CMES CMES CMES Computer Modeling in Engineering & Sciences 1526-1506 1526-1492 Tech Science Press USA 14393 10.32604/cmes.2021.014393 Article Power Aggregation Operators and Similarity Measures Based on Improved Intuitionistic Hesitant Fuzzy Sets and their Applications to Multiple Attribute Decision Making Power Aggregation Operators and Similarity Measures Based on Improved Intuitionistic Hesitant Fuzzy Sets and their Applications to Multiple Attribute Decision Making Power Aggregation Operators and Similarity Measures Based on Improved Intuitionistic Hesitant Fuzzy Sets and their Applications to Multiple Attribute Decision Making Mahmood Tahir 1 Ali Wajid 1 Ali Zeeshan 1 Chinram Ronnason 2 ronnason.c@psu.ac.th Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan Algebra and Applications Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla, Thailand *Corresponding Author: Ronnason Chinram. Email: ronnason.c@psu.ac.th 30 12 2020 126 3 1165 1187 23 09 2020 11 11 2020 © 2021 Mahmood et al. 2021 Mahmood et al. This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Intuitionistic hesitant fuzzy set (IHFS) is a mixture of two separated notions called intuitionistic fuzzy set (IFS) and hesitant fuzzy set (HFS), as an important technique to cope with uncertain and awkward information in realistic decision issues. IHFS contains the grades of truth and falsity in the form of the subset of the unit interval. The notion of IHFS was defined by many scholars with different conditions, which contain several weaknesses. Here, keeping in view the problems of already defined IHFSs, we will define IHFS in another way so that it becomes compatible with other existing notions. To examine the interrelationship between any numbers of IHFSs, we combined the notions of power averaging (PA) operators and power geometric (PG) operators with IHFSs to present the idea of intuitionistic hesitant fuzzy PA (IHFPA) operators, intuitionistic hesitant fuzzy PG (IHFPG) operators, intuitionistic hesitant fuzzy power weighted average (IHFPWA) operators, intuitionistic hesitant fuzzy power ordered weighted average (IHFPOWA) operators, intuitionistic hesitant fuzzy power ordered weighted geometric (IHFPOWG) operators, intuitionistic hesitant fuzzy power hybrid average (IHFPHA) operators, intuitionistic hesitant fuzzy power hybrid geometric (IHFPHG) operators and examined as well their fundamental properties. Some special cases of the explored work are also discovered. Additionally, the similarity measures based on IHFSs are presented and their advantages are discussed along examples. Furthermore, we initiated a new approach to multiple attribute decision making (MADM) problem applying suggested operators and a mathematical model is solved to develop an approach and to establish its common sense and adequacy. Advantages, comparative analysis, and graphical representation of the presented work are elaborated to show the reliability and effectiveness of the presented works.

Intuitionistic fuzzy sets intuitionistic hesitant fuzzy sets power aggregation operators similarity measures multiple attribute decision making
Introduction

In modern decision science, multi-attribute decision making (MADM) is a vital investigation area on how to choose the correct option corresponding to many prominent attributes . Usually, the decision-makers (DMs) utilize crisp figures to express the favorites regarding the alternative in conventional multi-attribute decision making difficulties. But, because of shortage of data, lack of time, deficiency of information and quality values, particularly, for subjective attribute values, usually may not be shown by real numbers, and few of them are simpler to be stated by fuzzy data. Since Zadeh  introduced the notion of fuzzy set, several expansions of fuzzy sets (FS) were presented by scholars . A FS contains an ordered pair of an element and a membership (MS) function, which gives grade of MS to every component of universal set X in the closed interval from 0 to 1. The model of fuzzy set is applied in many areas, mainly wherever traditional numerical methods restrict effectiveness, involving organic and social sciences, linguistics, psychology and mostly soft sciences. In these areas, variables are hard to evaluate and conditions among variables are so ill-defined. Further, Atanassov [8,9] gave the idea of intuitionistic fuzzy set IFS in 1986. IFS is an expansion of FS to cope with doubtful and complicated data. In IFS every object is indicated by an ordered pair set where every ordered pair set described a grade of MS as well as a grade of NMS.

The total of the grade of MS and the grade of NMS of each ordered pair set is smaller than or equivalent to 1 and greater than or equivalent to 0. The IFS has been receiving more consideration since its arrival . Intuitionistic fuzzy set is extra influential in managing with vagueness than fuzzy set which only provides a grade of MS to every component. Undoubtedly, IF data aggregation performs a crucial part in intuitionistic fuzzy set, that is an attractive study direction. Zhao et al.  established few elementary arithmetic aggregation operators, whereas IF weighted averaging operator, IF ordered weighted averaging operator and IF hybrid averaging operator for aggregating IFSs. Xu et al.  established few basic geometric aggregation operators whereas IF weighted geometric operator, IF ordered weighted geometric operator, and IF hybrid geometric operator and enforced them to MADM established on IFS. Furthermore, Torra et al. [23,24] presented the hesitant fuzzy set HFS. An HFS is a direct simplification of FS. The theory of hesitant fuzzy set is extensively utilized in many problems. Many researchers gave a serious analysis on HF information aggregation methods and their implications in decision making .

An HFS allows the MS taking a set of conceivable values for example, in order to obtain a sensible decision outcome, a decision association, containing many DMs, which is approved to assess the grade that an alternative should fulfill a criterion. Consider there are three situations, few DMs offer 0.2, few offer 0.4, and the rest offer 0.9, and these units may not convince one another, thus the grade that the alternative should fulfill the criterion can be signified by an HF {0.2,0.4,0.9}. It is observed that the HF {0.2,0.4,0.9} may define the above condition more quantitatively than the interval-valued FS [0.2,0.4], due to grades that alternatives fulfil the condition out of the convex of 0.2 and 0.9 or the interval between 0.2 and 0.9. Thereafter, several multi attribute decision making techniques  and procedures containing relationship, distance, and similarity have offered for hesitant fuzzy set by various investigators. Liao et al. [39,40] introduced the subtraction and division operations, hybrid arithmetical averaging for hesitant fuzzy sets, and hybrid arithmetical geometric for HFSs. Zhang  introduced power aggregation operators for HFS. In everyday life, DMs would think ranking among unlike conditions. To manage this type of position, Yager  established PA operator and implements it to multi-attribute decision making difficulties. Liu et al.  introduced POWA operator to manage the fuzzy data.

Enlarge several PA operators, as IHFPA operator, IHFPWA operator, IHFPOWA operator, IHFPHA operator, IHFPG operator, IHFPWG operator, IHFPOWG operator, IHFPHG operator and check their characteristics.

Explore the similarity measures based on IHFSs and justified with the help of numerical example.

Describe a new DM method consists over the proposal operations.

Provide some numerical to demonstrate the reliability and supremacy of described techniques.

The making of article is followed as in portion 2, it gives few fundamental notions as well as in this section we reviewed the definition of IHFS which are established by Beg et al.  and Geetha et al. . In Section 3, we established few IHF power aggregation operators and calculated their suitable characteristics. In Section 4, we explored the similarity measures based on IHFSs. In Section 5, we utilized these operators to establish few forms for multi attribute decision making challenges founded by IHFPWA operator and IHFPWG operator with intuitionistic hesitant fuzzy data. Additionally, we mentioned a practical problem for examining efficiency of the suggested operators. In Section 6, we summarized this article and wrote few comments.

Another View of Intuitionistic Hesitant Fuzzy Sets

In this study, we review the idea of IHFS which was established by Beg et al.  and established by Geetha et al. . Then we redefine IHFS to make it compatible with other existing notions .

Definition 1:  An IHFS on X are functions μ and v that when applied to X return the subsets of [0, 1], which can be represented as the following:

P={(x,μ(x),v(x))xX}

where μ(x) and v(x) are sets of some values in [0, 1], denoting the possible membership degrees and non-membership degrees of the element xX to the set P with the conditions: max(μ(x))+min(v(x))1 and min(μ(x))+max(v(x))1. For convenience, (μ(x),v(x)) is an intuitionistic hesitant fuzzy element (IHFE).

Definition 2:  An intuitionistic hesitant fuzzy set P on X is represented by using the two functions μ and v. Mathematically, it is represented by following expression:

P={(x,μ(x),v(x))xX}

where μ(x) and v(x) are sets of some values in [0, 1], denoting the possible membership degrees and non-membership degrees of the element xX to the set P with the condition that 0max(μ(x))+max(v(x))1. For convenience, (μ(x),v(x)) is an intuitionistic hesitant fuzzy element (IHFE).

Definition 3: An intuitionistic hesitant fuzzy set P on X is represented by using the two functions μ and v. Mathematically, it is represented by following expression:

P={(x,μ(x),v(x))xX}

where μ(x) and v(x) are sets of some values in [0, 1], denoting the possible membership degrees and non-membership degrees of the element xX to the set E with the condition that 0max(μ(x))+max(v(x))1. For convenience, (μ(x),v(x)) an intuitionistic hesitant fuzzy element (IHFE). In this manuscript we will follow throughout the IHFS:

P={(x,μ(x),v(x))xX}

satisfying 0max(μ(x))+max(v(x))1.

Definition 4: For any IHFE P=(μP,vP), the score function and accuracy function are stated by:

S(P)=(S(μP)-S(vP))2

H(P)=(S(μP)+S(vP))2

where (μP)=sumofallelementsin(μP)orderof(μP), S(vP)=sumofallelementsin(vP)orderof(vP), S(P)[-1,1], H(P)[0,1].

Definition 5: For any two IHFEs P1=(μ1,ν1) and P2=(μ2,ν2), then

P1P2= (l1εμ1l2εμ2m1εν1m2εν2 )({l1+l2-l1l2},{m1m2})

P1P2= (l1εμ1l2εμ2m1εν1m2εν2 )({l1l2},{m1+m2-m1m2})

0 \label{eqn-9} \end{equation}$$]]> λP1= (l1εμ1m1εv1 )(1-(1-l1)λ,m1λ),λ>0 0 \label{eqn-10} \end{equation}$$]]> P1λ= (l1εμ1m1εv1 )(l1λ,1-(1-m1)λ),λ>0 P1c=(l1,m1)

Definition 6: Power aggregation (PA) operator is defined as:

PA(P1,P2,Pn)=(i=1n(1+T(Pi))Pi)(i=1n(1+T(Pi)))

where

T(Pi)=j=1jinSup(Pi,Pj)

and Sup(P1,P2) is the Sup for P1 from P2, which meets the given properties: Sup(P1,P2)[0,1]Sup(P1,P2)=Sup(P2,P1)Sup(P1,P2)Sup(X,Y),if|P1-P2|<|X-Y|

The support (Sup) amount is basically a similarity indicator.

Definition 7: Power geometric (PG) operator is defined as:

PG(P1,P2,,Pn)= i=1nPi1+T(Pi)i=1n(1+T(Pi))

Based upon intuitionistic hesitant fuzzy PA operators and PG, we will describe few IHFPG aggregation operators. Next, we will establish few intuitionistic hesitant fuzzy power arithmetic aggregation operators.

Intuitionistic Hesitant Fuzzy Power Aggregation Operators

The purpose of this section is to establish few novel aggregation operators for IHFSs which are IHFPA, IHFPG, IHFPWA, IHFPWG, IHFPOWA, IHFPOWG, IHFPHA, IHFPHG operators and verify their fundamental properties. The mentioned operators are not only developed in this section but also their characteristics have been studied and their fitness is established using induction phenomenon.

Definition 8: Suppose Pj=(μj,νj) is a gathering of IHFSs, then we describe the IHFPA operator as follow:

IHFPA(P1,P2,,Pn)= j=1n(1+T(Pj)Pj)j=1n(1+T(Pj))

where

T(Pj)=i=1ijnSup(Pj,Pi)

and Sup(Pj,Pi) is the support for Pj from Pi, with the conditions:

Sup(Pj,Pi)[0,1];

Sup(Pj,Pi)=Sup(Pi,Pj);

Sup(Pj,Pi)Sup(Ps,Pt) if d(Pj,Pi)<d(Ps,Pt), wherever d be a distance measure.

If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (15) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade is zero then the Eq. (15) will be converted for hesitant fuzzy sets.

Theorem 1: The aggregated objects by utilizing intuitionistic hesitant fuzzy power average (IHFPA) operator is as well an IHFS, wherever

IHFPA(P1,P2,,Pn)=j=1n((1+T(Pj))Pj)j=1n(1+T(Pj))= ljεμjmjεvj(1- j=1n(1-(lj))(1+T(Pj))j=1n(1+T(Pj)),j=1n(mj)(1+T(Pj))j=1n(1+T(Pj)))

where

T(Pj)= μjεPjvjεPj(i=1ijnSup(Pj,Pi))

If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (17) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade is zero then the Eq. (17) will be converted for hesitant fuzzy sets.

Definition 9: Let Pj=(μj,νj) be a group of IHFS and ω=(ω1,ω2,,ωn)T is weight vector of Pj, 0$]]>ωj>0 and j=1nωj=1, (j=1,2,3,n) The IHFPWA operator is a function IHFPWA: PnP where IHFPWAω(P1,P2,,Pn)=j=1n(ωj(1+T(Pj)Pj))j=1nωj(1+T(Pj))= ljεμjmjεvj(1- j=1n(1-(lj))ωj(1+T(Pj))j=1nωj(1+T(Pj)),j=1n(mj)ωj(1+T(Pj))j=1nωj(1+T(Pj))) where T(Pj)= μjεPjvjεPj(i=1ijnωjSup(Pj,Pi)) If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (19) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade to be zero then the Eq. (19) will be converted for hesitant fuzzy sets. Property 1: (Idempotency) When Pj are equivalent, Pj = P for every j(j=1,2,3,n), then IHFPWAω(P1,P2,,Pn)=P Property 2: (Boundedness) Let Pj be a family of IHFSs, and allows P-=jminPj,P+=jmaxPj(j=1,2,,n) then P-IHFPWAω(P1,P2,,Pn)P+ Property 3: (Monotonicity) Let Pj and Pj be two sets of intuitionistic hesitant fuzzy sets (IHFSs), if PjPj for all j, then IHFPWAω(P1,P2,,Pn)IHFPWAω(P1,P2,,Pn) Further, we give an IHFPOWA operator as follows: Definition 10: Suppose Pj=(μj,νj) is family of IHFSs, the IHFPOWA operator of dimension n a function IHFPOWA: PnP, associated with weight vector ω=(ω1,ω2,,ωj)T such that 0$]]>ωj>0 and j=1nωj=1. Furthermore

IHFPOWAω(P1,P2,,Pn)=j=1n(ωj(1+T(Pσ(j))Pσ(j)))j=1nωj(1+T(Pσ(j)))= lσ(j)εμjmσ(j)εvj(1- j=1n(1-(lσ(j)))(ωj(1+T(Pσ(j))))j=1nωj(1+T(Pσ(j))), j=1n(mσ(j))(ωj(1+T(Pσ(j))))j=1nωj(1+T(Pσ(j))))

where σ(1),σ(2),,σ(n) indicates permutation of (1,2,,n), where Pσ(j-1)Pσ(j), ωj (j=1,2,,n) is family of weights in such a way that

ωj=g(RjTV)-g(Rj-1TV),Rj=i=1jVσ(i),TV=i=1nVσ(i),Vσ(i.)=1.+T(Pσ(i))

where T(Pσ(i)) implies the Sup of jth main IHFS T(Pσ(i)) by all the other (IHFSs), that is,

T(Pσ(i))= μσ(j)εPjvσ(j)εPj(i=1ijnSup(Pσ(j),Pσ(i)))

where μσ(j)εPjvσ(j)εPj(i=1ijnSup(Pσ(j),Pσ(i))) shows the Sup of jth is the biggest intuitionistic hesitant fuzzy set (IHFS) Pσ(j), for the ith largest intuitionistic hesitant fuzzy set (IHFS) Pσ(i). If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (24) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade is zero then the Eq. (24) will be converted for hesitant fuzzy sets. Some characteristics of IHFPOWA operator are as follow:

Property 4: (Idempotency) when each Pj is equivalent, which is, Pj = P for every j(j=1,2,3,n), so

IHFPOWAω(P1,P2,,Pn)=P

Property 5: (Boundedness) Suppose Pj is family of IHFSs, suppose P-=jminPj,P+=jmaxPj then

P-IHFPOWAω(P1,P2,,Pn)P+

Property 6: (Monotonicity) Let Pj and Pj be IHFSs, if PjPj for all j. Then

IHFPOWAω(P1,P2,,Pn)IHFPOWAω(P1,P2,,Pn)

Property 7: (Commutativity) Let Pj and Pj be IHFSs, if PjPj for all j. Then

IHFPOWAω(P1,P2,,Pn)=IHFPOWAω(P1,P2,,Pn)

where Pj be a permutation of Pj.

Definition 11: Let Pj=(μj,νj) be family of IHFSs, the intuitionistic hesitant fuzzy power hybrid averaging (IHFPHA) operator of elements n a function IHFPHA: PnP, such that

IHFPHAω(P1,P2,,Pn)=j=1n(ωj(1+T(Ṗσ(j))Ṗσ(j)))j=1nωj(1+T(Ṗσ(j)))= l̇σ(j)εμ̇jṁσ(j)εv̇j(1- j=1n(1-(l̇σ(j)))(ωj(1+T(Ṗσ(j))))j=1nωj(1+T(Ṗσ(j))), j=1n(ṁσ(j))(ωj(1+T(Ṗσ(j))))j=1nωj(1+T(Ṗσ(j))))

where ω=(ω1,ω2,,ωj)T is a mapped weight vector, such that ωj and j=1nωj=1 and Ṗσ(j) is the jth biggest element in intuitionistic hesitant fuzzy arguments Ṗj(Ṗ=(nωj)Pj,j=1,2,,n), ω=(ω1,ω2,,ωn) be the weighting vector of IHF arguments Pj(j=1,2,,n), ωj and j=1nωj=1. And ωj be a family such that

ωj=g(RjTV)-g(Rj-1TV),Rj=i=1jVσ(i),TV=i=1nVσ(i),Vσ(i)=1+T(Ṗσ(i))

where T(Ṗσ(i)) is the Sup of jth biggest IHFSs T(Ṗσ(i)) by all the other (IHFSs), that is,

T(Ṗσ(i))= μ̇σ(j)εṖjv̇σ(j)εṖj(i=1ijnSup(Ṗσ(j),Ṗσ(i)))

where μ̇σ(j)εṖjv̇σ(j)εṖj(i=1ijnSup(Ṗσ(j),Ṗσ(i))) shows the Sup of jth biggest IHFS Ṗσ(j), for the ith biggest IHFS Ṗσ(i). Particularly, IHFPHA is decreased to IHFPWA operator if ω=(1n,1n,,1n)T and IHFPHA is decreased to IHFPOWA operator if ω=(1n,1n,,1n). If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (31) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade is zero then the Eq. (31) will be converted for hesitant fuzzy sets.

Definition 12: Suppose Pj=(μj,νj) is family of IHFSs, intuitionistic hesitant fuzzy power geometric (IHFPG) operator defined as a function IHFPG: PnP where

IHFPG(P1,P2,,Pn)=j=1n(P)1+T(Pj)i=1n(1+T(Pj))

where

T(Pj)=i=1ijnSup(Pj,Pi)

where Sup(Pj,Pi) is the support for Pj from Pi, with the conditions

Sup(Pj,Pi)[0,1];

Sup(Pj,Pi)=Sup(Pi,Pj);

Sup(Pj,Pi)Sup(Ps,Pt) if d(Pj, Pi) < d(Ps, Pt), such that d is a distance measure.

If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (34) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade is zero then the Eq. (34) will be converted for hesitant fuzzy sets.

Theorem 2: The aggregated elements by utilizing IHFPG operator define an IHFS, wherever

IHFPG(P1,P2,,Pn)=j=1n(P)1+T(Pj)i=1n(1+T(Pj))= ljεμjmjεvj( j=1n(lj)(1+T(Pj))j=1n(1+T(Pj)),1-j=1n(1-(mj)(1+T(Pj))j=1n(1+T(Pj))))

where

T(Pj)= μjεPjvjεPj(i=1ijnSup(Pj,Pi))

Definition 13: Let Pj=(μj,νj) (j=1,2,,n) be family of IHFSs, ω=(ω1,ω2,,ωj)T is weight vector of Pj, 0, \sum\limits_{j=1}^{n}\omega_{j}=1$]]>ωj>0,j=1nωj=1. The IHFPWG operator defined as mapping IHFPWG: PnP where IHFPWG(P1,P2,,Pn)=j=1n(P)(ωj(1+T(Pj)))j=1nωj(1+T(Pj))= ljεμjmjεvj( j=1n(lj)ωj(1+T(Pj))j=1nωj(1+T(Pj)),1-j=1n(1-(mj))ωj(1+T(Pj))j=1nωj(1+T(Pj))) where T(Pj)= μjεPjvjεPj(i=1ijnωjSup(Pj,Pi)) If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (36) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade is zero then the Eq. (36) will be converted for hesitant fuzzy sets. IHFPWG operator has following characteristics. Property 8: (Idempotency) when every Pj (j=1,2,,n) is equivalent, such that Pj = P for every j, then IHFPWGω(P1,P2,,Pn)=P Property 9: (Boundedness) Suppose Pj is family of IHFSs, also suppose P-= minjPj,P+= maxjPj then P-IHFPWGω(P1,P2,,Pn)P+ Property 10: (Monotonicity) Let Pj (j=1,2,,n) and Pj be IHFSs, if PjPj for all j, then IHFPWGω(P1,P2,,Pn)IHFPWGω(P1,P2,,Pn) Further, we give an IHFPOWG operator below: Definition 14: Suppose Pj=(μj,νj) is family of IHFSs, IHFPOWG operator of dimension n is mapping IHFPOWG: PnP, with an associated weight vector ω=(ω1,ω2,,ωj)T such that$]]>ωj>0 and j=1nωj=1. Furthermore

IHFPOWGω(P1,P2,,Pn)=j=1n(Pσ(j))ωj(1+T(Pσ(j)))j=1nωj(1+T(Pσ(j)))= lσ(j)εμjmσ(j)εvj( j=1n(lσ(j))(ωj(1+T(Pσ(j))))j=1nωj(1+T(Pσ(j))),1- j=1n(1-(mσ(j)))ωj(1+T(Pσ(j)))j=1nωj(1+T(Pσ(j))))

Such that (σ(1),σ(2),,σ(n)) be permutation of 1,2,,n, where Pσ(j-1)Pσ(j) for every j=1,2,,n, ωj(j=1,2,,n) is family of weights where

ωj=g(RjTV)-g(Rj-1TV),Rj=i=1jVσ(i),TV=i=1nVσ(i),Vσ(i)=1+T(Pσ(i))

where T(Pσ(i)) indicates the sup of jth biggest intuitionistic hesitant fuzzy sets (IHFSs) T(Pσ(i)) by all the other (IHFSs), that is,

T(Pσ(i))= μσ(j)εPjvσ(j)εPj(i=1ijnSup(Pσ(j),Pσ(i)))

where i=1ijnSup(Pσ(j),Pσ(i)) shows the sup of jth biggest IHFS Pσ(j),for the ith biggest IHFS Pσ(i). If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (43) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade is zero then the Eq. (43) will be converted for hesitant fuzzy sets. IHFPOWG operator provides the following characteristics.

Property 11: (Idempotency) when every Pj (j=1,2,,n) is equivalent, such that, Pj = P for every j, so

IHFPOWGω(P1,P2,,Pn)=P

Property 12: (Boundedness) Suppose Pj is family of IHFSs, suppose P-= minjPj,P+= maxjPj then

P-IHFPOWGω(P1,P2,,Pn)P+

Property 13: (Monotonicity) Let Pj and Pj be of intuitionistic hesitant fuzzy sets (IHFSs), if PjPj for all j, then

IHFPOWGω(P1,P2,,Pn)IHFPOWGω(P1,P2,,Pn)

Property 14: (Commutativity) Let Pj and Pj be two intuitionistic hesitant fuzzy sets (IHFSs), if PjPj for all j, then

IHFPOWGω(P1,P2,,Pn)=IHFPOWGω(P1,P2,,Pn),(j=1,2,,n)

where Pj be permutation of Pj.

Definition 15: Let Pj=(μj,νj) be family of IHFSs, the IHFPHG operator of elements n is the function IHFPHG: PnP, where

IHFPHGω(P1,P2,,Pn)=j=1n(Ṗσ(j))j=1n(ωj(1+T(Ṗσ(j))Ṗσ(j)))j=1nωj(1+T(Ṗσ(j)))= l̇σ(j)εμ̇jṁσ(j)εv̇j( j=1n(l̇σ(j))ωj(1+T(Ṗσ(j)))j=1nωj(1+T(Ṗσ(j))),1- j=1n1-(ṁσ(j))ωj(1+T(Ṗσ(j)))j=1nωj(1+T(Ṗσ(j))))

where ω=(ω1,ω2,,ωj)T is a related weight vector, where ωj and j=1nωj=1. Ṗσ(j) is the jth biggest element of the intuitionistic hesitant fuzzy arguments Ṗj (Ṗ=(Pj)nωj, j=1,2,,n), ω=(ω1,ω2,,ωn) is weighting vector of IHF arguments Pj where ωj, j=1nωj=1 and n is a matching factor, and ωj(j=1,2,,n) is the collection of weights such that

ωj=g(RjTV)-g(Rj-1TV),Rj=i=1jVσ(i),TV=i=1nVσ(i),Vσ(i)=1+T(Ṗσ(i))

where T(Ṗσ(i)) indicates the Sup of jth biggest IHFSs T(Ṗσ(i)) by all the other (IHFSs), that is,

T(Ṗσ(i))= μ̇σ(j)εṖjv̇σ(j)εṖj(i=1ijnSup(Ṗσ(j),Ṗσ(i)))

where μ̇σ(j)εṖjv̇σ(j)εṖj(i=1ijnSup(Ṗσ(j),Ṗσ(i))) shows the Sup of jth biggest IHFS Ṗσ(j) and the ith biggest IHFS Ṗσ(i). Particularly, IHFPHA is decreased to the IHFPWA operator when ω=(1n,1n,,1n)T IHFPHA is decreased to the IHFPOWA operator when ω=(1n,1n,,1n). If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (50) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade to be zero then the Eq. (50) will be converted for hesitant fuzzy sets.

Similarity Measures Based on Intuitionistic Hesitant Fuzzy Sets

The VSM is one of the important tools for the similarity degree between objects. We straightforwardly utilized Jaccard, Dice and Cosine SM. Presently in this segment we characterize VSMs and weighted VSMs (WVSMs) for IHFSs.

Definition 16: Suppose that P=(μP,νP) and Q=(μQ,νQ) are two IHFSs on X, then the Jaccard similarity measure (JSM) between P and Q is denoted and defined as follows:

Jac(P,Q)=1nk=1n(1gp=1gμPp(xk)μQp(xk)+1hq=1hνPp(xk)νQp(xk)1gp=1gμPp2(xk)+1gp=1gμQp2(xk)+1hq=1hνPp2(xk)+1hq=1hνQp2(xk)-(1gp=1gμPp(xk)μQp(xk)+1hq=1hνPp(xk)νQp(xk)) )

JSMs fulfill the following axioms:

0Jac(P,Q)1;

Jac(P,Q)=Jac(Q,P);

Jac(P,Q)=1, if P = Q.

If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (53) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade to be zero then the Eq. (53) will be converted for hesitant fuzzy sets.

Definition 17: Suppose that P=(μP,νP) and Q=(μQ,νQ) are two IHFSs on X, then the weighted JSM (WJSM) between P and Q is denoted and defined as follows:

Jacw(P,Q)=k=1nwk(1gp=1gμPp(xk)μQp(xk)+1hq=1hνPp(xk)νQp(xk)1gp=1gμPp2(xk)+1gp=1gμQp2(xk)+1hq=1hνPp2(xk)+1hq=1hνQp2(xk)-(1gp=1gμPp(xk).μQp(xk)+1hq=1hνPp(xk).νQp(xk)) )

WJSMs fulfill the following axioms:

0Jacw(P,Q)1;

Jacw(P,Q)=Jacw(Q,P);

Jacw(P,Q)=1, if P = Q.

where w=(w1,w2,,wn)T speaks to the weight vector of every component xk(k=1,2,3,,n) contained in IHFS and the weight vector fulfills wk[0,1] for each k=1,2,3,,n, k=1nwk=1. When we assume the weight vector be w=(1n,1n,,1n)T, at that point the WJSM will change into JSM. Otherwise speaking when wk=1n, k=1,2,3,,n then Jacw(P,Q)=Jac(P,Q). If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (54) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade to be zero then the Eq. (54) will be converted for hesitant fuzzy sets.

Definition 18: Suppose that P=(μP,νP) and Q=(μQ,νQ) are two IHFSs on X, then the Dice similarity measure (DSM) between P and Q is denoted and defined as follows:

Dic(P,Q)=1nk=1n(2gp=1gμPp(xk)μQp(xk)+2hq=1hνPp(xk)νQp(xk)1gp=1gμPp2(xk)+1gp=1gμQp2(xk)+1hq=1hνPp2(xk)+1hq=1hνQp2(xk))

DSMs fulfills the following axioms:

0Dic(P,Q)1;

Dic(P,Q)=Dic(Q,P);

Dic(P,Q)=1, if P = Q.

If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (55) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade to be zero then the Eq. (55) will be converted for hesitant fuzzy sets.

Definition 19: Suppose that SP=(μP,νP) and Q=(μQ,νQ) are two IHFSs on X, then the weighted DSM (WDSM) between P and Q is denoted and defined as follows

Dicw(P,Q)=k=1nwk(2gp=1gμPp(xk)μQp(xk)+2hq=1hνPp(xk)νQp(xk)1gp=1gμPp2(xk)+1gp=1gμQp2(xk)+1hq=1hνPp2(xk)+1hq=1hνQp2(xk))

WDSMs fulfills the following axioms

0Dicw(P,Q)1;

Dicw(P,Q)=Dicw(Q,P);

Dicw(P,Q)=1, if P = Q.

where w=(w1,w2,,wn)T speaks to the weight vector of every component xk(k=1,2,3,,n) contained in IHFS and the weight vector fulfills wk[0,1] for each k=1,2,3,,n, k=1nwk=1. When we assume the weight vector be w=(1n,1n,,1n)T, at that point the WDSM will change into DSM. Otherwise speaking when wk=1n, k=1,2,3,,n then Dicw(P,Q)=Dic(P,Q). If we will choose the grade of truth and falsity in the form of singleton sets then the Eq. (56) will be converted for intuitionistic fuzzy sets. Similarly, if we choose the values of falsity grade to be zero then the Eq. (56) will be converted for hesitant fuzzy sets.

Multiple Attribute Decision Making Technique Based on Intuitionistic Hesitant Fuzzy Sets

In the portion, we use IHF power aggregation operators to multiple attribute DM through intuitionistic hesitant fuzzy data. Following hypotheses or concepts are utilized to signify the multiple attribute DM difficulties for possible calculation of developing technology commercialization with intuitionistic hesitant fuzzy data. Consider A={A1,A2,,Am} is distinct set of alternatives, G={G1,G2,,Gn} is the set of attributes. Consider ω=(ωj) (j=1,2,n) is weight vector of attributes, such that ωj0, j=1nωj=1. Then we will be going to apply the IHFPWA or PFPWG operator to the multiple attribute DM difficulties for possible calculation of developing technology commercialization by IHF data.

Step 1. Compute the supports:

Sup(Pij,Pik)=1-d(Pij,Pik),(j,k=1,2,,n),

where justify Sup terms (1)(3) in portion 3. Here, with no loss of generalization, we compute d(Pij, Pik) with the normalized Hamming distance

d(Pij,Pik)=1mi=1m(12fs=1f(|lijσ(s)-likσ(s)|+|mijσ(s)-mikσ(s)|))

where lijμij, likμik, mijνij and mikνik.

Step 2. Using the weights ωj of attribute Gj to compute weighted sup T(Pij) of the IHFS Pij by other IHFS Pik (j, k=1,2,,n, kj)

T(Pij)= (μijεPijμikεPikvijεPijvikεPik )(k=1kjnωjSup(Pij,Pik))

where calculated weight ξij is connected with the IHPFS Pij, (j=1,2,,n, i=1,2,,m)

ξij=ωj(1+T(Pij))j=1nωj(1+T(Pij))

where ξij0, j=1nξij=1.

Step 3. Use decision data provided in Tab. 1, IHFPWA operator

Pi=IHFPWAω(Pi1,Pi2,,Pin)=j=1n(ωj(1+T(Pij)Pij))j=1nωj(1+T(Pij))= lijεμijmijεvij(1- j=1n1-(lij)ωj(1+T(Pij))j=1nωj(1+T(Pij)),j=1n(mij)ωj(1+T(Pij))j=1nωj(1+T(Pij)))

or

Pi=IHFPWG(Pi1,Pi2,,Pin)=j=1n(Pij)(ωj(1+T(Pij)))j=1nωj(1+T(Pij))= lijεμijmijεvij( j=1n(lij)ωj(1+T(Pij))j=1nωj(1+T(Pij)),1-j=1n1-(mij)ωj(1+T(Pij))j=1nωj(1+T(Pij)))

to receive the total preference objects Pi of the alternative Ai(i=1,2,,m).

Step 4. Compute scores S(Pi) of the whole IHFSs Pi to rank each the Ai then to select the top one(s). If two scores S(Pi) and S(Pj) have no difference then we want to compute the accuracy grades H(Pi), H(Pj) of the whole IHFSs Pi, Pj, respectively, classify the alternatives Ai, Aj consistent with accuracy grades H(Pi) and H(Pj).

Step 5. Ranking whole alternatives Ai and choose the greatest one(s) in accord by (Pi) (i=1,2,,m).

Step 6. The end.

Example 1: Therefore, in the portion we give a mathematical model to illustrate the possible estimation of developing technology commercialization by intuition hesitant fuzzy data illustrating the technique recommended in this article. There is the board with five possible developing technologies enterprises Ai(i=1,,5) to choose. Specialists choose four attributes to calculate the five possible developing technology enterprises: (i) G1 is the technical development (ii) G2 is the potential market and market risk; (iii) G3 is the industrialized structure, human resource management, and economic circumstances (iv) G4 is the job creation and the development of science and technology. The five possible developing technology enterprises Ai(i = 1, 2, 3, 4, 5) are to be estimated utilizing the IHF data by the decision maker in accordance with proposed attributes and weighting vector ω=(0.4,0.2,0.1,0.3)T shown in Tab. 1.

Original decision matrix
G1 G2 G3 G4
{{0.1,0.3},{0.1,0.4}} {{0.0,0.3},{0.3,0.4}} {{0.0,0.3},{0.1,0.1}} {{0.2,0.4},{0.1,0.2}}
{{0.1,0.0},{0.2,0.2}} {{0.0,0.1},{0.1,0.2}} {{0.1,0.1},{0.1,0.3}} {{0.1,0.2},{0.1,0.3}}
{{0.3,0.2},{0.2,0.1}} {{0.1,0.2},{0.0,0.1}} {{0.2,0.5},{0.2,0.1}} {{0.0,0.6},{0.2,0.1}}
{{0.3,0.1},{0.2,0.5}} {{0.3,0.5},{0.1,0.1}} {{0.1,0.0},{0.1,0.2}} {{0.3,0.2},{0.2,0.2}}
{{0.2,0.1},{0.5,0.1}} {{0.1,0.4},{0.1,0.2}} {{0.2,0.2},{0.5,0.2}} {{0.3,0.5},{0.2,0.2}}

Next, we use the method established to indicate potential evaluation of developing technology commercialization of four possible developing technology enterprises, see Tab. 2.

Aggregated values by using the formulas of IHFPWA and IHFPWG
IHFPWA IHFPWG
A1 (0.3315,0.1203) (0.3269,0.1372)
A2 (0.0879,0.1377) (0,0.1476)
A3 (0.3707,0) (0.2965,0.1)
A4 (0.2877,0.1696) (0.2781,0.1773)
A5 (0.3012,0.1452) (0.2150,0.1552)

Step 1. Compute the weight ξij (i=1,,5, j=1,,4) that is related by IHFN p̃ (i=1,5, j=1,,4), that included R=(r̃ij)5×4 ξ=[0.46360.16830.06870.29920.46210.16890.06860.30010.46280.16850.06880.29970.46160.16840.06880.30090.46160.16940.06890.3000 ]

Step 2. Corresponding to ξ and IHFN p̃ij (i=1,,5, j=1,,4), compute the whole IHFNs p̃ij (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4) by utilizing the IHFPWA (IHFPWG) operator to get the whole IHFNs p̃i (i = 1, 2, 3, 4, 5) of the developing technology enterprise Ai. The aggregating values are reflected in Tab. 1.

By using the formula of score value, we examine the score values of the aggregated values of Tab. 2, see Tab. 3.

Step 1. In accordance with the aggregating values presented in Tab. 2 and the score functions of the developing technology enterprises are presented in Tab. 3.

Score values of the aggregated values
IHFPWA IHFPWG
A1 0.2112 0.1897
A2 −0.0498 −0.1476
A3 0.3706 0.196
A4 0.118 0.1008
A5 0.156 0.1196

Further, we examine ranking results of the score values.

Step 2. Approve the score functions presented in the Tab. 3, and compare the formula of score functions, the ranking of the developing technology enterprises as presented in Tab. 4. Remember this the greater than sign “ > ” implies “preference.”

Ranking results
Methods Ranking
IHFPWA A3 > A1 > A5 > A4 > A2
IHFPWG A3 > A1 > A5 > A4 > A2

From above analysis, we get as best option alternative A3.

Example 2: The amount from developments by an organization is legitimately corresponding to the standard of building substances they use. Appropriate review of building substance before development is the confirmation of good building measures. The building substances to be utilized ought to be carefully checked before applying. The best possible check and equalization arrangement of investigation approves the manufacturers to utilize the correct substances for developments to improve the standard of their task. Let five known building substances Pr(r=1,2,3,4,5) be as given in the IHFSs structure as follows: P1={(x1,{{0.2,0.3,0.5},{0.3,0.2}}),(x2,{{0.3},{0.6,0.4}}),(x3,{{0.6},{0.2}}),(x4,{{0.15,0.6},{0.35}})(x5,{{0.4},{0.3}}) }P2={(x1,{{0.25},{0.6,0.3}}),(x2,{{0.35},{0.5}}),(x3,{{0.35,0.45,0.6},{0.3,0.1}}),(x4,{{0.6,0.5},{0.2,0.25}})(x5,{{0.5},{0.25,0.4}}) }P3={(x1,{{0.3,0.2},{0.4,0.5}}),(x2,{{0.45,0.5},{0.15}}),(x3,{{0.6},{0.35}}),(x4,{{0.7},{0.25,0.15}})(x5,{{0.1,0.25,0.4},{0.5,0.2}}) }P4={(x1,{{0.4,0.6},{0.15,0.3}}),(x2,{{0.3},{0.6}}),(x3,{{0.4,0.2,0.6},{0.2,0.15}}),(x4,{{0.5,0.3},{0.4}})(x5,{{0.6,0.55,0.3},{0.35,0.15,0.1}}) }P5={(x1,{{0.3,0.2},{0.5,0.4}}),(x2,{{0.45,0.7},{0.25}}),(x3,{{0.2},{0.6}}),(x4,{{0.7},{0.15,0.25}})(x5,{{0.4,0.25,0.1},{0.5,0.5,0.3}}) } and P={(x1,{{{0.7,0.3},{0.2,0.1}}}),(x2,{{0.7},{0.2}}),(x3,{{0.6,0.5,0.4},{0.3,0.1}}),(x4,{0.4,0.6},{0.15})(x5,{{0.5,0.4,0.2},{0.3,0.1,0.2}}) }

By using the Eq. (56), we get the following values, which are summarized based on (0.1,0.15,0.3,0.2,0.25): Dicw(P1,P)=0.5723,Dicw(P2,P)=0.7724,Dicw(P3,P)=0.4536,Dicw(P4,P)=0.674,Dicw(P5,P)=0.6457

Ranking values of the above measures are summarized as follows: P2P4P5P1P3

Comparative analysis of the explored and existing measures
Methods Results Ranking
Peng et al.  Dicw(P1,P)=0.7165, P2P4P5P1P3
Dicw(P2,P)=0.8946,
Dicw(P3,P)=0.5674,
Dicw(P4,P)=0.8423,
Dicw(P5,P)=0.7728
Beg et al.  Dicw(P1,P)=0.7257, P2P4P5P1P3
Dicw(P2,P)=0.8979,
Dicw(P3,P)=0.5721,
Dicw(P4,P)=0.8472,
Dicw(P5,P)=0.7727
Explored measures Dicw(P1,P)=0.5723, P2P4P5P1P3
Dicw(P2,P)=0.7724,
Dicw(P3,P)=0.4536,
Dicw(P4,P)=0.674,
Dicw(P5,P)=0.6457

The best option is P2. Additionally, if we choose the intuitionistic hesitant fuzzy types of information’s with existing conditions that are the sum of the maximum (also for minimum) of the truth grade and minimum (also for maximum) of the falsity grade cannot exceed from unit interval, and the sum of the maximum of the truth grade and falsity grade exceeds the unit interval, then it is very difficult to cope with such types of issues. But, when we choose the condition as in this explorative study then the sum of the maximum of the truth grade and falsity grade cannot exceed from the unit interval. The theories of intuitionistic fuzzy set and hesitant fuzzy set describe the foundation of the intuitionistic hesitant fuzzy set. When we choose the intuitionistic fuzzy types of information’s or hesitant fuzzy types of information then the explored approach easily copes with it. But, if we choose the intuitionistic hesitant fuzzy types of information’s, then the existing types of theories are cannot able to cope with it.

The comparative analysis of the explored measures with selected existing measures are summarized in Tab. 5.

From above analysis, the three different measures above share the same ranking values and the best option is P2. The graphical representation for the information of Tab. 5, we explained with the help of Fig. 1.

Graphical representation of the explored and existing measures

Fig. 1 represents the family of proposed and existing ideas and contains five types of values for each operator showing the family of alternatives. The alternative two provides the best values for all operators. For simplicity we have drawn the Fig. 1.

From the above analysis, the explored measures and operators based on IHFSs are more perfect and more proficient then existing methods and measures.

Conclusion

We explored the improved intuitionistic hesitant fuzzy set with a new condition that is the sum of the maximum of the truth grade and maximum of the falsity grade which cannot exceed from the unit interval. Additionally, we examine the multi-attribute decision making challenge built upon power aggregating operators with IHF data. So, inspired from the model of power aggregating operators, we established few power aggregation operators for aggregating IHF information: IHFPA operator, IHFPG operator, IHFPWA operator, IHFPWG operator, IHFPOWA operator, IHFPOWG operator, IHFPHA operator, and IHFPHG operator. Additionally, some similarity measures based on IHFSs are also explored and their special cases discussed. Outstanding feature of these recommended operators are examined. So, we used operators to establish few methods to resolve the IHF multi attribute DM difficulties. A helpful example is presented to confirm the established methodology and to determine its practicability and efficiency. The advantages, comparative analysis, and geometrical representation of the presented works are also discussed in detailed.

Notably, that the article results of the article can be expanded to the IvIHF situation and further fuzzy situations. In superior study, it is enough to get the implementation of these operators to resolve the actual DM drawbacks as fuzzy investigation, unsure programming and image recognition, etc. We must also deal with few new operators for the foundation of PHFNs for example, modify them to complex q-rung fuzzy aggregation operators , complex Pythagorean fuzzy set , spherical fuzzy operators .

Data Availability: The data used in this article are artificial and hypothetical, and anyone can use these data before prior permission by just citing this article.

Funding Statement: This paper is supported by “Algebra and Applications Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla University”.

Conflicts of Interest: The authors declare that they have no conflicts of interest.

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