1301 0 −0.3820 −0.5666 0.0650 0.1454 0 −0.5918 0.2345 0.2157 0.2253 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ A1 A2 A3 A4 φ5 = A1 A2 A3 A4 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 −0.5078 0.1072 0.0619 0.0875 0 0.0619 0.0875 −0.6219 −0.3591 0 0.1072 −0.3591 −0.5078 −0.6219 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A1 A2 A3 A4 φ6 = A1 A2 A3 A4 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 −0.4957 −0.4293 −0.3505 0.0897 0 0.1002 0.0634 0.0777 −0.5542 0 0.0777 0.0634 −0.3505 −0.4293 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ A1 A2 A3 A4 φ7 = A1 A2 A3 A4 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 0 −0.6489 −0.3468 0 0 −0.6489 −0.3468 0.1199 0.1199 0 −0.6489 0.0641 0.0641 0.1199 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ A1 A2 A3 A4 φ8 = A1 A2 A3 A4 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 0.1515 0.1515 0.0875 −0.4401 0 −0.4401 −0.3593 −0.4401 0.1515 0 0.1237 −0.2541 0.1237 −0.3593 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ A1 A2 A3 A4 φ9 = A1 A2 A3 A4 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 −0.2746 −0.8683 −0.7766 0.0405 0 −0.8237 −0.7265 0.1280 0.1214 0 0.0572 0.1145 0.1071 −0.3883 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ A1 A2 A3 A4 φ10 = A1 A2 A3 A4 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 0.0596 0.0596 0.0843 −0.3727 0 0.0596 −0.5271 −0.3727 −0.3727 0 −0.6455 −0.5271 0.0843 0.1033 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Secondly, we calculate the overall dominance degree of the *Ai* over each alternative *Ak*:

$$
\delta = \begin{array}{c c c c c}
 & \mathbf{A\_1} & \mathbf{A\_2} & \mathbf{A\_3} & \mathbf{A\_4} \\
 \mathbf{A\_1} & 0 & -2.0371 & -3.4477 & -2.9572 \\
 \mathbf{A\_2} & -1.3312 & 0 & -2.9172 & -3.2547 \\
 -0.7359 & -1.0740 & 0 & -1.8458 \\
 -0.8460 & -0.7964 & -1.7408 & 0 \\
\end{array}
$$

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Finally, we can ge<sup>t</sup> the overall value of each alternative *Ai* by the follows:

$$
\xi(A\_1) = 0; \xi(A\_2) = 0.1856; \xi(A\_3) = 0.9461; \xi(A\_4) = 1; 2
$$

Thus, the ranking of alternatives is *A*4 > *A*3 > *A*2 > *A*1, and *A*4 has best credit.

#### *4.2. Comparative Analysis*

In this section, some comparative analysis will be performed to compare our proposed method with the other two operators defined by Wei and Lu [5].

4.2.1. Comparison with the Dual Hesitant Pythagorean Fuzzy Weighted Average Operator

Then, we contrast this method with a dual hesitant Pythagorean fuzzy weighted average (DHPFWA) operator proposed by Wei and Lu [5] as follows:

**Definition 8** ([5]). Let α*i* = (<sup>Γ</sup><sup>α</sup>*i*(*x*), <sup>Ψ</sup><sup>α</sup>*i*(*x*)) ∈ *DHPFNs*(*i* = 1, 2, ... , *n*) and ω = {<sup>ω</sup>1, ω2, ... , <sup>ω</sup>*n*}*<sup>T</sup>* be the attributes set's weight vector, with <sup>ω</sup>*j* ∈ [0, 1] and *n j*=1 ω*j* = 1, then

$$\begin{split} \text{DHPFWA}\_{\omega}(\alpha\_{1}, \alpha\_{2}, \dots, \alpha\_{n}) &= \omega\_{1}\alpha\_{1} \oplus \omega\_{2}\alpha\_{2} \oplus \dots \oplus \omega\_{n}\alpha\_{n} \\ &= \circ \underset{u\_{\mathrm{al}\_{i}} \in \Gamma\_{a\_{i}}}{\text{U}} \left\{ \sqrt{1 - \prod\_{i=1}^{n} \left(1 - u\_{\mathrm{al}\_{i}}^{2}\right)^{\omega\_{i}}} \right\}\_{v\_{\mathrm{al}\_{i}} \in \mathbb{V}\_{a\_{i}}} \left\{ \prod\_{i=1}^{n} v\_{\alpha\_{i}}^{\omega\_{i}} \right\} > \end{split} \tag{19}$$

We can get:


	- {0.5968, 0.5882, 0.5829, 0.5745, 0.5882, 0.5796, 0.5744, 0.5661, 0.5685, 0.5602, 0.5552, 0.5472, 0.5602, 0.5521, 0.5472, 0.5392, 0.5811, 0.5726, 0.5675, 0.5593, 0.5726, 0.5643, 0.5593, 0.5512, 0.5535,0.5454,0.5406,0.5327,0.5454,0.5375,0.5327,0.5250})

then, the score values of each alternative are as follows: *sA*1 = 0.3914, *sA*2 = 0.3878, *sA*3 = 0.4521, *sA*4 = 0.5080. We can ge<sup>t</sup> the ranking of alternatives according to the score value: *sA*4 > *sA*3 > *sA*1 > *sA*2 , and *A*4 has best credit.

4.2.2. Comparison with the Dual Hesitant Pythagorean Fuzzy Weighted Geometric Operator

Then, we contrast this method with dual hesitant Pythagorean fuzzy weighted geometric (DHPFWG) operator proposed by Wei and Lu [5] as follows:

**Definition 9** ([5]). Let α*i* = (<sup>Γ</sup><sup>α</sup>*i* (*x*), <sup>Ψ</sup><sup>α</sup>*i* (*x*)) ∈ *DHPFNs*(*i* = 1, 2, ... , *n*) and ω = {<sup>ω</sup>1, ω2, ... , <sup>ω</sup>*n*} *T* be the attributes set's weight vector, with <sup>ω</sup>*j* ∈ [0, 1] and *n j*=1 <sup>ω</sup>*j* = 1,then

$$\begin{split} \text{DHPFW}G\_{\boldsymbol{\alpha}}(\alpha\_{1}, \alpha\_{2}, \dots, \alpha\_{n}) &= \alpha\_{1}\alpha\_{1} \otimes \alpha\_{2}\alpha\_{2} \otimes \dots \otimes \alpha\_{n}\alpha\_{n} \\ &= \ll\_{\boldsymbol{u}\_{\boldsymbol{a}\_{i}} \in \Gamma\_{\boldsymbol{a}\_{i}}} \left\{ \prod\_{i=1}^{n} \boldsymbol{u}\_{\boldsymbol{a}\_{i}}^{\boldsymbol{\alpha}\_{i}} \right\}\_{\boldsymbol{v}\_{\boldsymbol{a}\_{i}} \in \Psi\_{\boldsymbol{a}\_{i}}} \left\{ \sqrt{1 - \prod\_{i=1}^{n} \left(1 - \boldsymbol{v}\_{\boldsymbol{a}\_{i}}^{2}\right)^{\boldsymbol{\omega}\_{i}}} \right\} > \\ &\tag{20} \end{split} \tag{20}$$

We can get:


then, the score values of each alternative are as follows: *sA*1 = 0.3587, *sA*2 = 0.3436, *sA*3 = 0.4122, *sA*3 = 0.4607. We can ge<sup>t</sup> the rank of all alternatives according to the score values: *sA*4 > *sA*3 > *sA*1 > *sA*2 , and *A*4 has best credit.

The results of comparative analyses by using di fferent methods are as follows:

As you can see from Table 3, the three methods have the same best choice *A*4 and their results are slightly di fferent. Obviously, the proposed method considers the attitude of decision makers towards risk, that is, people are more sensitive to loss than gain, which makes the results of decisions consistent with the expectations of decision makers. Each of these methods has its own advantages: (1) DHPFWA operator is a ffected by groups, (2) DHPFWG operator is a ffected by individuals, and (3) the dual hesitant Pythagoras fuzzy TODIM method reduces uncertainty while taking into account the psychological characteristics of lenders to avoid risk. Today's credit environment is full of risks, and the psychological

behaviors of decision makers are important factors that cannot be ignored. So, the dual Pythagorean hesitant fuzzy TODIM method is applicable for evaluating personal default risk.


**Table 3.** Rank of alternatives by using di fferent methods.

#### **5. Concluding Remarks and Suggestions**

In the decision-making environment full of risks, the TODIM method enables decision-makers' psychological behaviors to be described. But it doesn't work when it's used directly to solve the MADM problems with fuzzy information. Through existing research, we can find that the membership and non-membership of the dual hesitant Pythagoras fuzzy set are respectively represented by a set containing all possible values, this fuzzy set is a good tool for people to express hesitancy in daily life. Therefore, we have proposed a method named dual hesitant Pythagoras fuzzy TODIM to solve MADM problem with dual hesitant Pythagoras fuzzy information. One of the most important advantages of the proposed approach is that it can reduce uncertainty while taking the psychological characteristics of lenders into account to avoid risk. At the same time, this method can be further extended to other similar MADM problems with interdependent attribute under dual hesitant Pythagoras fuzzy environments, such as performance evaluation, risk investment and so on. At last, the dual hesitant Pythagoras fuzzy has made a grea<sup>t</sup> contribution to the expansion of fuzzy set and also plays a significant role in practical decision process.

In this paper, the dual hesitant Pythagoras fuzzy TODIM method is applied to the case of personal default risk evaluation in the P2P lending platforms, and some comparative analyses are performed to compare the dual hesitant Pythagorean fuzzy TODIM method with the other two integrated operators DHPFWA and DHPFWG. The advantage of the dual hesitant Pythagoras fuzzy TODIM method over these two operators is that it takes the decision maker's reference dependence and the psychological behavior characteristics of loss aversion into account. We have demonstrated that the proposed approach is applicable for evaluating personal default risk through the comparisons. More broadly, it can also be applied to solve more similar MADM problems. In the future, we shall continue studying the MADM problems with the application and extension of the developed operators to other domains [26–33].

Finally, in view of the macroeconomic downturn and strong financial supervision, we would like to put forward the following suggestions to prevent default risk of P2P online platforms: (1) Improve the information disclosure system to reduce information asymmetry. On the one hand, online loan platforms should actively participate in relevant regulatory associations and fully disclose the information of the borrowers; on the other hand, it is necessary and even desirable to examine the authenticity and legitimacy of the information and issue a reliable risk assessment report. (2) Establish a third-party depository system. On the one hand, the funds of the loan project should be managed by a special third-party financial institution like a bank, and the online loan platforms are only used as a medium for information interaction. On the other hand, it avoids short-term goals and extracts a certain percentage of risk reserves to protect the investment. This can improve the anti-risk ability of online loan platforms, and also enhance investors' confidence in the platform.

**Author Contributions:** Data curation, L.Y.; writing—review and editing, L.Y.; investigation, X.J.; writing—original draft preparation, X.J.; software, J.F. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Social Science Foundation the People's Republic of China (No. 19BJL025) and the Humanities and Social Sciences Foundation of the Ministry of Education of the People's Republic of China (No.17YJA790088)

**Conflicts of Interest:** The authors declare no conflicts of interest.
