**2. Preliminaries**

This section emphasizes on basic definitions regarding the DHFSs, PHFSs and PDHFSs.

**Definition 1.** *On the universal set X, Zhu et al. [26] defined dual hesitant fuzzy set as:*

$$\mathfrak{a} = \{ (\mathbf{x}, \mathfrak{h}(\mathbf{x}), \mathfrak{g}(\mathbf{x})) \mid \mathbf{x} \in X \}\tag{1}$$

*where the sets h*(*x*) *and g*(*x*) *have values in* [0, 1]*, which signifies possible membership and non-membership degrees for x* ∈ *X. Also,*

$$0 \le \gamma, \eta \le 1; 0 \le \gamma^+ + \eta^+ \le 1\tag{2}$$

*in which, γ* ∈ *h*(*x*); *η* ∈ *g*(*x*) *; γ*<sup>+</sup> ∈ *<sup>h</sup>*+(*x*) = *γ*∈*h*(*x*) max{*γ*} *and η*<sup>+</sup> ∈ *g*+(*x*) = *η*∈*g*(*x*) max{*η*}

**Definition 2.** *Let X be a reference set, then a probabilistic hesitant fuzzy set (PHFS) [39] P on X is given as*

$$P = \{ \langle \mathbf{x}, h\_{\mathbf{x}}(p\_{\mathbf{x}}) \rangle \mid \mathbf{x} \in X \} \tag{3}$$

*Here, the set hx contains several values in* [0, 1]*, and described by the probability distribution px. Also, hx denotes membership degree of x in X. For simplicity, hx*(*px*) *is called a probabilistic hesitant fuzzy element (PHFE), denoted as h*(*p*) *and is given as*

$$h(p) = \{\gamma\_i(p\_i) \mid i = 1, 2, \dots, \#H\}\_i$$

*where pi satisfying* #*H* ∑ *i*=1 *pi* ≤ 1*, is the probability of the possible value γi and* #*H is the number of all <sup>γ</sup>i*(*pi*)*.*

**Definition 3** ([49])**.** *A probabilistic dual hesitant fuzzy set (PDHFS) on X is defined as:*

$$\mathfrak{a} = \{ (\mathbf{x}, h(\mathbf{x}) | p(\mathbf{x}), \mathbf{g}(\mathbf{x}) | q(\mathbf{x})) \mid \mathbf{x} \in X \}\tag{4}$$

*Here, the sets h*(*x*)|*p*(*x*) *and g*(*x*)|*q*(*x*) *contains possible elements where h*(*x*) *and g*(*x*) *represent the hesitant fuzzy membership and non-membership degrees x* ∈ *X, respectively. Also, p*(*x*) *and q*(*x*) *are their associated probabilistic information. Moreover,*

$$0 \le \gamma, \eta \le 1; 0 \le \gamma^+ + \eta^+ \le 1\tag{5}$$

*and*

$$p\_i \in [0, 1], q\_j \in [0, 1], \sum\_{i=1}^{\#h} p\_i = 1, \sum\_{j=1}^{\#g} q\_j = 1 \tag{6}$$

*where γ* ∈ *h*(*x*); *η* ∈ *g*(*x*); *γ*<sup>+</sup> ∈ *<sup>h</sup>*+(*x*) = *γ*∈*h*(*x*) max{*γ*}; *η*<sup>+</sup> ∈ *g*+(*x*) = *η*∈*g*(*x*) max{*η*}*. The symbols* #*h and* #*g are total values in* (*h*(*x*)|*p*(*x*)) *and* (*g*(*x*)|*q*(*x*)) *respectively. For sake of convenience, we shall denote it as* (*h*|*p*, *g*|*q*) *and name it as probabilistic dual hesitant fuzzy element (PDHFE).*

**Definition 4** ([49])**.** *For a PDHFE α, defined over a universal set X, the complement is defined as*

$$\mathfrak{a}^c = \begin{cases} \bigcup\_{\gamma \in h, \eta \in \mathfrak{g}} \left( \{ \eta \mid q\_{\eta} \}, \{ \gamma \mid p\_{\gamma} \} \right), & \text{if} \quad \mathfrak{g} \neq \mathfrak{g} \text{ and} \quad \mathfrak{h} \neq \mathfrak{g} \\ \bigcup\_{\gamma \in h} \left( \{ 1 - \gamma \}, \{ \Phi \} \right), & \text{if} \quad \mathfrak{g} = \mathfrak{g} \quad \text{and} \quad \mathfrak{h} \neq \mathfrak{g} \\ \bigcup\_{\eta \in \mathfrak{g}} \left( \{ \Phi \}, \{ 1 - \eta \} \right), & \text{if} \quad \mathfrak{h} = \mathfrak{g} \quad \text{and} \quad \mathfrak{g} \neq \mathfrak{g} \end{cases} \tag{7}$$

**Definition 5** ([49])**.** *Let α* = (*h*|*p*, *g*|*q*) *be a PDHFE, then the score function is defined as:*

$$S(\mathfrak{a}) = \sum\_{i=1}^{\#h} \gamma\_i \cdot p\_i - \sum\_{j=1}^{\#g} \eta\_j \cdot q\_j \tag{8}$$

*where* #*h and* #*g are total numbers of elements in the components* (*h*|*p*) *and* (*g*|*q*) *respectively and γ* ∈ *h, η* ∈ *g. For two PDHFEs α*1 *and α*2*, if <sup>S</sup>*(*<sup>α</sup>*1) > *<sup>S</sup>*(*<sup>α</sup>*2), *then the PDHFE α*1 *is regarded more superior to α*2 *and is denoted as α*1 *α*2*.*

### **3. Proposed Distance Measures for PDHFEs**

In this section, we propose some measures to calculate the distance between two PDHFEs defined over a universal set *X* = {*<sup>x</sup>*1, *x*2, ... , *xn*}. Throughout this paper, the main notations used are listed below:


Let *A* = *<sup>x</sup>*, *hAi*(*x*)*pAi*(*x*), *gAj*(*x*)*qAj*(*x*) | *x* ∈ *X* and *B* = *<sup>x</sup>*, *hBi* (*x*)*pBi* (*x*), *gBj* (*x*)*qBj* (*x*) | *x* ∈ *X* where *i* = 1, 2, ... , *MA*; *j* = 1, 2, ... , *NA*; *i* = 1, 2, ... , *MB* and *j* = 1, 2, ... , *NB*, be two PDHFSs. Also, let *M* = max{*MA*, *MB*}, *N* = max{*NA*, *NB*}, be two real numbers, then for a real-number *λ* > 0, we define distance between *A* and *B* as:

$$d\_1(A,B) = \left(\sum\_{k=1}^n \frac{1}{n} \left(\frac{1}{M+N} \left(\sum\_{i=1}^M \left|\gamma\_{A\_i}(\mathbf{x}\_k) p\_{A\_i}(\mathbf{x}\_k) - \gamma\_{B\_i}(\mathbf{x}\_k) p\_{B\_i}(\mathbf{x}\_k)\right|^\lambda\right)\right)\right)^{\frac{1}{\lambda}}\tag{9}$$

where *γAi* ∈ *hAi* , *γBi* ∈ *hBi* , *ηAi* ∈ *gAi* , *ηBi* ∈ *gBi* . It is noticeable that, there may arise the cases in which *MA* = *MB* as well as *NA* = *NB*. Under such situations, for operating distance *d*1, the lengths of these elements should be equal to each other. To achieve this, under the hesitant environments, the experts repeat the least or the greatest values among all the hesitant values, in the smaller set, till the length of both *A* and *B* becomes equal. In other words, if *MA* > *MB*, then repeat the smallest value in set *hB* till *MB* becomes equal to *MA* and if *MA* < *MB*, then repeat the smallest value in set *hA* till *MA* becomes equal to *MB*. Alike the smallest values, the largest values may also be repeated. This choice of the smallest or largest value's repetition entirely depends on decision-makers optimistic or pessimistic approach. If the expert opts for the optimistic approach then he will expect the highest membership values and thus will repeat the largest values. However, if the expert chooses to follow the pessimistic approach, then he will expect the least favoring values and will go with repeating the smallest values till the same length is achieved. But sometimes, length of *A* and *B* cannot be matched

by increasing the numbers of elements, then in such cases, the distance *d*1 can be unappropriate for the data evaluations. To handle such cases, we propose another distance measure *d*2 in which there is no need to repeat the values for matching the length of the elements under consideration. This distance *d*2 is calculated as:

$$d\_{2}(A,B) = \left(\sum\_{k=1}^{n} \frac{1}{n} \left(\frac{\left|\frac{1}{M\_{A}}\sum\_{i=1}^{M\_{A}} \left(\gamma\_{A\_{i}}(\mathbf{x}\_{k})p\_{A\_{i}}(\mathbf{x}\_{k})\right) - \frac{1}{M\_{B}}\sum\_{j'=1}^{M\_{B}} \left(\gamma\_{B\_{j}'}(\mathbf{x}\_{k})p\_{B\_{j}'}(\mathbf{x}\_{k})\right)\right|^{\lambda}}{2}\right.\right)^{\frac{1}{\lambda}}\right)^{\frac{1}{\lambda}}\tag{10}$$

$$+\left(\frac{\frac{1}{N\_{A}}\sum\_{j=1}^{N\_{A}} \left(\eta\_{A\_{j}}(\mathbf{x}\_{k})q\_{A\_{j}}(\mathbf{x}\_{k})\right) - \frac{1}{N\_{B}}\sum\_{j'=1}^{N\_{B}} \left(\eta\_{B\_{j}'}(\mathbf{x}\_{k})q\_{B\_{j}'}(\mathbf{x}\_{k})\right)}{2}\right)^{\lambda}\tag{11}$$

The distance measures proposed above satisfy the axiomatic statement given below:

**Theorem 1.** *Let A and B be two PDHFSs, then the distance measure d*1 *satisfies the following conditions:*

*(P1)* 0 ≤ *d*1(*<sup>A</sup>*, *B*) ≤ 1*; (P2) d*1(*<sup>A</sup>*, *B*) = *d*1(*<sup>B</sup>*, *<sup>A</sup>*); *(P3) d*1(*<sup>A</sup>*, *B*) = 0 *if A* = *B*; *(P4) If A* ⊆ *B* ⊆ *C*, *then d*1(*<sup>A</sup>*, *B*) ≤ *d*1(*<sup>A</sup>*, *C*) *and d*1(*<sup>B</sup>*, *C*) ≤ *d*1(*<sup>A</sup>*, *<sup>C</sup>*).

**Proof.** Let *X* = {*<sup>x</sup>*1, *x*2, ... , *xn*} be the universal set and *A*, *B* be two PDHFSs defined over *X*. Then for each *xk*, *k* = 1, 2, . . . , *n*, we have

(P1) Since, 0 ≤ *γAi* (*xk*) ≤ 1 and 0 ≤ *pAi* (*xk*) ≤ 1, for all *i* = 1, 2, ... , *M*, this implies that 0 ≤ *γAi* (*xk*)*pAi* (*xk*) ≤ 1 and 0 ≤ *γBi* (*xk*)*pBi* (*xk*) ≤ 1. Thus, for any *λ* > 0, we have 0 ≤ *γAi* (*xk*)*pAi* (*xk*) − *γBi* (*xk*)*pBi* (*xk*) *λ* ≤ 1. Further, *M* ∑ *i*=1 0 ≤ *M* ∑ *i*=1 *γAi* (*xk*)*pAi* (*xk*) − *γBi* (*xk*)*pBi* (*xk*) *λ* ≤ *M* ∑ *i*=1 1 which leads to 0 ≤ *M* ∑ *i*=1 *γAi* (*xk*)*pAi* (*xk*) − *γBi* (*xk*)*pBi* (*xk*) *λ* ≤ *M*. Similarly, for *j* = 1, 2, ... , *N*, 0 ≤ *N* ∑ *j*=1 *ηAj* (*xk*)*qAj* (*xk*) − *ηBj* (*xk*)*qBj* (*xk*) *λ* ≤ *N* which yields 0 ≤ *M* ∑ *i*=1 *γAi* (*xk*)*pAi* (*xk*) − *γBi* (*xk*)*pBi* (*xk*) *λ* + *N* ∑ *j*=1 *ηAj* (*xk*)*qAj* (*xk*) − *ηBj* (*xk*)*qBj* (*xk*) *λ* ≤ *M* + *N*.

Thus,

$$0 \le \left( \sum\_{k=1}^{n} \frac{1}{n} \left( \frac{1}{\overline{M} + N} \left( \left. \sum\_{i=1}^{M} \left| \gamma\_{A\_i}(\mathbf{x}\_k) p\_{A\_i}(\mathbf{x}\_k) - \gamma\_{B\_i}(\mathbf{x}\_k) p\_{B\_i}(\mathbf{x}\_k) \right|^{\lambda} \right. \right) \right) \right)^{\frac{1}{\lambda}} \le 1, \sum\_{j=1}^{N} \frac{1}{n} \left( \left. \left. \gamma\_{A\_j}(\mathbf{x}\_k) q\_{A\_j}(\mathbf{x}\_k) - \gamma\_{B\_j}(\mathbf{x}\_k) q\_{B\_j}(\mathbf{x}\_k) \right|^{\lambda} \right) \right)^{\frac{1}{\lambda}} \le 1,$$

which clearly implies that 0 ≤ *d*1(*<sup>A</sup>*, *B*) ≤ 1. (P2) Since

$$\begin{split} d\_{1}(A,B) &= \left(\sum\_{k=1}^{n} \frac{1}{n} \left( \frac{1}{M+N} \left( \sum\_{i=1}^{M} \left| \gamma\_{A\_{i}}(\mathbf{x}\_{k}) p\_{A\_{i}}(\mathbf{x}\_{k}) - \gamma\_{B\_{i}}(\mathbf{x}\_{k}) p\_{B\_{i}}(\mathbf{x}\_{k}) \right|^{\lambda} \right) \right) \right)^{\frac{1}{\lambda}} \\ &= \left( \sum\_{k=1}^{n} \frac{1}{n} \left( \frac{1}{M+N} \left( \sum\_{i=1}^{M} \left| \gamma\_{B\_{i}}(\mathbf{x}\_{k}) p\_{B\_{i}}(\mathbf{x}\_{k}) - \gamma\_{A\_{i}}(\mathbf{x}\_{k}) p\_{A\_{i}}(\mathbf{x}\_{k}) \right|^{\lambda} \right) \right)^{\frac{1}{\lambda}} \\ &= \left( \sum\_{k=1}^{n} \frac{1}{n} \left( \frac{1}{M+N} \left( \sum\_{i=1}^{M} \left| \gamma\_{B\_{i}}(\mathbf{x}\_{k}) p\_{B\_{i}}(\mathbf{x}\_{k}) - \gamma\_{A\_{i}}(\mathbf{x}\_{k}) p\_{A\_{i}}(\mathbf{x}\_{k}) \right|^{\lambda} \right) \right)^{\frac{1}{\lambda}} \right)^{\frac{1}{\lambda}} \\ &= d\_{1}(B,A) \end{split}$$

Hence, the distance measure *d*1 possess a symmetric nature.

(P3) For *A* = *B*, we have *<sup>γ</sup>Ai*(*xk*) = *<sup>γ</sup>Bi*(*xk*) and *pAi*(*xk*) = *pBi*(*xk*). Also, *<sup>η</sup>Aj*(*xk*) = *<sup>η</sup>Bj*(*xk*) and *qAj*(*xk*) = *qBj*(*xk*). Thus, we have *<sup>γ</sup>Ai*(*xk*)*pAi*(*xk*) − *<sup>γ</sup>Ai*(*xk*)*pAi*(*xk*)*<sup>λ</sup>* = 0 and *<sup>η</sup>Aj*(*xk*)*qAj*(*xk*) − *<sup>η</sup>Aj*(*xk*)*qAj*(*xk*)*<sup>λ</sup>* = 0. Hence, it implies that

$$\begin{pmatrix} \sum\_{k=1}^{n} \frac{1}{n} \left( \frac{1}{M+N} \left( \sum\_{i=1}^{M} \left| \gamma\_{A\_i}(\mathbf{x}\_k) p\_{A\_i}(\mathbf{x}\_k) - \gamma\_{B\_i}(\mathbf{x}\_k) p\_{B\_i}(\mathbf{x}\_k) \right|^\lambda \right) \\\\ + \sum\_{j=1}^{N} \left| \eta\_{A\_j}(\mathbf{x}\_k) q\_{A\_j}(\mathbf{x}\_k) - \eta\_{B\_j}(\mathbf{x}\_k) q\_{B\_j}(\mathbf{x}\_k) \right|^\lambda \\\\ d\_1(A, B) = 0. \end{pmatrix} \right)^{\frac{1}{\lambda}} = 0$$

(P4) Since, *A* ⊆ *B* ⊆ *C*, then *<sup>γ</sup>Ai*(*xk*)*pAi*(*xk*) ≤ *<sup>γ</sup>Bi*(*xk*)*pBi*(*xk*) ≤ *<sup>γ</sup>Ci*(*xk*)*pCi*(*xk*) and *<sup>η</sup>Aj*(*xk*)*qAj*(*xk*) ≥ *<sup>η</sup>Bj*(*xk*)*qBj*(*xk*) ≥ *<sup>η</sup>Cj*(*xk*)*qCj*(*xk*). Further, *<sup>γ</sup>Ai*(*xk*)*pAi*(*xk*) − *<sup>γ</sup>Bi*(*xk*)*qBi*(*xk*)*<sup>λ</sup>* ≤ *<sup>γ</sup>Ai*(*xk*)*pAi*(*xk*) − *<sup>γ</sup>Ci*(*xk*)*qCi*(*xk*)*<sup>λ</sup>* and *<sup>η</sup>Aj*(*xk*)*qAj*(*xk*) − *<sup>η</sup>Bj*(*xk*)*qBj*(*xk*)*<sup>λ</sup>* ≥ *<sup>η</sup>Aj*(*xk*)*qAj*(*xk*) − *<sup>η</sup>Cj*(*xk*)*qCj*(*xk*)*<sup>λ</sup>*. Therefore, *d*1(*<sup>A</sup>*, *B*) ≤ *d*1(*<sup>A</sup>*, *C*) and *d*1(*<sup>B</sup>*, *C*) ≤ *d*1(*<sup>A</sup>*, *<sup>C</sup>*).

$$\mathbb{D}$$

**Theorem 2.** *Let A and B be two PDHFSs, then the distance measure d*2 *satisfies the following conditions:*

$$(P1) \quad 0 \le d\_2(A, B) \le 1;$$

*(P2) d*2(*<sup>A</sup>*, *B*) = *d*2(*<sup>B</sup>*, *<sup>A</sup>*);

⇒

*(P3) d*2(*<sup>A</sup>*, *B*) = 0 *if A* = *B*;

*(P4) If A* ⊆ *B* ⊆ *C*, *then d*2(*<sup>A</sup>*, *B*) ≤ *d*2(*<sup>A</sup>*, *C*) *and d*2(*<sup>B</sup>*, *C*) ≤ *d*2(*<sup>A</sup>*, *<sup>C</sup>*).

**Proof.** The proof is similar to Theorem 1, so we omit it here.

### **4. Einstein Aggregation Operational laws for PDHFSs**

In this section, we propose some operational laws and the investigate some of their properties associated with PDHFEs.

**Definition 6.** *Let α, α*1 *and α*2 *be three PDHFEs such that α* = *h*|*ph*, *<sup>g</sup>*|*qg, α*1 = *<sup>h</sup>*1|*ph*1 , *<sup>g</sup>*1|*qg*1 *and α*2 = *<sup>h</sup>*2|*ph*2, *<sup>g</sup>*2|*qg*2 *. Then, for λ* > 0*, we define the Einstein operational laws for them as follows:*

*Mathematics* **2018**, *6*, 280

$$\begin{split} (i) \quad a\_{1} \oplus a\_{2} &= \bigcup\_{\substack{\gamma\_{1} \in h\_{1}, \eta\_{1} \in \mathcal{S}\_{\mathcal{S}} \\ \gamma\_{2} \in h\_{2}, \eta\_{2} \in \mathcal{S}\_{\mathcal{S}} \end{subarray}} \left( \left\{ \frac{\gamma\_{1} + \gamma\_{2}}{1 + \gamma\_{1}\gamma\_{2}} \; \Big|\; p\_{\gamma\_{1}} p\_{\gamma\_{2}} \right\}, \left\{ \frac{\eta\_{1}\eta\_{2}}{1 + (1 - \eta\_{1})(1 - \eta\_{2})} \; \Big|\; q\_{\eta\_{1}} q\_{\eta\_{2}} \right\} \right\}; \\ (ii) \quad a\_{1} \otimes a\_{2} &= \bigcup\_{\substack{\gamma\_{1} \in h\_{1}, \eta\_{1} \in \mathcal{S}\_{\mathcal{S}} \\ \gamma\_{2} \in h\_{2}, \eta\_{2} \in \mathcal{S}\_{\mathcal{S}} \end{subarray}} \left( \left\{ \frac{\gamma\_{1}\gamma\_{2}}{1 + (1 - \gamma\_{1})(1 - \gamma\_{2})} \; \Big|\; p\_{\gamma\_{1}} p\_{\gamma\_{2}} \right\}, \left\{ \frac{\eta\_{1} + \eta\_{2}}{1 + \eta\_{1}\eta\_{2}} \; \Big|\; q\_{\eta\_{1}} q\_{\eta\_{2}} \right\} \right\}; \end{split}$$

$$\begin{array}{rcl} \text{(iii)} & \lambda \mathfrak{a} = & \bigcup\_{\substack{\gamma \in \mathfrak{h}\_r \\ \eta \in \mathfrak{g}}} \left( \left\{ \frac{(1+\gamma)^{\lambda} - (1-\gamma)^{\lambda}}{(1+\gamma)^{\lambda} + (1-\gamma)^{\lambda}} \; \middle| \; \mathcal{P}\gamma \right\} , \left\{ \frac{2(\eta)^{\lambda}}{(2-\eta)^{\lambda} + (\eta)^{\lambda}} \; \middle| \; \mathcal{q}\eta \right\} \right); \\ & & \eta \in \mathfrak{g} \end{array}$$

$$\begin{array}{rcl} \langle i\nu \rangle & \mathfrak{a}^{\lambda} = & \bigcup\_{\substack{\gamma \in \mathfrak{h}\_{\star} \\ \eta \in \mathfrak{g}}} \left( \left\{ \frac{2(\gamma)^{\lambda}}{(2-\gamma)^{\lambda}+(\gamma)^{\lambda}} \mid p\_{\gamma} \right\}, \left\{ \frac{(1+\eta)^{\lambda}-(1-\eta)^{\lambda}}{(1+\eta)^{\lambda}+(1-\eta)^{\lambda}} \mid q\_{\eta} \right\} \right) \\ & & \eta \in \mathfrak{g} \end{array}$$

**Theorem 3.** *For real value λ* > 0*, the operational laws for PDHFEs given in Definition 6 that is α*1 ⊕ *α*2 *, α*1 ⊗ *α*2*, λα*, *and α<sup>λ</sup> are also PDHFEs.*

**Proof.** For two PDHFEs *α*1 and *α*2, we have

$$a\_1 \oplus a\_2 = \bigcup\_{\substack{\gamma\_1 \in \mathfrak{h}\_1, \eta\_1 \in \mathfrak{g}\_1 \\ \gamma\_2 \in \mathfrak{h}\_2, \eta\_2 \in \mathfrak{g}\_2}} \left( \left\{ \frac{\gamma\_1 + \gamma\_2}{1 + \gamma\_1 \gamma\_2} \, \Big|\, p\_{\gamma\_1} p\_{\gamma\_2} \right\}, \left\{ \frac{\eta\_1 \eta\_2}{1 + (1 - \eta\_1)(1 - \eta\_2)} \, \Big|\, q\_{\eta\_1} q\_{\eta\_2} \right\} \right),$$

As 0 ≤ *γ*1, *γ*2, *η*1, *η*2 ≤ 1, thus it is evident that 0 ≤ *γ*1 + *γ*2 ≤ 2 and 1 ≤ 1 + *γ*1*γ*2 ≤ 2, thus it follows that 0 ≤ *γ*1+*γ*2 1+*γ*1*γ*2 ≤ 1. On the other hand, 0 ≤ *η*1*η*2 ≤ 1 and 1 ≤ 1 + (1 − *η*1)(<sup>1</sup> − *η*2) ≤ 2. Thus, 0 ≤ *η*1*η*2 <sup>1</sup>+(<sup>1</sup>−*η*1)(<sup>1</sup>−*η*2) ≤ 1 Also, since 0 ≤ *pγ*1 , *pγ*2 , *qη*1 , *qη*2 ≤ 1, thus 0 ≤ *pγ*1 *pγ*2 ≤ 1 and 0 ≤ *qη*1 *qη*2 ≤ 1. Similarly, *α*1 ⊗ *α*2 , *λα* and *α<sup>λ</sup>* are also PDHFEs.

**Theorem 4.** *Let α*1, *α*2, *α*3 *be three PDHFEs and λ*, *λ*1, *λ*2 > 0 *be three real numbers, then following results hold:*



**Proof.** Let *α*1 = - *<sup>h</sup>*1|*ph*1 , *<sup>g</sup>*1|*qg*1 , *α*2 = - *<sup>h</sup>*2|*ph*2 , *<sup>g</sup>*2|*qg*2 , *α*3 = - *<sup>h</sup>*3|*ph*3 , *<sup>g</sup>*3|*qg*3 be three PDHFEs. Then, we have

(i) For two PDHFEs *α*1 and *α*2, from Definition 6, we have

$$\begin{split} \mathfrak{a}\_{1} \oplus \mathfrak{a}\_{2} &= \bigcup\_{\substack{\gamma\_{1} \in h\_{1}, \eta\_{1} \in \mathfrak{z}\_{\mathcal{S}} \\ \gamma\_{2} \in h\_{2}, \eta\_{2} \in \mathfrak{z}\_{\mathcal{S}} \end{subarray}} \left( \left\{ \frac{\gamma\_{1} + \gamma\_{2}}{1 + \gamma\_{1}\gamma\_{2}} \; \big|\; p\_{\gamma\_{1}} p\_{\gamma\_{2}} \right\}, \left\{ \frac{\eta\_{1}\eta\_{2}}{1 + (1 - \eta\_{1})(1 - \eta\_{2})} \; \middle|\; q\_{\eta\_{1}} q\_{\eta\_{2}} \right\} \right\} \\ &= \bigcup\_{\substack{\gamma\_{1} \in h\_{1}, \eta\_{1} \in \mathfrak{z}\_{\mathcal{S}} \atop \gamma\_{2} \in h\_{2}, \eta\_{2} \in \mathfrak{z}\_{\mathcal{S}} \\ \varepsilon\_{1} \in h\_{2}, \eta\_{2} \in \mathfrak{z}\_{\mathcal{S}} \\ \end{split}} \left( \left\{ \frac{\gamma\_{2} + \gamma\_{1}}{1 + \gamma\_{2}\gamma\_{1}} \; \middle|\; p\_{\gamma\_{2}} p\_{\gamma\_{1}} \right\}, \left\{ \frac{\eta\_{2} \eta\_{1}}{1 + (1 - \eta\_{2})(1 - \eta\_{1})} \; \middle|\; q\_{\eta\_{2}} q\_{\eta\_{1}} \right\} \right\} \\ &= \mathfrak{a}\_{2} \oplus \mathfrak{a}\_{1} \end{split}$$

(ii) Proof is obvious so we omit it here. ### (iii) For three PDHFEs *α*1, *α*2 and *α*3, consider L.H.S. i.e.,

$$\begin{aligned} & (a\_1 \oplus a\_2) \oplus a\_3 \\ &= \left( \bigcup\_{\substack{\gamma\_1 \in \mathbb{N}\_1 \eta\_1 \gamma\_3 \in \mathbb{R}\_2 \\ \gamma\_2 \in \mathbb{N}\_2 \eta\_2 \gamma\_3 \in \mathbb{R}\_2}} \left( \left\{ \frac{\gamma\_1 + \gamma\_2}{1 + \gamma\_1 \gamma\_2} \, \Big|\, p\_{\gamma\_1} p\_{\gamma\_2} \right\}, \left\{ \frac{\eta\_1 \eta\_2}{1 + (1 - \eta\_1)(1 - \eta\_2)} \, \Big|\, q\_{\eta\_1} q\_{\eta\_2} \right\} \right) \right) \oplus a\_3 \\ &= \bigcup\_{\substack{\gamma\_1 \in \mathbb{N}\_1 \eta\_1 \gamma\_3 \in \mathbb{N}\_2 \\ \gamma\_2 \in \mathbb{N}\_2 \eta\_2 \gamma\_3 \in \mathbb{N}\_2 \\ \gamma\_3 \in \mathbb{N}\_3 \eta\_3 \in \mathbb{N}\_2 \\ \gamma\_3 \in \mathbb{N}\_3 \eta\_3 \in \mathbb{N}\_2}} \left( \left\{ \frac{\gamma\_1 + \gamma\_2 + \gamma\_3 + \gamma\_1 \gamma\_2 \gamma\_3}{1 + \gamma\_1 \gamma\_2 + \gamma\_3 \gamma\_1 + \gamma\_3 \gamma\_1} \, \Big|\, p\_{\gamma\_1} p\_{\gamma\_2} p\_{\gamma\_3} \right\}, \left\{ \frac{\eta\_1 \eta\_2 \eta\_3}{4 - 2\eta\_1 - 2\eta\_2 + \eta\_1 \eta\_2 + \eta\_2 \eta\_3 + \eta\_1 \eta\_3} \, \Big|\, q\_{\eta\_1} q\_{\eta\_2} q\_{\eta\_3} \right\} \right) \end{aligned} \tag{11}$$

Also, on considering the R.H.S., we have

$$\begin{aligned} \mathfrak{a}\_1 &\oplus (\mathfrak{a}\_2 \oplus \mathfrak{a}\_3) \\ \mathfrak{b}\_1 &= \mathfrak{a}\_1 \oplus \left( \bigcup\_{\substack{\gamma\_2 \in \mathfrak{b}\_2, \eta\_2 \in \mathfrak{g}\_2 \\ \gamma\_3 \in \mathfrak{b}\_3, \eta\_3 \in \mathfrak{g}\_3}} \left( \left\{ \frac{\gamma\_2 + \gamma\_3}{1 + \gamma\_2 \gamma\_3} \, \Big|\, p\_{\gamma\_2} p\_{\gamma\_3} \right\} , \left\{ \frac{\eta\_2 \eta\_3}{1 + (1 - \eta\_2)(1 - \eta\_3)} \, \Big|\, q\_{\eta\_2} q\_{\eta\_3} \right\} \right) \right) \end{aligned}$$

$$=\bigcup\_{\substack{\gamma\_{1}\in b\_{1},\eta\_{1}\in g\_{1} \\ \gamma\_{2}\in b\_{2},\eta\_{2}\in g\_{2} \\ \gamma\_{3}\in b\_{3},\eta\_{3}\in g\_{3}}} \left( \left\{ \frac{\gamma\_{1}+\gamma\_{2}+\gamma\_{3}+\gamma\_{1}\gamma\_{2}\gamma\_{3}}{1+\gamma\_{1}\gamma\_{2}+\gamma\_{2}\gamma\_{3}+\gamma\_{3}\gamma\_{1}} \; \middle|\; p\_{\gamma\_{1}}p\_{\gamma\_{2}}p\_{\gamma\_{3}} \right\} , \left\{ \frac{\eta\_{1}\eta\_{2}\eta\_{3}}{4-2\eta\_{1}-2\eta\_{2}-2\eta\_{3}+\eta\_{1}\eta\_{2}+\eta\_{2}\eta\_{3}+\eta\_{1}\eta\_{3}} \; \middle|\; q\_{\eta\_{1}}q\_{\eta\_{2}}q\_{\eta\_{3}} \right\} \right) \tag{12}$$

From Equations (11) and (12), the required result is obtained.


$$\lambda\left(a\_1\oplus a\_2\right) = \lambda\left(\bigcup\_{\substack{\gamma\_1\in b\_1, \eta\_1\in\underline{\eta}\underline{\eta}\\ \gamma\_2\in b\_2, \eta\_2\in\underline{\eta}\underline{\eta}}} \left( \left\{ \frac{(1+\gamma\_1)(1+\gamma\_2) - (1-\gamma\_1)(1-\gamma\_2)}{(1+\gamma\_1)(1+\gamma\_2) + (1-\gamma\_1)(1-\gamma\_2)} \; \middle|\; \begin{array}{c} p\_{\gamma\_1}p\_{\gamma\_2} \\ (\frac{2\eta\_1\eta\_2}{(2-\eta\_1)(2-\eta\_2)+\eta\_1\eta\_2} \; \middle|\; \begin{array}{c} q\_{\eta\_1}q\_{\eta\_2} \\ \end{array} \right\} \right) \right)$$

For sake of convenience, put (1 + *<sup>γ</sup>*1)(<sup>1</sup> + *<sup>γ</sup>*2) = *a* ; (1 − *<sup>γ</sup>*1)(<sup>1</sup> − *<sup>γ</sup>*2) = *b*; *η*1*η*2 = *c* and (2 − *η*1)(<sup>2</sup> − *η*2) = *d*. This implies

*<sup>λ</sup>*(*<sup>α</sup>*1 ⊕ *<sup>α</sup>*2) = *λ γ*1∈*h*1,*η*1∈*g*1 *γ*2∈*h*2,*η*2∈*g*2 *a* − *b a* + *b pγ*1 *pγ*2 , 2*c d* + *c qη*1 *<sup>q</sup>η*2 = *γ*1∈*h*1,*η*1∈*g*1 *γ*2∈*h*2,*η*2∈*g*2 ⎛⎜⎜⎜⎝⎧⎪⎪⎪⎨⎪⎪⎪⎩1 + *a* − *b a* + *bλ* − 1 − *a* − *b a* + *bλ* 1 + *a* − *b a* + *bλ* + 1 − *a* − *b a* + *bλ pγ*1 *pγ*2⎫⎪⎪⎪⎬⎪⎪⎪⎭, ⎧⎪⎪⎪⎨⎪⎪⎪⎩ 2 2*c d* + *cλ* 2 − 2*c d* + *cλ* + 2*c d* + *cλ qη*1 *qη*2⎫⎪⎪⎪⎬⎪⎪⎪⎭⎞⎟⎟⎟⎠ = *γ*1∈*h*1,*η*1∈*g*1 *γ*2∈*h*2,*η*2∈*g*2 ⎛⎜⎜⎜⎝⎧⎪⎪⎪⎨⎪⎪⎪⎩ 2*a a* + *bλ* − 2*b a* + *bλ* 2*a a* + *bλ* + 2*b a* + *bλ pγ*1 *pγ*2⎫⎪⎪⎪⎬⎪⎪⎪⎭, ⎧⎪⎪⎪⎨⎪⎪⎪⎩ 2 2*c d* + *cλ* 2*d d* + *cλ* + 2*a d* + *cλ qη*1 *qη*2⎫⎪⎪⎪⎬⎪⎪⎪⎭⎞⎟⎟⎟⎠ = *γ*1∈*h*1,*η*1∈*g*1 *γ*2∈*h*2,*η*2∈*g*2 (*a<sup>λ</sup>* − *b<sup>λ</sup>*) (*a<sup>λ</sup>* + *b<sup>λ</sup>*) *pγ*1 *pγ*2, 2*c<sup>λ</sup> d<sup>λ</sup>* + *c<sup>λ</sup> qη*1 *<sup>q</sup>η*2

Re-substituting *a*, *b*, *c* and *d* we have

$$\begin{array}{lcl} &=&\bigcup\_{\begin{subarray}{c}\gamma\_{1}\in\mathsf{b}\_{1},\mathsf{y}\_{1}\in\mathsf{g}\_{2}\end{subarray}}\left(\left\{\frac{(1+\gamma\_{1})^{\lambda}\left(1+\gamma\_{2}\right)^{\lambda}-(1-\gamma\_{1})^{\lambda}\left(1-\gamma\_{2}\right)^{\lambda}}{\left(1+\gamma\_{1}\right)^{\lambda}\left(1+\gamma\_{2}\right)^{\lambda}+\left(1-\gamma\_{1}\right)^{\lambda}\left(1-\gamma\_{2}\right)^{\lambda}}\,\Big|\,p\_{\gamma\_{1}}p\_{\gamma\_{2}}\right\},\left\{\frac{2\left(\eta\_{1}\eta\_{2}\right)^{\lambda}}{\left(2-\eta\_{1}\right)^{\lambda}\left(2-\eta\_{2}\right)^{\lambda}+\eta\_{1}\eta\_{2}}\,\Big|\,q\_{\eta\_{1}}q\_{\eta\_{2}}\right\}\right)\\ &=&\lambda\mathfrak{a}\_{1}\oplus\lambda\mathfrak{a}\_{2}\end{array}$$

(vi) For *λ* > 0,

$$\left(\left(\mathfrak{a}\_{1}\otimes\mathfrak{a}\_{2}\right)^{\lambda}\;\;\;=\;\left(\bigcup\_{\substack{\gamma\in\mathbb{N},\eta\_{1}\in\mathfrak{g}\_{2}\\ \gamma\_{1}\in\mathfrak{h}\_{\mathcal{I}},\eta\_{2}\in\mathfrak{g}\_{2}}}\left(\left\{\frac{2\gamma\_{1}\gamma\_{2}}{1+\left(1-\gamma\_{1}\right)\left(1-\gamma\_{2}\right)}\;\Big|\;\,p\_{\gamma\_{1}}p\_{\gamma\_{2}}\right\},\left\{\frac{\left(1+\eta\_{1}\right)\left(1+\eta\_{2}\right)-\left(1-\eta\_{1}\right)\left(1-\eta\_{2}\right)}{\left(1+\eta\_{1}\right)\left(1+\eta\_{2}\right)+\left(1-\eta\_{1}\right)\left(1-\eta\_{2}\right)}\;\middle|\;\;p\_{\gamma\_{1}}q\_{\gamma\_{2}}\right\}\right)\right)^{\lambda}$$

For sake of convenience, put

$$\gamma\_1 \gamma\_2 = a; \left(2 - \gamma\_1\right)\left(2 - \gamma\_2\right) = b; \left(1 + \eta\_1\right)\left(1 + \eta\_2\right) = c \text{ and } \left(1 - \eta\_1\right)\left(1 - \eta\_2\right) = d$$

So we obtain

(*<sup>α</sup>*1 ⊗ *<sup>α</sup>*2)*<sup>λ</sup>* = ⎛⎜⎜⎝ *γ*1∈*h*1,*η*1∈*g*1 *γ*2∈*h*2,*η*2∈*g*2 2*a b* + *a pγ*1 *pγ*2, *c* − *d c* + *d qη*1 *qη*2⎞⎟⎟⎠*λ* = *γ*1∈*h*1,*η*1∈*g*1 *γ*2∈*h*2,*η*2∈*g*2 ⎛⎜⎜⎜⎝⎧⎪⎪⎪⎨⎪⎪⎪⎩ 2 2*a b* + *aλ* 2 − 2*a b* + *aλ* + 2*a b* + *aλ pγ*1 *pγ*2⎫⎪⎪⎪⎬⎪⎪⎪⎭, ⎧⎪⎪⎪⎨⎪⎪⎪⎩1 + *c* − *d c* + *dλ* − 1 − *c* − *d c* + *dλ* 1 + *c* − *d c* + *dλ* + 1 − *c* − *d c* + *dλ qη*1 *qη*2⎫⎪⎪⎪⎬⎪⎪⎪⎭⎞⎟⎟⎟⎠ = *γ*1∈*h*1,*η*1∈*g*1 *γ*2∈*h*2,*η*2∈*g*2 ⎛⎜⎜⎜⎝⎧⎪⎪⎪⎨⎪⎪⎪⎩ 2 2*a b* + *aλ* 2*b b* + *aλ* + 2*a b* + *aλ pγ*1*γ*2⎫⎪⎪⎪⎬⎪⎪⎪⎭, ⎧⎪⎪⎪⎨⎪⎪⎪⎩ 2*c c* + *dλ* − 2*d c* + *dλ* 2*c c* + *dλ* + 2*d c* + *dλ qη*1 *qη*2⎫⎪⎪⎪⎬⎪⎪⎪⎭⎞⎟⎟⎟⎠ = *γ*1∈*h*1,*η*1∈*g*1 *γ*2∈*h*2,*η*2∈*g*2 2*a<sup>λ</sup> b<sup>λ</sup>* + *a<sup>λ</sup> pγ*1 *pγ*2, *cλ* − *d<sup>λ</sup> c<sup>λ</sup>* + *d<sup>λ</sup> qη*1 *<sup>q</sup>η*2

Re-substituting values of *a*, *b*, *c* and *d* we ge<sup>t</sup>

$$\begin{split} \mathcal{I} &= \bigcup\_{\substack{\gamma\_{1}\in\mathcal{Y}\_{1},\eta\_{1}\in\mathcal{Y}\_{1}\p\_{1}\in\mathcal{Y}\_{1} \\ \gamma\_{2}\in\mathfrak{h}\_{2},\eta\_{2}\in\mathcal{Y}\_{2}}} \left\{ \left\{ \frac{2\left(\gamma\_{1}\gamma\_{2}\right)^{\lambda}}{(2-\gamma\_{1})^{\lambda}\left(2-\gamma\_{2}\right)^{\lambda}+\left(\gamma\_{1}\gamma\_{2}\right)^{\lambda}} \; \middle|\; p\_{\gamma\_{1}}p\_{\gamma\_{2}} \right\}, \left\{ \frac{\left(1+\eta\_{1}\right)^{\lambda}\left(1+\eta\_{2}\right)^{\lambda}-\left(1-\eta\_{1}\right)^{\lambda}\left(1-\eta\_{2}\right)^{\lambda}}{\left(1+\eta\_{1}\right)^{\lambda}\left(1+\eta\_{2}\right)^{\lambda}+\left(1-\eta\_{1}\right)^{\lambda}\left(1-\eta\_{2}\right)^{\lambda}} \; \middle|\; q\_{\gamma\_{1}}q\_{\gamma\_{2}} \right\} \right\} \\ &= \mathcal{I}\_{1} \; \mathcal{I}\_{2} \; \mathcal{I} \otimes a\_{2}^{\lambda} \end{split}$$

**Theorem 5.** *Let α* = *h*|*ph*, *g*|*qg α*1 = *<sup>h</sup>*1|*ph*1 , *<sup>g</sup>*1|*qg*1 , *and α*2 = *<sup>h</sup>*2|*ph*2 , *<sup>g</sup>*2|*qg*2 *be three PDHFEs, and λ* > 0 *be a real number, then*

*(i)* (*α<sup>c</sup>*)*<sup>λ</sup>* = *λαc;*


**Proof.** (i) Let *α* = *h*|*ph*, *g*|*qg* be a PDHFE, then using Definition 4, the proof for the three possible cases is given as:

(Case 1) If *h* = *φ*; *g* = *φ* then for a PDHFE *α* = *h*|*ph*, *<sup>g</sup>*|*qg*, from Equation (7) we have

$$\begin{aligned} (\mathfrak{a}^c)^\lambda &= \left( \bigcup\_{\substack{\gamma \in h \\ \eta \in g}} \left( \left\{ \eta \; \middle| \; q\_\eta \right\} , \left\{ \gamma \; \middle| \; p\_\gamma \right\} \right) \right)^\lambda \\ &= \bigcup\_{\substack{\gamma \in h \\ \eta \in g}} \left( \left\{ \frac{2(\eta)^\lambda}{(2-\eta)^\lambda + (\eta)^\lambda} \; \middle| \; q\_{\eta \cdot} \right\} , \left\{ \frac{(1+\gamma)^\lambda - (1-\gamma)^\lambda}{(1+\gamma)^\lambda + (1-\gamma)^\lambda} \; \middle| \; p\_\gamma \right\} \right)^\lambda \end{aligned}$$

*Mathematics* **2018**, *6*, 280

$$\begin{aligned} &= \left( \bigcup\_{\substack{\gamma \in \mathbb{A} \\ \eta \in \mathfrak{g}}} \left( \left\{ \frac{(1+\gamma)^{\lambda} - (1-\gamma)^{\lambda}}{(1+\gamma)^{\lambda} + (1-\gamma)^{\lambda}} \; \middle|\; p\_{\gamma} \right\} , \left\{ \frac{2(\eta)^{\lambda}}{(2-\eta)^{\lambda} + (\eta)^{\lambda}} \; \middle|\; q\_{\eta} \right\} \right) \right)^{c} \\ &= \left( \lambda \left( \bigcup\_{\substack{\gamma \in \mathbb{A} \\ \eta \in \mathfrak{g}}} \left\{ \gamma \; \middle|\; p\_{\gamma} \right\} , \left\{ \eta \; \middle|\; q\_{\eta} \right\} \right) \right)^{c} = (\lambda a)^{c} \end{aligned}$$

(Case 2) If *g* = *φ*, *h* = *φ*, then

$$\begin{aligned} (a^c)^\lambda &= \left( \bigcup\_{\gamma \in h} \left( \left\{ 1 - \gamma \; \middle| \; p\_\gamma \right\}, \{\phi\} \right) \right)^\lambda \\ &= \bigcup\_{\gamma \in h} \left( \left\{ \frac{2(1 - \gamma)^\lambda}{(2 - (1 - \gamma))^\lambda + (1 - \gamma)^\lambda} \; \middle| \; p\_\gamma \right\}, \{\phi\} \right)^\lambda \\ &= \quad (\lambda a)^c \end{aligned}$$

(Case 3) If *h* = *φ*, *g* = *φ*, then

$$\begin{split} (\boldsymbol{a}^{c})^{\lambda} &= \quad \left( \bigcup\_{\eta \in \mathcal{S}} \left( \{\boldsymbol{\Phi}\}, \left\{ 1 - \eta \; \middle| \; q \; \middle\} \right\} \right)^{\lambda} \\ &= \quad \bigcup\_{\eta \in \mathcal{S}} \left( \{\boldsymbol{\Phi}\}, \left\{ \frac{(1 + (1 - \eta))^{\lambda} - (1 - (1 - \eta))^{\lambda}}{(1 + (1 - \eta))^{\lambda} + (1 - (1 - \eta))^{\lambda}} \; \middle\| \; q \;\right\} \right)^{\lambda} \\ &= \quad \left( \bigcup\_{\eta \in \mathcal{S}} \left( \left\{ \frac{(2 - \eta)^{\lambda} - (\eta)^{\lambda}}{(2 - \eta)^{\lambda} + (\eta)^{\lambda}} \; \middle\| \; q \;\right\} \right)^{c} \\ &= \quad \left( \lambda \bigcup\_{\eta \in \mathcal{S}} \left\{ (1 - \eta) \; \middle\| \; q \;\right\} \right)^{c} = (\lambda \boldsymbol{a})^{c} \end{split}$$


(Case 1) If *h*1 = *φ*, *g*1 = *φ*, *h*2 = *φ* and *g*2 = *φ*

*αc*1 ⊕ *αc*2 = *<sup>γ</sup>*1∈*h*1 *η*1∈*g*1 *<sup>η</sup>*1 *<sup>q</sup>η*1 , *γ*1 *<sup>p</sup>γ*1 ⊕ *<sup>γ</sup>*2∈*h*2 *η*2∈*g*2 *<sup>η</sup>*2 *<sup>q</sup>η*2 , *γ*2 *<sup>p</sup>γ*2 = *γ*1∈*h*1,*η*1∈*g*1 *γ*2∈*h*2,*η*2∈*g*2 *η*1 + *η*2 1 + *η*1*η*2 *qη*1 *qη*2 , *γ*1*γ*2 1 + (1 − *<sup>γ</sup>*1)(<sup>1</sup> − *<sup>γ</sup>*2) *pγ*1 *<sup>p</sup>γ*2 = *γ*1∈*h*1,*η*1∈*g*1 *γ*2∈*h*2,*η*2∈*g*2 *γ*1*γ*2 1 + (1 − *<sup>γ</sup>*1)(<sup>1</sup> − *<sup>γ</sup>*2) *pγ*1 *pγ*2 , *η*1 + *η*2 1 + *η*1*η*2 *qη*1 *<sup>q</sup>η*2*<sup>c</sup>* = (*<sup>α</sup>*1 ⊗ *<sup>α</sup>*2)*<sup>c</sup>*

(Case 2) If *h*1 = *φ*, *g*1 = *φ*, *h*2 = *φ* and *g*2 = *φ*, then

$$\begin{split} \mathfrak{a}\_{1}^{\varepsilon} \oplus \mathfrak{a}\_{2}^{\varepsilon} &= \bigcup\_{\begin{subarray}{c} \gamma\_{1} \in \mathbb{b}\_{1} \\ \eta\_{1} \in \mathfrak{g}\_{1} \end{subarray}} \left( \left\{ 1 - \gamma\_{1} \; \middle| \; \begin{subarray}{c} p\_{\gamma\_{1}} \end{subarray} \; \middle| \; \begin{subarray}{c} \{\Phi\} \end{subarray} \right\} \oplus \bigcup\_{\begin{subarray}{c} \gamma\_{2} \in \mathbb{b}\_{2} \\ \eta\_{2} \in \mathfrak{g}\_{2} \end{subarray}} \left( \left\{ 1 - \gamma\_{2} \; \middle| \; \begin{subarray}{c} p\_{\gamma\_{2}} \end{subarray} \; \middle| \; \begin{subarray}{c} p\_{\gamma\_{2}} \end{subarray} \right\} \right) \right) \\ &= \bigcup\_{\begin{subarray}{c} \gamma\_{1} \in \mathbb{b}\_{1}, \eta\_{1} \in \mathfrak{g}\_{1} \\ \gamma\_{2} \in \mathbb{b}\_{2}, \eta\_{2} \in \mathbb{g}\_{2} \\ \end{subarray}} \left( \left\{ \frac{(1 - \gamma\_{1}) + (1 - \gamma\_{2})}{1 + (1 - \gamma\_{1})(1 - \gamma\_{2})} \; \middle| \; \begin{subarray}{c} p\_{\gamma\_{1}} p\_{\gamma\_{2}} \end{subarray} \; \middle| \; \begin{subarray}{c} p\_{\gamma\_{1}} p\_{\gamma\_{2}} \end{subarray} \; \middle| \; \begin{subarray}{c} p\_{\delta} \end{subarray} \right\} \right) \\ &= \left( \mathfrak{a}\_{1} \otimes \mathfrak{a}\_{2} \right)^{c} \end{split}$$

(Case 3) If *h*1 = *φ*, *g*1 = *φ*, *h*2 = *φ*, *g*2 = *φ*

$$\begin{split} a\_1^{\varepsilon} \oplus a\_2^{\varepsilon} &= \bigcup\_{\substack{\gamma\_1 \in h\_1 \\ \eta\_1 \in \eta\_1}} \left( \{\emptyset\} \, \Big\{ 1 - \eta\_1 \, \Big\} \, q\_{\eta\_1} \right) \oplus \bigcup\_{\substack{\gamma\_2 \in h\_2 \\ \eta\_2 \in \eta\_2}} \left( \{\emptyset\} \, \Big\{ 1 - \eta\_2 \, \Big\} \, q\_{\eta\_2} \right) \\ &= \bigcup\_{\substack{\gamma\_1 \in h\_1 \eta\_1 \in \xi\_1 \\ \gamma\_2 \in h\_2, \eta\_2 \in \xi\_2}} \left( \{\emptyset\} \, \Big\{ \begin{array}{c} (1 - \eta\_1)(1 - \eta\_2) \\ 1 + \eta\_1 \eta\_2 \end{array} \Big\} \, q\_{\eta\_1} q\_{\eta\_2} \right) \\ &= \left( a\_1 \otimes a\_2 \right)^{\varepsilon} \end{split}$$

(iv) Similar, so we omit it here.

### **5. Probabilistic Dual Hesitant Weighted Einstein AOs**

In this section, we have defined some weighted aggregation operators by using aforementioned laws for a collection of PDHFEs. For it, let Ω be the family of PDHFEs.

**Definition 7.** *Let* Ω *be the family of PDHFEs αi* (*i* = 1, 2, ... , *n*) *with the corresponding weights ω* = (*<sup>ω</sup>*1, *ω*2,..., *<sup>ω</sup>n*)*T, such that ωi* > 0 *and n* ∑ *i*=1 *ωi* = 1*. If PDHFWEA:* Ω*n* → Ω, *is a mapping defined by*

$$\text{PDHFWEA}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n) \quad = \quad \omega\_1 \mathfrak{a}\_1 \oplus \omega\_2 \mathfrak{a}\_2 \oplus \dots \oplus \omega\_n \mathfrak{a}\_n \tag{13}$$

*then, PDHFWEA is called probabilistic dual hesitant fuzzy weighted Einstein average operator.*

**Theorem 6.** *For a family of PDHFEs αi* = *hi phi* , *gi qgi* ,(*<sup>i</sup>* = 1, 2, ... , *<sup>n</sup>*)*, the aggregated value obtained by using PDHFWEA operator is still a PDHFE and is given as*

$$PDHFWA(\mathbf{a}\_1, \mathbf{a}\_2, \dots, \mathbf{a}\_n) \quad \left( \bigcup\_{\substack{\gamma\_i \in \mathbb{N}\_i \\ \eta\_i \in \mathbb{N}\_i}} \left\{ \frac{\left\{ \begin{aligned} \frac{n}{\prod} (1 + \gamma\_i)^{\omega\_i} - \prod\_{i=1}^n (1 - \gamma\_i)^{\omega\_i} \\ \frac{n}{\prod} (1 + \gamma\_i)^{\omega\_i} + \prod\_{i=1}^n (1 - \gamma\_i)^{\omega\_i} \end{aligned} \right\} \prod\_{i=1}^n p\_{\gamma\_i} \right\}\right), \tag{14}$$
 
$$\left\{ \begin{aligned} \underbrace{2 \prod\_{i=1}^n (\eta\_i)^{\omega\_i}}\_{\prod\_{i=1}^n (1 - \eta\_i)^{\omega\_i} + \prod\_{i=1}^n (\eta\_i)^{\omega\_i}}\_{\prod\_{i=1}^n (1 - \eta\_i)^{\omega\_i} + \prod\_{i=1}^n (\eta\_i)^{\omega\_i}} \end{aligned} \tag{15}$$

*where ω* = (*<sup>ω</sup>*1, *ω*2,..., *<sup>ω</sup>n*)*<sup>T</sup> is a weight vector such that* ∑ *i*=1 *ωi* = 1 *where* 0 < *ωi* < 1.

**Proof.** We will prove the Equation (14) by following the steps mathematical induction on *n*, and the proof is executed as below:

(Step 1) For *n* = 2, we have *α*1 = *h*1 *ph*1 , *g*1 *qg*1 and *α*2 = *h*2 *ph*2 , *g*2 *qg*2 . Using operational laws on PDHFEs as stated in Definition 6 we ge<sup>t</sup>

$$\begin{array}{rcl}\omega\_{1}a\_{1} &=& \bigcup\_{\gamma\_{1} \in h\_{1}, \eta\_{1} \in \mathcal{g}\_{1}} \left( \left\{ \frac{(1+\gamma\_{1})^{\omega\_{1}} - (1-\gamma\_{1})^{\omega\_{1}}}{(1+\gamma\_{1})^{\omega\_{1}} + (1-\gamma\_{1})^{\omega\_{1}}} \; \middle| \; p\_{\gamma\_{1}} \right\} \right) \\ & & \left( \frac{2(\eta\_{1})^{\omega\_{1}}}{(2-\eta\_{1})^{\omega\_{1}} + (\eta\_{1})^{\omega\_{1}}} \; \middle| \; q\_{\eta\_{1}} \right) \\ \text{and} & \omega\_{2}a\_{2} &=& \bigcup\_{\gamma\_{2} \in h\_{2}, \eta\_{2} \in \mathcal{g}\_{2}} \left( \left\{ \frac{(1+\gamma\_{2})^{\omega\_{2}} - (1-\gamma\_{2})^{\omega\_{2}}}{(1+\gamma\_{2})^{\omega\_{2}} + (1-\gamma\_{2})^{\omega\_{2}}} \; \middle| \; p\_{\gamma\_{2}} \right\} \right) \\ & & & & \\ \end{array}$$

Hence, by addition of PDHFEs, we ge<sup>t</sup>

$$\begin{aligned} & \text{PDHFWEA}(\alpha\_1, \alpha\_2) = \omega\_1 \alpha\_1 \oplus \omega\_2 \alpha\_2 \\ &= \bigcup\_{\substack{\gamma\_1 \in \mathbb{H}\_1, \eta\_1 \in \mathcal{G}\_1 \\ \gamma\_2 \in \mathbb{H}\_2, \eta\_2 \in \mathcal{G}\_2}} \left( \left\{ \frac{\prod\_{i=1}^2 (1 + \gamma\_i)^{\omega\_i} - \prod\_{i=1}^2 (1 - \gamma\_i)^{\omega\_i}}{2} \, \middle| \, \prod\_{i=1}^2 p\_{\gamma\_i} \right\} \right) \\ &= \bigcup\_{\substack{\gamma\_1 \in \mathbb{H}\_1, \eta\_1 \in \mathcal{G}\_1 \\ \gamma\_2 \in \mathbb{H}\_2, \eta\_2 \in \mathcal{G}\_2}} \left( \left\{ \frac{2 \prod\_{i=1}^2 (\eta\_i)^{\omega\_i}}{\prod\_{i=1}^2 (\eta\_i)^{\omega\_i}} \, \middle| \, \prod\_{i=1}^2 q\_{\eta\_i} \right\} \right) \end{aligned}$$

Thus, the result holds for *n* = 2.

(Step 2) If Equation (14) holds for *n* = *k*, then for *n* = *k* + 1, we have

PDHFWEA(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>k*+<sup>1</sup>) = "*k i*=1 *ωiαi*! ⊕ (*<sup>ω</sup>k*+1*αk*+<sup>1</sup>) = *γi*<sup>∈</sup>*hi*,*ηi*∈*gi* ⎛⎜⎜⎜⎝⎧⎪⎪⎪⎨⎪⎪⎪⎩ *k*∏*<sup>i</sup>*=<sup>1</sup>(<sup>1</sup> + *<sup>γ</sup>i*)*<sup>ω</sup><sup>i</sup>* − *k*∏*<sup>i</sup>*=<sup>1</sup>(<sup>1</sup> − *<sup>γ</sup>i*)*<sup>ω</sup><sup>i</sup> k*∏*<sup>i</sup>*=<sup>1</sup>(<sup>1</sup> + *<sup>γ</sup>i*)*<sup>ω</sup><sup>i</sup>* + *k*∏*<sup>i</sup>*=<sup>1</sup>(<sup>1</sup> − *<sup>γ</sup>i*)*<sup>ω</sup><sup>i</sup> k*∏*i*=1 *pγi*⎫⎪⎪⎪⎬⎪⎪⎪⎭, ⎧⎪⎪⎪⎨⎪⎪⎪⎩ 2 *k*∏*<sup>i</sup>*=<sup>1</sup>(*ηi*)*<sup>ω</sup><sup>i</sup> k*∏*<sup>i</sup>*=<sup>1</sup>(<sup>2</sup> − *ηi*)*<sup>ω</sup><sup>i</sup>* + *k*∏*<sup>i</sup>*=<sup>1</sup>(*ηi*)*<sup>ω</sup><sup>i</sup> k*∏*i*=1 *qηi*⎫⎪⎪⎪⎬⎪⎪⎪⎭⎞⎟⎟⎟⎠ ⊕ *<sup>γ</sup>k*+1<sup>∈</sup>*hk*+1, *ηk*+1∈*gk*+<sup>1</sup> (1 + *<sup>γ</sup>k*+<sup>1</sup>)*<sup>ω</sup>k*+<sup>1</sup> − (1 − *<sup>γ</sup>k*+<sup>1</sup>)*<sup>ω</sup>k*+<sup>1</sup> (1 + *<sup>γ</sup>k*+<sup>1</sup>)*<sup>ω</sup>k*+<sup>1</sup> + (1 − *<sup>γ</sup>k*+<sup>1</sup>)*<sup>ω</sup>k*+<sup>1</sup> *pγk*+<sup>1</sup> , , <sup>2</sup>(*ηk*+<sup>1</sup>)*<sup>ω</sup>k*+<sup>1</sup> (2 − *ηk*+<sup>1</sup>)*<sup>ω</sup>k*+<sup>1</sup> + (*ηk*+<sup>1</sup>)*<sup>ω</sup>k*+<sup>1</sup> *<sup>q</sup>ηk*+<sup>1</sup> = *γi*<sup>∈</sup>*hi*,*ηi*∈*gi* ⎛⎜⎜⎜⎝⎧⎪⎪⎪⎨⎪⎪⎪⎩*k*+1 ∏*<sup>i</sup>*=<sup>1</sup>(<sup>1</sup> + *<sup>γ</sup>i*)*<sup>ω</sup><sup>i</sup>* − *k*+1 ∏*<sup>i</sup>*=<sup>1</sup>(<sup>1</sup> − *<sup>γ</sup>i*)*<sup>ω</sup><sup>i</sup> k*+1 ∏*<sup>i</sup>*=<sup>1</sup>(<sup>1</sup> + *<sup>γ</sup>i*)*<sup>ω</sup><sup>i</sup>* + *k*+1 ∏*<sup>i</sup>*=<sup>1</sup>(<sup>1</sup> − *<sup>γ</sup>i*)*<sup>ω</sup><sup>i</sup> k*+1 ∏*i*=1 *pγi*⎫⎪⎪⎪⎬⎪⎪⎪⎭, ⎧⎪⎪⎪⎨⎪⎪⎪⎩ 2 *k*+1 ∏*<sup>i</sup>*=<sup>1</sup>(*ηi*)*<sup>ω</sup><sup>i</sup> k*+1 ∏*<sup>i</sup>*=<sup>1</sup>(<sup>2</sup> − *ηi*)*<sup>ω</sup><sup>i</sup>* + *k*+1 ∏*<sup>i</sup>*=<sup>1</sup>(*ηi*)*<sup>ω</sup><sup>i</sup> k*+1 ∏*i*=1 *qηi*⎫⎪⎪⎪⎬⎪⎪⎪⎭⎞⎟⎟⎟⎠

Thus,

$$=\bigcup\_{\substack{\boldsymbol{\eta}\_{i}\in\boldsymbol{k}\_{i}\\ \boldsymbol{\eta}\_{i}\in\boldsymbol{g}\_{i}}}\left(\left\{\frac{\prod\_{i=1}^{n}(1+\gamma\_{i})^{\omega\_{i}}-\prod\_{i=1}^{n}(1-\gamma\_{i})^{\omega\_{i}}}{\prod\_{i=1}^{n}(1+\gamma\_{i})^{\omega\_{i}}+\prod\_{i=1}^{n}(1-\gamma\_{i})^{\omega\_{i}}}\,\middle|\,\prod\_{i=1}^{n}p\_{\gamma\_{i}}\right\}\right)$$

which completes the proof.

Further, it is observed that the proposed PDHFWEA operator satisfies the properties of boundedness and monotonicity, for a family of PDHFEs *<sup>α</sup>i*,(*<sup>i</sup>* = 1, 2, ... , *n*) which can be demonstrated as follows:

$$\begin{array}{lcl} \textbf{Property} & \textbf{1. (Boundaryness)} & \textbf{For } \ a\_{i} &= \left( \left. h\_{i} \right| \, \left| \, \eta\_{h\_{i}} \, \text{g}\_{i} \right| \, \left| \, q\_{\overline{g}\_{i}} \right| \right) \text{ where } i &= & (1, 2, \ldots, n), \\ \textbf{let } a^{-} = \left( \min(h\_{i}) \, \left| \, \min(p\_{h\_{i}}) \, \text{, } \max(\underline{g}\_{i}) \, \right| \, \left| \, \max(q\_{\overline{g}\_{i}}) \right| \right) = \left( \left\{ \gamma\_{\text{min}} \, \left| \, \begin{array}{l} p\_{\text{min}} \, \right| \, \left| \, q\_{\text{max}} \right| \, q\_{\text{max}} \right\} \right\} \text{ and }\\ a^{+} = \left( \max(h\_{i}) \, \left| \, \max(p\_{h\_{i}}) , \text{min}(\underline{g}\_{i}) \, \right| \, \left| \, \text{min}(q\_{\overline{g}\_{i}}) \right| \right) = \left( \left\{ \gamma\_{\text{max}} \, \left| \, p\_{\text{max}} \right\} , \left\{ \eta\_{\text{min}} \right| \, q\_{\text{min}} \right\} \right) \text{ be } PDHFS\_{s}, \\ \text{then } a^{-} \le PDHFWEA(a\_{1}, a\_{2}, \ldots, a\_{n}) \le a^{+}. \end{array}$$

**Proof.** Since each *αi* is a PDHFE, it is obvious that min(*hi*) ≤ *hi* ≤ max(*hi*), min(*gi*) ≤ *gi* ≤ max(*gi*), *p*min ≤ *pi* ≤ *p*max and *qmin* ≤ *qi* ≤ *q*max. Let *f*(*x*) = 1−*<sup>x</sup>* 1+*x* , *x* ∈ [0, 1], *f* (*x*) = −2 (<sup>1</sup>+*x*)<sup>2</sup> < 0 i.e., *f*(*x*) is a decreasing function. Since, *γ*min ≤ *γi* ≤ *γ*max, for all *i*, then *f*(*<sup>γ</sup>*max) ≤ *f*(*<sup>γ</sup>i*) ≤ *f*(*<sup>γ</sup>*min) i.e., 1−*γ*max 1+*γ*max ≤ 1−*γ<sup>i</sup>* 1+*γi* ≤ 1−*γ*max 1+*γ*max . Let *ω* = (*<sup>ω</sup>*1, *ω*2, ... , *<sup>ω</sup>n*)*<sup>T</sup>* be the weight vector of (*<sup>α</sup>*1, *α*2, ... , *<sup>α</sup>n*) such that each *ωi* ∈ (0, 1) and *n* ∑ *i*=1*ωi* = 1, then we have

$$\left(\frac{1-\gamma\_{\max}}{1+\gamma\_{\max}}\right)^{\omega\_{\hat{i}}} \le \left(\frac{1-\gamma\_{\hat{i}}}{1+\gamma\_{\hat{i}}}\right)^{\omega\_{\hat{i}}} \le \left(\frac{1-\gamma\_{\min}}{1+\gamma\_{\min}}\right)^{\omega\_{\hat{i}}}$$

Thus, we ge<sup>t</sup>

$$\begin{split} 1 + \left(\frac{1 - \gamma\_{\max}}{1 + \gamma\_{\max}}\right) &\leq 1 + \prod\_{i=1}^{n} \left(\frac{1 - \gamma\_{i}}{1 + \gamma\_{i}}\right)^{\omega\_{i}} \leq 1 + \left(\frac{1 - \gamma\_{\min}}{1 + \gamma\_{\min}}\right)^{\omega\_{i}} \\ \Rightarrow \frac{2}{1 + \gamma\_{\max}} &\leq \frac{\prod\_{i=1}^{n} (1 + \gamma\_{i})^{\omega\_{i}} + \prod\_{i=1}^{n} (1 - \gamma\_{i})^{\omega\_{i}}}{\prod\_{i=1}^{n} (1 + \gamma\_{i})^{\omega\_{i}}} \leq \frac{2}{1 + \gamma\_{\min}} \\ \Rightarrow \gamma\_{\min} &\leq \frac{1 - \prod\_{i=1}^{n} \left(\frac{1 - \gamma\_{i}}{1 + \gamma\_{i}}\right)^{\omega\_{i}}}{1 + \prod\_{i=1}^{n} \left(\frac{1 - \gamma\_{i}}{1 + \gamma\_{i}}\right)^{\omega\_{i}}} \leq \gamma\_{\max} \\ \Rightarrow \gamma\_{\min} &\leq \frac{\prod\_{i=1}^{n} (1 + \gamma\_{i})^{\omega\_{i}} - \prod\_{i=1}^{n} (1 - \gamma\_{i})^{\omega\_{i}}}{\prod\_{i=1}^{n} (1 + \gamma\_{i})^{\omega\_{i}} + \prod\_{i=1}^{n} (1 - \gamma\_{i})^{\omega\_{i}}} \leq \gamma\_{\max} \end{split}$$

Hence, we obtain the required result for membership values.

Now, for non-membership, let *c*(*y*) = 2−*y y* , *y* ∈ (0, 1], then *c* (*y*) < 0 i.e., *c*(*y*) is the decreasing function. Since, *η*min ≤ *ηi* ≤ *η*max, then for all *i*, we have *<sup>c</sup>*(*η*max) ≤ *<sup>c</sup>*(*ηi*) ≤ *<sup>c</sup>*(*η*min), that is 2−*η*max *η*max ≤ 2−*η<sup>i</sup> ηi* ≤ 2−*η*min *η*min . Let *ω* = (*<sup>ω</sup>*1, *ω*2, ... , *<sup>ω</sup>n*)*<sup>T</sup>* be the weight vector of (*<sup>α</sup>*1, *α*2, ... , *<sup>α</sup>n*) such that *ωi* ∈ (0, 1) and *n* ∑ *i*=1 *ωi* = 1, then

$$\begin{split} \left(\frac{2-\eta\_{\text{max}}}{\eta\_{\text{max}}}\right)^{\omega\_{i}} &\leq \left(\frac{2-\eta\_{i}}{\eta\_{i}}\right)^{\omega\_{i}} \leq \left(\frac{2-\eta\_{\text{min}}}{\eta\_{\text{min}}}\right)^{\omega\_{i}} \\ \text{Thus,} \quad \prod\_{i=1}^{n} \left(\frac{2-\eta\_{\text{max}}}{\eta\_{\text{max}}}\right)^{\omega\_{i}} &\leq \prod\_{i=1}^{n} \left(\frac{2-\eta\_{i}}{\eta\_{i}}\right)^{\omega\_{i}} \leq \prod\_{i=1}^{n} \left(\frac{2-\eta\_{\text{min}}}{\eta\_{\text{min}}}\right)^{\omega\_{i}} \\ &\Rightarrow \frac{2}{\eta\_{\text{min}}} \leq \frac{1}{1 + \prod\_{i=1}^{n} \left(\frac{2-\eta\_{i}}{\eta\_{i}}\right)^{\omega\_{i}}} \leq \frac{2}{\eta\_{\text{max}}} \\ &\Rightarrow \eta\_{\text{min}} \leq \frac{2\prod\_{i=1}^{n} (\eta\_{i})^{\omega\_{i}}}{\prod\_{i=1}^{n} (\eta\_{i})^{\omega\_{i}} + \prod\_{i=1}^{n} (2-\eta\_{i})^{\omega\_{i}}} \leq \eta\_{\text{max}} \end{split}$$

Hence, the required for non-membership values is obtained.

Now, for probabilities, since *p*min ≤ *pi* ≤ *p*max and *q*min ≤ *qi* ≤ *q*max this implies that *n* ∏ *i*=1 *p*min ≤ *n* ∏ *i*=1 *pi* ≤ *n* ∏ *i*=1 *p*max and *n* ∏ *i*=1 *q*min ≤ *n* ∏ *i*=1 *qi* ≤ *n* ∏ *i*=1 *q*max. According to the score function, as defined in Definition 5, we obtain *<sup>S</sup>*(*α*<sup>−</sup>) ≤ *<sup>S</sup>*(*α*) ≤ *<sup>S</sup>*(*α*+). Hence, from all the above notions, *α* − ≤ PDHFWEA(*<sup>α</sup>*1, *α*2,..., *<sup>α</sup>n*) ≤ *<sup>α</sup>*+.

**Property 2.** *(Monotonicity) Let αi* = *hi phi* , *gi qgi and α*∗*i* = *h*∗*i ph*<sup>∗</sup>*i* , *g*∗*i qg*<sup>∗</sup>*i , for all i* = (1, 2, . . . , *n*) *be two families of PDHFEs where for each element in αi and α*∗*i , there are γαi* ≤ *γα*<sup>∗</sup>*i and ηαi* ≥ *ηα*<sup>∗</sup>*i while the probabilities remain the same i.e., phi* = *ph*<sup>∗</sup>*i* , *qgi* = *qg*<sup>∗</sup>*i then PDHFWEA*(*<sup>α</sup>*1, *α*2, ... , *<sup>α</sup>n*) ≤ *PDHFWEA*(*α*<sup>∗</sup>1, *<sup>α</sup>*<sup>∗</sup>2,..., *<sup>α</sup>*<sup>∗</sup>*n*)*.*

**Proof.** Similar to that of Property 1, so we omit it here.

However, the PDHFWEA operator does not satisfy the idempotency. To illustrate this, we give the following example:

**Example 1.** *Let α*1 = *α*2 = 0.30.25, 0.40.75 , 0.20.4, 0.30.6 *be two PDHFEs and ω* = (0.2, 0.8)*<sup>T</sup> be the weight vector, then for* (*i* = 1, 2) *the aggregated value using PDHFWEA operator is obtained as*

$$\begin{split} PDHFWA(a\_{1},a\_{2}) &= \bigcup\_{\substack{\gamma\_{i}\in\mathbb{N}\_{i}\\ \eta\_{i}\in\mathbb{N}\_{i}}} \left( \left\{ \frac{\frac{2}{\prod\_{i=1}^{2}(1+\gamma\_{i})^{\omega\_{i}}-\prod\_{i=1}^{2}(1-\gamma\_{i})^{\omega\_{i}}}{\prod\_{i=1}^{n}(1+\gamma\_{i})^{\omega\_{i}}+\prod\_{i=1}^{2}(1-\gamma\_{i})^{\omega\_{i}}} \; \middle| \; \prod\_{i=1}^{n}p\_{\eta\_{i}} \right\} \right) \\ &= \left\{ \left\{ \frac{2\prod\_{i=1}^{2}(\eta\_{i})^{\omega\_{i}}}{\prod\_{i=1}^{n}(2-\eta\_{i})^{\omega\_{i}}+\prod\_{i=1}^{2}(\eta\_{i})^{\omega\_{i}}} \; \middle| \; \prod\_{i=1}^{n}q\_{\eta\_{i}} \right\} \\ &= \left( \begin{cases} \left\{ 0.3 \middle| 0.625, 0.3807 \middle| 0.1875, 0.3206 \middle| 0.1875, 0.41 \middle| 0.5625 \right\} \; \up| \; \mo{cases} \\ 0.2 \middle| 0.16, 0.2772 \middle| 0.24, 0.30 \middle| 0.36 \right\} \end{cases} \right) \end{split}$$

*which clearly shows that PDHFWEA*(*<sup>α</sup>*1, *<sup>α</sup>*1) = *α*1*. Thus, it does not satisfy idempotency.*

**Definition 8.** *Let αi*(*i* = 1, 2, . . . , *n*) *be the collection of PDHFEs, and PDHFOWEA:* Ω*n* → Ω*, if*

$$\text{PDHFOWEA}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n) \quad = \quad \omega\_1 \mathfrak{a}\_{\sigma(1)} \oplus \omega\_2 \mathfrak{a}\_{\sigma(2)} \oplus \dots \oplus \mathfrak{a}\_n \mathfrak{a}\_{\sigma(n)} \tag{15}$$

*where* Ω *is the set of PDHFEs and ω* = (*<sup>ω</sup>*1, *ω*2, ... , *<sup>ω</sup>n*)*<sup>T</sup> is the weight vector of αi such that ωi* > 0 *and n* ∑ *i*=1 *ωi* = 1*.* (*σ*(1), *<sup>σ</sup>*(2), ... , *σ*(*n*)) *is a permutation of* (1, 2, ... , *n*) *such that ασ*(*<sup>i</sup>*−<sup>1</sup>) ≥ *ασ*(*i*) *for* (*i* = 2, 3, . . . , *<sup>n</sup>*)*, then PDHFOWEA is called probabilistic dual hesitant fuzzy ordered weighted Einstein AO.*

**Theorem 7.** *For a family of PDHFEs αi* = *hi phi* , *gi qgi* ,(*<sup>i</sup>* = 1, 2, ... , *<sup>n</sup>*)*, the combined value obtained by using PDHFOWEA operator is given as*

$$\begin{split} PDHFOWA(\mathfrak{a}\_{1}, \mathfrak{a}\_{2}, \dots, \mathfrak{a}\_{n}) &= \bigcup\_{\substack{\gamma\_{\sigma(\varepsilon)} \in \mathfrak{b}\_{\sigma(\varepsilon)} \\ \mathfrak{a}\_{\sigma(\varepsilon)} \in \mathfrak{a}\_{\sigma(\varepsilon)}}} \left( \left\{ \frac{\prod\_{i=1}^{n} (1 + \gamma\_{\sigma(i)})^{\omega\_{\sigma(i)}} - \prod\_{i=1}^{n} (1 - \gamma\_{\sigma(i)})^{\omega\_{\sigma(i)}}}{\prod\_{i=1}^{n} (1 + \gamma\_{\sigma(i)})^{\omega\_{\sigma(i)}} + \prod\_{i=1}^{n} (1 - \gamma\_{\sigma(i)})^{\omega\_{\sigma(i)}}} \prod\_{i=1}^{n} p\_{\gamma\_{\sigma(i)}} \right\}, \\ & \left\{ \frac{2 \prod\_{i=1}^{n} (\eta\_{\sigma(i)})^{\omega\_{\sigma(i)}}}{\prod\_{i=1}^{n} (2 - \eta\_{\sigma(i)})^{\omega\_{\sigma(i)}} + \prod\_{i=1}^{n} (\eta\_{\sigma(i)})^{\omega\_{\sigma(i)}}} \; \middle| \; \prod\_{i=1}^{n} q\_{\eta\_{\sigma(i)}} \right\} \end{split} \tag{16}$$

*where ω* = (*<sup>ω</sup>*1, *ω*2,..., *<sup>ω</sup>n*)*<sup>T</sup> is a weight vector such that n* ∑ *i*=1 *ωi* = 1 *where* 0 < *ωi* < 1.

**Proof.** Similar to Theorem 6.

**Property 3.** *For all PDHFEs, αi* = *hi phi* , *gi qgi where i* = (1, 2, ... , *n*) *and for an associated weight vector ω* = (*<sup>ω</sup>*1, *ω*2,..., *<sup>ω</sup>n*)*T, such that each ωi* > 0 *and n* ∑ *i*=1 *ωi* = 1*, we have*


**Proof.** Similar to Properties 1 and 2.

**Definition 9.** *Let* Ω *be a family of all PDHFEs αi* (*i* = 1, 2, ... , *n*) *with the corresponding weights ω* = (*<sup>ω</sup>*1, *ω*2,..., *<sup>ω</sup>n*)*T, such that ωi* > 0 *and n* ∑ *i*=1 *ωi* = 1*. If PDHFWEG:* Ω*n* → Ω, *is a mapping defined by*

$$\text{PDHFWEG}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n) \quad = \quad \mathfrak{a}\_1^{\omega\_1} \otimes \mathfrak{a}\_2^{\omega\_2} \otimes \dots \otimes \mathfrak{a}\_n^{\omega\_n} \tag{17}$$

*then, PDHFWEG is called probabilistic dual hesitant fuzzy weighted Einstein geometric operator.*

**Theorem 8.** *For a collection of PDHFEs αi* = *hi phi* , *gi qgi* ,(*<sup>i</sup>* = 1, 2, ... , *<sup>n</sup>*)*, the combined value obtained by using PDHFWEG operator is still a PDHFE and is given as*

$$=\bigcup\_{\gamma\_{i}\in h\_{i},\boldsymbol{\eta}\_{i}\in\mathcal{G}\_{i}}\left(\left\{\frac{2\prod\_{i=1}^{n}(\gamma\_{i})^{\omega\_{i}}}{\prod\_{i=1}^{n}(2-\gamma\_{i})^{\omega\_{i}}+\prod\_{i=1}^{n}(\gamma\_{i})^{\omega\_{i}}}\;\middle|\;\prod\_{i=1}^{n}p\_{\gamma\_{i}}\right\}\right),\tag{18}$$

$$=\left(\begin{cases}\frac{n}{n}(1+\eta\_{i})^{\omega\_{i}}-\prod\_{i=1}^{n}(1-\eta\_{i})^{\omega\_{i}}\\\frac{n}{n}(1+\eta\_{i})^{\omega\_{i}}+\prod\_{i=1}^{n}(1-\eta\_{i})^{\omega\_{i}}\\\prod\_{i=1}^{n}(1+\eta\_{i})^{\omega\_{i}}+\prod\_{i=1}^{n}(1-\eta\_{i})^{\omega\_{i}}\end{cases}\right)\tag{19}$$

*where ω* = (*<sup>ω</sup>*1, *ω*2,..., *<sup>ω</sup>n*)*<sup>T</sup> is a weight vector such that n* ∑ *i*=1 *ωi* = 1 *where* 0 < *ωi* < 1.

**Proof.** Same as Theorem 6.

Also, it has been seen that the PDHFWEG operator satisfies the properties of boundedness and monotonicity.

**Definition 10.** *Let αi* (*i* = 1, 2, . . . , *n*) *be the family of PDHFEs, and PDHFOWEG:* Ω*n* → Ω*, if*

$$\text{PDHFOWEG}(\mathfrak{a}\_1, \mathfrak{a}\_2, \dots, \mathfrak{a}\_n) \quad = \ a\_{\sigma(1)}^{\omega\_1} \oplus a\_{\sigma(2)}^{\omega\_2} \dots \oplus a\_{\sigma(n)}^{\omega\_n} \tag{19}$$

*where* Ω *is the set of PDHFEs and ω* = (*<sup>ω</sup>*1, *ω*2, ... , *<sup>ω</sup>n*)*<sup>T</sup> is the weight vector of αi such that ωi* > 0 *and n* ∑ *i*=1 *ωi* = 1*.* (*σ*(1), *<sup>σ</sup>*(2), ... , *σ*(*n*)) *is a permutation of* (1, 2, ... , *n*) *such that ασ*(*<sup>i</sup>*−<sup>1</sup>) ≥ *ασ*(*i*) *for* (*i* = 2, 3, . . . , *<sup>n</sup>*)*, then PDHFOWEG is called probabilistic dual hesitant fuzzy ordered weighted Einstein geometric operator.*

**Theorem 9.** *For a family of PDHFEs αi* = *hi phi* , *gi qgi* ,(*<sup>i</sup>* = 1, 2, ... , *<sup>n</sup>*)*, the combined value obtained by using PDHFOWEG operator is given as*

$$\bigcup\_{\substack{\boldsymbol{\gamma}\_{\sigma(i)}\in\mathbb{A}\_{\sigma(i)}\\ \boldsymbol{\gamma}\_{\sigma(i)}\in\mathcal{E}\_{\sigma(i)}}} \left( \left\{ \frac{2\prod\_{i=1}^{n}(\gamma\_{\sigma(i)})^{\omega\_{\sigma(i)}}}{\prod\_{i=1}^{n}(2-\gamma\_{\sigma(i)})^{\omega\_{\sigma(i)}}+\prod\_{i=1}^{n}(\gamma\_{\sigma(i)})^{\omega\_{\sigma(i)}}} \; \bigg|\; \prod\_{i=1}^{n}p\_{\gamma\_{\sigma(i)}} \right\} \right) \; \begin{array}{c} \hline \\ \prod\_{i=1}^{n}p\_{\gamma\_{\sigma(i)}} \\ \left\{ \frac{\prod\_{i=1}^{n}(1+\eta\_{\sigma(i)})^{\omega\_{\sigma(i)}}-\prod\_{i=1}^{n}(1-\eta\_{\sigma(i)})^{\omega\_{\sigma(i)}}}{\prod\_{i=1}^{n}(1+\eta\_{\sigma(i)})^{\omega\_{\sigma(i)}}+\prod\_{i=1}^{n}(1-\eta\_{\sigma(i)})^{\omega\_{\sigma(i)}}} \; \bigg|\; \prod\_{i=1}^{n}q\_{\eta\_{\sigma(i)}} \right\} \end{array} \tag{20}$$

*where ω* = (*<sup>ω</sup>*1, *ω*2,..., *<sup>ω</sup>n*)*<sup>T</sup> is a weight vector such that n* ∑ *i*=1 *ωi* = 1 *where* 0 < *ωi* < 1.

### **Proof.** Similar to Theorem 6.

Also, it has been seen that the PDHFOWEG operator satisfies the properties of boundedness and monotonicity.

### **6. Maximum Deviation Method for Determination the Weights**

The choice of weights directly affects the performance of weighted aggregation operators. For this purpose, in this subsection, the effective maximizing deviation method is adapted to calculate the weights in MCDM when the weights are unknown or partially known.

*Mathematics* **2018**, *6*, 280

Given the set of alternatives *A* = {*<sup>A</sup>*1, *A*2, ... , *Am*} and the set of criteria *C* = {*<sup>C</sup>*1, *C*2, ... , *Ct*} which is being evaluated by a decision maker under the PDHFS environment over the universal set *X* = {*<sup>x</sup>*1, *x*2,..., *xn*}. Assume that the rating values corresponding to each alternative is expressed in terms of PDHFEs as

$$A\_r = \left\{ \left( \mathbb{C}\_1, \mathbf{s}\_{r1} \right), \left( \mathbb{C}\_2, \mathbf{s}\_{r2} \right), \dots, \left( \mathbb{C}\_{t'}, \mathbf{s}\_{rv} \right) \right\}, \tag{21}$$

where *srv* = *hrv*(*xk*)*prv*(*xk*), *grv*(*xk*)*qrv*(*xk*), where *r* = 1, 2, ... , *m*; *v* = 1, 2, ... , *t*, *k* = 1, 2, ... , *n*. Assume that the importance of each criterion are given in the form of weights as (*<sup>ω</sup>*1, *ω*2, ... , *<sup>ω</sup>t*) respectively such that 0 < *ωv* ≤ 1 and *t* ∑ *<sup>v</sup>*=1 *ωv* = 1. Now, by using the proposed distances *d*1 in Equation (9) or *d*2 in (10) ; the deviation measure between the alternative *Ar* and all other alternatives with respect to the criteria *Cv* is given as:

$$D\_{\rm rv}(\omega) = \sum\_{b=1}^{m} w\_{\rm v} D(s\_{\rm rv}, s\_{\rm bv}) \quad \text{ $r = 1$ ,  $2, \dots, m$ ;  $v = 1$ ,  $2, \dots, t$ }\tag{22}$$

In accordance to the notion of maximizing deviation method, if the distance between the alternatives is smaller for a criteria, then it should have smaller weight. This one shows that the alternatives are homologous to the criterion. Contrarily, it should have larger weights. Let,

$$D\_{\mathbf{v}}(\omega) = \sum\_{r=1}^{m} D\_{\text{rv}}(\omega) = \sum\_{r=1}^{m} \sum\_{b=1}^{m} w\_{\text{v}} D(s\_{\text{rv}}, s\_{\text{bv}}), \quad \mathbf{v} = 1, 2, \dots, t \tag{23}$$

Here *Dv*(*ω*) represents the distance of all the alternatives to the other alternatives under the criteria *Cv* ∈ *C*. Moreover, '*D*' represents either distance *d*1 or *d*2 as given in Equations (9) and (10) respectively. Based on the concept of maximum deviation, we have to choose a weight vector '*ω*' to maximize all the deviations measures for the criteria. For this, we construct a non-linear programming model as given below:

$$\begin{cases} \max \ D(\omega) = \sum\_{v=1}^{t} \sum\_{r=1}^{m} D\_{rv}(\omega) = \sum\_{v=1}^{t} \sum\_{r=1}^{m} \sum\_{b=1}^{m} D(s\_{rv}, s\_{bv}) \omega\_{v} \\ \text{s.t.} \quad \omega\_{v} > 0; \quad \sum\_{v=1}^{t} \omega\_{v} = 1; \quad v = 1, 2, \dots, t \end{cases} \tag{24}$$

where '*D*' can be either *d*1 or *d*2.

> If *D* = *d*1, then for *λ* > 0, we have

$$D(\omega) = \sum\_{v=1}^{l} \sum\_{r=1}^{m} \sum\_{b=1}^{m} \omega\_{v} \left( \sum\_{k=1}^{n} \frac{1}{n} \left( \frac{1}{M+N} \left( \sum\_{l=1}^{M} \left| (\gamma\_{A\_{l}}(\mathbf{x}\_{k}) p\_{A\_{l}}(\mathbf{x}\_{k}))(\mathbf{x}\_{lv}) - (\gamma\_{B\_{l}}(\mathbf{x}\_{k}) p\_{B\_{l}}(\mathbf{x}\_{k}))(\mathbf{x}\_{bv}) \right|^{\lambda} \right. \\ \left. + \sum\_{j=1}^{N} \left| (\eta\_{A\_{j}}(\mathbf{x}\_{k}) q\_{A\_{j}}(\mathbf{x}\_{k}))(\mathbf{x}\_{vv}) - (\eta\_{B\_{l}}(\mathbf{x}\_{k}) q\_{B\_{j}}(\mathbf{x}\_{k}))(\mathbf{x}\_{bv}) \right|^{\lambda} \right) \right)^{\frac{1}{\lambda}}$$

and if *D* = *d*2, then

$$D(\omega) = \sum\_{\nu=1}^{l} \sum\_{r=1}^{m} \sum\_{b=1}^{M} \omega\_{\nu} \left( \sum\_{k=1}^{n} \frac{1}{n} \left( \frac{\left| \frac{1}{M\_{A}} \sum\_{i=1}^{M\_{\mathcal{A}}} \left( \gamma\_{A\_{i}}(\mathbf{x}\_{k}) p\_{A\_{i}}(\mathbf{x}\_{k}) \right) (\mathbf{x}\_{r\mathcal{I}}) - \frac{1}{M\_{B}} \sum\_{i'=1}^{M\_{\mathcal{B}}} \left( \gamma\_{B\_{i'}}(\mathbf{x}\_{k}) p\_{B\_{i'}}(\mathbf{x}\_{k}) \right) (\mathbf{x}\_{\mathcal{I}\nu}) \right|^{\lambda} \right. \\ \left. \left. \left. \frac{1}{M\_{A}} \sum\_{i=1}^{M\_{\mathcal{A}}} \left( \eta\_{A\_{i}}(\mathbf{x}\_{k}) q\_{A\_{i}}(\mathbf{x}\_{k}) \right) (\mathbf{x}\_{\mathcal{I}\nu}) - \frac{1}{M\_{B}} \sum\_{j'=1}^{N\_{\mathcal{B}}} \left( \eta\_{B\_{j'}}(\mathbf{x}\_{k}) q\_{B\_{j'}}(\mathbf{x}\_{k}) \right) (\mathbf{x}\_{\mathcal{I}\nu}) \right) \right]^{\lambda} \right)^{-1}$$

*Mathematics* **2018**, *6*, 280

If the information about criteria weights is completely unknown, then another programming method can be established as:

$$\begin{cases} \max \ D(\omega) = \sum\_{v=1}^{t} \sum\_{r=1}^{m} D\_{rv}(\omega) = \sum\_{v=1}^{t} \sum\_{r=1}^{m} \sum\_{b=1}^{m} D(s\_{rv}, s\_{bv}) \omega\_{v} \\ \text{s.t.} \quad \omega\_{v} \ge 0; \quad \sum\_{v=1}^{t} \omega\_{v}^{2} = 1; \quad v = 1, 2, \dots, t \end{cases} \tag{25}$$

To solve this, a Lagrange's function is constructed as

$$L(\omega, \zeta) = \sum\_{v=1}^{t} \sum\_{r=1}^{m} \sum\_{b=1}^{m} D(s\_{\text{rv}}, s\_{\text{bv}}) \omega\_{\text{v}} + \frac{\zeta}{2} \left( \sum\_{v=1}^{t} \omega\_{\text{v}}^{2} - 1 \right) \tag{26}$$

where *ζ* is the Lagrange's parameter. Computing the partial derivatives of Lagrange's function w.r.t *ωv* as well as *ζ* and letting them equal to zero.

$$\begin{cases} \frac{\partial L}{\partial \omega\_{\upsilon}} = \sum\_{r=1}^{m} \sum\_{b=1}^{m} D(s\_{\tau \nu} s\_{b \upsilon}) + \zeta \omega\_{\upsilon} = 0; \quad \upsilon = 1, 2, \dots, t \\\frac{\partial L}{\partial \xi} = \sum\_{\upsilon=1}^{t} \omega\_{\upsilon}^{2} - 1 = 0 \end{cases} \tag{27}$$

Solving, Equation (27) we can obtain,

$$\omega\_{\upsilon} = \frac{\sum\_{r=1}^{m} \sum\_{b=1}^{m} D(s\_{r\upsilon}, s\_{b\upsilon})}{\sqrt{\sum\_{v=1}^{t} \left(\sum\_{r=1}^{m} \sum\_{b=1}^{m} D(s\_{r\upsilon}, s\_{b\upsilon})\right)^2}}; \quad \upsilon = 1, 2, \dots, t \tag{28}$$

Normalizing Equation (28) we ge<sup>t</sup>

$$\omega\_{\upsilon} = \frac{\sum\_{r=1}^{m} \sum\_{b=1}^{m} D(s\_{r\upsilon}, s\_{b\upsilon})}{\sum\_{v=1}^{t} \sum\_{r=1}^{m} \sum\_{b=1}^{m} D(s\_{r\upsilon}, s\_{b\upsilon})} \tag{29}$$

In DM process, the data values for evaluation are available as DHFSs or PDHFSs which are integrated to form the PDHFSs. In order to gather the information, the probability values are assigned to each possible membership or non-membership value. An algorithm followed for this information fusion is outlined in Algorithm 1.

**Algorithm 1** Aggregating probabilities for more than one Probabilistic fuzzy sets.

**Input:** *<sup>α</sup>*(1), *<sup>α</sup>*(2), ... , *α*(*d*) where *α*(*d*) = *h*(*d*)*p*(*d*) where *d* = 1, 2, ... , *D* such that *D* is the total number of elements to be fused together. 

**Output:** *α*(*out*) = *h*(*out*)*p*(*out*) 


To demonstrate the working of aforementioned algorithm, an example is given below.

**Example 2.** *Let α*(1) = -{0.10.1, 0.20.5, 0.30.4}, {0.51}*; α*(2) = -{0.20.4, 0.30.6} , {0.50.2, 0.60.8} *and α*(3) = -{0.10.4, 0.20.4, 0.60.2}, {0.11} *be three PDHFEs to be fused together. Since, h*(1), *p*(1) = -{0.10.1, 0.20.5, 0.30.4}*, h*(2), *p*(2) = -{0.20.4, 0.30.6} *and h*(3), *p*(3) = -{0.10.4, 0.20.4, 0.60.2}*, so we get M* = {0.10.1, 0.20.5, 0.30.4, 0.20.4, 0.30.6, 0.10.4, 0.20.4, 0.60.2} *where* #*L* = 8 *and thus l* = 1, 2, ... , 8*. Clearly, here D* = 3*. Now, by following Algorithm 1 for both membership and non-membership degrees, we obtained the final PDHFE as:*

$$\mathbf{a}^{(\text{out})} = \left( \{ 0.1 \vert 0.1667, 0.2 \vert 0.4333, 0.3 \vert 0.3333, 0.6 \vert 0.066 \}, \{ 0.5 \vert 0.4, 0.6 \vert 0.2666, 0.1 \vert 0.3333 \} \} \right)$$

### **7. Decision Making Approach Using the Proposed Operators**

In this section, a DM approach based on proposed AOs is given followed by a numerical example.

### *7.1. Approach Based on the Proposed Operators*

Consider a set of *m* alternatives *A* = {*<sup>A</sup>*1, *A*2, ... , *Am*} which are evaluated by the experts classified under criteria information *C* = {*<sup>C</sup>*1, *C*2, ... , *Ct*}. The ratings for each alternative in PDHFEs are given as:

$$A\_r = \left\{ (\mathbb{C}\_1, a\_{r1}), (\mathbb{C}\_2, a\_{r2}), \dots, (\mathbb{C}\_t, a\_{rv}) \right\},\tag{30}$$

where *αrv* = *hrvprv*, *grvqrv*, where *r* = 1, 2, ... , *m*; *v* = 1, 2, ... , *t*. In order to ge<sup>t</sup> the best alternative(s) for a problem, DM approach is summarized in the following steps by utilizing proposed AOs as:

Step 1: Construct decision matrices *R*(*d*) for '*d*' number of decision makers in form of PDHFEs as:

*R*(*d*) = *C*1 *C*2 ... *Ct* ⎛⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎠ *A*1 *h*(*d*) 11 *p*(*d*) 11 , *g*(*d*) 11 *q*(*d*) 11 *h*(*d*) 12 *p*(*d*) 12 , *g*(*d*) 12 *q*(*d*) 12 ... *h*(*d*) 1*t p*(*d*) 1*t* , *g*(*d*) 1*t q*(*d*) 1*t A*2 *h*(*d*) 21 *p*(*d*) 21 , *g*(*d*) 21 *q*(*d*) 21 *h*(*d*) 22 *p*(*d*) 22 , *g*(*d*) 22 *q*(*d*) 22 ... *h*(*d*) 2*t p*(*d*) 2*t* , *g*(*d*) 2*t q*(*d*) 2*t* ... ... ... ... ... *Am h*(*d*) *m*1 *p*(*d*) *m*1 , *g*(*d*) *m*1 *q*(*d*) *m*1 *h*(*d*) *m*2 *p*(*d*) *m*2 , *g*(*d*) *m*2 *q*(*d*) *m*2 ... *h*(*d*) *mt p*(*d*) *mt* , *g*(*d*) *mt q*(*d*) *mt* 

where *h*(*d*) *rv p*(*d*) *rv* , *g*(*d*) *rv q*(*d*) *rv* = *γ*(*d*) *rv p*(*d*) *rv* , *η*(*d*) *rv q*(*d*) *rv* , such that *r* = 1, 2, ... , *m* and *v*= 1, 2, . . . , *t* . 

 2: If *d* = 1, then *h*(*d*) *rv p*(*d*) *rv* , *g*(*d*) *rv q*(*d*) *rv* is equal to *hrvprv*, *grvqrv* , where *hrvprv*, *grvqrv* = *<sup>γ</sup>rvprv*, *<sup>η</sup>rvqrv*; such that *r* = 1, 2, ... , *m* and *v* = 1, 2, ... , *t* and goto Section 7.1 Step 3. If *d* ≥ 2, then a matrix is formed by combining the probabilities in accordance to the Algorithm 1. The comprehensive matrix so obtained is given as:

*R* = *C*1 *C*2 ... *Ct* ⎛⎜⎜⎝ ⎞⎟⎟⎠ *A*1 *<sup>h</sup>*11*<sup>p</sup>*11, *<sup>g</sup>*11*<sup>q</sup>*11 *<sup>h</sup>*12*<sup>p</sup>*12, *<sup>g</sup>*12*<sup>q</sup>*12 ... *<sup>h</sup>*1*t<sup>p</sup>*<sup>1</sup>*t*, *<sup>g</sup>*1*t<sup>q</sup>*1*t A*2 *<sup>h</sup>*21*<sup>p</sup>*21, *<sup>g</sup>*21*<sup>q</sup>*21 *<sup>h</sup>*22*<sup>p</sup>*22, *<sup>g</sup>*22*<sup>q</sup>*22 ... *<sup>h</sup>*2*t<sup>p</sup>*<sup>2</sup>*t*, *<sup>g</sup>*2*t<sup>q</sup>*2*t* ... ... ... ... ... *Am hm*1*pm*1, *gm*1*qm*1 *hm*2*pm*2, *gm*2*qm*2 ... *hmtpmt*, *gmtqmt*

where *hrvprv*, *grvqrv* = *<sup>γ</sup>rvprv*, *<sup>η</sup>rvqrv*, where *r* = 1, 2, . . . , *m* and *v* = 1, 2, . . . , *t*. Step 3: Choose the appropriate distance measure among *d*1 or *d*2 as given in Equations (9) and (10), on the basis of need the expert. If the repeated values of the largest or smallest dual-hesitant probabilistic values can be repeated according to the optimistic or pessimistic behavior of the expert then choose measure *d*1 otherwise choose measure *d*2 and determine the weights of different criteria using Equation (29).

Step 4: Compute the overall aggregated assessment '*Qr*' of alternatives using PDHFWEA or PDHFOWEA or PDHFWEG or PDHFOWEG operators as given below in Equations (31)–(34) respectively.

$$\begin{array}{rcl} Q\_{r} &=& \text{PDHFWE}(\boldsymbol{a}\_{r1}, \boldsymbol{a}\_{r2}, \dots, \boldsymbol{a}\_{rv}) \\ &=& \bigcup\_{\begin{subarray}{c} \gamma\_{r} \in \boldsymbol{b}\_{r} \\ \forall r \in \boldsymbol{b}\_{r} \end{subarray}} \left( \left\{ \frac{\prod\_{v=1}^{t} (1 + \gamma\_{rv})^{\omega\_{v}} - \prod\_{v=1}^{t} (1 - \gamma\_{rv})^{\omega\_{v}}}{\prod\_{v=1}^{t} (1 + \gamma\_{rv})^{\omega\_{v}} + \prod\_{v=1}^{t} (1 - \gamma\_{rv})^{\omega\_{v}}} \; \middle| \; \prod\_{v=1}^{t} p\_{\gamma\_{rv}} \right\} \right) \\ &=& \prod\_{\begin{subarray}{c} \gamma\_{v} = 1 \\ v = 1 \\ \hline \end{subarray}} \left( \frac{2 \prod\_{v=1}^{t} (\eta\_{rv})^{\omega\_{v}}}{v = 1} \left| \; \prod\_{v=1}^{t} q\_{\eta\_{rv}} \right. \\ & \left. \frac{1}{\prod\_{v=1}^{t} (2 - \eta\_{rv})^{\omega\_{v}}} + \prod\_{v=1}^{t} (\eta\_{rv})^{\omega\_{v}} \right| \; \middle| \; \prod\_{v=1}^{t} q\_{\eta\_{rv}} \right) \end{array} \tag{31}$$

Step

or

*Qr* = PDHFOWEA(*<sup>α</sup>r*1, *αr*2,..., *<sup>α</sup>rv*) = *γσ*(*rv*)<sup>∈</sup>*hσ*(*rv*) *ησ*(*rv*)<sup>∈</sup>*gσ*(*rv*) ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎧⎪⎪⎨⎪⎪⎩ *t*∏*<sup>v</sup>*=<sup>1</sup>(<sup>1</sup> + *γσ*(*rv*))*ωσ*(*v*) − *t*∏*<sup>v</sup>*=<sup>1</sup>(<sup>1</sup> − *γσ*(*rv*))*ωσ*(*v*) *t*∏*<sup>v</sup>*=<sup>1</sup>(<sup>1</sup> + *γσ*(*rv*))*ωσ*(*v*) + *t*∏*<sup>v</sup>*=<sup>1</sup>(<sup>1</sup> − *γσ*(*rv*))*ωσ*(*v*) *t* ∏*<sup>v</sup>*=1 *pγσ*(*rv*)⎫⎪⎪⎬⎪⎪⎭ , ⎧⎪⎪⎨⎪⎪⎩ 2 *t*∏*<sup>v</sup>*=<sup>1</sup>(*ησ*(*rv*))*ωσ*(*v*) *t*∏*<sup>v</sup>*=<sup>1</sup>(<sup>2</sup> − *ησ*(*rv*))*ωσ*(*v*) + *t*∏*<sup>v</sup>*=<sup>1</sup>(*ησ*(*rv*))*ωσ*(*v*) *t* ∏*<sup>v</sup>*=1 *qησ*(*rv*)⎫⎪⎪⎬⎪⎪⎭ ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (32)

or

$$\begin{array}{rcl} Q\_{\boldsymbol{r}} &=& \text{PDHF} \text{EG} \left( a\_{r1}, a\_{r2}, \dots, a\_{\boldsymbol{r}\boldsymbol{v}} \right) \\ &=& \bigcup\_{\begin{subarray}{c} \boldsymbol{\eta} = \boldsymbol{v} \succeq \boldsymbol{\eta} \boldsymbol{v} \\ \boldsymbol{\eta} = \boldsymbol{v} \boldsymbol{v} \end{subarray}} \left( \left\{ \begin{subarray}{c} 2 \prod\_{v=1}^{t} (\gamma\_{\boldsymbol{r}\boldsymbol{v}})^{\omega\_{\boldsymbol{v}}} \\ \left\{ \prod\_{v=1}^{t} (2 - \gamma\_{\boldsymbol{r}\boldsymbol{v}})^{\omega\_{\boldsymbol{v}}} + \sum\_{v=1}^{t} (\gamma\_{\boldsymbol{r}\boldsymbol{v}})^{\omega\_{\boldsymbol{v}}} \right\} \left| \prod\_{v=1}^{t} p\_{\gamma\_{\boldsymbol{r}\boldsymbol{v}}} \right. \\ \left\{ \begin{subarray}{c} \prod\_{v=1}^{t} (1 + \eta\_{\boldsymbol{r}\boldsymbol{v}})^{\omega\_{\boldsymbol{v}}} - \prod\_{v=1}^{t} (1 - \eta\_{\boldsymbol{r}\boldsymbol{v}})^{\omega\_{\boldsymbol{v}}} \\ \prod\_{v=1}^{t} (1 + \eta\_{\boldsymbol{r}\boldsymbol{v}})^{\omega\_{\boldsymbol{v}}} + \prod\_{v=1}^{t} (1 - \eta\_{\boldsymbol{r}\boldsymbol{v}})^{\omega\_{\boldsymbol{v}}} \end{subarray} \; \middle| \; \prod\_{v=1}^{t} q\_{\eta\_{\boldsymbol{r}\boldsymbol{v}}} \right\} \end{array} \tag{33}$$

or

*Qr* = PDHFOWEG(*<sup>α</sup>r*1, *αr*2,..., *<sup>α</sup>rv*) = *γσ*(*rv*)<sup>∈</sup>*hσ*(*rv*) *ησ*(*rv*)<sup>∈</sup>*gσ*(*rv*) ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎧⎪⎪⎨⎪⎪⎩ 2 *t*∏*<sup>v</sup>*=<sup>1</sup>(*γσ*(*rv*))*ωσ*(*v*) *t*∏*<sup>v</sup>*=<sup>1</sup>(<sup>2</sup> − *γσ*(*rv*))*ωσ*(*v*) + *t*∏*<sup>v</sup>*=<sup>1</sup>(*γσ*(*rv*))*ωσ*(*v*) *t* ∏*<sup>v</sup>*=1 *pγσ*(*rv*)⎫⎪⎪⎬⎪⎪⎭ , ⎧⎪⎪⎨⎪⎪⎩ *t*∏*<sup>v</sup>*=<sup>1</sup>(<sup>1</sup> + *ησ*(*rv*))*ωσ*(*v*) − *t*∏*<sup>v</sup>*=<sup>1</sup>(<sup>1</sup> − *ησ*(*rv*))*ωσ*(*v*) *t*∏*<sup>v</sup>*=<sup>1</sup>(<sup>1</sup> + *ησ*(*rv*))*ωσ*(*v*) + *t*∏*<sup>v</sup>*=<sup>1</sup>(<sup>1</sup> − *ησ*(*rv*))*ωσ*(*v*) *t* ∏*<sup>v</sup>*=1 *qησ*(*rv*)⎫⎪⎪⎬⎪⎪⎭ ,⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34)

Step 5: Utilize Definition 5 to rank the overall aggregated values and select the most desirable alternative(s).
