A Novel Multi-Attribute Group Decision-Making Approach in the Framework of Proportional Dual Hesitant Fuzzy Sets

Abstract

Making decisions are very common in the modern socio-economic environments. However, with the increasing complexity of the social, today’s decision makers (DMs) face such problems in which they hesitate and irresolute to provide their views. To cope with these uncertainties, many generalizations of fuzzy sets are designed, among them dual hesitant fuzzy set (DHFS) is quite resourceful and efficient in solving problems of a more vague nature. In this article, a novel concept called proportional dual hesitant fuzzy set (PDHFS) is proposed to further improve DHFS. The PDHFS is a flexible tool composed of some possible membership values and some possible non-membership values along with their associated proportions. In the theme of PDHFS, the proportions of membership values and non-membership values are considered to be independent. Some basic operations, properties, distance measure and comparison method are studied for the proposed set. Thereafter, a novel approach based on PDHFSs is developed to solve problems for multi-attribute group decision-making (MAGDM) in a fuzzy situation. It is totally different from the traditional approach. Finally, a practical example is given in order to elaborate the proposed method for the selection of the best alternative and detailed comparative analysis is given in order to validate the practicality.

1. Introduction

In our daily life, there are many problems where uncertainty is involved in the data associated with them. These problems are very common in engineering, economics, medicine and social sciences. There always occurs difficulties in addressing such problems by applying classical mathematical techniques because such techniques are structured for clear situations. One of the best known theories to deal with uncertainty is a fuzzy set theory. Following the pioneering work of Zadeh [1], the fuzzy set theory has expanded in a numerous directions, such as intuitionistic fuzzy set [2], hesitant fuzzy set [3], hesitant probabilistic fuzzy set [4], expanded hesitant fuzzy set [5], and necessary and possible hesitant fuzzy set [6]. Atanassov [2] generalized the notion of fuzzy set to intuitionistic fuzzy set (IFS) by adding the concept of a non-membership function to a fuzzy set. IFS is a powerful tool used to handle the problems of decision-making that a fuzzy set could not address properly. Yager [7] suggested the concept of Pythagorean fuzzy subsets which are the most effective ones that relate to the representation of the membership degrees of the underlying fuzzy set. Torra [3] introduced a new generalized form of fuzzy set called hesitant fuzzy set (HFS), which opens a new dimension of further research in decision theory.

HFS has a lot of merits over the traditional fuzzy set and its extensions, especially in group decision-making with anonymity. HFS has attracted several researchers’ attention. Xu and Xia [8,9] initially provided the mathematical expression of HFS and investigated the distance, similarity and correlation measure for HFSs. Torra also set up the relation between HFS and IFS, according to which Xia and Xu [8] defined some basic operational laws for HFSs. In addition, many extensions of HFS [10,11,12,13] were proposed to model the fuzzy hesitant problems from different prospectives. The HFS is a widely used fuzzy set in decision-making, especially, in those situations in which DMs hesitate to give their preferences. Anyhow, it has the following two major deficiencies:

• HFS provides only the membership information of DMs’ opinions and pay no attention to the non-membership information. In order to get accurate results, DMs should be allowed to provide the membership as well as the non-membership information.
• HFS ignores the importance of multiple times occurring opinions, which may lead to unfruitful decision ranking results.

To handle real life scenarios more accurately, Zhu et al. [13] proposed a dual hesitant fuzzy set (DHFS) that permits both the membership and non-membership grades of an element to a given set having different values. The DHFS is a more flexible tool for DMs to assign their views for each element in the domain. Pathinathan and Savarimuthu [14] extended the TOPSIS method for DHFS and describe the application of this new type of set in the framework of decision making. Wang et al. [15] investigated the correlation measure of DHFS and develop a direct transfer algorithm regarding the problem of complex operation of matrix synthesis. Zhu and Xu [16] extended the DHFS to typical dual hesitant fuzzy set (TDHFS) and construct a number of operations and properties to generalize the operations of this new type of set. After that, the distance and similarity measure [17,18], the entropy measure [19] and the evaluation model [20] regarding DHFS have been developed to handle the MAGDM problems. For the purpose of comparison of DHFSs, Ren et al. [21] studied the score function and distance measure. In addition to this, they also extended the VIKOR method for DHFSs.

Anyhow, DHFSs, as well as its extensions, cover the deficiency 1 of HFS, but they are not suitable for the situation where DMs could not reach agreement. For example, assume that a decision is made by a decision-making group that contains no less than two DMs. Different DMs have different opinions on the membership and non-membership degrees and it is possible that some DMs opinions may coincide. Thus, repeated rating values occur in data. This repetition is important and removal of this may lead to unfruitful decision results. Recently, Xiong et al. [22] defined a proportional hesitant fuzzy set (PHFS) to handle deficiency 2 of HFS. This is a new approach for handling the imprecise data with more care.

In order to cover both the aforementioned deficiencies of HFS, in this paper, the concept of PDHFS is developed, in which the membership grades as well as the non-membership grades are presented by several possible numbers along with their proportions. The PHFS enclosed HFS, DHFS and PHFS as special cases, making it a more powerful tool to handle uncertainty and complexity of real life.

We highlight the article by the following pioneering work:

• A novel concept known as PDHFS is proposed along with practical example which encompasses DHFS and its extensions.
• For the purpose of comparison of PDHFSs, score function and deviation degree of PDHFSs are designed.
• Some basic operations, main results and normalized Hamming distance on DHFSs are extended to the PDHFS setting.
• A comprehensive MAGDM TOPSIS method with a proportional dual hesitant fuzzy decision information is introduced and applied to the university faculty evaluation problem.
• The detailed comparative analysis with other MAGDM methods is done in order to show the advantages of the proposed set and methodology.

The remainder of the paper is structured as follows: in Section 2, some basic concepts are provided, which is essential for the understanding of our research work. In Section 3, we first define the concept of PDHFS and then prove some of its interesting mathematical properties. In Section 4, distance measure is proposed between two PDHFSs in order to study a comparison between them. Section 5 is dedicated to presenting the working algorithm for TOPSIS. Section 6 gives an application to elaborate our proposed methodology. The comparative analysis is studied in Section 7. Section 8 makes the concluding remarks and indicates future work lines.

2. Literature Review and Preliminaries

In this section, we briefly review the concepts and notions that will be used in the coming sections.

2.1. Hesitant Fuzzy Set

Torra [3] proposed the concept of a hesitant fuzzy set which covers arguments with a set of possible values.

Definition 1

([3]). Let X be a fixed set; then, a hesitant fuzzy set A on X is represented by the following mathematical symbol:

$$A=\left\{\left(x,h_A(x)\mid x\in X\right)\right\},$$

(1)

where $h_A(x)$ is a set of values in $[0,1]$, denoting the possible membership values of the element $x\in X$ to the set A. For convenience, we call $h_A(x)$ a hesitant fuzzy element (HFE) denoted by h.

2.2. Dual Hesitant Fuzzy Set

Zhu et al. [13], generalized HFS and defined a DHFS in terms of two functions that return two sets of membership values and non-membership values respectively for every element in the domain as follows:

Definition 2

([13]). Let X be a fixed set, and then a DHFS on X is described by the following mathematical symbol:

$$D=\left\{⟨x,h(x),g(x)⟩\mid x\in X\right\},$$

(2)

in which $h(x)$ and $g(x)$ are two sets of some possible values in $[0,1],$ denoting the membership degrees and the non-membership degrees, respectively, of the element $x\in X$ to the set D, satisfying

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

where ${\gamma }^{+}={\bigcup }_{\gamma \in h(x)}max\{\gamma \},$ and ${\eta }^{+}={\bigcup }_{\eta \in g(x)}max\{\eta \}.$

For convenience, $d(x)=(h(x),g(x))$ is called a dual hesitant fuzzy element (DHFE) denoted by $d=(h,g)$.

Wang et al. [23] proposed the following simple operational laws for DHFEs.

Definition 3

([23]). Let $d=(h_d,g_d),d_1=(h_{d_1},g_{d_1})\mathrm{and}{d}_2=(h_{d_2},g_{d_2})$ be three DHFEs such that $l(h_d)=l(h_{d_1})=l(h_{d_2})\mathrm{and}l(g_d)=l(g_{d_1})=l(g_{d_2})$, then:

• $d\oplus d_1=\left(h_d\oplus h_{d_1},g_d\otimes g_{d_1}\right),$
  $=\left(\{h_d^{\sigma (j)}+h_{d_1}^{\sigma (j)}-h_d^{\sigma (j)}{h}_{d_1}^{\sigma (j)}\mid j=1,2,0\dots ,l(h_d)\},\{g_d^{\sigma (t)}{g}_{d_1}^{\sigma (t)}\mid t=1,2,\dots ,l(g_d)\}\right),$
• $d\otimes d_1=\left(h_d\otimes h_{d_1},g_d\oplus g_{d_1}\right),$
  $=\left(\{h_d^{\sigma (j)}{h}_{d_1}^{\sigma (j)}\mid j=1,2,0\dots ,l(h_d)\},\{g_d^{\sigma (t)}+g_{d_1}^{\sigma (t)}-g_d^{\sigma (t)}{g}_{d_1}^{\sigma (t)}\mid t=1,2,\dots ,l(g_d)\}\right),$
• $d^{\alpha }=\left(\{{(h_d^{\sigma (j)})}^{\alpha }\mid j=1,2,0\dots ,l(h_d)\},\{1-{(1-h_d^{\sigma (t)})}^{\alpha }\mid t=1,2,\dots ,l(g_d)\}\right),$
• $\alpha d=(\{1-{(1-h_d^{\sigma (j)})}^{\alpha }\mid j=1,2,0\dots ,l(h_d)\},\{{(g_d^{\sigma (t)})}^{\alpha }\mid t=1,2,\dots ,l(g_d)\}),$

where α is a positive integral, $l(h_d)$ and $l(g_d)$ denote the number of values in $h_d$ and $g_d$, $h_d^{\sigma (j)}$ is the jth smallest value in $h_d$ and $g_d^{\sigma (t)}$ is the tth smallest value in $g_d$.

Remark 1.

Here, it is assumed that all the DHFEs have the same number of values for membership degrees and non-membership degrees. However, if they are different, then we need to equalize them according to a transformation and regulation method [23].

Definition 4.

[24] Let A and B be any dual hesitant fuzzy sets of the same length; then, a dual hesitant normalized Hamming distance between A and B is given as:

$$d_{dnh}(A,B)=\sum _{i=1}^n\left(\frac{1}{n}\left(\frac{1}{l(x_i)}\left(\sum _{j=1}^{l(h(x_i))}\left|{\gamma }_A^{\rho (j)}(x_i)-{\gamma }_B^{\rho (j)}(x_i)\right|+\sum _{k=1}^{l(g(x_i))}\left|{\eta }_A^{\rho (k)}(x_i)-{\eta }_B^{\rho (k)}(x_i)\right|\right)\right)\right).$$

(3)

$l(x_i)=l(h(x_i))+l(g(x_i));l(h(x_i))\mathrm{and}l(g(x_i))$ denote the number of elements in h and g, respectively, ${\gamma }_A^{\rho (j)}(x_i)$ is the $jth$ largest value in $h_A(x_i)$ and ${\eta }_A^{\rho (k)}(x_i)$ is the $kth$ largest value in $g_A(x_i)$.

2.3. Proportional Hesitant Fuzzy Set

In 2018, Xiong et al. [22] proposed the concept of PHFS.

Definition 5.

[22] Let X be a fixed set, and the proportional hesitant fuzzy set (PHFS) on X is described as follows:

$$E=\left\{\left(x,p{h}_E(x)\mid x\in X\right)\right\}=\left\{\left(x,\left(h_E(x),p_E(x)\right)\mid x\in X\right)\right\},$$

(4)

where

• $h_E(x)=\left\{{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\right\}$ is a set of n values in $[0,1]$, presenting possible membership values of the element $x\in X$ to set E, and
• $p_E(x)=\left\{{\tau }_1,{\tau }_2,\dots ,{\tau }_n\right\}$ is a set of n values in $[0,1]$, where ${\tau }_i\left(i=1,2,\dots ,n\right)$ denotes the proportion of membership values ${\gamma }_i\left(i=1,2,\dots ,n\right)$ such that ${\sum }_i^n{\tau }_i=1$.

For convenience and simplicity, we call $ph=p{h}_E(x)$ a proportional hesitant fuzzy element (PHFE).

Definition 6.

Let $ph$ and $p{h}_1$ be two PHFEs on the fixed set X, then:

• $ph\bigcup p{h}_1={\bigcup }_{(\gamma ,\tau )\in ph,({\gamma }_1,{\tau }_1)\in p{h}_1}\left\{\left(\gamma \vee {\gamma }_1,\tau {\tau }_1\right)\right\}$;
• $ph\bigcap p{h}_1={\bigcup }_{(\gamma ,\tau )\in ph,({\gamma }_1,{\tau }_1)\in p{h}_1}\left\{\left(\gamma \wedge {\gamma }_1,\tau {\tau }_1\right)\right\}$.

Definition 7.

Let $ph$ be a PHFE. Then, its score and deviation values are defined as follows:

$$s(ph)=\sum _{\left(\gamma ,\tau \right)\in ph}\gamma \tau ;$$

(5)

$$t(ph)=\sum _{\left(\gamma ,\tau \right)\in ph}\tau {\left(\gamma -s(ph)\right)}^2.$$

(6)

3. The Notion of the Proportional Dual Hesitant Fuzzy Set

Inspired by the idea behind proportional hesitant fuzzy set, this section puts forward a novel concept called PDHFS to model and compute the group decision making (GDM) problems with more care and produce more accurate results. The basic operations and properties are also studied in this section.

Definition 8.

Let X be a universe of discourse, and a proportional dual hesitant fuzzy set (PDHFS) on X is a triplet described as

$$S=\left\{⟨x,ph(x),qg(x)⟩\mid x\in X\right\}=\left\{⟨x,\left\{\left(h(x),p(x)\right)\right\},\left\{\left(g(x),q(x)\right)\right\}⟩\mid x\in X\right\},$$

(7)

in which $h(x)=\left\{{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\right\}$ and $g(x)=\left\{{\eta }_1,{\eta }_2,\dots ,{\eta }_m\right\}$ are two sets in $[0,1]$, presenting n kinds of possible membership and m kinds of possible non-membership degrees of the element $x\in X$ to the set S, respectively, and $p(x)=\left\{{\tau }_1,{\tau }_2,\dots ,{\tau }_n\right\}$, $q(x)=\left\{{\tau }_1^{\prime },{\tau }_2^{\prime },\dots ,{\tau }_m^{\prime }\right\}$ are sets in $[0,1]$, presenting the proportion of membership degrees and non-membership degrees respectively, with the conditions:

$$\sum _i^n{\tau }_i=1,\sum _j^m{\tau }_j^{\prime }=1;$$

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

where ${\gamma }^{+}={\bigcup }_{\gamma \in h(x)}max\{\gamma \}$ and ${\eta }^{+}={\bigcup }_{\eta \in g(x)}max\{\eta \}$.

For convenience and simplicity, we call the pair $pd(x)=⟨ph(x),qg(x)⟩$ as a proportional dual hesitant fuzzy element (PDHFE) and also will be written $pd=⟨ph,qg⟩$ in short.

For clarification, we provide the following example.

Example 1.

Let $X=\left\{x_1,x_2,x_3\right\}$. Consider

$$\begin{array}{c}S=\{⟨x_1,\{(0.4,0.8),(0.5,0.2)\},\{(0.5,1)\}⟩,⟨x_2,\{(0.4,0.3),(0.6,0.5),(0.7,0.2)\},\{(0.3,0.6),(0.2,0.4)\}⟩\hfill \\ \hfill ,⟨x_3,\{(0.4,0.3),(0.5,0.4),(0.3,0.3)\},\{(0.5,0.4),(0.4,0.6)\}⟩\}.\end{array}$$

Then, it is easy to check the conditions of PDHFS and

• $⟨x_1,\{(0.4,0.8),(0.5,0.2)\},\{(0.5,1)\}⟩,$
• $⟨x_2,\{(0.4,0.3),(0.6,0.5),(0.7,0.2)\},\{(0.3,0.6),(0.2,0.4)\}⟩,$
• $⟨x_3,\{(0.4,0.3),(0.5,0.4),(0.3,0.3)\},\{(0.5,0.4),(0.4,0.6)\}⟩$ are PDHFEs.

Remark 2.

Particularly, if the proportion values of all terms in membership part and non-membership part become equal, then PDHFS reduces to DHFS. Then, in that case, PDHFS reduces to HFS. Thus, the definition of PDHFS encompasses these fuzzy set extensions.

In order to illustrate the PDHFS more straightforwardly, in the following, a practical life example is provided to depict the difference between PDHFS and DHFS:

Example 2.

Take the evaluation of a university teacher based on the attribute/criteria like research skill as an example. Three experts provide the following evaluation values to measure his performance.

$$\left({\mu }_1,{\nu }_1\right)=\left(0.6,0.4\right),\left({\mu }_2,{\nu }_2\right)=\left(0.3,0.4\right)\mathit{and}\left({\mu }_3,{\nu }_3\right)=\left(0.6,0.2\right)$$

in terms of DHFS the above evaluation data can be expressed as

$$E=\left\{\{0.6,0.3\}\{0.4,0.2\}\right\}.$$

The proportion of the membership value $0.6$ is $\frac{2}{3}$. Similarly, we can find the proportion of all other membership and non-membership values. The overall evaluation can be represented by PDHFE as follows:

$$pd=⟨\left\{(0.6,\frac{2}{3}),(0.3,\frac{1}{3})\right\},\left\{(0.2,\frac{1}{3}),(0.4,\frac{2}{3})\right\}⟩.$$

In the DHFE, a value repeated more times has no more importance than less repeated values, which is unrealistic because a higher repeated value should be more important in aggregating DMs’ opinions than the other values. Thus, DHFE fails to express the evaluation of a teacher more reasonably. Therefore, to get more reasonable results, it is better that the experts provide their evaluations in terms of PDHFEs because PDHFE gives importance to the repeated values.

Definition 9.

Let X be a universe of discourse, for any $x\in X$, we call:

• $pd(x)=⟨\left\{(0,1)\right\},\left\{(1,1)\right\}⟩$ as the empty proportional dual hesitant fuzzy element (EPDHFE).
  A PDHFS whose all PDHFEs are EPDHFEs is called an empty proportional dual hesitant fuzzy set (EPDHFS), denoted by Θ;
• $pd(x)=⟨\left\{(1,1)\right\},\left\{(1,1)\right\}⟩$ as the fully proportional dual hesitant fuzzy element (FPDHFE).
  A PDHFS whose all PDHFEs are FPDHFEs is called fully proportional dual hesitant fuzzy set (FPDHFS), denoted by ϝ.

Definition 10.

For a given PDHFE $pd=⟨\{ph\},\{qg\}⟩,$ its complement is defined as follows:

$$p{d}^c=\left\{\begin{array}{cc}\bigcup _{(\gamma ,{\tau }_{\gamma })\in ph,(\eta ,{\tau }_{\eta }^{\prime })\in qg}⟨\left\{(\eta ,{\tau }_{\eta }^{\prime })\right\},\left\{(\gamma ,{\tau }_{\gamma })\right\}⟩\hfill & ifh\ne \varnothing ,g\ne \varnothing \hfill \\ \bigcup _{(\gamma ,{\tau }_{\gamma })\in ph}⟨\left\{(1-\gamma ,{\tau }_{\gamma })\right\},\varnothing ⟩\hfill & ifh\ne \varnothing ,g=\varnothing \hfill \\ \bigcup _{(\eta ,{\tau }_{\eta }^{\prime })\in qg}⟨,\varnothing ,\left\{(1-\eta ,{\tau }_{\eta }^{\prime })\right\}⟩\hfill & ifh=\varnothing ,g\ne \varnothing .\hfill \end{array}\right.$$

(8)

Proposition 1.

The complement is involuative, i.e., ${\left(p{d}^c\right)}^c=pd$

Proof. 

It trivially follows from the Definition 10. □

Next, the union and intersection of PDHFSs are defined which are based on the corresponding lower and upper bounds of membership and non-membership PHFSs. For any two PDHFSs say, $p{d}_1$ and $p{d}_2$, one should note that $p{h}^{-}={\bigcup }_{(\gamma ,\tau )\in ph}min\left\{\gamma ·\tau \right\}$, $p{h}^{+}={\bigcup }_{(\gamma ,\tau )\in ph}max\left\{\gamma ·\tau \right\}$, $q{g}^{-}={\bigcup }_{(\eta ,{\tau }^{\prime })\in qg}min\left\{\eta ·{\tau }^{\prime }\right\}$ and $q{g}^{+}={\bigcup }_{(\eta ,{\tau }^{\prime })\in qg}max\left\{\eta ·{\tau }^{\prime }\right\}$ represent the corresponding lower and upper bounds to $ph$ and $qg$, respectively.

Definition 11.

Let X be a universe of discourse and $p{d}_1$, $p{d}_2$ be two PDHFEs. Then,

• $p{d}_1\bigcup p{d}_2=$
  $⟨\left\{ph\in \left(p{h}_1\bigcup p{h}_2\right)\mid ph\ge max\left(p{h}_1^{-},p{h}_2^{-}\right)\right\},\left\{qg\in \left(q{g}_1\bigcap q{g}_2\right)\mid qg\le min\left(q{g}_1^{+},q{g}_2^{+}\right)\right\}⟩$;
• $p{d}_1\bigcap p{d}_2=$
  $⟨\left\{ph\in \left(p{h}_1\bigcap p{h}_2\right)\mid ph\le min\left(p{h}_1^{+},p{h}_2^{+}\right)\right\},\left\{qg\in \left(q{g}_1\bigcup q{g}_2\right)\mid qg\ge max\left(q{g}_1^{-},q{g}_2^{-}\right)\right\}⟩$.

$p{h}_1\bigcup p{h}_2$ and $q{g}_1\bigcap q{g}_2$ are computed according to Definition 6.

Let $l_h(pd)$ and $l_g(pd)$ represent the number of values in $ph$ and $qg$, respectively, of a PDHFE $pd=(ph,qg)$. To prove this, $l(pd)=\left(l_h(pd),l_g(pd)\right)$ in most cases $l(p{d}_A)\ne l(p{d}_B)$ i.e., $l_h(p{d}_A)\ne l_h(p{d}_B),l_g(p{d}_A)\ne l_g(p{d}_B)$. To overcome this issue of ordering and adding, a method for PDHFEs is proposed in which the following two simple steps are performed:

Step $(1)$: Ordering;

Arrange the elements in $p{d}_A=\left(p{h}_A,q{g}_A\right)$ and $p{d}_B=\left(p{h}_B,q{g}_B\right)$ in decreasing order according to the product values of the assessment and their associated proportions.

Step $(2)$: Adding;

Extended the shorter one by adding the element $(0,0)$ several times until their lengths becomes the same.

Definition 12.

Let A and B be any two PDHFSs on a fixed set X. Then, they are said to be of equal length if their corresponding PDHFEs are of equal length.

Note: One can generalize the above definition to n PDHFSs.

The following operations are defined for PDHFSs in order to use them in practical decision-making problems:

Definition 13.

Let X be a universe of discourse, $p{d}_1$ and $p{d}_2$ be two PDHFEs of equal length; then, the following operations are defined:

• $\oplus -union;p{d}_1\oplus p{d}_2=⟨p{h}_1\oplus p{h}_2,q{g}_1\otimes q{g}_2⟩$
  $=⟨\begin{array}{c}\left\{\left({\gamma }_1^{\sigma (j)}+{\gamma }_2^{\sigma (j)}-{\gamma }_1^{\sigma (j)}{\gamma }_2^{\sigma (j)},{\tau }_1^{\sigma (j)}{\tau }_2^{\sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_1)\right\},\\ \left\{\left({\eta }_1^{\sigma (t)}{\eta }_2^{\sigma (t)},{\tau }_1^{\prime \sigma (t)}{\tau }_2^{\prime \sigma (t)}\right)\mid t=1,2,\dots ,l(q{g}_1)\right\}\end{array}⟩;$
• $\otimes -intersection;p{d}_1\otimes p{d}_2=⟨p{h}_1\otimes p{h}_2,q{g}_1\oplus q{g}_2⟩$
  $=⟨\begin{array}{c}\left\{\left({\gamma }_1^{\sigma (j)}{\gamma }_2^{\sigma (j)},{\tau }_1^{\sigma (j)}{\tau }_2^{\sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_1)\right\},\\ \left\{\left({\eta }_1^{\sigma (t)}+{\eta }_2^{\sigma (t)}-{\eta }_1^{\sigma (t)}{\eta }_2^{\sigma (t)},{\tau }_1^{\prime \sigma (t)}{\tau }_2^{\prime \sigma (t)}\right)\mid t=1,2,\dots ,l(q{g}_1)\right\}\end{array}⟩;$
• $\alpha p{d}_1=⟨\begin{array}{c}\left\{\left(1-{\left(1-{\gamma }_1^{\sigma (j)}\right)}^{\alpha },{\tau }_1^{\sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_1)\right\},\\ \left\{\left({\left({\eta }_1^{\sigma (t)}\right)}^{\alpha },{\tau }_1^{\prime \sigma (t)}\right)\mid t=1,2,\dots ,l(q{g}_1)\right\}\end{array}⟩;$
• ${\left(p{d}_1\right)}^{\alpha }=⟨\begin{array}{c}\left\{\left({\left({\gamma }_1^{\sigma (j)}\right)}^{\alpha },{\tau }_1^{\sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_1)\right\},\\ \left\{\left(1-{\left(1-{\eta }_1^{\sigma (t)}\right)}^{\alpha },{\tau }_1^{\prime \sigma (t)}\right)\mid t=1,2,\dots ,l(q{g}_1)\right\}\end{array}⟩;$

where α is a positive integral, $\left({\gamma }_1^{\sigma (j)},{\tau }_1^{\sigma (j)}\right)$ and $\left({\eta }_1^{\sigma (t)},{\tau }_1^{\prime \sigma (t)}\right)$ are chosen such that ${\gamma }_1^{\sigma (j)}·{\tau }_1^{\sigma (j)}$ and ${\eta }_1^{\sigma (t)}·{\tau }_1^{\prime \sigma (t)}$ are the jth and tth smallest product values in PDHFEs $p{d}_1$ and $p{d}_2$, respectively.

Remark 3.

Here, we have assumed that all PDHFEs are of equal length. However, in most cases, they are unequal, which produces problems in basic operations and distance measure. Then, in that situation, their lengths are equalized according to the ordering and adding method as discussed above.

Theorem 1.

Let $A,B\mathrm{and}C$ be any three PDHFSs on X. Suppose $\alpha \ge 0$; then:

• $A\oplus B=B\oplus A$, $A\otimes B=B\otimes A;$
• $\left(A\oplus B\right)\oplus C=A\oplus \left(B\oplus C\right),$$\left(A\otimes B\right)\otimes C=A\otimes \left(B\otimes C\right);$
• $\alpha \left(A\oplus B\right)=\alpha A\oplus \alpha B$, ${\left(A\otimes B\right)}^{\alpha }=A^{\alpha }\otimes B^{\alpha };$
• ${\left(A\oplus B\right)}^c=A^c\otimes B^c$, ${\left(A\otimes B\right)}^c=A^c\oplus B^c.$

Proof. 

$(1):$

$$\begin{array}{cc}\hfill p{d}_A\oplus p{d}_B& =⟨\left\{\left({\gamma }_A^{\sigma (j)}+{\gamma }_B^{\sigma (j)}-{\gamma }_A^{\sigma (j)}{\gamma }_B^{\sigma (j)},{\tau }_A^{\sigma (j)}{\tau }_B^{\sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_1)\right\},\hfill \\ & \left\{\left({\eta }_A^{\sigma (t)}{\eta }_B^{\sigma (t)},{\tau }_A^{\prime \sigma (t)}{\tau }_B^{\prime \sigma (t)}\right)\mid t=1,2,\dots ,l(q{g}_1)\right\}⟩\hfill \\ & =⟨\left\{\left({\gamma }_B^{\sigma (j)}+{\gamma }_A^{\sigma (j)}-{\gamma }_B^{\sigma (j)}{\gamma }_A^{\sigma (j)},{\tau }_B^{\sigma (j)}{\tau }_A^{\sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_1)\right\},\hfill \\ & \left\{\left({\eta }_B^{\sigma (t)}{\eta }_A^{\sigma (t)},{\tau }_B^{\prime \sigma (t)}{\tau }_A^{\prime \sigma (t)}\right)\mid t=1,2,\dots ,l(q{g}_1)\right\}⟩\hfill \\ & =p{d}_B\oplus p{d}_A.\hfill \end{array}$$

Then, $A\oplus B=B\oplus A$. Similarly, one can show that $A\otimes B=B\otimes A$.

$(2):$ Since

$$\begin{array}{cc}\hfill (p{d}_A\oplus p{d}_B)\oplus p{d}_C& =⟨\left\{\left({\gamma }_A^{\sigma (j)}+{\gamma }_B^{\sigma (j)}-{\gamma }_A^{\sigma (j)}{\gamma }_B^{\sigma (j)},{\tau }_A^{\sigma (j)}{\tau }_B^{\sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_A)\right\},\hfill \\ & \left\{\left({\eta }_A^{\sigma (t)}{\eta }_B^{\sigma (t)},{\tau }_A^{\prime \sigma (t)}{\tau }_B^{\prime \sigma (t)}\right)\right\}⟩\oplus ⟨p{h}_C,q{g}_C⟩\hfill \\ & =⟨\left\{\right(\left({\gamma }_A^{\sigma (j)}+{\gamma }_B^{\sigma (j)}-{\gamma }_A^{\sigma (j)}{\gamma }_B^{\sigma (j)}\right)+{\gamma }_C^{\sigma (j)}-\left({\gamma }_A^{\sigma (j)}+{\gamma }_B^{\sigma (j)}-{\gamma }_A^{\sigma (j)}{\gamma }_B^{\sigma (j)}\right){\gamma }_C^{\sigma (j)},\hfill \\ & \left({\tau }_A^{\sigma (j)}{\tau }_B^{\sigma (j)}\right){\tau }_C^{\sigma (j)})\mid j=1,2,0\dots ,l(p{h}_A)\},\{\left(\left({\eta }_A^{\sigma (t)}{\eta }_B^{\sigma (t)}\right){\eta }_C^{\sigma (t)},\left({\tau }_A^{\prime \sigma (t)}{\tau }_B^{\prime \sigma (t)}\right){\tau }_C^{\prime \sigma (t)}\right)\hfill \\ & \mid t=1,2,\dots ,l(q{g}_A)\}⟩\hfill \\ & =p{d}_A\oplus (p{d}_B\oplus p{d}_C),\hfill \end{array}$$

then $\left(A\oplus B\right)\oplus C=A\oplus \left(B\oplus C\right)$.

Similarly, $\left(A\otimes B\right)\otimes C=A\otimes \left(B\otimes C\right)$

$(3):$ Since

$$\begin{array}{cc}\hfill \alpha \left(p{d}_A\oplus p{d}_B\right)& =⟨\left\{\left({\gamma }_A^{\sigma (j)}-{\left(1-\left({\gamma }_A^{\sigma (j)}+{\gamma }_B^{\sigma (j)-{\gamma }_A^{\sigma (j)}{\gamma }_B^{\sigma (j)}}\right)\right)}^{\alpha },{\tau }_A^{\prime \sigma (j)}{\tau }_B^{\prime \sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_A)\right\},\hfill \\ & \left\{\left({\left({\eta }_A^{\sigma (t)}{\eta }_B^{\sigma (t)}\right)}^{\alpha },{\tau }_A^{\prime \sigma (t)}{\tau }_B^{\prime \sigma (t)}\right)\mid t=1,2,\dots ,l(q{g}_A)\right\}⟩\hfill \\ & =⟨\left\{\left(1-{\left(1-{\gamma }_A^{\sigma (j)}\right)}^{\alpha }{\left(1-{\gamma }_B^{\sigma (j)}\right)}^{\alpha },{\tau }_A^{\prime \sigma (j)}{\tau }_B^{\prime \sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_A)\right\},\hfill \\ & \left\{\left({\left({\eta }_A^{\sigma (t)}\right)}^{\alpha }{\left({\eta }_B^{\sigma (t)}\right)}^{\alpha },{\tau }_A^{\prime \sigma (t)}{\tau }_B^{\prime \sigma (t)}\right)\mid t=1,2,\dots ,l(q{g}_A)\right\}⟩\hfill \\ & =\alpha p{d}_A\oplus \alpha p{d}_B,\hfill \end{array}$$

then $\alpha \left(A\oplus B\right)=\alpha A\oplus \alpha B$.

Similarly, one can show that ${\left(A\otimes B\right)}^{\alpha }=A^{\alpha }\otimes B^{\alpha }$.

$(4)$ Since ${\left(p{d}_A\oplus p{d}_B\right)}^c$

Case(I); If $h\ne \varnothing \mathrm{and}g\ne \varnothing$

$$\begin{array}{cc}\hfill {\left(p{d}_A\oplus p{d}_B\right)}^c& ={⟨\left\{\begin{array}{c}\left({\gamma }_A^{\sigma (j)}+{\gamma }_B^{\sigma (j)}-{\gamma }_A^{\sigma (j)}{\gamma }_B^{\sigma (j)},{\tau }_A^{\sigma (j)}{\tau }_B^{\sigma (j)}\right)\\ \mid j=1,2,\dots ,l(p{h}_A)\end{array}\right\},\left\{\begin{array}{c}\left({\eta }_A^{\sigma (t)}{\eta }_B^{\sigma (t)},{\tau }_A^{\prime \sigma (t)}{\tau }_B^{\prime \sigma (t)}\right)\\ \mid t=1,2,\dots ,l(q{g}_1)\end{array}\right\}⟩}^c\hfill \\ & =⟨\left\{\begin{array}{c}\left({\eta }_A^{\sigma (t)}{\eta }_B^{\sigma (t)},{\tau }_A^{\prime \sigma (t)}{\tau }_B^{\prime \sigma (t)}\right)\\ \mid t=1,2,\dots ,l(q{g}_1)\end{array}\right\},\left\{\begin{array}{c}\left({\gamma }_A^{\sigma (j)}+{\gamma }_B^{\sigma (j)}-{\gamma }_A^{\sigma (j)}{\gamma }_B^{\sigma (j)},{\tau }_A^{\sigma (j)}{\tau }_B^{\sigma (j)}\right)\\ \mid j=1,2,\dots ,l(p{h}_A)\end{array}\right\}⟩\hfill \\ & =\bigcup _{\left({\eta }_A,{\tau }_A\right)\in p{h}_A,\left({\eta }_A,{\tau }_A^{\prime }\right)\in q{g}_A}⟨\begin{array}{c}\left\{\left({\eta }_A,{\tau }_A^{\prime }\right)\right\},\\ \left\{\left({\gamma }_A,{\tau }_A\right)\right\}\end{array}⟩\otimes \bigcup _{\left({\eta }_B,{\tau }_B\right)\in p{h}_B,\left({\eta }_B,{\tau }_B^{\prime }\right)\in q{g}_B}⟨\begin{array}{c}\left\{\left({\eta }_B,{\tau }_B^{\prime }\right)\right\},\\ \left\{\left({\gamma }_B,{\tau }_B\right)\right\}\end{array}⟩\hfill \\ & =\bigcup _{\left({\gamma }_A,{\tau }_A\right)\in p{h}_A,\left({\eta }_A,{\tau }_A^{\prime }\right)\in q{g}_A}{⟨\begin{array}{c}\left\{\left({\gamma }_A,{\tau }_A\right)\right\},\\ \left\{\left({\eta }_A,{\tau }_A^{\prime }\right)\right\}\end{array}⟩}^c\otimes \bigcup _{\left({\gamma }_B,{\tau }_B\right)\in p{h}_B,\left({\eta }_B,{\tau }_B^{\prime }\right)\in q{g}_B}{⟨\begin{array}{c}\left\{\left({\gamma }_B,{\tau }_B\right)\right\},\\ \left\{\left({\eta }_B,{\tau }_B^{\prime }\right)\right\}\end{array}⟩}^c\hfill \\ \hfill {\left(p{d}_A\oplus p{d}_B\right)}^c& =p{d}_A^c\otimes p{d}_B^c.\hfill \end{array}$$

Case(II); If $h\ne \varnothing \mathrm{and}g=\varnothing$

$$\begin{array}{cc}\hfill {\left(p{d}_A\oplus p{d}_B\right)}^c& ={⟨\left\{\left({\gamma }_A^{\sigma (j)}+{\gamma }_B^{\sigma (j)}-{\gamma }_A^{\sigma (j)}{\gamma }_B^{\sigma (j)},{\tau }_A^{\sigma (j)}{\tau }_B^{\sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_A)\right\},\varnothing ⟩}^c\hfill \\ & ={⟨\left\{\left(1-\left({\gamma }_A^{\sigma (j)}+{\gamma }_B^{\sigma (j)}-{\gamma }_A^{\sigma (j)}{\gamma }_B^{\sigma (j)}\right),{\tau }_A^{\sigma (j)}{\tau }_B^{\sigma (j)}\right)\mid j=1,2,\dots ,l(p{h}_A)\right\},\varnothing ⟩}^c\hfill \\ & =\bigcup _{({\gamma }_A,{\tau }_A)\in p{h}_A}⟨\left(\left\{(1-{\gamma }_A,{\tau }_A)\right\},\varnothing \right)⟩\otimes \bigcup _{({\gamma }_B,{\tau }_B)\in p{h}_B}⟨\left(\left\{(1-{\gamma }_B,{\tau }_B)\right\},\varnothing \right)⟩\hfill \\ & =\bigcup _{({\gamma }_A,{\tau }_A)\in p{h}_A}{⟨\left(\left\{({\gamma }_A,{\tau }_A)\right\},\varnothing \right)⟩}^c\otimes \bigcup _{({\gamma }_B,{\tau }_B)\in p{h}_B}{⟨\left(\left\{({\gamma }_B,{\tau }_B)\right\},\varnothing \right)⟩}^c\hfill \\ & =p{d}_A^c\otimes p{d}_B^c.\hfill \end{array}$$

Case(III); If $h=\varnothing \mathrm{and}g\ne \varnothing$

$$\begin{array}{cc}\hfill {\left(p{d}_A\oplus p{d}_B\right)}^c& ={⟨\varnothing ,\left\{\left({\eta }_A^{\sigma (t)}{\eta }_B^{\sigma (t)},{\tau }_A^{\prime \sigma (t)}{\tau }_B^{\prime \sigma (t)}\right)\mid t=1,2,\dots ,l(q{g}_A)\right\}⟩}^c\hfill \\ & =⟨\varnothing ,\left\{\left(1-{\eta }_A^{\sigma (t)}{\eta }_B^{\sigma (t)},{\tau }_A^{\prime \sigma (t)}{\tau }_B^{\prime \sigma (t)}\right)\mid t=1,2,\dots ,l(q{g}_A)\right\}⟩\hfill \\ & =\bigcup _{\left({\eta }_A,{\tau }_A^{\prime }\right)\in q{g}_A}⟨\varnothing ,\left\{\left(1-{\eta }_A,{\tau }_A^{\prime }\right)\right\}⟩\otimes \bigcup _{\left({\eta }_B,{\tau }_B^{\prime }\right)\in q{g}_B}⟨\varnothing ,\left\{\left(1-{\eta }_B,{\tau }_B\right)\right\}⟩\hfill \\ & =\bigcup _{({\eta }_A,{\tau }_A^{\prime })\in q{g}_A}{⟨\varnothing ,\left\{\left({\eta }_A,{\tau }_A^{\prime }\right)\right\}⟩}^c\otimes \bigcup _{\left({\eta }_B,{\tau }_B^{\prime }\right)\in q{g}_B}{⟨\varnothing ,\left\{\left({\eta }_B,{\tau }_B\right)\right\}⟩}^c\hfill \\ & =p{d}_A^c\otimes p{d}_B^c.\hfill \end{array}$$

Then, ${\left(A\oplus B\right)}^c=A^c\otimes B^c$.

Similarly, one can show that ${\left(A\otimes B\right)}^c=A^c\oplus B^c$. □

4. The Comparison Notions between PDHFSs Based on PDHFEs

A distance measure is designed for determining the degree of distance between two sets in the fuzzy set theory, which has received attention from scholars [25,26,27]. This section first defines the distance between two PDHFSs. Based on it, the comparison law is developed for PDHFSs.

Based on Definition 4, we come up with the following definition.

Definition 14.

Let A and B be any two PDHFSs of equal length on $X=\left\{x_1,x_2,\dots ,x_n\right\}$; then, the proportional dual hesitant normalized Hamming distance is

$$\begin{array}{c}{d}_{pdnh}\left(A,B\right)=\sum _{i=1}^n\left(\frac{1}{n}\right(\frac{1}{2l(x_i)}(\sum _{j=1}^{l(h{x}_i)}\left(\left|{\gamma }_A^{\rho (j)}(x_i){\tau }_A^{\rho (j)}(x_i)-{\gamma }_B^{\rho (j)}(x_i){\tau }_B^{\rho (j)}(x_i)\right|+\left|{\tau }_A^{\rho (j)}(x_i)-{\tau }_B^{\rho (j)}(x_i)\right|\right)\hfill \\ +\sum _{k=1}^{l(g{x}_i)}\left(\left|{\eta }_A^{\rho (k)}(x_i){\tau }_A^{\prime \rho (k)}(x_i)-{\eta }_B^{\rho (k)}(x_i){\tau }_B^{\prime \rho (k)}(x_i)\right|+\left|{\tau }_A^{\prime \rho (k)}(x_i)-{\tau }_B^{\prime \rho (k)}(x_i)\right|\right)\left)\right)),\hfill \end{array}$$

(9)

where $l(x_i)=l(h{x}_i)+l(g{x}_i)$, $l(h{x}_i)$ and $l(g{x}_i)$ represent the number of elements in h and g, respectively, ${\gamma }_A^{\rho (j)}(x_i){\tau }_A^{\rho (j)}(x_i)$ is the $jth$ largest product value in $p{h}_A(x_i)$ and ${\eta }_A^{\rho (k)}(x_i){\tau }_A^{\prime \rho (k)}(x_i)$ is the $kth$ largest product value in $q{g}_A(x_i)$.

One can check easily that the following properties hold for $d_{pdnh}$.

• $0\le d\left(A,B\right)\le 1$;
• $d\left(A,B\right)=0$ if and only if $A=B$;
• $d\left(A,B\right)=d\left(B,A\right)$.

For the purpose of ranking alternatives, in the following, we develop the comparison method for PDHFSs.

Definition 15.

Let $pd$ be a PDHFE on $X=\left\{x_1,x_2,\dots ,x_n\right\}$; then, the score function of $pd$ is defined as follows:

$$s\left(pd\right)=\sum _{\left(\gamma ,\tau \right)\in ph}\gamma \tau -\sum _{\left(\eta ,{\tau }^{\prime }\right)\in qg}\eta {\tau }^{\prime }$$

(10)

and the deviation function of $pd$ is defined as follows:

$$\sigma \left(pd\right)={\left(\sum _{\left(\gamma ,\tau \right)\in ph}\tau {\left(\gamma -s\left(pd\right)\right)}^2+\sum _{\left(\eta ,{\tau }^{\prime }\right)\in qg}{\tau }^{\prime }{\left(\eta -s\left(pd\right)\right)}^2\right)}^{\frac{1}{2}}.$$

(11)

Combining with the distance measure, the comparison method for PDHFEs can be defined as follows:

Definition 16.

Let $p{d}_l\left(l=1,2\right)$ be two PDHFEs, $s\left(p{d}_l\right)\left(l=1,2\right)$ and $\sigma \left(p{d}_l\right)\left(l=1,2\right)$ be their score and deviation values, respectively.

• if $s\left(p{d}_l\right)>s\left(p{d}_2\right)$, then $p{d}_l>p{d}_2$; on the contrary, there is $p{d}_l<p{d}_2.$
• if $s\left(p{d}_l\right)=s\left(p{d}_2\right)$ and $\sigma \left(p{d}_l\right)<\sigma \left(p{d}_2\right)$, then $p{d}_l>p{d}_2.$
• if $s\left(p{d}_l\right)=s\left(p{d}_2\right),\sigma \left(p{d}_l\right)=\sigma \left(p{d}_2\right)$,
  (a) and $d_{pdnh}\left(\{p{d}_1,ϝ\}\right)=d_{pdnh}\left(\{p{d}_2,ϝ\}\right)$, then $p{d}_1=p{d}_2$,
  (b) and $d_{pdnh}\left(\{p{d}_1,ϝ\}\right)<d_{pdnh}\left(\{p{d}_2,ϝ\}\right)$, then $p{d}_1>p{d}_2$,

where ϝ is the fully proportional dual hesitant fuzzy set and $d_{pdnh}\left(A,B\right)$ is the distance between A and B.

Example 3.

Let $p{d}_1=⟨\left\{(0.5,0.2),(0.4,0.8)\right\},\left\{(0.5,0.6),(0.4,0.4)\right\}⟩$ and $p{d}_2=⟨\left\{(0.4,0.6),(0.6,0.4)\right\},\left\{(0.4,1)\right\}⟩$ be two PDHFEs. Then, $s(p{d}_1)=-0.04$ and $s(p{d}_2)=0.08$. According to Definition 16, $p{d}_2>p{d}_1$.

5. A Group Decision-Making Approach for Proportional Dual Hesitant Fuzzy Information

This section concentrates on introducing some theory in order to construct an algorithm to model real world problems.

5.1. Algorithm Based on Proportional Dual Hesitant Fuzzy MAGDM

The multi attribute group decision making (MAGDM) problem under the proportional dual hesitant fuzzy environment is described as follows:

Consider the MAGDM problem in which there is a set of m alternatives $x=\left\{x_1,x_2,\dots ,x_m\right\}$ and a set of attributes $c=\left\{c_1,c_2,\dots ,c_n\right\}$. Furthermore, let there be ‘l’ DMs $m=\left\{m_1,m_2,0\dots ,m_l\right\}$. The DMs offer their evaluation values in the form of classical intuitionistic fuzzy numbers (IFNs) keeping in mind all the alternatives and attributes and thus k evaluation matrices $R_k={\left[⟨{\mu }_{ij}^k,{\nu }_{ij}^k⟩\right]}_{m\times n};k=1,2,\dots ,l$ are constructed with the classical IFNs. Then, the overall evaluation matrix R is computed as follows:

$$R={\left[p{d}_{ij}\right]}_{m\times n}=[⟨\{h_{ij},p_{ij}\},\{g_{ij},q_{ij}\}⟩],$$

(12)

where

$$h_{ij}=\bigcup _{k=1}^l{\mu }_{ij}^k,g_{ij}=\bigcup _{k=1}^l{\nu }_{ij}^k.$$

Furthermore, $p_{ij}$ and $q_{ij}$ contain the proportional (repetition) values of the members of $h_{ij}$ and $g_{ij}$, respectively, for $i=1,2,\dots ,m,j=1,2,\dots ,n$.

It is noted that the overall values of R are PDHFEs.

Definition 17.

Let $R={\left[p{d}_{ij}\right]}_{m\times n}$ be the overall evaluation matrix. Then, $R^{+}=\left\{p{d}_1^{+},p{d}_2^{+},\dots ,p{d}_n^{+}\right\}$ is referred as the positive ideal solution (PIS) of alternatives, where

$$p{d}_j^{+}=\left\{\begin{array}{cc}⟨\underset{1\le i\le m}{max}\{p{d}_{ij}\}⟩\hfill & \mathit{for}\mathit{the}\mathit{benefit}\mathit{criteria}{c}_j;j=1,2,\dots ,n\hfill \\ ⟨\underset{1\le i\le m}{min}\{p{d}_{ij}\}⟩\hfill & \mathit{for}\mathit{the}\mathit{cos}t\mathit{criteria}{c}_j;j=1,2,\dots ,n.\hfill \end{array}\right.$$

Definition 18.

Let $R={\left[p{d}_{ij}\right]}_{m\times n}$ be the overall evaluation matrix. Then, $R^{-}=\left\{p{d}_1^{-},p{d}_2^{-},\dots ,p{d}_n^{-}\right\}$ is referred to as the negative ideal solution (NIS) of alternatives, where

$$p{d}_j^{-}=\left\{\begin{array}{cc}⟨\underset{1\le i\le m}{min}\{p{d}_{ij}\}⟩\hfill & \mathit{for}\mathit{the}\mathit{benefit}\mathit{criteria}{c}_j;j=1,2,\dots ,n\hfill \\ ⟨\underset{1\le i\le m}{max}\{p{d}_{ij}\}⟩\hfill & \mathit{for}\mathit{the}\mathit{cost}\mathit{criteria}{c}_j;j=1,2,\dots ,n.\hfill \end{array}\right.$$

It is noted that, in the above definitions, max and min are taken based on Definition 7.

Definition 19.

The separation of each alternative from the PIS is defined with the help of proportional dual hesitant normalized Hamming distance as follows:

$$s_i^{+}=d_{pdnh}\left(R^{+},R_i\right),i=1,2,\dots ,m,$$

(13)

where $R_i=\left\{p{d}_{i1},p{d}_{i2},\dots ,p{d}_{in}\right\}$.

Definition 20.

The separation of each alternative from the NIS is defined as follows:

$$s_i^{-}=d_{pdnh}\left(R^{-},R_i\right),i=1,2,\dots ,m,$$

(14)

where $R_i=\left\{p{d}_{i1},p{d}_{i2},\dots ,p{d}_{in}\right\}$.

Clearly, the smaller the distance $d_{pdnh}\left(R^{+},R_i\right)$ the better the alternative $x_i$ and the larger distance $d_{pdnh}\left(R^{-},R_i\right)$ implies the better alternative $x_i$.

After obtaining the separations of alternatives, the closeness coefficient are calculated by the following formula to rank all the alternatives:

$$cl(x_i)=\frac{s_i^{-}}{s_i^{+}+s_i^{-}};i=1,2,\dots ,m,$$

(15)

where $cl(x_i)\le 0;(i=1,2,\dots ,m)$ represents the closeness coefficient of each alternative. The larger the $cl(x_i)$, the better the alternative $x_i$. Therefore, all the alternatives can be ranked by using the closeness coefficient and thus the best alternative

$$x^b=\left\{x_i\mid \underset{1\le i\le m}{max}cl(x_i)\right\}$$

(16)

can be chosen.

5.2. TOPSIS

To solve decision-making problems with PDHFS, our choice is to design a TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method that uses PDHFEs to express the opinions of DMs. The TOPSIS was constructed to address problems of multi-attribute decision-making with uncertain and complex data. It is a practical and effective technique for ranking and choosing of the best alternative in a group decision environment. For better visibility, in this technique, the given data of alternatives judged against each criteria provided by DMs are placed in the evaluation matrices and then these preferences of DMs are aggregated into the overall evaluation matrix. Normalization process is done, if needed. After that, positive ideal solution and negative ideal solution are computed by applying specific formulae. Then, the separation of each alternative from the positive ideal solution and the negative ideal solution are calculated by using distance formula. In the end of TOPSIS, a closeness coefficient is computed in order to find a scenario that is closest to the positive ideal solution and farthest from the negative ideal solution to classify the alternatives from best to worst. Unlike other developments, this scheme is indeed a unified mechanism and is easily applicable to many real life decision-making problems without enlarging the computational burden. Motivated by [14,22], in the following, we design the TOPSIS in the PDHFS regard. This decision raking procedure involves the following steps.

Step no 1: Analyze the given MAGDM problem and determine a set of alternatives $x=\left\{x_1,x_2,\dots ,x_n\right\}$, a set of attributes $c=\left\{c_1,c_2,\dots ,c_n\right\}$ and DMs provide evaluation matrices $R_k={\left[⟨{\mu }_{ij}^k,{\nu }_{ij}^k⟩\right]}_{m\times n};k=1,2,\dots ,l$.

Step no 2: Calculate the overall evaluation information matrix

$R={\left[p{d}_{ij}\right]}_{m\times n};p{d}_{ij}\left(i=1,2,\dots ,m,j=1,2,\dots ,n\right)$ is a PDHFE.

Step no 3: Since all the values in the overall evaluation matrix R are PDHFEs, there is no need to normalize them.

Step no 4: Compute the PIS $R^{+}$ and the NIS $R^{-}$ of all the alternatives by using Definitions 17 and 18, respectively.

Step no 5: Definitions 19 and 20 are utilized to find $s_i^{+}$ and $s_i^{-}$ of each alternative.

Step no 6: Obtain the closeness coefficient $cl$ of each alternative by solving Equation (15).

Step no 7: Rank all alternatives and pick the best one according to the closeness coefficient $cl$.

Step no 8: End.

The Figure 1 presents the flowchart of proposed decision methodology.

Figure 1. Proportional dual hesitant fuzzy TOPSIS approach.

6. A Case Study

In order to demonstrate the applicability of proposed scheme, we choose the case study proposed by Katz et al. [28]. They studied the promotion criteria of university professors and used multiple regression analysis to choose the factors that play important roles in the professors’ promotion while making decisions at the university level and build a more reasonable method for rewarding and evaluating university professors. Actually, they were inspired by the fact that there is an unfair way under which professors get a promotion from assistant to associate and associate to full professor. They said that a professor’s promotion runs through an irrational and chaotic way and a more justifiable approach could be designed to improve the decision ranking process. In this section, we depict our proposed methodology by evaluating the university faculty in Pakistan adapted from [29].

The problem is formulated as follows: Let $X=\{x_1,x_2,x_3,x_4\}$ be a set of four faculty professors (alternatives) to be evaluated using the IFNs. Suppose that we use three criteria $c_1=$ teaching, $c_2=$ research and $c_3=$ service to evaluate the faculty professors for tenure and promotion. Based on these criteria, a committee of four experts $m_1$, $m_2$ and $m_3$ and $m_4$ provides their opinions in Table 1, Table 2, Table 3 and Table 4, respectively, in the form of IFNs.

Table 1. $R_1$: opinion of expert $m_1$.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$⟨0.22,0.65⟩$	$⟨0.36,0.3⟩$	$⟨0.54,0.26⟩$
$x_2$	$⟨0.32,0.22⟩$	$⟨0.44,0.5⟩$	$⟨0.22,0.56⟩$
$x_3$	$⟨0.54,0.17⟩$	$⟨0.46,0.3⟩$	$⟨0.3,0.32⟩$
$x_4$	$⟨0.26,0.44⟩$	$⟨0.16,0.22⟩$	$⟨0.14,0.34⟩$

Table 2. $R_2$: opinion of expert $m_2$.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$⟨0.12,0.65⟩$	$⟨0.65,0.3⟩$	$⟨0.44,0.22⟩$
$x_2$	$⟨0.32,0.15⟩$	$⟨0.22,0.5⟩$	$⟨0.44,0.17⟩$
$x_3$	$⟨0.81,0.17⟩$	$⟨0.36,0.46⟩$	$⟨0.35,0.32⟩$
$x_4$	$⟨0.44,0.16⟩$	$⟨0.26,0.32⟩$	$⟨0.26,0.37⟩$

Table 3. $R_3$: opinion of expert $m_3$.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$⟨0.35,0.14⟩$	$⟨0.65,0.15⟩$	$⟨0.36,0.22⟩$
$x_2$	$⟨0.36,0.17⟩$	$⟨0.44,0.22⟩$	$⟨0.22,0.56⟩$
$x_3$	$⟨0.81,0.19⟩$	$⟨0.36,0.46⟩$	$⟨0.3,0.36⟩$
$x_4$	$⟨0.32,0.26⟩$	$⟨0.16,0.54⟩$	$⟨0.26,0.34⟩$

Table 4. $R_4$: opinion of expert $m_4$.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$⟨0.12,0.16⟩$	$⟨0.44,0.15⟩$	$⟨0.44,0.36⟩$
$x_2$	$⟨0.3,0.15⟩$	$⟨0.5,0.22⟩$	$⟨0.32,0.17⟩$
$x_3$	$⟨0.26,0.17⟩$	$⟨0.3,0.54⟩$	$⟨0.46,0.3⟩$
$x_4$	$⟨0.32,0.16⟩$	$⟨0.46,0.32⟩$	$⟨0.65,0.16⟩$

TOPSIS Method for the Illustrative Example

We employ the proposed TOPSIS method designed in Section 5 to handle the above problem as follows:

Step 1: Based on the evaluation matrices $R_k(k=1,2,3,4)$, we compute the overall evaluation matrix $R={[p{d}_{ij}]}_{4\times 3}$ given in the Table 5.

Table 5. The overall evaluation matrix R.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$⟨\{(0.12,0.5),(0.35,0.25),(0.22,0.25)\},\{(0.65,0.5),(0.14,0.25),(0.16,0.25)\}⟩$	$⟨\{(0.65,0.5),(0.44,0.25),(0.36,0.25)\},\{(0.3,0.5),(0.15,0.5)\}⟩$	$⟨\{(0.44,0.5),(0.36,0.25),(0.54,0.25)\},\{(0.22,0.5),(0.36,0.25),(0.26,0.25)\}⟩$
$x_2$	$⟨\{(0.32,0.5),(0.36,0.25),(0.3,0.25)\},\{(0.15,0.5),(0.17,0.25),(0.22,0.25)\}⟩$	$⟨\{(0.22,0.25),(0.44,0.5),(0.5,0.25)\},\{(0.5,0.5),(0.22,0.5)\}⟩$	$⟨\{(0.44,0.25),(0.22,0.5),(0.32,0.25)\},\{(0.17,0.5),(0.56,0.5)\}⟩$
$x_3$	$⟨\{(0.81,0.5),(0.26,0.25),(0.54,0.25)\},\{(0.17,0.75),(0.19,0.25)\}⟩$	$⟨\{(0.36,0.5),(0.3,0.25),(0.46,0.25)\},\{(0.46,0.5),(0.54,0.25),(0.3,0.25)\}⟩$	$⟨\{(0.35,0.25),(0.3,0.5),(0.46,0.25)\},\{(0.32,0.5),(0.36,0.25),(0.3,0.25)\}⟩$
$x_4$	$⟨\{(0.44,0.25),(0.32,0.5),(0.26,0.25)\},\{(0.16,0.5),(0.26,0.25),(0.44,0.25)\}⟩$	$⟨\{(0.26,0.25),(0.16,0.5),(0.46,0.25)\},\{(0.32,0.5),(0.54,0.25),(0.22,0.25)\}⟩$	$⟨\{(0.26,0.5),(0.65,0.25),(0.14,0.25)\},\{(0.3,0.5),(0.34,0.25),(0.16,0.25)\}⟩$

• where $p{d}_{11}=⟨\{(0.12,0.5),(0.35,0.25),(0.22,0.25)\},\{(0.65,0.5),(0.14,0.25),(0.16,0.25)\}⟩,$
• pd₁₂ = ⟨{(0.65,0.5),(0.44,0.25),(0.36,0.25)},{(0.3,0.5),(0.15,0.5)}⟩⟩,
• pd₁₃ = ⟨{(0.44,0.5),(0.36,0.25),(0.54,0.25)},{(0.22,0.5),(0.36,0.25),(0.26,0.25)}⟩,
• pd₂₁ = ⟨{(0.32,0.5),(0.36,0.25),(0.3,0.25)},{(0.15,0.5),(0.17,0.25),(0.22,0.25)}⟩,
• pd₂₂ = ⟨{(0.22,0.25),(0.44,0.5),(0.5,0.25)},{(0.5,0.5),(0.22,0.5)}⟩,
• pd₂₃ = ⟨{(0.44,0.25),(0.22,0.5),(0.32,0.25)},{(0.17,0.5),(0.56,0.5)}⟩,
• pd₃₁ = ⟨{(0.81,0.5),(0.26,0.25),(0.54,0.25)},{(0.17,0.75),(0.19,0.25)}⟩,
• pd₃₂ = ⟨{(0.36,0.5),(0.3,0.25),(0.46,0.25)},{(0.46,0.5),(0.54,0.25),(0.3,0.25)}⟩,
• pd₃₃ = ⟨{(0.35,0.25),(0.3,0.5),(0.46,0.25)},{(0.32,0.5),(0.36,0.25),(0.3,0.25)}⟩,
• pd₄₁ = ⟨{(0.44,0.25),(0.32,0.5),(0.26,0.25)},{(0.16,0.5),(0.26,0.25),(0.44,0.25)}⟩,
• pd₄₂ = ⟨{(0.26,0.25),(0.16,0.5),(0.46,0.25)},{(0.32,0.5),(0.54,0.25),(0.22,0.25)}⟩,
• pd₄₃ = ⟨{(0.26,0.5),(0.65,0.25),(0.14,0.25)},{(0.3,0.5),(0.34,0.25),(0.16,0.25)}⟩.

Step 2: The score and deviation values of the PDHFEs in the overall evaluation matrix R are computed according to Equations (10) and (11) and displayed in Table 6.

Table 6. Score and deviation values of PDHFEs in matrix R.

		${\mathit{c}}_1$			${\mathit{c}}_2$			${\mathit{c}}_3$
	Score	Deviation	Rank	Score	Deviation	Rank	Score	Deviation	Rank
$x_1$	$-0.1975$	$0.76712$	4	$0.3$	$0.27982$	1	$0.18$	$0.29120$	1
$x_2$	$0.1525$	$0.17734$	2	$0.04$	$0.51284$	2	$-0.065$	$0.60361$	4
$x_3$	$0.43$	$0.38412$	1	$-0.07$	$0.68162$	3	$0.0275$	$0.44595$	3
$x_4$	$0.08$	$0.33615$	3	$-0.09$	$0.58720$	4	$0.0525$	$0.40850$	2

Step 3: Since all criteria are benefit criteria, following Definitions 17 and 18, the positive ideal solution is

$$R^{+}=\left\{p{d}_{31},p{d}_{12},p{d}_{13}\right\}$$

and the negative ideal solution is

$$R^{-}=\left\{p{d}_{11},p{d}_{42},p{d}_{23}\right\}.$$

Step 4: According to Definition 19, the separation of each alternative from the PIS are $s_1^{+}=0.03383,s_2^{+}=0.03676,s_3^{+}=0.00958,s_4^{+}=0.04389,$ and according to Definition 20, the separation of each alternative from the NIS are $s_1^{-}=-0.02778,s_2^{-}=-0.02826,s_3^{-}=-0.05028,s_4^{-}=0.02694.$

Step 5: Compute the closeness coefficient $cl(x_i);i=1,2,3,4:$ $cl(x_1)=0.45090,$ $cl(x_2)=0.43463,$ $cl(x_3)=0.83996,$ $cl(x_4)=0.38035.$

Step 6: The ranking result of all professors on their performance is

$$x_3>x_1>x_2>x_4.$$

Therefore, professor $x_3$ achievement is the top of the range.

7. Comparative Analysis of the Proposed Decision-Making Scheme with Existing Ones on the Basis of Case Study

In what follows, the detailed comparative analysis is presented with the traditional approach and Ref. [22] approach to illustrate the essentiality of our proposed scheme.

7.1. Dual Hesitant Fuzzy TOPSIS Approach for MAGDM

Neglecting the proportional figures, the major steps involved in the traditional TOPSIS approach are:

Step $1^{\prime }$: DMs provide evaluation matrices $R_k={\left[⟨{\mu }_{ij}^k,{\nu }_{ij}^k⟩\right]}_{m\times n};k=1,2,\dots ,l$ with the traditional IFNs.

Step $2^{\prime }$: Compute the overall evaluation information matrix $R^{\prime }={\left[d_{ij}\right]}_{m\times n};d_{ij}\left(i=1,2,\dots ,m,j=1,2,\dots ,n\right)$ is a DHFE.

Step $3^{\prime }$: The positive ideal solution (PIS) of alternatives is defined as follows:

$$R^{+}=\left\{d_1^{+},d_2^{+},\dots ,d_n^{+}\right\},$$

(17)

where

$$d_j^{+}=\left\{\begin{array}{cc}⟨\underset{1\le i\le m}{max}\{d_{ij}\}⟩\hfill & \mathrm{for}\mathrm{the}\mathrm{benefit}\mathrm{criteria}{c}_j;j=1,2,\dots ,n\hfill \\ ⟨\underset{1\le i\le m}{min}\{d_{ij}\}⟩\hfill & \mathrm{for}\mathrm{the}\cos \mathrm{t}\mathrm{criteria}{c}_j;j=1,2,\dots ,n\hfill \end{array}\right.$$

Similarly, the negative ideal solution (NIS) of alternatives is defined as follows:

$$R^{-}=\left\{d_1^{-},d_2^{-},\dots ,d_n^{-}\right\},$$

(18)

where

$$d_j^{-}=\left\{\begin{array}{cc}⟨\underset{1\le i\le m}{min}\{d_{ij}\}⟩\hfill & \mathrm{for}\mathrm{the}\mathrm{benefit}\mathrm{criteria}{c}_j;j=1,2,\dots ,n\hfill \\ ⟨\underset{1\le i\le m}{max}\{d_{ij}\}⟩\hfill & \mathrm{for}\mathrm{the}\cos \mathrm{t}\mathrm{criteria}{c}_j;j=1,2,\dots ,n\hfill \end{array}\right.$$

Step $4^{\prime }:$ Based on Definition 4, measure the separation of each alternative $x_i$ from the PIS as follows:

$$s_i^{+}=d_{dnh}\left(R^{+},R_i\right),i=1,2,\dots ,m,$$

(19)

where $R_i=\left\{d_{i1},d_{i2},\dots ,d_{in}\right\}$.

Similarly, measure the separation of each alternative $x_i$ from the NIS as follows:

$$s_i^{-}=d_{dnh}\left(R^{-},R_i\right),i=1,2,\dots ,m,$$

(20)

where $R_i=\left\{d_{i1},d_{i2},\dots ,d_{in}\right\}$.

Step $5^{\prime }$: Obtain the closeness coefficient of each alternative $x_i$ with respect to the PIS and NIS as:

$$cl(x_i)=\frac{s_i^{-}}{s_i^{+}+s_i^{-}};i=1,2,\dots ,m.$$

(21)

Step $6^{\prime }$: Classify all alternatives on the basis of their closeness coefficient. The greater the closeness coefficient $cl(x_i)$, the more acceptable the alternative $x_i$.

7.2. Handling the Case Study through the Dual Hesitant Fuzzy TOPSIS Approach

In order to conduct a comparative analysis, the illustrative example is again solved according to the above dual hesitant fuzzy TOPSIS approach as follows:

Step $1^{\prime }$: We compute the overall evaluation dual hesitant fuzzy information matrix $R={[d_{ij}]}_{4\times 3}$ given in the Table 7.

Table 7. The overall evaluation matrix R.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$⟨\{0.12,0.35,0.22\},\{0.65,0.14,0.16\}⟩$	$⟨\{0.65,0.44,0.36\},\{0.3,0.15\}⟩$	$⟨\{0.44,0.36,0.54\},\{0.22,0.36,0.26\}⟩$
$x_2$	$⟨\{0.32,0.36,0.3\},\{0.15,0.17,0.22\}⟩$	$⟨\{0.22,0.44,0.5\},\{0.5,0.22\}⟩$	$⟨\{0.44,0.22,0.32\},\{0.17,0.56\}⟩$
$x_3$	$⟨\{0.81,0.26,0.54\},\{0.17,0.19\}⟩$	$⟨\{0.36,0.3,0.46\},\{0.46,0.54,0.3\}⟩$	$⟨\{0.35,0.3,0.46\},\{0.32,0.36,0.3\}⟩$
$x_4$	$⟨\{0.44,0.32,0.26\},\{0.16,0.26,0.44\}⟩$	$⟨\{0.26,0.16,0.46\},\{0.32,0.54,0.22\}⟩$	$⟨\{0.26,0.65,0.14\},\{0.3,0.34,0.16\}⟩$

• where $d_{11}=⟨\{0.12,0.35,0.22\},\{0.65,0.14,0.16\}⟩,$
• d₁₂ = ⟨{0.65,0.44,0.36,},{0.3,0.15}⟩,
• d₁₃ = ⟨{0.44,0.36,0.54},{0.22,0.36,0.26}⟩,
• d₂₁ = ⟨{0.32,0.36,0.3},{0.15,0.17,0.22}⟩,
• d₂₂ = ⟨{0.22,0.44,0.5},{0.5,0.22}⟩,
• d₂₃ = ⟨{0.44,0.22,0.32,},{0.17.56}⟩,
• d₃₁ = ⟨{0.81,0.26,0.54},{0.17,0.19}⟩,
• d₃₂ = ⟨{0.36,0.3,0.46},{0.46,0.54,0.3}⟩,
• d₃₃ = ⟨{0.35,0.3,0.46},{0.32,0.36,0.3}⟩,
• d₄₁ = ⟨{0.44,0.32,0.26},{0.16,0.26,0.44}⟩,
• d₄₂ = ⟨{0.26,0.16,0.46},{0.32,0.54,0.22}⟩,
• d₄₃ = ⟨{0.26,0.65,0.14},{0.3,0.34,0.16}⟩.

Step $2^{\prime }$: Based on the Definition $3.6$ of Reference [13], the score and deviation values of DHFEs in overall evaluation matrix R are computed and displayed in Table 8.

Table 8. Score and deviation values of DHFEs in matrix R.

		${\mathit{c}}_1$			${\mathit{c}}_2$			${\mathit{c}}_3$
	Score	Deviation	Rank	Score	Deviation	Rank	Score	Deviation	Rank
$x_1$	$-0.08667$	$0.54667$	4	$0.25833$	$0.70833$	1	$0.16667$	$0.72667$	1
$x_2$	$0.14667$	$0.50667$	2	$0.02667$	$0.74667$	2	$-0.03833$	$0.69167$	4
$x_3$	$0.35667$	$0.71667$	1	$-0.06$	$0.80667$	3	$0.04333$	$0.69667$	3
$x_4$	$0.05333$	$0.62667$	3	$-0.06667$	$0.65333$	4	$0.08333$	$0.61667$	2

Step $3^{\prime }$: Since all criteria are benefit criteria, the positive ideal solution is therefore

$$R^{+}=\left\{d_{31},d_{12},d_{13}\right\}$$

and the negative ideal solution is

$$R^{-}=\left\{d_{11},d_{42},d_{23}\right\}.$$

Step $4^{\prime }$: The separation of each alternative from the PIS are $s_1^{+}=0.01717,$ $s_2^{+}=0.02287,$ $s_3^{+}=0.00958,$ $s_4^{+}=0.01611,$ while the separation of each alternative from the NIS are $s_1^{-}=0.01389,$ $s_2^{-}=0.01437,$ $s_3^{-}=0.0225,$ $s_4^{-}=0.01305.$

Step $5^{\prime }$: Compute the closeness coefficient $cl(x_i);i=1,2,3,4:$$cl(x_1)=0.44719,$$cl(x_2)=0.38587,$$cl(x_3)=0.70137,$$cl(x_4)=0.44753.$

Step $6^{\prime }$: The ranking result of all professors on their performance is

$$x_3>x_4>x_1>x_2.$$

Therefore, professor $x_3$ is the top of the range.

7.3. Handling the Case Study through the Proportional Hesitant Fuzzy TOPSIS Approach

Assume that DMs provide their decision information in the form of proportional hesitant fuzzy sets rather than proportional dual hesitant fuzzy elements. Then, according to proportional hesitant fuzzy TOPSIS method [22], we solve the fuzzy MAGDM as follows:

First, we build the following evaluation matrices by neglecting the non-membership parts from Table 1, Table 2, Table 3 and Table 4.

Step $1^{″}$: Based on the evaluation matrices $R_k(k=7,8,9,10)$ given in Table 9, Table 10, Table 11 and Table 12, we compute the overall evaluation matrix $R={[p{h}_{ij}]}_{4\times 3}$ displayed in the Table 13.

Table 9. $R_1$: opinion of expert $m_1$.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$0.22$	$0.36$	$0.54$
$x_2$	$0.32$	$0.44$	$0.22$
$x_3$	$0.54$	$0.46$	$0.3$
$x_4$	$0.26$	$0.16$	$0.14$

Table 10. $R_2$: opinion of expert $m_2$.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$0.12$	$0.65$	$0.44$
$x_2$	$0.32$	$0.22$	$0.44$
$x_3$	$0.81$	$0.36$	$0.35$
$x_4$	$0.44$	$0.26$	$0.26$

Table 11. $R_3$: opinion of expert $m_3$.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$0.35$	$0.65$	$0.36$
$x_2$	$0.36$	$0.44$	$0.22$
$x_3$	$0.81$	$0.36$	$0.3$
$x_4$	$0.32$	$0.16$	$0.26$

Table 12. $R_4$: opinion of expert $m_4$.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$0.12$	$0.44$	$0.44$
$x_2$	$0.3$	$0.5$	$0.32$
$x_3$	$0.26$	$0.3$	$0.46$
$x_4$	$0.32$	$0.46$	$0.65$

Table 13. The overall evaluation matrix R.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$
$x_1$	$\{(0.12,0.5),(0.35,0.25),(0.22,0.25)\}$	$\{(0.65,0.5),(0.44,0.25),(0.36,0.25)\}$	$\{(0.44,0.5),(0.36,0.25),(0.54,0.25)\}$
$x_2$	$\{(0.32,0.5),(0.36,0.25),(0.3,0.25)\}$	$\{(0.22,0.25),(0.44,0.5),(0.5,0.25)\}$	$\{(0.44,0.25),(0.22,0.5),(0.32,0.25)\}$
$x_3$	$\{(0.81,0.5),(0.26,0.25),(0.54,0.25)\}$	$\{(0.36,0.5),(0.3,0.25),(0.46,0.25)\}$	$\{(0.35,0.25),(0.3,0.5),(0.46,0.25)\}$
$x_4$	$\{(0.44,0.25),(0.32,0.5),(0.26,0.25)\}$	$\{(0.26,0.25),(0.16,0.5),(0.46,0.25)\}$	$\{(0.26,0.5),(0.65,0.25),(0.14,0.25)\}$

• where $p{h}_{11}=\{(0.12,0.5),(0.35,0.25),(0.22,0.25)\},$
• ph₁₂ = ⟨{(0.65,0.5),(0.44,0.25),(0.36,0.25)}⟩,
• ph₁₃ = ⟨{(0.44,0.5),(0.36,0.25),(0.54,0.25)}⟩,
• ph₂₁ = ⟨{(0.32,0.5),(0.36,0.25),(0.3,0.25)}⟩,
• ph₂₂ = ⟨{(0.22,0.25),(0.44,0.5),(0.5,0.25)}⟩,
• ph₂₃ = ⟨{(0.44,0.25),(0.22,0.5),(0.32,0.25)}⟩,
• ph₃₁ = ⟨{(0.81,0.5),(0.26,0.25),(0.54,0.25)}⟩,
• ph₃₂ = ⟨{(0.36,0.5),(0.3,0.25),(0.46,0.25)}⟩,
• ph₃₃ = ⟨{(0.35,0.25),(0.3,0.5),(0.46,0.25)}⟩,
• ph₄₁ = ⟨{(0.44,0.25),(0.32,0.5),(0.26,0.25)}⟩,
• ph₄₂ = ⟨{(0.26,0.25),(0.16,0.5),(0.46,0.25)}⟩,
• ph₄₃ = ⟨{(0.26,0.5),(0.65,0.25),(0.14,0.25)}⟩.

Step $2^{″}$: The score and deviation values of PHFEs in overall evaluation matrix R are computed according to Equations (5) and (6) and displayed in Table 14.

Table 14. Score and deviation values of PHFEs in matrix R.

		${\mathit{c}}_1$			${\mathit{c}}_2$			${\mathit{c}}_3$
	Score	Deviation	Rank	Score	Deviation	Rank	Score	Deviation	Rank
$x_1$	$0.2025$	$0.00892$	4	$0.525$	$0.01642$	1	$0.445$	$0.0047$	1
$x_2$	$0.325$	$0.00047$	2	$0.4$	$0.0114$	2	$0.3$	$0.0082$	4
$x_3$	$0.605$	$0.05183$	1	$0.37$	$0.0033$	3	$0.3525$	$0.00426$	2
$x_4$	$0.335$	$0.00427$	3	$0.26$	$0.015$	4	$0.3275$	$0.03707$	3

Step $3^{″}$: The PIS of alternatives is

$$R^{+}=\left\{p{h}_{31},p{h}_{12},p{h}_{13}\right\}$$

and the NIS of alternatives is

$$R^{-}=\left\{p{h}_{11},p{h}_{42},p{h}_{23}\right\}.$$

Step $4^{″}$: The separation of each alternative from the PIS are $s_1^{+}=0.03153,$$s_2^{+}=0.03944,$$s_3^{+}=0.01194,$$s_4^{+}=0.05625,$ and the separation of each alternative from the NIS are $s_1^{-}=0.04555,$$s_2^{-}=0.03764,$$s_3^{-}=0.06514,$$s_4^{-}=0.02083.$

Step $5^{″}$: Compute the closeness coefficient $cl(x_i);i=1,2,3,4:cl(x_1)=0.59094,$$cl(x_2)=0.48832,$$cl(x_3)=0.84553,$$cl(x_4)=0.27024.$

Step $6^{″}$: The ranking result of all professors on their performance is

$$x_3>x_1>x_2>x_4.$$

From the above ranking, one can see that professor $x_3$ achievement is again the top of the range.

7.4. Discussion

The major advantages of the proposed set and methodology over the existing studies are summarized below.

• From Table 7, the disadvantages of DHFS are apparent. As in DHFS, the proportion figures of the assessments are ignored, which means that all possible assessments in DHFS have the same occurrence probability, which is unrealistic because those values that occur multiple times have lost their importance. On the other hand, the inspection of Table 5 shows that PDHFS contains not only the membership and non-membership parts, but also take into account their associated proportion figures, and, thus PDHFS constitutes an extension of DHFS.
• Table 13 indicates that PHFS only assigns to each element a membership grades along with proportion, while PDHFS assigns to each element the membership grades as well as the non-membership grades along with their associated proportions. Thus, PDHFS provides a more flexible expression way for DMs to express their opinions from different prospectives in a more conformed to the reality approach.
• The score function and the deviation degree results of PDHFS shown in Table 6 are quite similar to those displayed in Table 8 for DHFS. However, from Table 14, we observe that these values are different from those derived for PHFS. The main reason of this difference is that the non-membership part of DM opinions is ignored in PHFS and thus partial information is lost. This comparison guarantees that the values of the score function and the deviation degree of PDHFS are reasonable and interpretable.
• From Table 15, we can see that the ranking order of traditional approach is different from the proposed method and the method in Ref. [22]. This difference is due to the effect of proportions. Although neglecting the proportional figures reduce the complexity of calculations, but its neglect leads to unsatisfactory ranking results.
  Table 15. Results comparison.

  Traditional TOPSIS	${\mathit{x}}_3>{\mathit{x}}_4>{\mathit{x}}_1>{\mathit{x}}_2$
  TOPSIS [22]	$x_3>x_1>x_2>x_4$
  Proposed TOPSIS	$x_3>x_1>x_2>x_4$

Various remarkable similarities and dissimilarities of the novel approach with other approaches are summarized in Table 16.

Table 16. Similarities and dissimilarities of the novel approach with other approaches.

Observation	Novel Approach	Reference [22] Approach	Traditional Approach
Expression of the initial decision judgment.	IFNs	FNs	IFNs
Expression of the collective decision judgment.	PDHFEs	PHFEs	IFNs
Additional information demanded for converting the MAGDM into MADM.	No	No	No
Can naturally reveal the preference judgment of the DMs group.	Yes	partially	No
Adapted distance measure for the separation of each alternative from the PIS/NIS.	Proportional dual hesitant normalized Hamming distance	Proportional hesitant normalized Hamming distance	Dual hesitant normalized Hamming distance

8. Conclusions

To model uncertainty problems, the past studies cannot reasonably handle the case where the repetition occurs in DM membership and non-membership opinions because the proportions of the membership and non-membership degrees are important as shown by the comparative analysis. To avoid the loss of these proportions, this study proposed a novel concept PDHFS, which is a comprehensive set encompassing several existing sets and whose membership and non-membership grades are represented by several possible values along with their associated proportions, taking much more information given by DMs into account and thus provide the best way to show the uncertainty in decision ranking problems. To ease the calculations of PDHFSs, an operational law is given that is much simpler to use, especially when there are a lot of possible membership and non-membership values with their associated proportions. Distance measure and comparison method were also studied in detail. TOPSIS for PDHFSs has also been developed for MAGDM problems. In addition, a practical life example has been provided to show the applicability of the proposed set. Finally, we have deduced that the concept of this paper has opened a new platform for study in decision theory. In future work, we will address the following problems.

(a)

Similarity and entropy measure for PDHFS.

(b)

Aggregation operators based on PDHFS.