Theory and Application of Hypersoft Set

EDITION

Theory and Application of Hypersoft Set

AUTHOR Prof. Dr. Florentin Smarandache Prof. Dr. Muhammad Saeed Muhammad Saqlain Dr. Mohamed Abdel-Baset

Theory and Application of Hypersoft Set

(Editors)

Prof. Dr. Muhammad Saeed Muhammad Saqlain Dr. Mohamed Abdel-Baset

Prof. Dr. Florentin Smarandache

ISBN 978-1-59973-699-0

Pons Publishing House Brussels, 2021

Neutrosophic Science International Association (NSIA)

University of New Mexico 705 Gurley Ave., Gallup, NM 87301, USA

Theory and Application Of Hypersoft Set

Prof. Dr. Xiao Long Xin School of Mathematics, Northwest university, Xian, China.

Peer-Reviewers

Associate Prof. Dr. Irfan Deli Muallim Rıfat Faculty of Education, 7 Aralık University, 79000 Kilis, Turkey.

Assistant Prof. Dr. Muhammad Riaz Department of Mathematics, University of Punjab, Lahore Pakistan.

Pons Publishing House / Pons asbl Quai du Batelage, 5 1000 -Bruxelles Belgium

DTP: George Lukacs ISBN 978-1-59973-699-0

Neutrosophic Science International Association (NSIA)

University of New Mexico 705 Gurley Ave., Gallup, NM 87301, USA

Table of Contents

Aims and Scope…………………………………………………………………………..I

Chapter 1

Muhammad Saeed, Atiqe Ur Rahman, Muhammad Ahsan, Florentin Smarandache

An Inclusive Study on Fundamentals of Hypersoft Set . ..….1

Chapter 2

Irfan Deli

Hybrid set structures under uncertainly parameterized hypersoft sets: Theory and applications .………..….24

Chapter 3

Adem Yolcu, Taha Yasin Ozturk Fuzzy Hypersoft Sets and It’s Application to Decision-Making……………………..…….…………50

Chapter 4

Muhammad Naveed Jafar, Muhammad Saeed, Muhammad Haseeb, Ahtasham Habib

Matrix Theory for Intuitionistic Fuzzy Hypersoft Sets and its application in Multi-Attributive Decision-Making Problems …………………………………………………………………...……65

Chapter 5

Rana Muhammad Zulqarnain, Xiao Long Xin, Muhammad Saeed A Development of Pythagorean fuzzy hypersoft set with basic operations and decision-making approach based on the correlation coefficient…………………………………….………………...…..85

Chapter 6

Adeel Saleem, Muhammad Saqlain, Sana Moin

Development of TOPSIS using Similarity Measures and Generalized weighted distances for Interval Valued Neutrosophic Hypersoft Matrices along with Application in MAGDM Problems .107

Chapter 7

Muhammad Umer Farooq, Muhammad Saqlain, Zaka-ur-Rehman

The Application of the Score Function of Neutrosophic Hypersoft Set in the Selection of SiC as Gate Dielectric For MOSFET………………………………………………………………………….…138

Chapter 8

Mahrukh Irfan, Maryam Rani, Muhammad Saqlain, Muhammad Saeed Tangent, Cosine, and Ye Similarity Measures of m-Polar Neutrosophic Hypersoft Sets………....155

Chapter 9

Muhammad Saeed, Muhammad Ahsan, Atiqe Ur Rahman

A Novel Approach to Mappings on Hypersoft Classes with Application…………………………175

Chapter 10

Atiqe Ur Rahman, Abida Hafeez, Muhammad Saeed, Muhammad Rayees Ahmad, Ume-e-Farwa Development of Rough Hypersoft Set with Application in Decision Making for the Best Choice of Chemical Material…………………………………………………………………………………...……192

Chapter 11

Muhammad Saeed, Muhammad Khubab Siddique, Muhammad Ahsan, Muhammad Rayees Ahmad, Atiqe Ur Rahman A Novel Approach to the Rudiments of Hypersoft Graphs…………………………………………203

Chapter 12

Taha Yasin Ozturk and Adem Yolcu On Neutrosophic Hypersoft Topological Spaces………………………...……………………………215

Aims and Scope

Florentin Smarandache generalize the soft set to the hypersoft set by transforming the function �� into a multi-argument function. This extension reveals that the hypersoft set with neutrosophic, intuitionistic, and fuzzy set theory will be very helpful to construct a connection between alternatives and attributes. Also, the hypersoft set will reduce the complexity of the case study. The Book “Theory and Application of Hypersoft Set” focuses on theories, methods, algorithms for decision making and also applications involving neutrosophic, intuitionistic, and fuzzy information. Our goal is to develop a strong relationship with the MCDM solving techniques and to reduce the complexion in the methodologies. It is interesting that the hypersoft theory can be applied on any decision-making problem without the limitations of the selection of the values by the decision-makers. Some topics having applications in the area: Multi-criteria decision making (MCDM), Multi-criteria group decision making (MCGDM), shortest path selection, employee selection, e-learning, graph theory, medical diagnosis, probability theory, topology, and some more. (Editors)

An Inclusive Study on Fundamentals of Hypersoft Set

Muhammad Saeed1 , Atiqe Ur Rahman2* , Muhammad Ahsan3, Florentin Smarandache4

1,2,3 University of Management and Technology, Lahore, Pakistan. E-mail: muhammad.saeed@umt.edu.pk, E-mail: aurkhb@gmail.com, E-mail: ahsan1826@gmail.com 4 Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA. E-mail: smarand@unm.edu

Abstract: Smarandache developed hypersoft set theory as an extension of soft set theory, to adequate the existing concepts for multi-attribute function. In this study, essential elementary properties e.g. not set, subset, absolute set, and aggregation operations e.g. union, intersection, complement, AND, OR, restricted union, extended intersection, relevant complement, restricted difference, restricted symmetric difference, are characterized under hypersoft set environment with illustrated examples. New notions of relation, function and their basic properties are also discussed for hypersoft sets. Moreover, matrix representation of hypersoft set is presented along with different operations.

Keywords: Hypersoft Set, Hypersoft Relation, Hypersoft Function, Hypersoft Matrix.

1. Introduction

The theories like theory of probability, theory of fuzzy sets, and the interval mathematics, are considered as mathematical means to tackle many intricate problems involving various uncertainties in different fields of mathematical sciences. These theories have their own complexities which restrain them to solve these problems successfully. The reason for these hurdles is, possibly, the inadequacy of the parametrization tool. A mathematical tool is needed for dealing with uncertainties which should be free of all such impediments. In 1999, Molodtsov [1] introduced a mathematical tool called soft sets in literature as a new parameterized family of subsets of the universe of discourse. In 2003, Maji et al. [2] extended the concept and introduced some fundamental terminologies and operations like equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, absolute soft set, AND, OR and also the operations of union and intersection. They verified De Morgan's laws and a number of other results In 2005, Pei et al. [3] discussed the relationship between soft sets and information systems. They showed the soft sets as a class of special information systems. In 2009, Ali et al. [4] pointed out several assertions in previous work of Maji et al. and proposed new notions such as the restricted intersection, the restricted union, the restricted difference and the extended intersection of two soft sets. In 2010, 2011, Babitha et. al. [5,6] introduced

conceptofsoftsetrelationasasubsoftsetoftheCartesianproductofthesoftsetsandalso discussedmanyrelatedconceptssuchasequivalentsoftsetrelation,partition,composition andfunction.In2011,Sezginetal.[7],Geetal.[8],Fuli[9]gavesomemodificationsinthe workofMajietal.andalsoestablishedsomenewresults.In2020,Saeedetal.[10]performed anextensiveinspectionoftheconceptofsoftelementsandsoftmembersinsoftsets.Many researchers[11–22]developedcertainhybridswithsoftsetstogetmoregeneralizedresultsfor implementationindecisionmakingandotherrelateddisciplines.In2018,Smarandache[23] introducedtheconceptofhypersoftsetasageneralizationofsoftset.

Inthispaper,someessentialfundamentals(i.e.elementaryproperties,settheoreticoperations, basiclaws,setrelations,setfunctionandmatrixrepresentation)areconceptualizedunderhypersoftsetenvironment.Therestofthisarticleisstructuredasfollows:Section2givessome basicdefinitionsandresultsonhypersoftsets.Section3presentselementarypropertiesof hypersoftsets.Section4describessettheoreticoperationsofhypersoftsets.Section5providessomebasicproperties,resultsandlawsonhypersoftsets.Section6discusseshypersoft relationsandhypersoftfunctions.Section7presentsthematrixrepresentationofhypersoft setswithsomeoperations.Section8presentssomehybridsofhypersoftsetsandthenlast section9concludesthepaper.

2. Preliminaries

Herewerecallsomebasicterminologiesregardingsoftsetandhypersoftset.Throughout thepaper, U denotestheuniverseofdiscourse. Definition2.1. [1]

Definition2.4. [2]

Intersectionoftwosoftsets(ζS1 , Λ1)and(ζS2 , Λ2)isasoftset(ζS3 , Λ3)withΛ3 =Λ1 ∩ Λ2 andfor ω ∈ Λ3, ζS3 (ω)= ζS1 (ω) ∩ ζS2 (ω)

Formoredetailsonsoftsetcanbefoundin[1–9]

3. HypersoftSet(HS-set)

Inthissection,somefundamentalsofhypersoftsetarepresented.Someofthedefinitions givenin[24]aremodified.

Definition3.1. [21]

Thepair(Ψ,G)iscalleda hypersoftset over U ,where G isthecartesianproductofndisjoint attribute-valuedsets G1,G2,G3,....,Gn correspondingtondistinctattributes g1,g2,g3,....,gn respectivelyandΨ: G → P (U ).ThecollectionofallhypersoftsetsisdenotedbyΩ(Ψ,G) Example3.2. SupposethatMr.Xwantstobuyamobilefromamobilemarket.Thereareeightkindsofmobiles(options)whichformthesetofdiscourse U = {m1,m2,m3,m4,m5,m6,m7,m8}.Thebestselectionmaybeevaluatedbyobservingtheattributesi.e. a1 =Company, a2 =CameraResolution, a3 =Size, a4 =RAM,and a5 =Battery power.Theattribute-valuedsetscorrespondingtotheseattributesare:

(g1, {m1,m2}) , (g2, {m1,m2,m3}) , (g3, {m2,m3,m4}) , (g4, {m4,m5,m6}) , (g5, {m6,m7,m8}) , (g6, {m2,m3,m4,m7}) , (g7, {m1,m3,m5,m6}) , (g8, {m2,m3,m6,m7}) , (g9, {m2,m3,m6,m7,m8}) , (g10, {m1,m3,m6,m7,m8}) , (g11, {m2,m4,m6,m7,m8}) , (g12, {m1,m2,m3,m6,m7,m8}) , (g13, {m2,m3,m5,m7,m8}) , (g14, {m1,m3,m5,m7,m8}) , (g15, {m1,m2,m3,m5,m7,m8}) , (g16, {m4,m5,m6,m7,m8})

Definition3.3. Let F (U )bethecollectionofallfuzzysetsover U .Let a1,a2,a3,.....,an, for n ≥ 1,bendistinctattributes,whosecorrespondingattributevaluesarerespectivelythe



sets G1,G2,G3,.....,Gn,with Gi ∩ Gj = ∅,for i = j,and i,j ∈{1, 2, 3,...,n}. Thena fuzzy hypersoftset (Ψfhs,G)over U isdefinedbythesetoforderedpairsasfollows, (Ψfhs,G)= {(g, Ψfhs(g)): g ∈ G, Ψfhs(g) ∈F (U )} whereΨfhs : G →F(U )andforall g ∈ G = G1 × G2 × G3 × × Gn Ψfhs(g)= {µΨfhs(g)(u)/u : u ∈U ,µΨfhs(g)(u) ∈ [0, 1]} isafuzzysetover U . Abovedefinitionisamodifiedversionoffuzzyhypersoftsetgivenin[21]and[23]. Example3.4. Consideringtheexample3.2,wehave Fuzzyhypersoftset(Ψfhs,G)isgivenas (Ψfhs,G)=

(g1, {0.1/m1, 0.2/m2}) , (g2, {0.1/m1, 0.2/m2, 0.3/m3}) , (g3, {0 2/m2, 0 3/m3, 0 4/m4}) , (g4, {0 4/m4, 0 5/m5, 0 6/m6}) , (g5, {0.6/m6, 0.7/m7, 0.8/m8}) , (g6, {0.2/m2, 0.3/m3, 0.4/m4, 0.7/m7}) , (g7, {0 1/m1, 0 3/m3, 0 5/m5, 0 6/m6}) , (g8, {0 2/m2, 0 3/m3, 0 6/m6, 0 7/m7}) , (g9, {0.2/m2, 0.3/m3, 0.6/m6, 0.7/m7, 0.8/m8}) , (g10, {0 1/m1, 0 3/m3, 0 6/m6, 0 7/m7, 0 8/m8}) , (g11, {0 2/m2, 0 4/m4, 0 6/m6, 0 7/m7, 0 8/m8}) , (g12, {0 1/m1, 0 2/m2, 0 3/m3, 0 6/m6, 0 7/m7, 0 8/m8}) , (g13, {0.2/m2, 0.3/m3, 0.5/m5, 0.7/m7, 0.8/m8}) , (g14, {0 1/m1, 0 3/m3, 0 5/m5, 0 7/m7, 0 8/m8}) , (g15, {0 1/m1, 0 2/m2, 0 3/m3, 0 5/m5, 0 7/m7, 0 8/m8}) , (g16, {0.4/m4, 0.5/m5, 0.6/m6, 0.7/m7, 0.8/m8})

Definition3.7. Aset G = G1 × G2 × G3 × ..... × Gn inhypersoftset(Ψ,G)issaidtobe Not set ifithastherepresentationas

G = { g1, g2, g3, g4,....., gm} where m = n i=1 |Gi|,each gi,i =1, 2,...,m,isa Not n-tupleelement.

Example3.8. Takingsets G1,G2,G3,G4,,G5 fromexample3.2,wehave G = { g1, g2, g3, g4,....., g16} whereeach gi,i =1, 2,..., 16,isa Not 5-tupleelement.

Definition3.9. Ahypersoftset(Ψ,G1)iscalleda relativenullhypersoftset w.r.t G1 ⊆ G, denotedby(Ψ,G1)Φ ,ifΨ(g)= ∅, ∀g ∈ G1.

Example3.10. Consideringexample3.2,if (Ψ,G1)Φ = (g1, ∅) , (g2, ∅) , (g3, ∅) where G1 ⊆ G.

Definition3.11. Ahypersoftset(Ψ,G1)iscalleda relativewholehypersoftset w.r.t G1 ⊆ G, denotedby(Ψ,G1)U ,ifΨ(g)= U , ∀g ∈ G1.

Example3.12. Consideringexample3.2,if (Ψ,G1)U = (g1, U ) , (g2, U ) , (g3, U ) where G1 ⊆ G.

Definition3.13. Ahypersoftset(Ψ,G)iscalleda absolutewholehypersoftset over U , denotedby(Ψ,G)U ,ifΨ(g)= U , ∀g ∈ G.

(g1, U ) , (g2, U ) , (g3, U ) , (g4, U ) , (g5, U ) , (g6, U ) , (g7, U ) , (g8, U ) , (g9, U ) , (g10, U ) , (g11, U ) , (g12, U ) , (g13, U ) , (g14, U ) , (g15, U ) , (g16, U )



Proposition3.15. Let (Ψ1,G1), (Ψ2,G2), (Ψ3,G3) ∈ Ω(Ψ,G) with G1,G2,G3 ⊆ G then (i) (Ψ1,G1) ⊆ (Ψ1,G1)U (ii) (Ψ1,G1)Φ ⊆ (Ψ1,G1) (iii) (Ψ1,G1) ⊆ (Ψ1,G1) (iv) If (Ψ1,G1) ⊆ (Ψ2,G2) and (Ψ2,G2) ⊆ (Ψ3,G3) then (Ψ1,G1) ⊆ (Ψ3,G3)

(v) If (Ψ1,G1)=(Ψ2,G2) and (Ψ2,G2)=(Ψ3,G3) then (Ψ1,G1)=(Ψ3,G3) Definition3.16. The complement ofahypersoftset(Ψ,G),denotedby(Ψ,G) ,isdefined as (Ψ,G) =(Ψ , G) where Ψ : G → P (U ) with Ψ ( g)= U\ Ψ(g), ∀g ∈ G

Example3.17. Assumingdatafromexample3.2,wehave (Ψ,G)

( g1, {m3,m4,m5,m6,m7,m8}) , ( g2, {m4,m5,m6,m7,m8}) , ( g3, {m1,m5,m6,m7,m8}) , ( g4, {m1,m2,m3,m7,m8}) , ( g5, {m1,m2,m3,m4,m5}) , ( g6, {m2,m3,m4,m7}) , ( g7, {m2,m4,m7,m8}) , ( g8, {m1,m4,m5,m8}) , ( g9, {m1,m4,m5}) , ( g10, {m2,m4,m5}) , ( g11, {m1,m3,m5}) , ( g12, {m4,m5}) , ( g13, {m1,m4,m6}) , ( g14, {m2,m4,m6}) , ( g15, {m4,m6}) , ( g16, {m1,m2,m3})

Definition3.18. The relativecomplement ofahypersoftset(Ψ,G),denotedby(Ψ,G) ,is definedas (Ψ,G) =(Ψ ,G) where Ψ : G → P (U ) with Ψ (g)= U\ Ψ(g), ∀g ∈ G Example3.19. Assumingdatafromexample3.2,wehave (Ψ,G)

(g1, {m3,m4,m5,m6,m7,m8}) , (g2, {m4,m5,m6,m7,m8}) , (g3, {m1,m5,m6,m7,m8}) , (g4, {m1,m2,m3,m7,m8}) , (g5, {m1,m2,m3,m4,m5}) , (g6, {m2,m3,m4,m7}) , (g7, {m2,m4,m7,m8}) , (g8, {m1,m4,m5,m8}) , (g9, {m1,m4,m5}) , (g10, {m2,m4,m5}) , (g11, {m1,m3,m5}) , (g12, {m4,m5}) , (g13, {m1,m4,m6}) , (g14, {m2,m4,m6}) , (g15, {m4,m6}) , (g16, {m1,m2,m3})

4.

(i) ((Ψ,G) ) =(Ψ,G) (ii) ((Ψ,G) ) =(Ψ,G) (iii) ((Ψ1,G1)U ) =(Ψ1,G1)Φ =((Ψ1,G1)U ) where G1 ⊆ G (iv) ((Ψ1,G1)Φ) =(Ψ1,G1)U =((Ψ1,G1)Φ) where G1 ⊆ G

SetTheoreticOperationsonHypersoftSet

Inthissection,settheoreticoperationsi.e.union,intersection,difference,AND,ORetc., arediscussedunderhypersoftsetenvironment.

Definition4.1. Unionoftwohypersoftsets (π,G1)and(λ,G2),denotedby(π,G1)∪(λ,G2), isahypersoftset(µ,G3)with G3 = G1 ∪ G2 andfor g ∈ G3, µ(g)=      π(g) λ(g) π(g) ∪ λ(g)

g ∈ (G1 \ G2) g ∈ (G2 \ G1) g ∈ (G1 ∩ G2)

Example4.2. Let (π,G1)= (g1, {m1,m2}) , (g2, {m1,m2,m3}) , (g3, {m2,m3,m4}) (λ,G2)= (g3, {m1,m2}) , (g4, {m4,m5,m6}) , (g5, {m2,m4,m6}) then (µ,G3)= (g1, {m1,m2}) , (g2, {m1,m2,m3}) , (g3, {m1,m2,m3,m4}) , (g4, {m4,m5,m6}) , (g5, {m2,m4,m6})

Definition4.3. Intersectionoftwohypersoftsets (π,G1)and(λ,G2),denotedby(π,G1) ∩ (λ,G2),isahypersoftset(µ,G3)with G3 = G1 ∩ G2 andfor g ∈ G3, µ(g)= π(g) ∩ λ(g).

Example4.4. Considertheexample4.2,wehave (µ,G3)= (g3, {m2})

Definition4.5. ExtendedIntersectionoftwohypersoftsets (π,G1)and(λ,G2),denotedby (π,G1) ∩ε (λ,G2),isahypersoftset(µ,G3)with G3 = G1 ∪ G2 andfor g ∈ G3, µ(g)=      π(g) λ(g) π(g) ∩ λ(g)

g ∈ (G1 \ G2) g ∈ (G2 \ G1) g ∈ (G1 ∩ G2)

Example4.6. Assumingsetsgiveninexample4.2,wehave (µ,G3)= (g1, {m1,m2}) , (g2, {m1,m2,m3}) , (g3, {m2}) , (g4, {m4,m5,m6}) , (g5, {m2,m4,m6})

Definition4.7. AND-operationoftwohypersoftsets (π,G1)and(λ,G2),denotedby (π,G1) (λ,G2),isahypersoftset(µ,G3)with G3 = G1 × G2 andfor(gi,gj ) ∈ G3,gi ∈ G1,gj ∈ G2, µ(gi,gj )= π(gi) ∪ λ(gj ).



(h1, {m1,m2}) , (h2, {m1,m2,m4,m5,m6}) , (h3, {m1,m2,m4,m6}) , (h4, {m1,m2,m3}) , (h5, {m1,m2,m3,m4,m5,m6}) , (h6, {m1,m2,m3,m4,m6}) , (h7, {m1,m2,m3,m4}) , (h8, {m2,m3,m4,m5,m6}) , (h9, {m2,m3,m4,m6}) ,





Example4.10. Consideringdatafromexamples4.2and4.10,wehave (µ,G3)= (h1, {m1,m2}) , (h2, {}) , (h3, {m2}) , (h4, {m1,m2}) , (h5, {}) , (h6, {m2}) , (h7, {m2}) , (h8, {m4}) , (h9, {m2,m4}) ,

Definition4.11. RestrictedUnionoftwohypersoftsets (π,G1)and(λ,G2),denotedby (π,G1) ∪R (λ,G2),isahypersoftset(µ,G3)with G3 = G1 ∩ G2 andfor g ∈ G3,

µ(g)= π(g) ∪ λ(g).

Example4.12. Forsetsgiveninexample4.2,wehave

(µ,G3)= (g3, {m1,m2,m3,m4})

Definition4.13. RestrictedDifferenceoftwohypersoftsets (π,G1)and(λ,G2),denotedby (π,G1) \R (λ,G2),isahypersoftset(µ,G3)with G3 = G1 ∩ G2 andfor g ∈ G3,

µ(g)= π(g) λ(g)

Example4.14. Forsetsgiveninexample4.2,wehave

(µ,G3)= (g3, {m3,m4})

Definition4.15. RestrictedSymmetricDifferenceoftwohypersoftsets (π,G1)and(λ,G2), denotedby(π,G1) (λ,G2),isahypersoftset(µ,G3)definedby (µ,G3)= ((π,G1) ∪R (λ,G2)) \R ((π,G1) ∩ (λ,G2)) or

(µ,G3)= ((π,G1) \R (λ,G2)) ∪R ((λ,G2) \R (π,G1))

Example4.16. Forsetsgiveninexample4.2,wehave

((π,G1) \R (λ,G2))= (g3, {m3,m4}) and ((λ,G2) \R (π,G1))= (g3, {m1}) then (µ,G3)= (g3, {m1,m3,m4})

5. BasicPropertiesandLawsofHypersoftSetOperations

Inthissection,somebasicpropertiesandlawsarediscussedforhypersoftsettheoretic operations.AllhypersoftsetsinΩ(ψ,G) satisfythefollowingproperties,resultsandlaws.

(a) IdempotentLaws

(i) (ψ, G) ∪ (ψ, G)=(ψ, G)=(ψ, G) ∪R (ψ, G) (ii) (ψ, G) ∩ (ψ, G)=(ψ, G)=(ψ, G) ∩ε (ψ, G)

(b) IdentityLaws

(i) (ψ, G) ∪ (ψ, G)Φ =(ψ, G)=(ψ, G) ∪R (ψ, G)Φ (ii) (ψ, G) ∩ (ψ, G)U =(ψ, G)=(ψ, G) ∩ε (ψ, G)U (iii) (ψ, G) \R (ψ, G)Φ =(ψ, G)=(ψ, G) (ψ, G)Φ (iv) (ψ, G) \R (ψ, G)=(ψ, G)Φ =(ψ, G) (ψ, G)

(c) DominationLaws

(i) (ψ, G) ∪ (ψ, G)U =(ψ, G)U =(ψ, G) ∪R (ψ, G)U (ii) (ψ, G) ∩ (ψ, G)Φ =(ψ, G)Φ =(ψ, G) ∩ε (ψ, G)Φ

(d) PropertyofExclusion

(ψ, G) ∪ (ψ, G) =(ψ, G)U =(ψ, G) ∪R (ψ, G)

(e) PropertyofContradiction

(ψ, G) ∩ (ψ, G) =(ψ, G)Φ =(ψ, G) ∩ε (ψ, G)

(f) AbsorptionLaws

(i) (ζ, G1) ∪ ((ζ, G1) ∩ (ξ, G2))=(ζ, G1) (ii) (ζ, G1) ∩ ((ζ, G1) ∪ (ξ, G2))=(ζ, G1)

(iii) (ζ, G1) ∪R ((ζ, G1) ∩ε (ξ, G2))=(ζ, G1)

(iv) (ζ, G1) ∩ε ((ζ, G1) ∪R (ξ, G2))=(ζ, G1)

(g) CommutativeLaws

(i) (ζ, G1) ∪ (ξ, G2)=(ξ, G2) ∪ (ζ, G1)

(ii) (ζ, G1) ∪R (ξ, G2)=(ξ, G2) ∪R (ζ, G1)

(iii) (ζ, G1) ∩ (ξ, G2)=(ξ, G2) ∩ (ζ, G1)

(iv) (ζ, G1) ∩ε (ξ, G2)=(ξ, G2) ∩ε (ζ, G1)

(v) (ζ, G1) (ξ, G2)=(ξ, G2) (ζ, G1)

(h) AssociativeLaws

(i) (ζ, G1) ∪ ((ξ, G2) ∪ (ψ, G3))=((ζ, G1) ∪ (ξ, G2)) ∪ (ψ, G3)

(ii) (ζ, G1) ∪R ((ξ, G2) ∪R (ψ, G3))=((ζ, G1) ∪R (ξ, G2)) ∪R (ψ, G3) (iii) (ζ, G1) ∩ ((ξ, G2) ∩ (ψ, G3))=((ζ, G1) ∩ (ξ, G2)) ∩ (ψ, G3) (iv) (ζ, G1) ∩ε ((ξ, G2) ∩ε (ψ, G3))=((ζ, G1) ∩ε (ξ, G2)) ∩ε (ψ, G3) (v) (ζ, G1) ((ξ, G2) (ψ, G3))=((ζ, G1) (ξ, G2)) (ψ, G3) (vi) (ζ, G1) ((ξ, G2) (ψ, G3))=((ζ, G1) (ξ, G2)) (ψ, G3)

(i) DeMorgansLaws

(i) ((ζ, G1) ∪ (ξ, G2)) =(ζ, G1) ∩ε (ξ, G2) (ii) ((ζ, G1) ∩ε (ξ, G2)) =(ζ, G1) ∪ (ξ, G2) (iii) ((ζ, G1) ∪R (ξ, G2)) =(ζ, G1) ∩ (ξ, G2) (iv) ((ζ, G1) ∩ (ξ, G2)) =(ζ, G1) ∪R (ξ, G2) (v) ((ζ, G1) (ξ, G2)) =(ζ, G1) (ξ, G2) (vi) ((ζ, G1) (ξ, G2)) =(ζ, G1) (ξ, G2) (vii) ((ζ, G1) (ξ, G2)) =(ζ, G1) (ξ, G2) (viii) ((ζ, G1) (ξ, G2)) =(ζ, G1) (ξ, G2)

(j) DistributiveLaws

(i) (ζ, G1) ∪ ((ξ, G2) ∩ (ψ, G3))=((ζ, G1) ∪ (ξ, G2)) ∩ ((ζ, G1) ∪ (ψ, G3)) (ii) (ζ, G1) ∩ ((ξ, G2) ∪ (ψ, G3))=((ζ, G1) ∩ (ξ, G2)) ∪ ((ζ, G1) ∩ (ψ, G3)) (iii) (ζ, G1) ∪R ((ξ, G2) ∩ε (ψ, G3))=((ζ, G1) ∪R (ξ, G2)) ∩ε ((ζ, G1) ∪R (ψ, G3))

(iv) (ζ, G1) ∩ε ((ξ, G2) ∪R (ψ, G3))=((ζ, G1) ∩ε (ξ, G2)) ∪R ((ζ, G1) ∩ε (ψ, G3))

(v) (ζ, G1) ∪R ((ξ, G2) ∩ (ψ, G3))=((ζ, G1) ∪R (ξ, G2)) ∩ ((ζ, G1) ∪R (ψ, G3))

(vi) (ζ, G1) ∩ ((ξ, G2) ∪R (ψ, G3))=((ζ, G1) ∩ (ξ, G2)) ∪R ((ζ, G1) ∩ (ψ, G3))

6. RelationsandFunctionsonHypersoftSets

Herewepresentthenotionsofrelationsandfunctionsforhypersoftsets.

Definition6.1. CartesianProduct oftwohypersoftsets(Ψ1,G1)and(Ψ2,G2),denotedby (Ψ1,G1) × (Ψ2,G2),isahypersoft(Ψ3,G3)where G3 = G1 × G2 and Ψ3 : G3 → P (U×U )

definedby Ψ3(gi,gj )=Ψ1(gi) × Ψ2(gj ) ∀ (gi,gj ) ∈ G3 thatis Ψ3(gi,gj )= {(hi,hj ): hi ∈ Ψ1(gi),hj ∈ Ψ2(gj )}

Definition6.2. If(Ψ1,G1), (Ψ2,G2) ∈ Ω(Ψ,G) thenarelationfrom(Ψ1,G1)to(Ψ2,G2)is called hypersoftsetrelation (R,G4)(simply R)whichisthehypersoftsubsetof(Ψ1,G1) × (Ψ2,G2)where G4 ⊆ G1 × G2 and ∀ (h1,h2) ∈ G4, R(h1,h2)=Ψ3(h1,h2),where(Ψ3,G3)= (Ψ1,G1) × (Ψ2,G2).

Definition6.3. Let R beahypersoftsetrelationfrom(Ψ1,G1)to(Ψ2,G2)suchthat (Ψ3,G3)=(Ψ1,G1) × (Ψ2,G2).Then

(i) Domainof R (Dom R)isahypersoftset(ψ,K) ⊂ (Ψ1,G1)where K = {gi ∈ G1 :Ψ3(gi,gj ) ∈ R forsomegj ∈ G2} and ψ(g1)=Ψ1(g1), ∀ g1 ∈ K

(ii) Rangeof R (Range R)isahypersoftset(ξ,L) ⊂ (Ψ2,G2)where L ⊂ G2 and L = {gj ∈ G2 :Ψ3(gi,gj ) ∈ R forsomegi ∈ G1} and ξ(g2)=Ψ1(g2), ∀ g2 ∈ L

(iii) Theinverseof R (R 1)isahypersoftsetrelationfrom(Ψ2,G2)to(Ψ1,G1)defined by R 1 = {Ψ2(qj ) × Ψ1(qi):Ψ1(qi)RΨ2(qj )}

Example6.4.

(i) Dom R =(ψ,K)where K = {g1,g2,g3}⊆ G1 and ψ(gi)=Ψ1(gi)∀ gi ∈ K

(ii) Range R =(ξ,L)where L = {g4,g6}⊂ G2 and ξ(gj )=Ψ2(gj )∀ gj ∈ L

(iii) R 1 =

(Ψ2(g4) × Ψ1(g1)), (Ψ2(g6) × Ψ1(g1)), (Ψ2(g6) × Ψ1(g2)), (Ψ2(g6) × Ψ1(g3))

Definition6.5. Let R and S aretwohypersoftsetrelationsonhypersoftset(Ψ,K),then wehave

(i) R ⊂ S,ifforall u,v ∈ K, Ψ(u) × Ψ(v) ∈ R thenΨ(u) × Ψ(v) ∈ S

(ii) TheComplementof R ,denotedby R c ,isdefinedas

R c = {Ψ(u) × Ψ(v):Ψ(u) × Ψ(v) / ∈ R, ∀ u,v ∈ K}

(iii) Theunionof R and S,denotedby R ∪ S,definedas

R ∪ S = {Ψ(u) × Ψ(v):Ψ(u) × Ψ(v) ∈ R or Ψ(u) × Ψ(v) ∈ S, ∀ u,v ∈ K}

(iv) Theintersectionof R and S,denotedby R ∩ S,definedas

R ∩ S = {Ψ(u) × Ψ(v):Ψ(u) × Ψ(v) ∈ R and Ψ(u) × Ψ(v) ∈ S, ∀ u,v ∈ K}

Example6.6. Let (Ψ,K)= Ψ(g1), Ψ(g2), Ψ(g3)

(Ψ,K) × (Ψ,K)=      (Ψ(g1) × Ψ(g1)), (Ψ(g1) × Ψ(g2)), (Ψ(g1) × Ψ(g3)), (Ψ(g2) × Ψ(g1)), (Ψ(g2) × Ψ(g2)), (Ψ(g2) × Ψ(g3)), (Ψ(g3) × Ψ(g1)), (Ψ(g3) × Ψ(g2)), (Ψ(g3) × Ψ(g3))

     thenwehave

R = (Ψ(g1) × Ψ(g1)), (Ψ(g1) × Ψ(g3)), (Ψ(g2) × Ψ(g3)), (Ψ(g3) × Ψ(g3)) and S = (Ψ(g1) × Ψ(g1)), (Ψ(g1) × Ψ(g2)), (Ψ(g2) × Ψ(g2)), (Ψ(g3) × Ψ(g2)) now (1)

R c = (Ψ(g1) × Ψ(g2)), (Ψ(g2) × Ψ(g1)), (Ψ(g2) × Ψ(g2)), (Ψ(g3) × Ψ(g1)), (Ψ(g3) × Ψ(g2)) S c = (Ψ(g1) × Ψ(g3)), (Ψ(g2) × Ψ(g1)), (Ψ(g2) × Ψ(g3)), (Ψ(g3) × Ψ(g3)) (2)

R ∪ S = (Ψ(g1) × Ψ(g1)), (Ψ(g1) × Ψ(g2)), (Ψ(g1) × Ψ(g3)), (Ψ(g2) × Ψ(g2)), (Ψ(g2) × Ψ(g3)), (Ψ(g3) × Ψ(g2)), (Ψ(g3) × Ψ(g3)) (3) R ∩ S = (Ψ(g1) × Ψ(g1))

Definition6.7. Let R beahypersoftsetrelationon(Ψ,K),then

(i) R is reflexive ifΨ(u) × Ψ(u) ∈ R forall u ∈ K,e.g.

R = (Ψ(g1) × Ψ(g1))

(ii) R is symmetric ifΨ(u) × Ψ(v) ∈ R thenΨ(v) × Ψ(u) ∈ R forall u,v ∈ K,e.g.

R = (Ψ(g1) × Ψ(g2)), (Ψ(g2) × Ψ(g1))

(iii) R is transitive ifΨ(u) × Ψ(v) ∈ R andΨ(v) × Ψ(w) ∈ R thenΨ(u) × Ψ(w) ∈ R for all u,v,w ∈ K,e.g.

R = (Ψ(g1) × Ψ(g2)), (Ψ(g1) × Ψ(g3)), (Ψ(g2) × Ψ(g3))

(iv) R iscalled equivalencerelation ifitisreflexive,symmetricandtransitive.e.g.

R = (Ψ(g1) × Ψ(g1)), (Ψ(g1) × Ψ(g2)), (Ψ(g2) × Ψ(g1)), (Ψ(g2) × Ψ(g2))

(v) R iscalled identity ifΨ(u) × Ψ(v) ∈ R then u = v forall u,v ∈ K,e.g. R = (Ψ(g1) × Ψ(g1)), (Ψ(g2) × Ψ(g2)), (Ψ(g3) × Ψ(g3)) Definition6.8. If R isahypersoftsetrelationfrom(Ψ1,G1)to(Ψ2,G2)and S isahypersoft setrelationfrom(Ψ2,G2)to(Ψ3,G3)thencompositionof R and S,denotedby R ◦ S,isalso ahypersoftsetrelation T from(Ψ1,G1)to(Ψ3,G3)definedas ifΨ1(u) ∈ (Ψ1,G1)andΨ3(w) ∈ (Ψ3,G3)thenΨ1(u) × Ψ3(w) ∈ R ◦ S i.e. Ψ1(u) × Ψ3(w) ∈ R ◦ S iffΨ1(u) × Ψ2(v) ∈ R andΨ2(v) × Ψ3(w) ∈ R Example6.9. Let R = (Ψ1(g1) × Ψ2(g1)), (Ψ1(g1) × Ψ2(g3)), (Ψ1(g2) × Ψ2(g3)), (Ψ1(g3) × Ψ2(g3)) and S = (Ψ2(g1) × Ψ3(g1)), (Ψ2(g1) × Ψ3(g2)), (Ψ2(g2) × Ψ3(g2)), (Ψ2(g3) × Ψ3(g2)) then R ◦ S = (Ψ1(g1) × Ψ3(g1)), (Ψ1(g1) × Ψ3(g2)), (Ψ1(g2) × Ψ3(g2)), (Ψ1(g3) × Ψ3(g2))

Definition6.10. Ahypersoftsetrelation F from(Ψ1,G1)to(Ψ2,G2),representedby F : (Ψ1,G1) → (Ψ2,G2),issaidtobehypersoftfunctionif (i) domainof F = G1 (ii) thereisnorepetitionofelementsin Dom F (iii) thereisauniqueelementin Range F correspondingtoeveryelementin Dom F. i.e.ifΨ1(u) F Ψ2(v)(orΨ1(u) × Ψ2(v) ∈ F)then F(Ψ1(u))=Ψ2(v).

Example6.11. Let G1 = {u1,u2,u3} and G2 = {v1,v2,v3,v4} then (Ψ1,G1)= Ψ1(u1), Ψ1(u2), Ψ1(u3) (Ψ2,G2)= Ψ2(v1), Ψ2(v2), Ψ2(v3), Ψ2(v4) sohypersoftfunctionsis

F = (Ψ1(u1) × Ψ2(v1)), (Ψ1(u2) × Ψ2(v3)), (Ψ1(u3) × Ψ2(v4))

Definition6.12. Ahypersoftfunction F :(Ψ1,G1) → (Ψ2,G2)issaidtobe

(i) into-hypersoftfunction if Range F ⊂ G2 e.g.Let G1 = {u1,u2,u3} and G2 = {v1,v2,v3,v4} then F = (Ψ1(u1) × Ψ2(v1)), (Ψ1(u2) × Ψ2(v3)), (Ψ1(u3) × Ψ2(v4))

(ii) into-hypersoftfunction(orsurjectivehypersoftfunction) if Range F = G2 e.g.Let G1 = {u1,u2,u3,u4} and G2 = {v1,v2,v3,v4} then F = (Ψ1(u1) × Ψ2(v1)), (Ψ1(u2) × Ψ2(v3)), (Ψ1(u3) × Ψ2(v4)), (Ψ1(u4) × Ψ2(v2))

(iii) one-to-onehypersoftfunction(orinjectivehypersoftfunction) ifΨ1(u1) =Ψ1(u2)then

F(Ψ1(u1)) = F(Ψ1(u2)). e.g.

F = (Ψ1(u1) × Ψ2(v1)), (Ψ1(u2) × Ψ2(v4)), (Ψ1(u3) × Ψ2(v2)), (Ψ1(u4) × Ψ2(v3))

(iv) bijectivehypersoftfunction(orone-to-onehypersoftcorrespondence) ifitisbothinjectiveandsurjective. e.g.

F = (Ψ1(u1) × Ψ2(v1)), (Ψ1(u2) × Ψ2(v2)), (Ψ1(u3) × Ψ2(v3)), (Ψ1(u4) × Ψ2(v4))

Definition6.13. Theidentityhypersoftsetfunctiononhypersoftsoftset(Ψ,L)isdefined by I :(Ψ,L) → (Ψ,L)suchthat I(Ψ(l))=Ψ(l) ∀ Ψ(l) ∈ (Ψ,L). e.g.Let L = {l1,l2,l3,l4} then I = (Ψ(l1) × Ψ(l1)), (Ψ(l2) × Ψ(l2)), (Ψ(l3) × Ψ(l3)), (Ψ(l4) × Ψ(l4))

Inthissection,matrixrepresentationofhypersoftsetispresented. Definition7.1. (i) Let(ζ,H)beahypersoftsetover U .Asubset RH of U× H issaidtoberelationform of(ζ,H)ifitisuniquelyrepresentedas RH = (u,h): h ∈ H,u ∈ ζ(h) (ii) Thecharacteristicfunction XRH isdefinedby XRH : U× H →{0, 1}, where XRH (u,h)= 1;(u,h) ∈ RH 0;(u,h) / ∈ RH

(iii) If |U| = m and |H| = n thenhypersoftset(ζ,H)canberepresentedbyamatrix(αij ) calledan m × n hypersoftmatrixof(ζ,H)over U asgivenbelow (αij )m×n =

Note:Thecollectionofall m × n hypersoftmatricesover U isdenotedby HSM (U )m×n. Example7.2. Let U = {u1,u2,u3,u4,u5} and H = {h1,h2,h3,h4,h5} whereeach hi isa ith tuple,for i =numberofattribute-valuedsets.Then ζ(h1)= {u1,u2},ζ(h2)= ∅,ζ(h3)= {u4,u5},ζ(h4)= {u2,u3,u4, },ζ(h5)= ∅, thereforewehave (ζ,H)= (h1, {u1,u2}), (h3, {u4,u5}), (h4, {u2,u3,u4, }) and RH = (u1,h1), (u2,h1), (u4,h3), (u5,h3), (u2,h4), (u3,h4), (u4,h4) .

Hencehypersoftmatrixisgivenas (αij )5×5 =

i,j =1, 2, 3, 4, 5 Definition7.3. Let(αij )m×n ∈ HSM (U )m×n then(αij )m×n issaidtobe: (i) Azero(ornull)hypersoftmatrix,denotedby(0)m×n,if αij =0 ∀ i,j e.g. (0)5×5 =

i,j =1, 2, 3, 4, 5 (ii) An H1-universalhypersoftmatrix,denotedby(αij )H1 m×n,if αij =1, ∀j ∈ JH1 = {j : hj ∈ H1} and i. e.g.Let H beasgivenin7.2and H1 = {h2,h4,h5}⊆ H with ζ(h2)= ζ(h4)= ζ(h5)=

Note:Wemayalsosaythat L1 isdominatedby L2 or L2 dominates L1.

L1 and L2 aresaidtobecomparable,denotedby L1 L2,if L1 ⊆ L2 or L2 ⊆ L1

L1 issaidtobe properhypersoftsub-matrix of L2,denotedby L1 ⊂ L2 ifforatleast oneterm

Proposition7.5. For X =(

ij )

,Y =(βij )

,Z =(

ij )m

n ∈ HSM (U )m×n,wehave followingcharacteristicsproperties,operationsandlaws: (i) X ∪ X = X,X ∩ X = X (ii) X ∪ (0)m×n = X,X ∩ (αij )U m×n = X (iii) X ∩ (0)m×n =(0)m×n,X ∪ (αij )U m×n =(αij )U m×n (iv) ((0)m×n) c =(αij )U m×n , ((αij )U m×n) c =(0)m×n (v) X ∪ X c =(αij )U m×n ,X ∩ X c =(0)m×n (vi) (X ∪ Y ) c = X c ∩ Y c , (X ∩ Y ) c = X c ∪ Y c (vii) (X c ) c = X (viii) X ∪ Y = Y ∪ X,X ∩ Y = Y ∩ X (ix) X ∪ (Y ∪ Z)=(X ∪ Y ) ∪ Z,X ∩ (Y ∩ Z)=(X ∩ Y ) ∩ Z (x) X ∪ (Y ∩ Z)=(X ∪ Y ) ∩ (X ∪ Z),X ∩ (Y ∪ Z)=(X ∩ Y ) ∪ (X ∩ Z)

Definition7.6. Let P =(pij )m×n,Q =(qik)m×n ∈ HSM (U )m×n,then

(i) AND-product of P and Q,denotedby P ∧ Q,isdefinedas

∧ : HSM (U )m×n × HSM (U )m×n → HSM (U )m×n2 with(pij ) ∧ (qik)=(ril)where ril = min{pij ,qik} and l = n(j 1)+ k.

(ii) OR-product of P and Q,denotedby P ∨ Q,isdefinedas

∨ : HSM (U )m×n × HSM (U )m×n → HSM (U )m×n2 with(pij ) ∨ (qik)=(ril)where ril = max{pij ,qik} and l = n(j 1)+ k.

(iii) AND-NOT-product of P and Q,denotedby P Q,isdefinedas : HSM (U )m×n × HSM (U )m×n → HSM (U )m×n2 with(pij ) (qik)=(ril)where ril = min{pij , 1 qik} and l = n(j 1)+ k.

(iv) OR-NOT-product of P and Q,denotedby P Q,isdefinedas : HSM (U )m×n × HSM (U )m×n → HSM (U )m×n2 with(pij ) (qik)=(ril)where ril = max{pij , 1 qik} and l = n(j 1)+ k.

8. HybridsofHypersoftSets

(j1, {[0.1, 0.2]/u1, [0.2, 0.3]/u2, [0.4, 0.5]/u4, [0.5, 0.6]/u5}) , (j2, {[0 1, 0 3]/u1, [0 2, 0 4]/u2, [0 3, 0 4]/u3, [0 6, 0 8]/u6}) , (j3, {[0 2, 0 3]/u2, [0 3, 0 4]/u3, [0 4, 0 5]/u4,, [0 5, 0 7]/u5}) ,

(j4, {[0.4, 0.5]/u4, [0.5, 0.6]/u5, [0.6, 0.7]/u6, [0.7, 0.8]/u7}) ,

(j5, {[0.3, 0.6]/u3, [0.6, 0.7]/u6, [0.7, 0.8]/u7, [0.8, 0.9]/u8}) ,



Example8.2. Let U = {u1,u2,u3,u4,u5,u6,u7,u8} and J = {j1,j2,j3,j4,j5,j6,j7,j8} where each ji is rth-tuple, r istheproductofordersof Ji,wehave Interval-Valuedfuzzyhypersoftset(Γ, J )isgivenas (Γ, J )=



     Definition8.5. An interval-valuedfuzzyparameterizedhypersoftset (iv-fphs-set)(E, J )over U isdefinedas (E, J )= (ηFiv (j)/j,γFiv (j)): j ∈J ,γFiv (j) ∈ P (U ),ηFiv (j) ∈ [0, 1] where Fiv isaninterval-valuedfuzzysetwith ηFiv : J→ [0, 1]asmembershipfunctionof fphs-setand γFiv : J→ P (U )iscalledapproximatefunction. Example8.6. Fromexample8.2,wehave (E, J )=   

     Definition8.7. An intuitionisticfuzzyparameterizedhypersoftset (ifphs-set)(H, J )over U isdefinedas (H, J )= (<ηT (j),ηF (j) >/j,γIF (j)): j ∈J ,γIF (j) ∈ P (U ),ηT (j) ∈ [0, 1],ηF (j) ∈ [0, 1] where IF isanintuitionisticfuzzysetwith ηT (j),ηF (j): J→ [0, 1]astruthandfalsity membershipfunctionsof ifphs-setand γIF : J→ P (U )iscalledapproximatefunction.





    







(< 0 1, 0 2, 0 2 >/j1, {u1,u2}) , (< 0 2, 0 3, 0 3 >/j2, {u1,u2,u3}) , (< 0 3, 0 4, 0 4 >/j3, {u2,u3,u4}) , (< 0 4, 0 5, 0 5 >/j4, {u4,u5,u6}) , (< 0.5, 0.6, 0.6 >/j5, {u6,u7,u8}) , (< 0.6, 0.7, 0.7 >/j6, {u2,u3,u4,u7}) , (< 0.7, 0.5, 0.8 >/j7, {u1,u3,u5,u6}) , (< 0.8, 0.4, 0.9 >/j8, {u2,u3,u6,u7})



 



Definition8.11. Ahypersoftset(B, J )issaidtobe bijectivehypersoftset (bhs-set)over U if (i) j∈J B(j)= U (ii) for jp,jq ∈J ,p = q, B(jp) ∩B(jq )= ∅

Definition8.15. Aninterval-valuedfuzzyhypersoftset(Bivf , J )issaidtobe bijective interval-valuedfuzzyhypersoftset (biv-fhs-set)over U if

(i) j∈J Bivf (j)= U with u∈U Sup(µf (u)) ∈ [0, 1]where µf (u)isaninterval-valuedfuzzy membershipforeach u ∈U (ii) for jp,jq ∈J ,p = q, Bivf (jp) ∩Bivf (jq )= ∅

Example8.16. Supposesetsgiveninexample8.2,wehave (Bivf , J )= (j1, {[0 01, 0 1]/u1}) , (j2, {[0 02, 0 2]/u2}) , (j3, {[0 03, 0 13]/u3}) , (j4, {[0 04, 0 14]/u4}) , (j5, {[0.03, 0.05]/u5}) , (j6, {[0.02, 0.06]/u6}) , (j7, {[0.03, 0.07]/u7}) , (j8, {[0.04, 0.08]/u8})

Definition8.17. Anintuitionisticfuzzyhypersoftset(Bif , J )issaidtobe bijectiveintuitionisticfuzzyhypersoftset (bifhs-set)over U if (i) j∈J Bif (j)= U with u∈U Tif (u) ∈ [0, 1]and u∈U Fif (u) ∈ [0, 1]where Tif (u)and Fif (u)aretruthandfalsemembershipforeach u ∈U (ii) for jp,jq ∈J ,p = q, Bif (jp) ∩Bif (jq )= ∅

Example8.18. Letthesetsprovidedinexample8.2,wehave (Bif , J )= 



 (j1, {< 0 01, 0 1 >/u1}) , (j2, {< 0 02, 0 2 >/u2}) , (j3, {< 0 03, 0 13 >/u3}) , (j4, {< 0.04, 0.14 >/u4}) , (j5, {< 0.03, 0.05 >/u5}) , (j6, {< 0.02, 0.06 >/u6}) , (j7, {< 0.03, 0.07 >/u7}) , (j8, {< 0.04, 0.08 >/u8})

    

(j1, {< 0 01, 0 02, 0 1 >/u1}) , (j2, {< 0 02, 0 03, 0 2 >/u2}) , (j3, {< 0 03, 0 04, 0 13 >/u3}) , (j4, {< 0 04, 0 05, 0 14 >/u4}) , (j5, {< 0.03, 0.04, 0.05 >/u5}) , (j6, {< 0.02, 0.05, 0.06 >/u6}) , (j7, {< 0 03, 0 04, 0 07 >/u7}) , (j8, {< 0 04, 0 05, 0 08 >/u8})







ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

References

1. Molodtsov,D.,SoftSetTheory-FirstResults, Comput.Math.withAppl. 1999,37,19-31.

2. Maji,P.K.;Biswas,R.,andRoy,A.R.,SoftSetTheory, Comput.Math.withAppl. 2003,45,555-562.

3. Pei.D.,andMiao.D.,Fromsoftsettoinformationsystem, Ininternationalconferenceofgranularcomputing,IEEE 2005,2,617-621.

4. Ali,M.I.;Feng,F.;Liu,X.;Min,W.K.,andSabir,M.,Onsomenewoperationsinsoftsettheory, Comput. Math.withAppl. 2009,57,1547-1553.

5. BabithaK.V.,andSunilJ.J.,Softsetrelationsandfunctions, Comput.Math.withAppl. 2010,60,18401849.

6. BabithaK.V.,andSunilJ.J.,Transitiveclosureandorderinginsoftset, Comput.Math.withAppl. 2011, 61,2235-2239.

7. Sezgin,A.,andAtag¨ un,A.O.,Onoperationsofsoftsets, Comput.Math.withAppl. 2011,61(5),1457-1467.

8. GeX.,andYangS.,Investigationsonsomeoperationsofsoftsets, WorldAcademyofScienceEngineering andTechnology 2011,75,1113-1116.

9. Li,F.,Notesonsoftsetoperations, ARPNJournalofsystemsandsoftwares 2011,205-208.

10. Saeed,M.,Hussain,M.,andMughal,A.A.,AStudyofSoftSetswithSoftMembersandSoftElements: ANewApproach, PunjabUniversityJournalofMathematics 2020,52(8).

11. Khalid,A.andAbbas,M.,Distancemeasuresandoperationsinintuitionisticandinterval-valuedintuitionisticfuzzysoftsettheory, IntJFuzzySyst. 2005,17(3),490-497.

12. Hassan,N.,Sayed,O.R.,Khalil,A.M.andGhany,M.A.,Fuzzysoftexpertsysteminpredictionof coronaryarterydisease, IntJFuzzySyst. 2017,19(5),1546-1559.

13. Feng,F.,Li,C.,Davvaz,B.andAli,M.I.,Softsetscombinedwithfuzzysetsandroughsets:atentative approach, SoftComputing 2010,14(9),899-911.

14. Guan,X.,Li,Y.andFeng,F.,Aneworderrelationonfuzzysoftsetsanditsapplication, SoftComputing 2013,17(1),63-70.

15. Khameneh,A.Z.andKilicman,A.,Parameterreductionoffuzzysoftsets:Anadjustableapproachbased onthethree-waydecision, IntJFuzzySyst. 2018,20(3),928-942.

16. Zhan,J.andZhu,K.,Anovelsoftroughfuzzyset:Z-softroughfuzzyidealsofhemiringsandcorresponding decisionmaking, SoftComputing 2017,21(8),1923-1936.

17. Paik,B.andMondal,S.K.,Adistance-similaritymethodtosolvefuzzysetsandfuzzysoftsetsbased decision-makingproblems, SoftComputing 2020,24(7),5217-5229.

18. Akram,M.,Ali,G.andAlcantud,J.C.R.,Newdecision-makinghybridmodel:intuitionisticfuzzyN-soft roughsets, SoftComputing 2019,23(20),9853-9868.

19. Zhang,J.,Wu,X.andLu,R.,DecisionAnalysisMethodsCombiningQuantitativeLogicandFuzzySoft Sets, IntJFuzzySyst. 2020,1-14.

20. Alshehri,H.A.,Abujabal,H.A.andAlshehri,N.O.,NewtypesofhesitantfuzzysoftidealsinBCKalgebras, SoftComputing 2018,22(11),3675-3683.

21. C¸a˘ gman,N.,C¸itak,F.,andEngino˘ glu,S.,Fuzzyparameterizedfuzzysoftsettheoryanditsapplications, TurkishJournalofFuzzySystem 2010,1(1),21-35.

22. Engino˘ glu,S.andC¸a˘ gman,N.,Fuzzyparameterizedfuzzysoftmatricesandtheirapplicationindecisionmaking, TWMSJournalofAppliedandEngineeringMathematics 2020,10(4),1105-1115.

23. Smarandache,F.,ExtensionofSoftSetofHypersoftSet,andthentoPlithogenicHypersoftSet, NeutrosophicSetsSyst. 2018,22,168-170.

24. Saeed,M.,Ahsan,M.,Siddique,M.k.andAhmad,M.R.,AStudyoftheFundamentalsofHypersoftSet Theory, InternationalJournalofScientificandEngineeringResearch 2020,11.

Hybrid set structures under uncertainly parameterized hypersoft sets: Theory and applications

Irfan Deli

Muallim Rıfat Faculty of Education, 7 Aralık University, 79000 Kilis, Turkey. E-mail: irfandeli@kilis.edu.tr

Abstract: In this paper, we firstly introduced a hybrid set structure with hypersoft sets under uncertainty, vagueness and indeterminacy, called neutrosophic valued n−attribute neutrosophic hypersoft set, which is a combination of neutrosophic sets and hypersoft sets. The neutrosophic valued n−attribute neutrosophic hypersoft set (n − NNHS−set) generalizes the following hybrid sets as; n−attribute Hypersoft Set, n−attribute Fuzzy Hypersoft Set, n−attribute Intuitionistic Fuzzy Hypersoft Set, n−attribute Neutrosophic Hypersoft Set , Fuzzy Valued n−attribute Hypersoft Set, Fuzzy Valued n−attribute Fuzzy Hypersoft Set, Fuzzy Valued n−attribute Intuitionistic Fuzzy Hypersoft Set, Fuzzy Valued n−attribute Neutrosophic Hypersoft Set, Intuitionistic Fuzzy Valued n−attribute Hypersoft Set, Intuitionistic Fuzzy Valued n−attribute Fuzzy Hypersoft Set, Intuitionistic Fuzzy Valued n−attribute Intuitionistic Fuzzy Hypersoft Set, Intuitionistic Fuzzy Valued n−attribute Neutrosophic Hypersoft Set, Neutrosophic Valued n−attribute Hypersoft Set, Neutrosophic Valued n−attribute Fuzzy Hypersoft Set, Neutrosophic Valued n−attribute Intuitionistic Fuzzy Hypersoft Set, neutrosophic parameterized neutrosophic soft set, and so on. Then, we introduce some definitions and operations on n − NNHS−set and some properties of the sets which are connected to operations. Finally, we proposed the decision-making method on the n−NNHS−set and presented a numerical example to show that this method can be successfully applied.

Keywords: Fuzzy set, Intuitionistic Fuzzy set, Neutrosophic set, Soft set, hypersoft sets, n − NNHS−set, decision making

1. Introduction

To handle uncertainties or indeterminate information, fuzzy sets [34] in [0,1], intuitionistic fuzzy sets [2] in [0,1]×[0,1], neutrosophic sets [25,32] in [0,1]×[0,1]×[0,1] and soft sets [22] in parameterize set and universe set are consistently being defined as efficient mathematical tools. Recently, soft sets have been developed by embedding the idea of fuzzy set, intuitionistic fuzzy set, neutrosophic set to expand the field of application of soft sets. For example, fuzzy soft sets [19], intuitionistic fuzzy soft sets [6], neutrosophic soft sets [8], fuzzy parameterized soft sets [5], fuzzy parameterized fuzzy soft sets [12], fuzzy parameterized intuitionistic fuzzy soft sets [10], fuzzy parameterized neutrosophic soft sets (Special version of [13]), intuitionistic fuzzy parameterized soft sets [9], intuitionistic fuzzy parameterized fuzzy soft sets [33], intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets [11], neutrosophic parameterized soft sets [12] and neutrosophic parameterized neutrosophic soft sets [13],

aresomeofthestudies.Also,softsetsgeneralizedtothehypersoftset[31]byusingamulti-argument functionand n distinctattributesets.Also,somestudieshavemadebythemanyauthors,forexample, onbasicoperationsonhypersoftsetsandhypersoftpoint[1],onm-Polarandm-Polarintervalvalued neutrosophichypersoftsets[26],onaggregateoperatorsofneutrosophichypersoftset[27],ondecision makingbasedonTOPSISunderneutrosophicsoftset[28]-[24],ongeneralizedaggregateoperatorson neutrosophichypersoftset[35],ongeneralizationofTOPSISforneutrosophichypersoftset[29],on singleandmulti-valuedneutrosophichypersoftset[30],onmulti-valuedintervalneutrosophiclinguistic softset[15],andsonon.

Inthischapter,weintroducedhybridsetstructurewithhypersoftsetsunderuncertainty,vaguenessandindeterminacy,calledneutrosophicvalued n attributeneutrosophichypersoftset,whichis acombinationofneutrosophicsets[25]andhypersoftsets[31].Theneutrosophicvalued n attribute neutrosophichypersoftsetgeneralizesthefollowingsets:

(1) n attributeHypersoftSet[31], (2) n attributeFuzzyHypersoftSet[31], (3) n attributeIntuitionisticFuzzyHypersoftSet[31], (4) n attributeNeutrosophicHypersoftSet[31], (5) FuzzyValued n attributeHypersoftSet(Specialversionofthisstudy), (6) FuzzyValued n attributeFuzzyHypersoftSet(Specialversionofthisstudy), (7) FuzzyValued n attributeIntuitionisticFuzzyHypersoftSet(Specialversionofthisstudy), (8) FuzzyValued n attributeNeutrosophicHypersoftSet(Specialversionofthisstudy), (9) IntuitionisticFuzzyValued n attributeHypersoftSet(Specialversionofthisstudy), (10) IntuitionisticFuzzyValued n attributeFuzzyHypersoftSet(Specialversionofthisstudy), (11) IntuitionisticFuzzyValued n attributeIntuitionisticFuzzyHypersoftSet(Specialversionof thisstudy), (12) IntuitionisticFuzzyValued n attributeNeutrosophicHypersoftSet(Specialversionofthis study), (13) NeutrosophicValued n attributeHypersoftSet(Specialversionofthisstudy), (14) NeutrosophicValued n attributeFuzzyHypersoftSet(Specialversionofthisstudy), (15) NeutrosophicValued n attributeIntuitionisticFuzzyHypersoftSet(Specialversionofthis study), (16) Softsets[4,22], (17) fuzzysoftsets[19], (18) intuitionisticfuzzysoftsets[6,18,21], (19) neutrosophicsoftsets[8,16,17,20], (20) fuzzyparameterizedsoftsets[5], (21) fuzzyparameterizedfuzzysoftsets[12],

(22) fuzzyparameterizedintuitionisticfuzzysoftsets[10], (23) fuzzyparameterizedneutrosophicsoftsets(Specialversionof[13]), (24) intuitionisticfuzzyparameterizedsoftsets[9], (25) intuitionisticfuzzyparameterizedfuzzysoftsets[33], (26) intuitionisticfuzzyparameterizedintuitionisticfuzzysoftsets[11], (27) intuitionisticfuzzyparameterizedneutrosophicsoftsets(Specialversionof[13]), (28) neutrosophicparameterizedsoftsets[3], (29) neutrosophicparameterizedfuzzysoftsets(Specialversionof[13]), (30) neutrosophicparameterizedintuitionisticfuzzysoftsets(Specialversionof[13]), (31) neutrosophicparameterizedneutrosophicsoftsets[13], Therefore,theneutrosophicvalued n attributeneutrosophichypersoftsetisapowerfulgeneral formalframeworkandismostcomprehensiveofabovesetstohandletheindeterminateinformation andinconsistentinformationwhichexistcommonlyinrealsituations.Researchers,canbejuststudy onneutrosophicvalued n attributeneutrosophichypersoftsetswhicharethemostgeneralformofthe abovehybridsetsin(1-31).Thus,insteadofworkingseparately,itwillsavetimeanditwillbeeasier toresearchandfindsolutionsofproblemswhichcontainuncertaintiesorindeterminateinformation. Todothis,therestofthispaperisstructuredasfollows:Insection2,wegivethebasicdefinitions andresultsoffuzzysettheory,intuitionisticfuzzysettheory,neutrosophicsettheory,softsettheory, neutrosophicparameterizedneutrosophicsoftsettheoryandhypersoftsettheory.Insection3,we developtheHybridSetStructurewithHypersoftSetsunderUncertainty,VaguenessandIndeterminacy, calledneutrosophicvalued n attributeneutrosophichypersoftsets,includingsomespecialhybridset structures,NPNSS-aggregationoperator,decisionmakingalgorithm.Insection4,wepresentasection ofconclusions.

2. Preliminary

Inthissection,wegivethebasicdefinitionsandresultsoffuzzysettheory,intuitionisticfuzzyset theory,neutrosophicsettheory,softsettheory,neutrosophicparameterizedneutrosophicsoftsettheory andhypersoftsettheorythatareusefulforsubsequentdiscussions.Formoredetails,thereadercould referto[2,8,9,13,14,16,17,20,22,23,25,31,32,34].

Definition2.1.

1].Itcanbewrittenas

1] × [0, 1]into[0, 1].Thesepropertiesareformulatedwiththefollowingconditions: ∀a,b,c,d ∈ [0, 1],

(1) t(0, 0)=0and t(a, 1)= t(1,a)= a, (2) If a ≤ c and b ≤ d,then t(a,b) ≤ t(c,d)

(3) t(a,b)= t(b,a) (4) t(a,t(b,c))= t(t(a,b,c)) Definition2.3. [14] t-conorms(s-norm)areassociative,monotonicandcommutativetwoplaced functions s whichmapfrom[0, 1] × [0, 1]into[0, 1].Thesepropertiesareformulatedwiththefollowing conditions: ∀a,b,c,d ∈ [0, 1],

(1) s(1, 1)=1 ands(a, 0)= s(0,a)= a, (2) if a ≤ c and b ≤ d,then s(a,b) ≤ s(c,d) (3) s(a,b)= s(b,a) (4) s(a,s(b,c))= s(s(a,b,c) t-normand t-conormarerelatedinasenseoflogicalduality.Typicaldualpairsofnonparametrized t-normand t-conormarecompliedbelow:

(1) Drasticproduct: tw (a,b)= min{a,b},max{ab} =1 0,otherwise (2) Drasticsum: sw (a,b)= max{a,b},min{ab} =0 1,otherwise

(3) Boundedproduct: t1 (a,b)= max{0,a + b 1} (4) Boundedsum: s1 (a,b)= min{1,a + b} (5) Einsteinproduct: t1 5 (a,b)= a.b 2 [a + b a.b] (6) Einsteinsum: s1 5 (a,b)= a + b 1+ a.b

(7) Algebraicproduct: t2 (a,b)= a.b

(8) Algebraicsum: s2 (a,b)= a + b a.b

(9) Hamacherproduct: t2 5 (a,b)= a.b a + b a.b

(10) Hamachersum: s2 5 (a,b)= a + b 2.a.b 1 a.b

(11) Minumum: t3 (a,b)= min{a,b}

(12) Maximum: s3 (a,b)= max{a,b}

Definition2.4. [2]LetUbeaspaceofpoints(objects),withagenericelementinUdenotedbyu. Aintuitionisticfuzzyset(IF-set) IF inUischaracterizedbyamembershipfunction µIF : U → [0, 1] andanon-membershipfunction νIF : U → [0, 1].Itcanbewrittenas

IF = {<u, (µIF (u),νIF (u)) >: u ∈ U,µIF (u),µIF (u) ∈ [0, 1]}. where0 ≤ µIF (u)+ νIF (u) ≤ 1. Definition2.5. [32]LetUbeaspaceofpoints(objects),withagenericelementinUdenotedby u.Aneutrosophicset N inUischaracterizedbyatruth-membershipfunction TN : U → [0, 1],a indeterminacy-membershipfunction IN : U → [0, 1]andafalsity-membershipfunction FN : U → [0, 1]. Itcanbewrittenas N = {<u, (TN (u),IN (u),FN (u)) >: u ∈ U }. Thereisnorestrictiononthesumof TN (u); IN (u)and FN (u),so0 ≤ TN (u)+ IN (u)+ FN (u) ≤ 3. Also,forspecialcases,wehaveinthefollowinghybridsetstructures:

(1) Aneutrosophicsetfor IN (u)=0and FN (u)=1 TN (u)reducedtofuzzyset[34]as; N = {<u, (TN (u), 0, 1 TN (u))) >: u ∈ U }.

(2) Aneutrosophicsetfor IN (u)=0and0 ≤ FN (u)+ TN (u) ≤ 1reducedtointuitionisticfuzzy set[2]as;

N = {<u, (TN (u), 0,FN (u))) >: u ∈ U }.

Definition2.6. [22]Let U beaninitialuniverse, P (U )bethepowersetof U , E beasetofall parameters.Thenasoftset S over U isasetdefinedbyafunctionrepresentingamapping fS : E → P (U )Here, fX iscalledapproximatefunctionofthesoftset S,andthevalue fS (x)isasetcalled x-element ofthesoftsetforall x ∈ E.Thus,asoftsetover U canberepresentedbythesetofordered pairs S = {(x,fS (x)): x ∈ E,fS (x) ∈ P (U )}

Definition2.7. [31]Let U beauniverse, E beasetofattributesthataredescribetheelementsof U , Ei ⊆ E(i ∈{1, 2,...,n})be i setofattributessuchthat Ei ∩ Ej = ∅, forall i = j,and i,j ∈{1, 2,...,n}

Then,an n attributeHypersoftSet(n HS set) Sn HS over U isasetdefinedbyasetvaluedfunction fSn HS representingamapping

fSn HS : E1 × E2 × × En → P (U )

where fSn HS iscalledapproximatefunctionofthe Sn HS .For(x1 ,x2 ,...xn) ∈ E1 × E2 × ... × En, theset fSn HS (x1 ,x2 ,...xn)iscalled(x1 ,x2 ,...xn) approximationofthe Sn HS .Itcanbewrittenas asetoforderedpairs, Sn HS = ((x1 ,x2 ,...xn),fSn HS (x1 ,x2 ,...xn)):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En,

fSn HS (x1 ,x2 ,...xn) ∈ U

Definition2.8. [13]Let U beauniverse, N (U )bethesetofallneutrosophicsetsonU, E bea setofparametersthataredescribetheelementsof U and K beaneutrosophicsetover E.Then,a neutrosophicparameterizedneutrosophicsoftset(npn softset) SNpN over U isasetdefinedbyaset valuedfunction fSNpN representingamapping fSNpN : K → N (U ) where fSNpN iscalledapproximatefunctionofthe npn softset SNpN .For x ∈ E,theset fSNpN (x)is calledx-approximationofthe npn softset SNpN .Itcanbewrittenasetoforderedpairs, SNpN = (<x,TSNpN (x),ISNpN (x),FSNpN (x) >, {<u,TfSNpN (x) (u),IfSNpN (x) (u),FfSNpN (x) (u) >: u ∈ U }): x ∈ E where FSNpN (x),ISNpN (x),TSNpN (x),TSNpN (u),ISNpN (u),FSNpN (u) ∈ [0, 1]

3. HybridSetStructurewithHypersoftSetsunderUncertainty,VaguenessandIndeterminacy

Inthissection,wecombinedtheconceptofHypersoftset[31]andneutrosophicset[32],forgeneralizing npn softset[13],byintroducinganewhybridsetstructurecalledneutrosophicvalued n attribute neutrosophicHypersoftset(n NNHS set).Then,weintroducesomedefinitionsandoperationson n NNHS setsandsomepropertiesofthe n NNHS setswhichareconnectedtooperationshave beenproposed.Someofitisquotedorinspiredorgeneralizedfrom[8,9,13,16,17,20,23,25,31].

3.1. NeutrosophicValued n attributeNeutrosophicHypersoftSets

Definition3.1. Let U beauniverse, N (U )bethesetofallneutrosophicsetsonU, E beasetof attributesthataredescribedtheelementsof U , Ei ⊆ E(i ∈{1, 2,...,n})be i setofattributessuch that Ei ∩ Ej = ∅, forall i = j,and i,j ∈{1, 2,...,n} and Ki beaneutrosophicsetover Ei.Then,a

NeutrosophicValued n attributeNeutrosophicHypersoftSet(n NNHS set) Sn NNHS over U isa setdefinedbyasetvaluedfunction fSn NNHS representingamapping fSn NNHS : K1 × K2 × × Kn → N (U ) where fSn NNHS iscalledapproximatefunctionofthe Sn NNHS .For(x1 ,x2 ,...xn) ∈ E1 × E2 × ... × En,theset fSn NNHS (x1 ,x2 ,...xn)iscalled(x1 ,x2 ,...xn) approximationofthe Sn NNHS .Itcanbe writtenasasetoforderedpairs, Sn NNHS = ((<x1 ,TSn NNHS (x1 ),ISn NNHS (x1 ),FSn NNHS (x1 ) >,<x2 ,TSn NNHS (x2 ), ISn NNHS (x2 ),FSn NNHS (x2 ) >,...,<xn,TSn NNHS (xn),ISn NNHS (xn), FSn NNHS (xn) >), {<u,TSn NNHS (x1 ,x2 ,...xn ) (u),ISn NNHS (x1 ,x2 ,...xn ) (u), FSn NNHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × ... × En where

TSn NNHS (xi)(i ∈{1, 2,...,n}),ISn NNHS (xi)(i ∈{1, 2,...,n}),FSn NNHS (xi) ∈ [0, 1](i ∈ {1, 2,...,n}), and TSn NNHS (x1 ,x2 ,...xn ) (u),ISn NNHS (x1 ,x2 ,...xn ) (u),FSn NNHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1] suchthat 0 ≤ TSn NNHS (xi)+ ISn NNHS (xi),FSn NNHS (xi) ≤ 3(i ∈{1, 2,...,n}) and 0 ≤ TSn NNHS (x1 ,x2 ,...xn ) (u)+ ISn NNHS (x1 ,x2 ,...xn ) (u)+ FSn NNHS (x1 ,x2 ,...xn ) (u) ≤ 3

Notethatifthesetofattributes Ei issinglethenthe n NNHS setsreducedto npn softsets[13]. Therefore,the n NNHS setsgeneralizesthefollowingsets: (1) Softsets[22], (2) fuzzysoftsets[19], (3) intuitionisticfuzzysoftsets[6], (4) neutrosophicsoftsets[8], (5) fuzzyparameterizedsoftsets[5], (6) fuzzyparameterizedfuzzysoftsets[12], (7) fuzzyparameterizedintuitionisticfuzzysoftsets[10], (8) fuzzyparameterizedneutrosophicsoftsets(Specialversionof[13]), (9) intuitionisticfuzzyparameterizedsoftsets[9], (10) intuitionisticfuzzyparameterizedfuzzysoftsets[33], (11) intuitionisticfuzzyparameterizedintuitionisticfuzzysoftsets[11], (12) intuitionisticfuzzyparameterizedneutrosophicsoftsets(Specialversionof[13]), (13) neutrosophicparameterizedsoftsets[3],

(14) neutrosophicparameterizedfuzzysoftsets(Specialversionof[13]), (15) neutrosophicparameterizedintuitionisticfuzzysoftsets(Specialversionof[13]), (16) neutrosophicparameterizedneutrosophicsoftsets[13],

Definition3.2. Let Sn NNHS bean n NNHS set.Then,thecomplementof Sn NNHS denoted by Sc n NNHS andisdefinedby

Sc n NNHS = ((<x1 ,FSn NNHS (x1 ), 1 ISn NNHS (x1 ),TSn NNHS (x1 ) >,<x2 ,FSn NNHS (x2 ), 1 ISn NNHS (x2 ),TSn NNHS (x2 ) >,...,<xn,FSn NNHS (xn), 1 ISn NNHS (xn), TSn NNHS (xn) >), {<u,FSn NNHS (x1 ,x2 ,...xn ) (u), 1 ISn NNHS (x1 ,x2 ,...xn ) (u), TSn NNHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En

Definition3.3. Let S 1 n NNHS and S 2 n NNHS betwo n NNHS sets.Then, S 1 n NNHS issaidto be n NNHS subsetof S 2 n NNHS ,isdenotedby S 1 n NNHS ˜ ⊆S 2 n NNHS ,if

TS 1 n NNHS (xi) ≤ TS 2 n NNHS (xi),TS 1 n NNHS (x1 ,x2 ,...xn ) (u) ≤ TS 2 n NNHS (x1 ,x2 ,...xn ) (u), IS 1 n NNHS (xi) ≥ IS 2 n NNHS (xi),I 1 Sn NNHS (x1 ,x2 ,...xn ) (u) ≥ IS 2 n NNHS (x1 ,x2 ,...xn ) (u), FS 1 n NNHS (xi) ≥ FS 2 n NNHS (xi),FS 1 n NNHS (x1 ,x2 ,...xn ) (u) ≥ FS 2 n NNHS (x1 ,x2 ,...xn ) (u) forall i ∈{1, 2,...,n}. Also,If S 1 n NNHS is n NNHS subsetof S 2 n NNHS and S 2 n NNHS is n NNHS subsetof S 1 n NNHS ,then S 2 n NNHS and S 2 n NNHS isequalandwedenoteitwith S 1 n NNHS = S 1 n NNHS

Proposition3.4. Let S 1 n NNHS , S 2 n NNHS and S 3 n NNHS beanythree n NNHS sets.Then, (1) S 1 n NNHS ⊆S 1 n NNHS (2) S 1 n NNHS ⊆S 2 n NNHS and S 2 n NNHS ⊆S 3 n NNHS ⇒ S 1 n NNHS ⊆S 3 n NNHS ) (3) S 1 n NNHS = S 2 n NNHS and S 2 n NNHS = S 3 n NNHS ⇒ S 1 n NNHS = S 3 n NNHS )

Definition3.5. Let S 1 n NNHS bean n NNHS set.Then, S 1 n NNHS iscallednull n NNHS set, denotedby S∅,if

TS 1 n NNHS (xi)=0,TS 1 n NNHS (x1 ,x2 ,...xn ) (u)=0,

IS 1 n NNHS (xi)=1,IS 1 n NNHS (x1 ,x2 ,...xn ) (u)=1,

FS 1 n NNHS (xi)=1,FS 1 n NNHS (x1 ,x2 ,...xn ) (u)=1.

forall i ∈{1, 2,...,n} (or

TS 1 n NNHS (xi)=0,TS 1 n NNHS (x1 ,x2 ,...xn ) (u)=0,

IS 1 n NNHS (xi)=0,IS 1 n NNHS (x1 ,x2 ,...xn ) (u)=0,

FS 1 n NNHS (xi)=1,FS 1 n NNHS (x1 ,x2 ,...xn ) (u)=1

forall i ∈{1, 2,...,n}.(Thedefinitiondonotuseinthisstudy))

Definition3.6. Let S 1 n NNHS bean n NNHS set.Then, S 1 n NNHS iscalleduniversal n NNHS set,denotedby SU ,if

TS 1 n NNHS (xi)=1,TS 1 n NNHS (x1 ,x2 ,...xn ) (u)=1,

IS 1 n NNHS (xi)=0,I 1 Sn NNHS (x1 ,x2 ,...xn ) (u)=0,

FS 1 n NNHS (xi)=0,FS 1 n NNHS (x1 ,x2 ,...xn ) (u)=0 (or

TS 1 n NNHS (xi)=1,TS 1 n NNHS (x1 ,x2 ,...xn ) (u)=1,

IS 1 n NNHS (xi)=1,I 1 Sn NNHS (x1 ,x2 ,...xn ) (u)=1,

FS 1 n NNHS (xi)=0,FS 1 n NNHS (x1 ,x2 ,...xn ) (u)=0 forall i ∈{1, 2,...,n}.(Thedefinitiondonotuseinthisstudy))

Proposition3.7. Let S 1 n NNHS , S 2 n NNHS and S 3 n NNHS beanythree n NNHS sets.Then, (1) (SU )c = S∅ (2) (S∅)c = SU (3) S 1 n NNHS ⊆SU (4) ((S 1 n NNHS )c)c = S 1 n NNHS (5) S∅⊆S 1 n NNHS Definition3.8. Let S 1 n NNHS and S 2 n NNHS betwo n NNHS sets.Then,theunion(orsum) of S 1 n NNHS and S 2 n NNHS isdenotedby S 3 n NNHS = S 1 n NNHS ∪S 2 n NNHS (or S 3 n NNHS = S 1 n NNHS +S 2 n NNHS )andisdefinedby S 3 n NNHS = ((<x1 ,TS 3 n NNHS (x1 ),IS 3 n NNHS (x1 ),FS 3 n NNHS (x1 ) >,<x2 ,TS 3 n NNHS (x2 ), IS 3 n NNHS (x2 ),FS 3 n NNHS (x2 ) >,...,<xn,TS 3 n NNHS (xn),IS 3 n NNHS (xn), FS 3 n NNHS (xn) >), {<u,TS 3 n NNHS (x1 ,x2 ,...xn ) (u),IS 3 n NNHS (x1 ,x2 ,...xn ) (u), FS 3 n NNHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En

where TS 3 n NNHS (xi)= s(TS 1 n NNHS (xi),TS 2 n NNHS (xi)),

TS 3 n NNHS (x1 ,x2 ,...xn ) (u)= s(TS 1 n NNHS (x1 ,x2 ,...xn ) (u),TS 2 n NNHS (x1 ,x2 ,...xn ) (u)),

IS 3 n NNHS (xi)= t(IS 1 n NNHS (xi),IS 2 n NNHS (xi)),

IS 3 n NNHS (x1 ,x2 ,...xn ) (u)= t(IS 1 n NNHS (x1 ,x2 ,...xn ) (u),IS 2 n NNHS (x1 ,x2 ,...xn ) (u)),

FS 3 n NNHS (xi)= t(FS 1 n NNHS (xi),FS 2 n NNHS (xi)),

FS 3 n NNHS (x1 ,x2 ,...xn ) (u)= t(FS 1 n NNHS (x1 ,x2 ,...xn ) (u),FS 2 n NNHS (x1 ,x2 ,...xn ) (u))

forall i ∈{1, 2,...,n}.

Proposition3.9. Let S 1 n NNHS , S 2 n NNHS and S 3 n NNHS beanythree n NNHS sets.Then, (1) S 1 n NNHS ∪SU = SU (2) S 1 n NNHS ∪S∅ = S 1 n NNHS (3) S 1 n NNHS ∪(S 1 n NNHS =(S 1 n NNHS (4) S 1 n NNHS ∪S 2 n NNHS = S 2 n NNHS ∪S 1 n NNHS (5) S 1 n NNHS ∪(S 2 n NNHS ∪S 3 n NNHS )=(S 1 n NNHS ∪S 2 n NNHS )∪S 3 n NNHS

Definition3.10. Let S 1 n NNHS and S 2 n NNHS betwo n NNHS sets.Then,theintersection(or product)of S 1 n NNHS and S 2 n NNHS isdenotedby S 4 n NNHS = S 1 n NNHS ∩S 2 n NNHS (or S 4 n NNHS = S 1 n NNHS ˜ ×S 2 n NNHS )andisdefinedby S 4 n NNHS = ((<x1 ,TS 4 n NNHS (x1 ),IS 4 n NNHS (x1 ),FS 4 n NNHS (x1 ) >,<x2 ,TS 4 n NNHS (x2 ), IS 4 n NNHS (x2 ),FS 4 n NNHS (x2 ) >,...,<xn,TS 4 n NNHS (xn),IS 4 n NNHS (xn), FS 4 n NNHS (xn) >), {<u,TS 4 n NNHS (x1 ,x2 ,...xn ) (u),IS 4 n NNHS (x1 ,x2 ,...xn ) (u), FS 4 n NNHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En

where

TS 4 n NNHS (xi)= t(TS 1 n NNHS (xi),TS 2 n NNHS (xi)),

TS 4 n NNHS (x1 ,x2 ,...xn ) (u)= t(TS 1 n NNHS (x1 ,x2 ,...xn ) (u),TS 2 n NNHS (x1 ,x2 ,...xn ) (u)),

IS 4 n NNHS (xi)= s(IS 1 n NNHS (xi),IS 2 n NNHS (xi)),

IS 4 n NNHS (x1 ,x2 ,...xn ) (u)= s(IS 1 n NNHS (x1 ,x2 ,...xn ) (u),IS 2 n NNHS (x1 ,x2 ,...xn ) (u)),

FS 4 n NNHS (xi)= s(FS 1 n NNHS (xi),FS 2 n NNHS (xi)),

FS 4 n NNHS (x1 ,x2 ,...xn ) (u)= s(FS 1 n NNHS (x1 ,x2 ,...xn ) (u),FS 2 n NNHS (x1 ,x2 ,...xn ) (u)). forall i ∈{1, 2,...,n}.

Proposition3.11. Let S 1 n NNHS , S 2 n NNHS and S 3 n NNHS beanythree n NNHS sets.Then, (1) S 1 n NNHS ∩SU = S 1 n NNHS (2) S 1 n NNHS ∩S∅ = S∅ (3) S 1 n NNHS ∩S 1 n NNHS = S 1 n NNHS (4) S 1 n NNHS ∩S 2 n NNHS = S 2 n NNHS ∩S 1 n NNHS (5) S 1 n NNHS ∩(S 2 n NNHS ∩S 3 n NNHS )=(S 1 n NNHS ∩S 2 n NNHS )∩S 3 n NNHS

Definition3.12. Let S 1 n NNHS bean n NNHS setand γ =0.Then, S 5 n NNHS = γ(S 1 n NNHS ) isdefinedas; S 5 n NNHS = ((<x1 ,TS 5 n NNHS (x1 ),IS 5 n NNHS (x1 ),FS 5 n NNHS (x1 ) >,<x2 ,TS 5 n NNHS (x2 ), IS 5 n NNHS (x2 ),FS 5 n NNHS (x2 ) >,...,<xn,TS 5 n NNHS (xn),IS 5 n NNHS (xn), FS 5 n NNHS (xn) >), {<u,TS 5 n NNHS (x1 ,x2 ,...xn ) (u),IS 5 n NNHS (x1 ,x2 ,...xn ) (u), FS 5 n NNHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × ... × En where

TS 5 n NNHS (xi)=1 (1 TS 1 n NNHS (xi))γ ,

TS 5 n NNHS (x1 ,x2 ,...xn ) (u)=1 (1 TS 1 n NNHS (x1 ,x2 ,...xn ) (u))γ ,

IS 5 n NNHS (xi)=(IS 1 n NNHS (xi))γ ,

I 5 Sn NNHS (x1 ,x2 ,...xn ) (u)=(IS 1 n NNHS (x1 ,x2 ,...xn ) (u))γ ,

FS 5 n NNHS (xi)=(FS 1 n NNHS (xi))γ ,

FS 5 n NNHS (x1 ,x2 ,...xn ) (u)=(FS 1 n NNHS (x1 ,x2 ,...xn ) (u))γ

forall i ∈{1, 2,...,n}

Definition3.13. Let S 1 n NNHS bean n NNHS setand γ =0.Then, S 6 n NNHS =(S 1 n NNHS )γ isdefinedas; S 6 n NNHS = ((<x1 ,TS 6 n NNHS (x1 ),IS 6 n NNHS (x1 ),FS 6 n NNHS (x1 ) >,<x2 ,TS 6 n NNHS (x2 ),

IS 6 n NNHS (x2 ),FS 6 n NNHS (x2 ) >,...,<xn,TS 6 n NNHS (xn),IS 6 n NNHS (xn), FS 6 n NNHS (xn) >), {<u,TS 6 n NNHS (x1 ,x2 ,...xn ) (u),IS 6 n NNHS (x1 ,x2 ,...xn ) (u), FS 6 n NNHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En

where

TS 6 n NNHS (xi)=(TS 1 n NNHS (xi))γ ,

TS 6 n NNHS (x1 ,x2 ,...xn ) (u)=(TS 1 n NNHS (x1 ,x2 ,...xn ) (u))γ ,

IS 6 n NNHS (xi)=1 (1 IS 1 n NNHS (xi))γ ,

I 6 Sn NNHS (x1 ,x2 ,...xn ) (u)=1 (1 IS 1 n NNHS (x1 ,x2 ,...xn ) (u))γ ,

FS 6 n NNHS (xi)=1 (1 FS 1 n NNHS (xi))γ ,

FS 6 n NNHS (x1 ,x2 ,...xn ) (u)=1 (1 FS 1 n NNHS (x1 ,x2 ,...xn ) (u))γ forall i ∈{1, 2,...,n}.

Proposition3.14. Let S 1 n NNHS , S 2 n NNHS and S 3 n NNHS beanythree n NNHS setand γ1 ,γ2 ,γ =0.Then, (1) S 1 n NNHS ∪(S 2 n NNHS ∩S 3 n NNHS )=(S 1 n NNHS ∪S 2 n NNHS )∩(S 1 n NNHS ∪S 3 n NNHS ) (2) S 1 n NNHS ∩(S 2 n NNHS ∪S 3 n NNHS )=(S 1 n NNHS ∩S 2 n NNHS )∪(S 1 n NNHS ∩S 3 n NNHS ) (3) (S 1 n NNHS ∩S 2 n NNHS )c =(S 1 n NNHS )c∪(S 2 n NNHS )c (4) (S 1 n NNHS ∪S 2 n NNHS )c =(S 1 n NNHS )c∩(S 2 n NNHS )c

NotethattheproofsofProposition3.4-3.14canbeeasilyobtainedaccordingtoDefinition3.1-3.13

S 1 3 NNHS = ((<x1 , (0 1, 0 5, 0 4) >,<x4 , (0 2, 0 8, 0 9) >,<x6 , (0 1, 0 2, 0 7) >); {<u1 , (0.2, 0.7, 0.4) >,<u2 , (0.7, 0.2, 0.3) >,<u3 , (0.5, 0.6, 0.2) >}), ((<x1 , (0 4, 0 8, 0 9) >,<x5 , (0 4, 0 8, 0 7) >,<x6 , (0 7, 0 4, 0 9) >); {<u1 , (0 5, 0 4, 0 7) >,<u2 , (0 2, 0 3, 0 7) >,<u3 , (0 4, 0 3, 0 7) >}), ((<x2 , (0 7, 0 8, 0 4) >,<x4 , (0 3, 0 5, 0 1) >,<x6 , (0 4, 0 5, 0 1) >); {<u1 , (0 4, 0 5, 0 3) >,<u2 , (0 2, 0 2, 0 2) >,<u3 , (0 8, 0 7, 0 4) >}), ((<x2 , (0.4, 0.4, 0.9) >,<x5 , (0.4, 0.5, 0.2) >,<x6 , (0.2, 0.5, 0.8) >);

{<u1 , (0 5, 0 6, 0 7) >,<u2 , (0 8, 0 5, 0 5) >,<u3 , (0 5, 0 3, 0 2) >}), ((<x3 , (0 6, 0 7, 0 8) >,<x4 , (0 5, 0 7, 0 1) >,<x6 , (0 4, 0 1, 0 8) >);

{<u1 , (0.1, 0.2, 0.8) >,<u2 , (0.5, 0.3, 0.3) >,<u3 , (0.5, 0.4, 0.6) >}), ((<x3 , (0 7, 0 4, 0 1) >,<x5 , (0 9, 0 7, 0 5) >,<x6 , (0 5, 0 2, 0 2) >); {<u1 , (0 8, 0 3, 0 7) >,<u2 , (0 6, 0 7, 0 1) >,<u3 , (0 7, 0 5, 0 7) >}) and S 2 3 NNHS = ((<x1 , (0 2, 0 3, 0 4) >,<x4 , (0 3, 0 1, 0 9) >,<x6 , (0 2, 0 5, 0 5) >); {<u1 , (0.5, 0.2, 0.3) >,<u2 , (0.7, 0.4, 0.1) >,<u3 , (0.2, 0.5, 0.3) >}), ((<x1 , (0 5, 0 8, 0 8) >,<x5 , (0 5, 0 9, 0 7) >,<x6 , (0 7, 0 5, 0 9) >); {<u1 , (0 5, 0 5, 0 7) >,<u2 , (0 6, 0 7, 0 7) >,<u3 , (0 5, 0 7, 0 7) >}), ((<x2 , (0.7, 0.8, 0.5) >,<x4 , (0.7, 0.5, 0.1) >,<x6 , (0.5, 0.5, 0.1) >); {<u1 , (0 5, 0 5, 0 7) >,<u2 , (0 6, 0 6, 0 6) >,<u3 , (0 9, 0 7, 0 5) >}), ((<x2 , (0 5, 0 5, 0 9) >,<x5 , (0 5, 0 5, 0 6) >,<x6 , (0 6, 0 5, 0 9) >); {<u1 , (0.5, 0.6, 0.7) >,<u2 , (0.9, 0.5, 0.5) >,<u3 , (0.5, 0.7, 0.6) >}), ((<x3 , (0 6, 0 7, 0 9) >,<x4 , (0 5, 0 7, 0 1) >,<x6 , (0 5, 0 1, 0 9) >); {<u1 , (0.1, 0.6, 0.8) >,<u2 , (0.5, 0.7, 0.7) >,<u3 , (0.5, 0.5, 0.6) >}), ((<x3 , (0 7, 0 5, 0 1) >,<x5 , (0 9, 0 7, 0 5) >,<x6 , (0 5, 0 6, 0 6) >);

{<u1 , (0.8, 0.7, 0.7) >,<u2 , (0.6, 0.7, 0.1) >,<u3 , (0.7, 0.5, 0.7) >})

Therefore, (1) Sc n NNHS iscomputedas;

S 1 c n NNHS = ((<x1 , (0 4, 0 5, 0 1) >,<x4 , (0 9, 0 2, 0 2) >,<x6 , (0 7, 0 8, 0 1) >); {<u1 , (0 4, 0 3, 0 2) >,<u2 , (0 3, 0 8, 0 7) >,<u3 , (0 2, 0 4, 0 5) >}), ((<x1 , (0.9, 0.2, 0.4) >,<x5 , (0.7, 0.2, 0.4) >,<x6 , (0.9, 0.6, 0.7) >); {<u1 , (0 7, 0 6, 0 5) >,<u2 , (0 7, 0 7, 0 2) >,<u3 , (0 7, 0 7, 0 4) >}), ((<x2 , (0.4, 0.2, 0.7) >,<x4 , (0.1, 0.5, 0.3) >,<x6 , (0.1, 0.5, 0.4) >); {<u1 , (0 3, 0 5, 0 4) >,<u2 , (0 2, 0 8, 0 2) >,<u3 , (0 4, 0 3, 0 8) >}), ((<x2 , (0 9, 0 6, 0 4) >,<x5 , (0 2, 0 5, 0 4) >,<x6 , (0 8, 0 5, 0 2) >); {<u1 , (0.7, 0.4, 0.5) >,<u2 , (0.5, 0.5, 0.8) >,<u3 , (0.2, 0.7, 0.5) >}), ((<x3 , (0 8, 0 3, 0 6) >,<x4 , (0 1, 0 3, 0 5) >,<x6 , (0 8, 0 9, 0 4) >); {<u1 , (0 8, 0 8, 0 1) >,<u2 , (0 3, 0 7, 0 5) >,<u3 , (0 6, 0 6, 0 5) >}), ((<x3 , (0.1, 0.6, 0.7) >,<x5 , (0.5, 0.3, 0.9) >,<x6 , (0.2, 0.8, 0.5) >); {<u1 , (0.7, 0.7, 0.8) >,<u2 , (0.1, 0.3, 0.6) >,<u3 , (0.7, 0.5, 0.7) >}) (2) S 3 n NNHS = S 1 n NNHS ˜ ∪S 2 n NNHS iscomputedas;

S 3 n NNHS = ((<x1 , (0 28, 0 15, 0 24) >,<x4 , (0 44, 0 08, 0 81) >,<x6 , (0 28, 0 10, 0 35) >); {<u1 , (0.60, 0.14, 0.12) >,<u2 , (0.91, 0.08, 0.03) >,<u3 , (0.60, 0.30, 0.06) >}), ((<x1 , (0 70, 0 64, 0 72) >,<x5 , (0 70, 0 72, 0 49) >,<x6 , (0 91, 0 20, 0 81) >); {<u1 , (0.75, 0.20, 0.49) >,<u2 , (0.68, 0.21, 0.49) >,<u3 , (0.70, 0.21, 0.49) >}), ((<x2 , (0 91, 0 64, 0 20) >,<x4 , (0 79, 0 25, 0 01) >,<x6 , (0 70, 0 25, 0 01) >); {<u1 , (0 70, 0 25, 0 21) >,<u2 , (0 68, 0 12, 0 12) >,<u3 , (0 98, 0 49, 0 20) >}), ((<x2 , (0.70, 0.20, 0.81) >,<x5 , (0.70, 0.25, 0.12) >,<x6 , (0.68, 0.25, 0.72) >);

{<u1 , (0 75, 0 36, 0 49) >,<u2 , (0 98, 0 25, 0 25) >,<u3 , (0 75, 0 21, 0 12) >}), ((<x3 , (0 84, 0 49, 0 72) >,<x4 , (0 75, 0 49, 0 01) >,<x6 , (0 70, 0 01, 0 72) >);

{<u1 , (0.19, 0.12, 0.64) >,<u2 , (0.75, 0.21, 0.21) >,<u3 , (0.75, 0.20, 0.36) >}),

((<x3 , (0 91, 0 20, 0 01) >,<x5 , (0 99, 0 49, 0 25) >,<x6 , (0 75, 0 12, 0 12) >);

{<u1 , (0 96, 0 21, 0 49) >,<u2 , (0 84, 0 49, 0 01) >,<u3 , (0 91, 0 25, 0 49) >})

S 4 n NNHS = ((<x1 , (0.02, 0.65, 0.76) >,<x4 , (0.06, 0.82, 0.99) >,<x6 , (0.02, 0.60, 0.85) >);

{<u1 , (0 10, 0 76, 0 58) >,<u2 , (0 49, 0 52, 0 37) >,<u3 , (0 10, 0 80, 0 44) >}),

((<x1 , (0 20, 0 96, 0 98) >,<x5 , (0 20, 0 98, 0 91) >,<x6 , (0 49, 0 70, 0 99) >);

{<u1 , (0 25, 0 70, 0 91) >,<u2 , (0 12, 0 79, 0 91) >,<u3 , (0 20, 0 79, 0 91) >}),

((<x2 , (0 49, 0 96, 0 70) >,<x4 , (0 21, 0 75, 0 19) >,<x6 , (0 20, 0 75, 0 19) >);

{<u1 , (0.20, 0.75, 0.79) >,<u2 , (0.12, 0.68, 0.68) >,<u3 , (0.72, 0.91, 0.70) >}),

((<x2 , (0 20, 0 70, 0 99) >,<x5 , (0 20, 0 75, 0 68) >,<x6 , (0 12, 0 75, 0 98) >);

{<u1 , (0 25, 0 84, 0 91) >,<u2 , (0 72, 0 75, 0 75) >,<u3 , (0 25, 0 79, 0 68) >}),

((<x3 , (0.36, 0.91, 0.98) >,<x4 , (0.25, 0.91, 0.19) >,<x6 , (0.20, 0.19, 0.98) >);

{<u1 , (0 01, 0 68, 0 96) >,<u2 , (0 25, 0 79, 0 79) >,<u3 , (0 25, 0 70, 0 84) >}), ((<x3 , (0 49, 0 70, 0 19) >,<x5 , (0 81, 0 91, 0 75) >,<x6 , (0 25, 0 68, 0 68) >); {<u1 , (0 64, 0 79, 0 91) >,<u2 , (0 36, 0 91, 0 19) >,<u3 , (0 49, 0 75, 0 91) >})

S 5 3 NNHS = ((<x1 , (0.19, 0.25, 0.36) >,<x4 , (0.36, 0.64, 0.81) >,<x6 , (0.19, 0.04, 0.49) >);

{<u1 , (0 36, 0 49, 0 16) >,<u2 , (0 91, 0 04, 0 09) >,<u3 , (0 75, 0 36, 0 04) >}), ((<x1 , (0 64, 0 64, 0 81) >,<x5 , (0 64, 0 64, 0 49) >,<x6 , (0 91, 0 16, 0 81) >);

{<u1 , (0.75, 0.16, 0.49) >,<u2 , (0.36, 0.09, 0.49) >,<u3 , (0.64, 0.09, 0.49) >}), ((<x2 , (0 91, 0 64, 0 16) >,<x4 , (0 51, 0 25, 0 01) >,<x6 , (0 64, 0 25, 0 01) >); {<u1 , (0 64, 0 25, 0 09) >,<u2 , (0 36, 0 04, 0 04) >,<u3 , (0 96, 0 49, 0 16) >}),

((<x2 , (0 64, 0 16, 0 81) >,<x5 , (0 64, 0 25, 0 04) >,<x6 , (0 36, 0 25, 0 64) >);

{<u1 , (0 75, 0 36, 0 49) >,<u2 , (0 96, 0 25, 0 25) >,<u3 , (0 75, 0 09, 0 04) >}),

((<x3 , (0.75, 0.49, 0.01) >,<x4 , (0.75, 0.49, 0.01) >,<x6 , (0.64, 0.01, 0.64) >);

{<u1 , (0 19, 0 04, 0 64) >,<u2 , (0 75, 0 09, 0 09) >,<u3 , (0 75, 0 16, 0 36) >}),

((<x3 , (0 91, 0 16, 0 01) >,<x5 , (0 99, 0 49, 0 25) >,<x6 , (0 75, 0 04, 0 04) >);

{<u1 , (0 96, 0 09, 0 49) >,<u2 , (0 84, 0 49, 0 01) >,<u3 , (0 91, 0 25, 0 49) >})

(4) S 5 n NNHS =2(S 1 n NNHS )iscomputedas;

(5) S 6 n NNHS =(S 1 n NNHS )2 iscomputedas;

S 6 3 NNHS = ((<x1 , (0 01, 0 75, 0 84) >,<x4 , (0 04, 0 96, 0 99) >,<x6 , (0 01, 0 36, 0 91) >);

{<u1 , (0.04, 0.91, 0.64) >,<u2 , (0.49, 0.36, 0.51) >,<u3 , (0.25, 0.84, 0.36) >}),

((<x1 , (0 16, 0 96, 0 99) >,<x5 , (0 16, 0 96, 0 91) >,<x6 , (0 49, 0 64, 0 99) >);

{<u1 , (0 25, 0 64, 0 91) >,<u2 , (0 04, 0 51, 0 91) >,<u3 , (0 16, 0 51, 0 91) >}),

((<x2 , (0 49, 0 96, 0 64) >,<x4 , (0 09, 0 75, 0 19) >,<x6 , (0 16, 0 75, 0 19) >);

{<u1 , (0 16, 0 75, 0 51) >,<u2 , (0 04, 0 36, 0 36) >,<u3 , (0 64, 0 91, 0 64) >}),

((<x2 , (0.16, 0.64, 0.99) >,<x5 , (0.16, 0.75, 0.36) >,<x6 , (0.04, 0.75, 0.96) >);

{<u1 , (0 25, 0 84, 0 91) >,<u2 , (0 64, 0 75, 0 75) >,<u3 , (0 25, 0 51, 0 36) >}),

((<x3 , (0 36, 0 91, 0 96) >,<x4 , (0 25, 0 91, 0 19) >,<x6 , (0 16, 0 19, 0 96) >);

{<u1 , (0.01, 0.36, 0.96) >,<u2 , (0.25, 0.51, 0.51) >,<u3 , (0.25, 0.64, 0.84) >}),

((<x3 , (0 49, 0 64, 0 19) >,<x5 , (0 81, 0 91, 0 75) >,<x6 , (0 25, 0 36, 0 36) >);

{<u1 , (0 64, 0 51, 0 91) >,<u2 , (0 36, 0 91, 0 19) >,<u3 , (0 49, 0 75, 0 91) >})

3.2. Hybridsetstructures

Let Sn NNHS an n NNHS setover U as;

Sn NNHS = ((<x1 ,TSn NNHS (x1 ),ISn NNHS (x1 ),FSn NNHS (x1 ) >,<x2 ,TSn NNHS (x2 ), ISn NNHS (x2 ),FSn NNHS (x2 ) >,...,<xn,TSn NNHS (xn),ISn NNHS (xn), FSn NNHS (xn) >), {<u,TSn NNHS (x1 ,x2 ,...xn ) (u),ISn NNHS (x1 ,x2 ,...xn ) (u), FSn NNHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En where

TSn NNHS (xi)(i ∈{1, 2,...,n}),ISn NNHS (xi)(i ∈{1, 2,...,n}),FSn NNHS (xi)(i ∈{1, 2,...,n}) ∈ [0, 1], and TSn NNHS (x1 ,x2 ,...xn ) (u),ISn NNHS (x1 ,x2 ,...xn ) (u),FSn NNHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1] suchthat 0 ≤ TSn NNHS (xi)+ ISn NNHS (xi),FSn NNHS (xi) ≤ 3(i ∈{1, 2,...,n}) and 0 ≤ TSn NNHS (x1 ,x2 ,...xn ) (u)+ ISn NNHS (x1 ,x2 ,...xn ) (u)+ FSn NNHS (x1 ,x2 ,...xn ) (u) ≤ 3 Then,wehaveinthefollowinghybridsetstructures

(1) an n attributeHypersoftSet(n HS set) Sn HS over U canbewrittenasasetofordered pairsas;

Sn HS = ((<x1 ,TSn HS (x1 ), 0, 1 TSn HS (x1 ) >,<x2 ,TSn HS (x2 ), 0, 1 TSn HS (x2 ) >,...,<xn,TSn HS (xn), 0, 1 TSn HS (xn) >), {<u,TSn HS (x1 ,x2 ,...xn ) (u), 0, 1 TSn HS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × ... × En where TSn HS (xi)(i ∈{1, 2,...,n}) ∈{0, 1} and TSn HS (x1 ,x2 ,...xn ) (u) ∈{0, 1}

(2) an n attributeFuzzyHypersoftSet(n FHS set) Sn FHS over U canbewrittenasasetof orderedpairsas;

Sn FHS = ((<x1 ,TSn FHS (x1 ), 0, 1 TSn FHS (x1 ) >,<x2 ,TSn FHS (x2 ), 0, 1 TSn FHS (x2 ) >,...,<xn,TSn FHS (xn), 0, 1 TSn FHS (xn) >), {<u,TSn FHS (x1 ,x2 ,...xn ) (u), 0, 1 TSn FHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × ... × En where TSn FHS (xi)(i ∈{1, 2,...,n}) ∈{0, 1} and TSn FHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1].

(3) an n attributeIntuitionisticFuzzyHypersoftSet(n IFHS set) Sn IFHS over U canbe writtenasasetoforderedpairsas;

Sn IFHS = ((<x1 ,TSn IFHS (x1 ), 0, 1 TSn IFHS (x1 ) >,<x2 ,TSn IFHS (x2 ), 0, 1 TSn IFHS (x2 ) >,...,<xn,TSn IFHS (xn), 0, 1 TSn IFHS (xn) >), {<u,TSn IFHS (x1 ,x2 ,...xn ) (u), 0,FSn IFHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × ... × En where TSn IFHS (xi)(i ∈{1, 2,...,n}) ∈{0, 1},and TSn IFHS (x1 ,x2 ,...xn ) (u), FSn IFHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1] suchthat0 ≤ TSn IFHS (x1 ,x2 ,...xn ) (u)+ FSn IFHS (x1 ,x2 ,...xn ) (u) ≤ 1

where TSn NHS (xi)(i ∈{1, 2,...,n}) ∈{0, 1} and TSn NHS (x1 ,x2 ,...xn ) (u),ISn NHS (x1 ,x2 ,...xn ) (u),FSn NHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1] suchthat 0 ≤ TSn NHS (x1 ,x2 ,...xn ) (u)+ ISn NHS (x1 ,x2 ,...xn ) (u)+ FSn NHS (x1 ,x2 ,...xn ) (u) ≤ 3.

(5) aFuzzyValued n attributeHypersoftSet(n FHS set) Sn FHS over U canbewrittenasa setoforderedpairsas;

Sn FHS = ((<x1 ,TSn FHS (x1 ), 0, 1 TSn FHS (x1 ) >,<x2 ,TSn FHS (x2 ), 0, 1 TSn FHS (x2 ) >,...,<xn,TSn FHS (xn), 0, 1 TSn FHS (xn) >), {<u,TSn FHS (x1 ,x2 ,...xn ) (u), 0, 1 TSn FHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En where TSn FHS (xi)(i ∈{1, 2,...,n}) ∈ [0, 1]and TSn FHS (x1 ,x2 ,...xn ) (u) ∈{0, 1}

(6) aFuzzyValued n attributeFuzzyHypersoftSet(n FFHS set) Sn FFHS over U canbe writtenasasetoforderedpairsas;

Sn FFHS = ((<x1 ,TSn FFHS (x1 ), 0, 1 TSn FFHS (x1 ) >,<x2 ,TSn FFHS (x2 ), 0, 1 TSn FFHS (x2 ) >,...,<xn,TSn FFHS (xn), 0, 1 TSn FFHS (xn) >), {<u,TSn FFHS (x1 ,x2 ,...xn ) (u), 0, 1 TSn FFHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En where TSn FFHS (xi)(i ∈{1, 2,...,n}) ∈ [0, 1]and TSn FFHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1].

(7) aFuzzyValued n attributeIntuitionisticFuzzyHypersoftSet(n FIFHS set) Sn FIFHS over U canbewrittenasasetoforderedpairsas;

Sn FIFHS = ((<x1 ,TSn FIFHS (x1 ), 0, 1 TSn FIFHS (x1 ) >,<x2 ,TSn FIFHS (x2 ), 0, 1 TSn FIFHS (x2 ) >,...,<xn,TSn FIFHS (xn), 0, 1 TSn FIFHS (xn) >), {<u,TSn FIFHS (x1 ,x2 ,...xn ) (u), 0,FSn FIFHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En where TSn FIFHS (xi)(i ∈{1, 2,...,n}) ∈ [0, 1],and TSn FIFHS (x1 ,x2 ,...xn ) (u), FSn FIFHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1] suchthat

0 ≤ TSn FIFHS (xi) ≤ 1(i ∈{1, 2,...,n}) and 0 ≤ TSn FIFHS (x1 ,x2 ,...xn ) (u)+ FSn FIFHS (x1 ,x2 ,...xn ) (u) ≤ 1.

(8) aFuzzyValued n attributeNeutrosophicHypersoftSet(n FNHS set) Sn FNHS over U canbewrittenasasetoforderedpairsas; Sn FNHS = ((<x1 ,TSn FNHS (x1 ), 0, 1 TSn FNHS (x1 ) >,<x2 ,TSn FNHS (x2 ), 0, 1 TSn FNHS (x2 ) >,...,<xn,TSn FNHS (xn), 0, 1 TSn FNHS (xn) >), {<u,TSn FNHS (x1 ,x2 ,...xn ) (u),ISn FNHS (x1 ,x2 ,...xn ) (u),FSn FNHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En where TSn FNHS (xi)(i ∈{1, 2,...,n}) ∈ [0, 1] and TSn FNHS (x1 ,x2 ,...xn ) (u),ISn FNHS (x1 ,x2 ,...xn ) (u),FSn FNHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1] suchthat 0 ≤ TSn FNHS (xi) ≤ 1(i ∈{1, 2,...,n}) and 0 ≤ TSn FNHS (x1 ,x2 ,...xn ) (u)+ ISn FNHS (x1 ,x2 ,...xn ) (u)+ FSn FNHS (x1 ,x2 ,...xn ) (u) ≤ 3.

(9) anIntuitionisticFuzzyValued n attributeHypersoftSet(n IFHS set) Sn IFHS over U canbewrittenasasetoforderedpairsas;

Sn IFHS = ((<x1 ,TSn IFHS (x1 ), 0,FSn IFHS (x1 ) >,<x2 ,TSn IFHS (x2 ), 0,FSn IFHS (x2 ) >,...,<xn,TSn IFHS (xn), 0, FSn IFHS (xn) >), {<u,TSn IFHS (x1 ,x2 ,...xn ) (u), 0, 1 TSn IFHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En where TSn IFHS (xi)(i ∈{1, 2,...,n}),FSn IFHS (xi)(i ∈{1, 2,...,n}) ∈ [0, 1],and TSn IFHS (x1 ,x2 ,...xn ) (u) ∈{0, 1} suchthat 0 ≤ TSn IFHS (xi)+ FSn IFHS (xi) ≤ 1(i ∈{1, 2,...,n}).

(10) anIntuitionisticFuzzyValued n attributeFuzzyHypersoftSet(n IFFHS set) Sn IFFHS over U canbewrittenasasetoforderedpairsas;

Sn IFFHS = ((<x1 ,TSn IFFHS (x1 ), 0,FSn IFFHS (x1 ) >,<x2 ,TSn IFFHS (x2 ), 0,FSn IFFHS (x2 ) >,...,<xn,TSn IFFHS (xn), 0, FSn IFFHS (xn) >), {<u,TSn IFFHS (x1 ,x2 ,...xn ) (u), 0, 1 TSn IFFHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En where TSn IFFHS (xi)(i ∈{1, 2,...,n}),FSn IFFHS (xi)(i ∈{1, 2,...,n}) ∈ [0, 1],and TSn IFFHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1] suchthat 0 ≤ TSn IFFHS (xi)+ FSn IFFHS (xi) ≤ 1(i ∈{1, 2,...,n}) and 0 ≤ TSn IFFHS (x1 ,x2 ,...xn ) (u) ≤ 1 (11) anIntuitionisticFuzzyValued n attributeIntuitionisticFuzzyHypersoftSet(n IFIFHS set) Sn IFIFHS over U canbewrittenasasetoforderedpairsas; Sn IFIFHS = ((<x1 ,TSn IFIFHS (x1 ), 0,FSn IFIFHS (x1 ) >,<x2 ,TSn IFIFHS (x2 ), 0,FSn IFIFHS (x2 ) >,...,<xn,TSn IFIFHS (xn), 0, FSn IFIFHS (xn) >), {<u,TSn IFIFHS (x1 ,x2 ,...xn ) (u), 0,FSn IFIFHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En where TSn IFIFHS (xi)(i ∈{1, 2,...,n}),FSn IFIFHS (xi)(i ∈{1, 2,...,n}) ∈ [0, 1],and TSn IFIFHS (x1 ,x2 ,...xn ) (u),FSn IFIFHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1] suchthat 0 ≤ TSn IFIFHS (xi)+ FSn IFIFHS (xi) ≤ 1(i ∈{1, 2,...,n}) and 0 ≤ TSn IFIFHS (x1 ,x2 ,...xn ) (u)+ FSn IFIFHS (x1 ,x2 ,...xn ) (u) ≤ 1.

(12) anIntuitionisticFuzzyValued n attributeNeutrosophicHypersoftSet(n IFNHS set) Sn IFNHS over U canbewrittenasasetoforderedpairsas;

Sn IFNHS = ((<x1 ,TSn IFNHS (

and TSn IFNHS (x1 ,x2 ,...xn ) (u),ISn IFNHS (x1 ,x2 ,...xn ) (u),FSn IFNHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1] suchthat 0 ≤ TSn IFNHS (xi)+ FSn IFNHS (xi) ≤ 1(i ∈{1, 2,...,n}) and 0 ≤ TSn IFNHS (x1 ,x2 ,...xn ) (u)+ ISn IFNHS (x1 ,x2 ,...xn ) (u)+ FSn IFNHS (x1 ,x2 ,...xn ) (u) ≤ 3.

(13) aNeutrosophicValued n attributeHypersoftSet(n NHS set) Sn NHS over U canbe writtenasasetoforderedpairsas;

Sn NHS = ((<x1 ,TSn NHS (x1 ),ISn NHS (x1 ),FSn NHS (x1 ) >,<x2 ,TSn NHS (x2 ), ISn NHS (x2 ),FSn NHS (x2 ) >,...,<xn,TSn NHS (xn),ISn NHS (xn), FSn NHS (xn) >), {<u,TSn NHS (x1 ,x2 ,...xn ) (u), 0, 1 TSn NHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × ... × En where TSn NHS (xi)(i ∈{1, 2,...,n}),ISn NHS (xi)(i ∈{1, 2,...,n}),FSn NHS (xi)(i ∈{1, 2,...,n}) ∈ [0, 1],and TSn NHS (x1 ,x2 ,...xn ) (u) ∈{0, 1} suchthat 0 ≤ TSn NHS (xi)+ ISn NHS (xi),FSn NHS (xi) ≤ 3(i ∈{1, 2,...,n}).

(14) aNeutrosophicValued n attributeFuzzyHypersoftSet(n NFHS set) Sn NFHS over U canbewrittenasasetoforderedpairsas;

Sn NFHS = ((<x1 ,TSn NFHS (x1 ),ISn NFHS (x1 ),FSn NFHS (x1 ) >,<x2 ,TSn NFHS (x2 ), ISn NFHS (x2 ),FSn NFHS (x2 ) >,...,<xn,TSn NFHS (xn),ISn NFHS (xn), FSn NFHS (xn) >), {<u,TSn NFHS (x1 ,x2 ,...xn ) (u), 0, 1 TSn NFHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En where TSn NFHS (xi)(i ∈{1, 2,...,n}),ISn NFHS (xi)(i ∈{1, 2,...,n}),FSn NFHS (xi)(i ∈ {1, 2,...,n})

Sn NIFHS = ((<x1 ,TSn NIFHS (x1 ),ISn NIFHS (x1 ),FSn NIFHS (x1 ) >,<x2 ,TSn NIFHS (x2 ), ISn NIFHS (x2 ),FSn NIFHS (x2 ) >,...,<xn,TSn NIFHS (xn),ISn NIFHS (xn), FSn NIFHS (xn) >), {<u,TSn NIFHS (x1 ,x2 ,...xn ) (u), 0,FSn NIFHS (x1 ,x2 ,...xn ) (u) >: u ∈ U }):(x1 ,x2 ,...xn) ∈ E1 × E2 × × En where TSn NIFHS (xi)(i ∈{1, 2,...,n}),ISn NIFHS (xi)(i ∈{1, 2,...,n}),FSn NIFHS (xi)(i ∈ {1, 2,...,n}) ∈ [0, 1],and TSn NIFHS (x1 ,x2 ,...xn ) (u),FSn NIFHS (x1 ,x2 ,...xn ) (u) ∈ [0, 1] suchthat 0 ≤ TSn NIFHS (xi)+ ISn NIFHS (xi),FSn NIFHS (xi) ≤ 3(i ∈{1, 2,...,n}) and 0 ≤ TSn NIFHS (x1 ,x2 ,...xn ) (u)+ FSn NIFHS (x1 ,x2 ,...xn ) (u) ≤ 1.

3.3. NPNSS-aggregationoperator

Definition3.16. Let U beauniverse, NS(U )bethesetofallneutrosophicsoftsetsonU, E beaset ofattributesthataredescribetheelementsof U , Ei ⊆ E(i ∈{1, 2,...,n})be i setofattributessuch that Ei ∩ Ej = ∅, forall i = j,and i,j ∈{1, 2,...,n} and N (Ei)bethesetofallneutrosophicseton Ei.Also,let Sn NNHS beanya n NNHS set.ThenNNHS-aggregationoperatorof Sn NNHS , denotedby NNHSagg ,isdefinedby NNHSagg :(N (E1 ) × N (E2 ) × ... × N (En)) × NS(U ) → F (U ) NPNSSagg (N (E1 ) × N (E2 ) × ... × N (En),NS(U ))= S∗ where

ISn NNHS (x1 ,x2 ,...xn)= t(ISn NNHS (x1 ),ISn NNHS (x2 ),...,ISn NNHS (xn)) and

FSn NNHS (x1 ,x2 ,...xn)= t(FSn NNHS (x1 ),FSn NNHS (x2 ),...,FSn NNHS (xn))

3.4. Algorithm

Let U beauniversesetand E beaattributessetsuchthat Ei ⊆ E(i =1, 2,...,n)thataredescribed bytheelementsof U .Thealgorithmforthesolutionisgivenbelow;

Step1:Constructfeasiblean n NNHS set Sn NNHS over U , Step2:Findtheaggregatefuzzyset S∗ of Sn NNHS , Step3: Findthelargestmembershipgrade max{µS∗ n NNHS (u): u ∈ U }

Example3.17. Let U = {u1 ,u2 ,u3 } beauniverse, E1 ,E2 ,E3 ⊆ E = {x1 ,x2 ,x3 ,x4 ,x5 ,x6 } be attributessetssuchthat

E1 =Foreignlanguage= {x1 =English,x2 =French,x3 =German}

E2 =Workexperience= {x4 = computer,x5 =E-marketing}

E3 =Nationality= {x6 =Turkish} andletusconsiderthet-norm t(a,b)= a.b ands-norm s(a,b)= a + b a.b.Then,supposethata company’sboardofdirectorsplanstogetanexpertpersonalforonlinetradingtheduringCOVID-19 pandemic.Theprocessissummarizedbasedon n NNHS setsasfollows;

Step1:Weconstructedfeasiblea3 NNHS set S3 NNHS over U as, S3 NNHS = ((<x1 , (0.1, 0.5, 0.4) >,<x4 , (0.2, 0.8, 0.9) >,<x6 , (0.1, 0.2, 0.7) >); {<u1 , (0 2, 0 7, 0 4) >,<u2 , (0 7, 0 2, 0 3) >,<u3 , (0 5, 0 6, 0 2) >}), ((<x1 , (0 4, 0 8, 0 9) >,<x5 , (0 4, 0 8, 0 7) >,<x6 , (0 7, 0 4, 0 9) >); {<u1 , (0.5, 0.4, 0.7) >,<u2 , (0.2, 0.3, 0.7) >,<u3 , (0.4, 0.3, 0.7) >}),

((<x2 , (0 7, 0 8, 0 4) >,<x4 , (0 3, 0 5, 0 1) >,<x6 , (0 4, 0 5, 0 1) >); {<u1 , (0 4, 0 5, 0 3) >,<u2 , (0 2, 0 2, 0 2) >,<u3 , (0 8, 0 7, 0 4) >}), ((<x2 , (0.4, 0.4, 0.9) >,<x5 , (0.4, 0.5, 0.2) >,<x6 , (0.2, 0.5, 0.8) >);

{<u1 , (0 5, 0 6, 0 7) >,<u2 , (0 8, 0 5, 0 5) >,<u3 , (0 5, 0 3, 0 2) >}), ((<x3 , (0.6, 0.7, 0.8) >,<x4 , (0.5, 0.7, 0.1) >,<x6 , (0.4, 0.1, 0.8) >); {<u1 , (0 1, 0 2, 0 8) >,<u2 , (0 5, 0 3, 0 3) >,<u3 , (0 5, 0 4, 0 6) >}), ((<x3 , (0 7, 0 4, 0 1) >,<x5 , (0 9, 0 7, 0 5) >,<x6 , (0 5, 0 2, 0 2) >); {<u1 , (0 8, 0 3, 0 7) >,<u2 , (0 6, 0 7, 0 1) >,<u3 , (0 7, 0 5, 0 7) >})

Step2:Wefoundtheaggregatefuzzyset S∗ of Sn NNHS as, S∗ n NNHS = {u1 /0 1581,u2 /0 2139,u3 /0 2001}

Step3: Finally,sincethe u2 hasthelargestmembershipgrade,itisthebestexpertpersonalfor onlinetradingon U

4. Conclusion

Inthisstudy,hybridsetstructurewithhypersoftsetsunderuncertainty,vaguenessandindeterminacy,calledneutrosophicvalued n attributeneutrosophichypersoftset,whichisacombinationof neutrosophicsets[25]andhypersoftsets[31]waspresented.Theneutrosophicvalued n attributeneutrosophichypersoftset(n NNHS set)generalizesthefollowinghybridsetsas; n attributeHypersoftSet, n attributeFuzzyHypersoftSet, n attributeIntuitionisticFuzzyHypersoftSet, n attribute NeutrosophicHypersoftSet,FuzzyValued n attributeHypersoftSet,FuzzyValued n attributeFuzzy HypersoftSet,FuzzyValued n attributeIntuitionisticFuzzyHypersoftSet,FuzzyValued n attribute NeutrosophicHypersoftSet,IntuitionisticFuzzyValued n attributeHypersoftSet,IntuitionisticFuzzy Valued n attributeFuzzyHypersoftSet,IntuitionisticFuzzyValued n attributeIntuitionisticFuzzy HypersoftSet,IntuitionisticFuzzyValued n attributeNeutrosophicHypersoftSet,NeutrosophicValued n attributeHypersoftSet,NeutrosophicValued n attributeFuzzyHypersoftSet,Neutrosophic Valued n attributeIntuitionisticFuzzyHypersoftSet,neutrosophicparameterizedneutrosophicsoft sets[13],andsoon.Then,somedefinitionsandoperationson n NNHS setandsomeproperties ofthesetswhichareconnectedtooperationshavebeengiven.Finally,adecisionmakingmethodon the n NNHS settoshowthatthismethodcanbesuccessfullyworkedwasdeveloped.Inthefuture works, n NNHS setscanbeexpandedwithnewresearchsubjectsas;Matrices,neutrosophicnumbersandarithmeticaloperations,relationalstructures,relationalequations,entropy,similaritymeasure, distancemeasuress,orderings,probability,logicaloperations,programming,implicators,multi-valued mappings,mathematicalmorphology,algebraicstructures,models,topology,cognitivemaps,matrix, graph,fusionrules,relationalmaps,relationaldatabases,imageprocessing,linguisticvariables,decisionmakingbasedonVIKOR,TOPSIS,AHPandELECTREI-II-III,preferencestructures,expert systems,reliabilitytheory,softcomputingtechniques,gametheoryandsoon. ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.Thefundershadnoroleinthedesign ofthestudy;inthecollection,analyses,orinterpretationofdata;inthewritingofthemanuscript,or inthedecisiontopublishtheresults.

References

1. M.Abbas,G.Murtaza,F.Smarandache.Basicoperationsonhypersoftsetsandhypersoftpoint.Neutrosophicsets andsystem,35,(2020)407-421.

2. K.T.Atanassov,IntuitionisticFuzzySets,Pysica-VerlagASpringer-VerlagCompany,NewYork(1999).

3. S.Broumi,I.Deli,F.Smarandache,NeutrosophicParametrizedSoftSetTheoryandItsDecisionMaking,InternationalFrontierScienceLetters,1(1)(2014)01–11.

4. N.C¸a˘gmanandS.Engino˘glu,Softsettheoryanduni-intdecisionmaking,EuropeanJournalofOperationalResearch, DOI:10.1016/j.ejor.2010.05.004.

5. Ca˘gman,N.C¸ıtak,F.andEngino˘glu,S.FP-softsettheoryanditsapplications,AnnalsofFuzzyMathematicsand Informatics2/2,219-226,2011.

6. N.Ca˘gman,S.Karata¸s,Intuitionisticfuzzysoftsettheoryanditsdecisionmaking,JournalofIntelligentandFuzzy Systems24/4(2013)829–836.

7. I.DeliandS.Broumi,Neutrosophicsoftrelationsandsomeproperties,AnnalsofFuzzyMathematicsandInformatics 9(1)(2015)169–182.

8. I.Deli,S.Broumi,NeutrosophicSoftMatricesandNSM-decisionMaking,JournalofIntelligentandFuzzySystems, 28(2015)2233-2241.

9. I.DeliandN.Ca˘gman,Intuitionisticfuzzyparameterizedsoftsettheoryanditsdecisionmaking,AppliedSoft Computing28(2015)109–113.

10. E.Sulukan,N.Ca˘gman,T.Aydn,FuzzyParameterizedIntuitionisticFuzzySoftSetsandTheirApplicationtoa Performance-BasedValueAssignmentProblem,JournalofNewTheory,29(2019)79–88.

11. F.Karaaslan,IntuitionisticFuzzyParameterizedIntuitionisticFuzzySoftSetswithApplicationsinDecisionMaking, AnnalsofFuzzyMathematicsandInformatics11(4)(2016)607619.

12. N.C . a˘gman,F.C . tak,S.Enginoglu,FuzzyParameterizedFuzzySoftSetTheoryandItsApplica-tions,Turkish JournalofFuzzySystems1(1)(2010)2135.

13. I.Deli, npn SoftSetsTheoryandApplications,AnnalsofFuzzyMathematicsandInformatics,10/6(2015)847862.

14. Dubois,D.andPrade,H.FuzzySetandSystems:TheoryandApplications,AcademicPress,NewYork,1980.

15. L.N.Kamal.M.,Abdullah.L.andSaqlain.M.Multi-ValuedIntervalNeutrosophicLinguisticSoftSetTheoryand ItsApplicationinKnowledgeManagement,IETCAAITransactionsonIntelligenceTechnology,(2020)1-13.

16. F.Karaaslan,NeutrosophicSoftSetswithApplicationsinDecisionMaking,InternationalJournalofInformation ScienceandIntelligentSystem,4(2)(2015)1–20.

17. P.K.Maji,Neutrosophicsoftset,ComputersandMathematicswithApplications,45(2013)555-562.

18. P.K.Maji,A.R.Roy,R.Biswas,OnIntuitionisticFuzzySoftSets.TheJournalofFuzzyMathematics,12(3)(2004) 669-683.

19. P.K.Maji,R.BiswasandA.R.Roy,Fuzzysoftsets,JournalofFuzzyMathematics,9(3)(2001)589-602.

20. P.K.Maji,Aneutrosophicsoftsetapproachtoadecisionmakingproblem,AnnalsofFuzzyMathematicsand Informatics,3/2,(2012),313–319.

21. P.K.Maji,R.BiswasA.R.Roy,IntuitionisticFuzzySoftSets.TheJournalofFuzzyMathematics,9(3)(2001)677-692.

22. D.A.Molodtsov,Softsettheory-firstresults,Comput.Math.Appl.37(1999)19-31.

23. J.J.Peng,J.Q.Wang,X.H.Wu,J.Wang,X.H.Chen,Multi-valuedNeutrosophicSetsandPowerAggregation OperatorswithTheirApplicationsinMulti-criteriaGroupDecision-makingProblems,InternationalJournalofComputationalIntelligenceSystems,8/2(2015)345–363.

24. M.Riaz,M.SaqlainandM.Saeed.ApplicationofGeneralizedFuzzyTOPSISinDecisionMakingforNeutrosophic SoftsettoPredicttheChampionofFIFA2018:AMathematicalAnalysis,PunjabUniversityJournalofMathematics, 51(8)(2019)111-126.

25. F.Smarandache,”AUnifyingFieldinLogics.Neutrosophy:NeutrosophicProbability,SetandLogic”.Rehoboth: AmericanResearchPress,(1998).

26. M.Saqlain,L.X.Xin,IntervalValued,m-Polarandm-PolarIntervalValuedNeutrosophicHypersoftSets,NeutrosophicSetsandSystems,36,(2020)389-399.

27. M.Saqlain,S.Moin,N.Jafar,M.Saeed,andF.Smarandache,AggregateOperatorsofNeutrosophicHypersoftSet,

Neutrosophic Sets and Systems, 32 (2020) 294-306.

28. M.Saqlain,N.Jafar,M.Riaz,ANewApproachofNeutrosophicSoftSetwithGeneralizedFuzzyTOPSISin ApplicationofSmartPhoneSelection,NeutrosophicSetsandSystems,32(2020)307-316.

29. M.Saqlain,M.Saeed,M.R.Ahmad,F.Smarandache,GeneralizationofTOPSISforNeutrosophicHypersoftset usingAccuracyFunctionanditsApplication,NeutrosophicSetsandSystems),27(2019)131-137.

30. M.Saqlain,M.Sana,N.Jafar,M.Saeed,BS.Broumi.SingleandMulti-valuedNeutrosophicHypersoftsetand TangentSimilarityMeasureofSinglevaluedNeutrosophicHypersoftSets,NeutrosophicSetsandSystems,32:(2020) 317-329.

31. F.Smarandache,ExtensionofSoftSettoHypersoftSet,andthentoPlithogenicHypersoftSet,NeutrosophicSets andSystems,22(2018)168–170.

32. H.Wang,F.Smarandache,Y.Q.ZhangandR.Sunderraman,Singlevaluedneutrosophicsets,Multispaceand Multistructure4410–413,2010.

33. E.E.Yagubi,A.R.Salleh,IntuitionisticFuzzyparameterizedfuzzysoftset,JournalofQualityMeasurementand Analysis,9(2)(2013)73-81

34. L.A.Zadeh,FuzzySets,Inform.andControl8(1965)338-353.

35. M.R.Zulqarnain,M.Saqlain,L.X.Xin,F.Smarandache,GeneralizedaggregateoperatorsonNeutrosophicHypersoft set,NeutrosophicSetsandSystems,36(2020)271-281.

Fuzzy Hypersoft Sets and It’s Application to Decision-Making

Adem Yolcu 1,∗, Taha Yasin Ozturk 2

1 Department of Mathematic, Kafkas University, Kars, Turkey. E-mail: yolcu.adem@gmail.com

2 Department of Mathematic, Kafkas University, Kars, Turkey. E-mail: taha36100@hotmail.com

Abstract: Multi-criteria decision-making (MCDM) is concerned with coordinating and taking care of matters of preference and preparation, including multi-criteria. Fuzzy soft set environments cannot be used to solve certain types of problems if attributes are more than one and further bifurcated. Therefore, there was a serious need to identify a new approach to solve such problems, so a new setting, namely the Fuzzy Hypersoft Sets (FHSS), is established for this reason. In this paper, we introduced some notions like union, intersection, subset, equal set, complement, null set, absolute set etc. of Fuzzy Hypersoft sets. The study has been enriched with many suitable examples to support the accuracy of the defined concepts. We also present an object recognition system from an imprecise data on a multiobserver. Decision-making system involves the construct of Comparison Table from a fuzzy hypersoft set in a parametric sense for decision-making.

Keywords: Soft sets, fuzzy sets, Hypersoft sets, Fuzzy hypersoft sets, Decision-making problem

1. Introduction

Numerous complicated problems include unclear data in social sciences, economics, medical sciences, engineering and other fields. These problems, with which one is faced in life, cannot be solved classical mathematical tools. In classical mathematics, a model is designed and the exact solution of this model is calculated. However, if the given situation contains uncertainty, solving this model with classical mathematical method is very complex. Fuzzy set theory [1], rough set theory [2] and other theories have been described to solve situations involving uncertainty. The fuzzy set theory introduced by Zadeh has become very popular for uncertainty problems and has been a suitable construct for representing uncertain concepts as it allows for the partial membership function. Mathematicians and computer scientists have worked on fuzzy sets and over the years many useful applications of fuzzy set structure have emerged such as fuzzy control systems, fuzzy automata, fuzzy logic, fuzzy topology.

somestructuraldifficultiesoffuzzysettheoryandothertheories,asMolodtsov[3]introduced in1999whenhepresentedtheideaofsoftsettheorywhichisacompletelynewapproachto modelinguncertainty.Furthergeneralizationsandextensionshavebeenthesubjectofmany effortsofthesoftsetsofMolodtsov.Majietal.[4]bycombiningthefuzzysetandsoftset structure,ithascreatedthefuzzysoftclusterstructure,whichisahybridstructure.Inother words,adegreeisaddedtotheparameterisationoffuzzysetswhendefiningafuzzysoftset. Thefuzzysoftsetstructure,whichisacombinationofsoftsetstructureandfuzzysetstructure, hasbeenactivelyusedbyresearchersandmanystudieshavebeenaddedtotheliterature[5–8]. Asaresultoftheintenseinterestofresearchersinthissubject,importantdevelopments havebeenrealizedregardingtheuseoffuzzysoftsetstructureindecision-makingproblems. Becausethedefinitionofunrealobjectsinsoftsetsisnotrestricted,researcherscanchoosethe parameterformattheyneed,whichgreatlysimplifiesthedecision-makingprocessandmakesit moreefficientwhenpartialinformationismissing.SoftsetsusedMajiandRoy[9]forthefirst timeindecision-makingproblems.Chen[10]describedthereductionoftheparameterisation ofthesoftsetanddiscusseditsapplicationoftheproblemofdecision-making.Cagmanand Enginoglu[11,12]discussedtheoryofsoftmatrixanduni-intdecision-making,choosinga collectionofoptimalelementsfromvariousalternatives.Studiesondecision-makingproblems havetakenplacewidelyintheliteratureandhavebeenstudiedbymanyresearchers[13–23].

Smarandache[24]introducednewtechniquehandlinguncertainty.Bytransformingthefunctionalityintoamulti-decisionfunction,hegeneralizedthesoftsettohypersoftset.Although Hypersoftsettheoryismorerecent,ithasattractedgreatattentionfromresearchers[25–30].

Inthispaper,wehavedefinedsomebasicconceptssuchassubset,equalset,union,intersection,complement,nullset,absoluteset,AND,ORoperationsonthefuzzyhypersoftset structure.Wealsoappliedfuzzyhypersoftsetstosolvethedecision-makingproblem.We havepresentedanappropriatedecisionproblembyapplyingRoyandMaji’smethod[15]to fuzzyhypersoftsets.Thisstudyhasthebasicstudyfeatureinwhichthefuzzyhypersoftset structureisintroduced.Thereforeplaysanimportantroleformanysubsequentstudies.

2. Preliminaries

Definition2.3. [4]Let U beainitialuniverse, E beasetofparametersand FP (U )betheset ofallfuzzysetsin U. Thenapair(f,E)iscalledafuzzysoftsetover U, where f : E → FP (U ) isamapping.

Definition2.4. [24]Let U betheuniversalsetand P (U )bethepowersetof U .Consider e1,e2,e3,...,en for n ≥ 1,be n well-definedattributes,whosecorrespondingattributevaluesare resspectivelythesets E1,E2,...,En with Ei ∩ Ej = ∅, for i = j and i,j ∈{1, 2,...,n}, thenthe pair(Θ,E1×E2× ×En)issaidtobeHypersoftsetover U whereΘ: E1×E2× ×En → P (U )

3. FuzzyHypersoftSets

Definition3.1. Let U betheuniversalsetand FP (U )beafamilyofallfuzzysetover U and E1,E2,...,En thepairwisedisjointsetsofparameters.Let Ai bethenonemptysubsetof Ei foreach i =1, 2,...,n. Afuzzyhypersoftsetdefinedasthepair(Θ,A1 × A2 × × An)where; Θ: A1 × A2 × ... × An → FP (U )and Θ(A1 × A2 × ... × An)= {<u, Θ(α)(u) >: u ∈ U,α ∈ A1 × A2 × ... × An ⊆ E1 × E2 × ... × En} Forsakeofsimplicity,wewritethesymbolsΣfor E1 × E2 × ... × En, Γfor A1 × A2 × ... × An and α foranelementofthesetΓ. Thesetofallfuzzyhypersoftsetsover U willbedenoted by FHS(U, Σ) Hereafter,FHSwillbeusedforshortinsteadoffuzzyhypersoftsets.

Definition3.2. i)Afuzzyhypersoftset(Θ, Γ)overtheuniverse U issaidtobenullfuzzy hypersoftsetanddenotedby0(UFH ,Σ) ifforall u ∈ U and ε ∈ Γ, Θ(ε)(u)=0. ii)Afuzzyhypersoftset(Θ, Γ)overtheuniverse U issaidtobeabsolutefuzzyhypersoft setanddenotedby1(UFH ,Σ) ifforall u ∈ U and ε ∈ Γ, Θ(ε)(u)=1.

Supposethat

(Θ1, Γ1)= < (α2,β2,γ1), { u1 0,3 , u2 0,4 } >,< (α2,β2,γ2), { u1 0,2 , u2 0,5 , u3 0,1 } >, < (α2,β3,γ1), { u1 0,6 , u3 0,7 } >,< (α2,β3,γ2), { u2 0,4 , u3 0,5 } > , (Θ2, Γ2)= < (α1,β1,γ1), { u1 0,4 , u3 0,7 } >,< (α1,β2,γ1), { u2 0,3 , u3 0,6 } >, < (α2,β1,γ1), { u1 0,7 , u3 0,8 } >,< (α2,β2,γ1), { u2 0,5 , u3 0,7 } >

Corollary3.4. Itisclearthateachfuzzyhypersoftsetisalsofuzzysoftset.Anexampleof thissituationisprovidedbelow.

Example3.5. WeconsiderthatExample-3.3.Ifweselecttheparametersfromasingle attributesetsuchas E1 whilecreatingthefuzzyhypersoftset,thentheresultingsetbecomes thefuzzysoftset.Therefore,itisclearthateachfuzzyhypersoftsetisalsofuzzysoftset.That is,thefuzzyhypersoftsetstructureisthegeneralizedversionofthefuzzysoftsets.

Definition3.6. Let U beaninitialuniverseset(Θ1, Γ1), (Θ2, Γ2)betwofuzzyhypersoftsets overtheuniverse U .Wesaythat(Θ1, Γ1)isafuzzyhypersoftsubsetof(Θ2, Γ2)anddenote (Θ1, Γ1)⊆(Θ2, Γ2)if i)Γ1 ⊆ Γ2 ii)Forany ε ∈ Γ1, Θ1(ε) ⊆ Θ2(ε).

Definition3.7. Let U beaninitialuniverseset(Θ1, Γ1), (Θ2, Γ2)betwofuzzyhypersoft setsovertheuniverse U. Wesaythat(Θ1, Γ1)isfuzzyhypersoftequalto(Θ2, Γ2)anddenote (Θ1, Γ1)=(Θ2, Γ2)if(Θ1, Γ1)isafuzzyhypersoftsubsetof(Θ2, Γ2)and(Θ2, Γ2)isafuzzy hypersoftsubsetof(Θ1, Γ1)

Theorem3.8. Let U beaninitialuniversesetand (Θ1, Γ1), (Θ2, Γ2), (Θ3, Γ3) befuzzyhypersoftsetsovertheuniverse U. Then, i) (Θ, Γ1)⊆1(UFH ,Σ), ii) 0(UFH ,Σ)⊆(Θ, Γ1), iii) (Θ1, Γ1)⊆(Θ2, Γ2) and (Θ2, Γ2)⊆(Θ3, Γ3) ⇒ (Θ1, Γ1)⊆(Θ3, Γ3).

Proof. Straightforward.

Definition3.9. Thecomplementoffuzzyhypersoftset(Θ, Γ)overtheuniverse U isdenoted by(Θ, Γ)c anddefinedas(Θ, Γ)c =(Θc , Γ), whereΘc(ε)iscomplementofthesetΘ(ε), for ε ∈ Γ

Example3.10. AccordingtoExample-3.3,considerthefuzzyhypersoftset(Θ1, Γ1)overthe universe U = {u1,u2,u3}

(Θ1, Γ1)= < (α2,β2,γ1), { u1 0,3 , u2 0,4 } >,< (α2,β2,γ2), { u1 0,2 , u2 0,5 , u3 0,1 } >, < (α2,β3,γ1), { u1 0,6 , u3 0,7 } >,< (α2,β3,γ2), { u2 0,4 , u3 0,5 } > , Thenthecomplementof(Θ1, Γ1)iswrittenas; (Θ1, Γ1)c = < (α2,β2,γ1), { u1 0,7 , u2 0,6 , u3 1 } >,< (α2,β2,γ2), { u1 0,8 , u2 0,5 , u3 0,9 } >, < (α2,β3,γ1), { u1 0,4 , u2 1 u3 0,3 } >,< (α2,β3,γ2), { u1 1 , u2 0,6 , u3 0,5 } > ,

Theorem3.11. Let (Θ, Γ) beanyfuzzyhypersoftsetovertheuniverse U. Then, i) ((Θ, Γ)c)c =(Θ, Γ) ii) 0c (UFH ,Σ) =1(UFH ,Σ) iii) 1c (UFH ,Σ) =0(UFH ,Σ) Proof. Proofsaretrivial.

Definition3.12. Let U beaninitialuniversesetand(Θ1, Γ1), (Θ2, Γ2)betwofuzzyhypersoft setsovertheuniverse U. Theunionof(Θ1, Γ1)and(Θ2, Γ2)isdenotedby(Θ1, Γ1)∪(Θ2, Γ2)= (Θ3, Γ3)whereΓ3 =Γ1 ∪ Γ2 and Θ3(ε)=      Θ1(ε)if ε ∈ Γ1 Γ2 Θ2(ε)if ε ∈ Γ2 Γ1 max{ Θ1(ε), Θ2(ε)} if ε ∈ Γ1 ∩ Γ2

Theorem3.13. Let U beaninitialuniversesetand (Θ1, Γ1), (Θ2, Γ2), (Θ3, Γ3) betwofuzzy hypersoftsetsovertheuniverse U Then; i) (Θ1, Γ1)∪(Θ1, Γ1)=(Θ1, Γ1) ii) 0(UFH ,Σ) ∪(Θ1, Γ1)=(Θ1, Γ1) iii) (Θ1, Γ1)˜∪1(UFH ,Σ) =1(UFH ,Σ) iv) (Θ1, Γ1)∪(Θ2, Γ2)=(Θ2, Γ2)∪(Θ1, Γ1) v) ((Θ1, Γ1)∪(Θ2, Γ2)) ∪(Θ3, Γ3)=(Θ1, Γ1)∪ ((Θ2, Γ2)∪(Θ3, Γ3)) Proof.

Definition3.14. Let U beaninitialuniversesetand(Θ1, Γ1), (Θ2, Γ2)befuzzyhypersoftsets overtheuniverse U. Theintersectionof(Θ1, Γ1)and(Θ2, Γ2)isdenotedby(Θ1, Γ1)∩(Θ2, Γ2)= (Θ3, Γ3)whereΓ3 =Γ1 ∩ Γ2 andeach ε ∈ Γ3, Θ3(ε)(u)=min{ Θ1(ε)(u), Θ2(ε)(u)}

Theorem3.15. Let U beaninitialuniversesetand (Θ1, Γ1), (Θ2, Γ2), (Θ3, Γ3) befuzzyhypersoftsetsovertheuniverse U. Then; i) (Θ1, Γ1)˜∩(Θ1, Γ1)=(Θ1, Γ1)

ii) 0(UFH ,Σ) ∩(Θ1, Γ1)=0(UFH ,Σ) iii) (Θ1, Γ1)˜∩1(UFH ,Σ) =(Θ1, Γ1) iv) (Θ1, Γ1)∩(Θ2, Γ2)=(Θ2, Γ2)∩(Θ1, Γ1) v) ((Θ1, Γ1)∩(Θ2, Γ2)) ∩(Θ3, Γ3)=(Θ1, Γ1)∩ ((Θ2, Γ2)∩(Θ3, Γ3))

Proof. Proofsaretrivial.

Definition3.16. Let U beaninitialuniversesetand(Θ1, Γ1), (Θ2, Γ2)betwofuzzyhypersoftsetsovertheuniverse U. Thedifferenceof(Θ1, Γ1)and(Θ2, Γ2)isdenotedby (Θ1, Γ1)\(Θ2, Γ2)=(Θ3, Γ3)andisdefinedby(Θ1, Γ1)∩(Θ2, Γ2)c =(Θ3, Γ3)whereΓ3 = Γ1 ∪ Γ2

Example3.17. WeconsiderthatattributesinExample-3.3.Thenthefuzzyhypersoftsets (Θ1, Γ1)and(Θ1, Γ1)definedasfollows; (Θ1, Γ1)= < (α2,β2,γ1), { u1 0,3 , u2 0,4 } >,< (α2,β2,γ2), { u1 0,2 , u2 0,5 , u3 0,1 } >, < (α2,β3,γ1), { u1 0,6 , u3 0,7 } >,< (α2,β3,γ2), { u2 0,4 , u3 0,5 } > , (Θ2, Γ2)= < (α1,β1,γ1), { u1 0,4 , u3 0,7 } >,< (α1,β2,γ1), { u2 0,3 , u3 0,6 } >, < (α2,β1,γ1), { u1 0,7 , u3 0,8 } >,< (α2,β2,γ1), { u2 0,5 , u3 0,7 } >





< (α2,β2,γ1), { u1 0,3 , u2 0,5 , u3 0,7 } >,< (α2,β2,γ2), { u1 0,2 , u2 0,5 , u3 0,1 } >, < (α2,β3,γ1), { u1 0,6 , u3 0,7 } >,< (α2,β3,γ2), { u2 0,4 , u3 0,5 } >, < (α1,β1,γ1), { u1 0,4 , u3 0,7 } >,< (α1,β2,γ1), { u2 0,3 , u3 0,6 } >, < (α2,β1,γ1), { u1 0,7 , u3 0,8 } >

 

  

   

Case2: ε ∈ Γ2 Γ1. Then K(ε)=Θc 2(ε)= I(ε).

Case3: ε ∈ Γ1 ∩ Γ2 Then K(ε)=(σΘ1 (ε)(u) ∩ σΘ1 (ε)(u),θΘ2 (ε)(u) ∪ θΘ2 (ε)(u))=Θc 1(ε) ∩ Θc 2(ε)= I(ε)

Therefore, K and I aresameoperators,andso((Θ1, Γ1)∪(Θ2, Γ2))c =(Θ1, Γ1)c∩(Θ2, Γ2)c .

Definition3.19. Let U beaninitialuniversesetand(Θ1, Γ1), (Θ2, Γ2)befuzzyhypersoft setsovertheuniverse U. The”AND”operationonthemisdenotedby(Θ1, Γ1) ∧ (Θ2, Γ2)= (Υ, Γ1 × Γ2)isgivenas; (Υ, Γ1 × Γ2)= {(ε1,ε2),<u, Υ(ε1,ε2)(u) >: u ∈ U, (ε1,ε2) ∈ Γ1 × Γ2} where Υ(ε1,ε2)(u)= {<u, min {Θ1(ε1)(u), Θ2(ε2)(u)} >}

Definition3.20. Let U beaninitialuniversesetand(Θ1, Γ1),(Θ2, Γ2)befuzzyhypersoft setsovertheuniverse U. The”OR”operationonthemisdenotedby(Θ1, Γ1) ∨ (Θ2, Γ2)= (Υ, Γ1 × Γ2)isgivenas; (Υ, Γ1 × Γ2)= {(ε1,ε2),<u, Υ(ε1,ε2)(u) >: u ∈ U, (ε1,ε2) ∈ Γ1 × Γ2} where Υ(ε1,ε2)(u)= {<u, max {Θ1(ε1)(u), Θ2(ε2)(u)} >}

Theorem3.21. Let U beaninitialuniversesetand (Θ1, Γ1), (Θ2, Γ2), (Θ3, Γ3) befuzzyhypersoftsetsovertheuniverse U. Then;

i) (Θ1, Γ1) ∨ [(Θ2, Γ2) ∨ (Θ3, Γ3)]=[(Θ1, Γ1) ∨ (Θ2, Γ2)] ∨ (Θ3, Γ3) ii) (Θ1, Γ1) ∧ [(Θ2, Γ2) ∧ (Θ3, Γ3)]=[(Θ1, Γ1) ∧ (Θ2, Γ2)] ∧ (Θ3, Γ3)

Proof. Straightforward.

Theorem3.22. Let U beaninitialuniversesetand (Θ1, Γ1), (Θ2, Γ2) befuzzyhypersoftsets overtheuniverse U Then; i) [(Θ1, Γ1) ∨ (Θ2, Γ2)]c =(Θ1, Γ1)c ∧ (Θ2, Γ2)c ii) [(Θ1, Γ1) ∧ (Θ2, Γ2)]c =(Θ

Ontheotherhand, (Θ1, Γ1)c = {<u, 1 Θ1(ε1)(u) >: u ∈ U,ε1 ∈ Γ1} (Θ2, Γ2)c = {<u, 1 Θ2(ε2)(u) >: u ∈ U,ε2 ∈ Γ2}

Then, (Θ1, Γ1)c ∧ (Θ2, Γ2)c = {<u, min {1 Θ1(ε1)(u), 1 Θ2(ε2)(u)} >} =[(Θ1, Γ1) ∨ (Θ2, Γ2)]c Hence,[(Θ1, Γ1) ∨ (Θ2, Γ2)]c =(Θ1, Γ1)c ∧ (Θ2, Γ2)c isobtained.

Example3.23. WeconsiderthatattributesinExample-3.3.Thenthefuzzyhypersoftsets (Θ1, Γ1)and(Θ2, Γ2)definedasfollows; (Θ1, Γ1)= < (α2,β2,γ1), { u1 0,3 , u2 0,4 } >,< (α2,β2,γ2), { u1 0,2 , u2 0,5 , u3 0,1 } >, < (α2,β3,γ1), { u1 0,6 , u3 0,7 } >,< (α2,β3,γ2), { u2 0,4 , u3 0,5 } > , (Θ2, Γ2)= < (α1,β1,γ1), { u1 0,4 , u3 0,7 } >,< (α1,β2,γ1), { u2 0,3 , u3 0,6 } >, < (α2,β1,γ1), { u1 0,7 , u3 0,8 } >,< (α2,β2,γ1), { u2 0,5 , u3 0,7 } >

Let’sassume(α2,β2,γ1)= η1,(α2,β2,γ2)= η2,(α2,β3,γ1)= η3,(α2,β3,γ2)= η4 in(Θ1, Γ1) and(α1,β1,γ1)= k1,(α1,β2,γ1)= k2,(α2,β1,γ1)= k3, (α2,β2,γ1)= k4 in(Θ2, Γ2)foreasier operation.Thetabularformsofthesesetsareasfollows. (Θ1, Γ1) u1 u2 u3 η1 0, 30, 40 η2 0, 20, 50, 1 η3 0, 600, 7 η4 00, 40, 5

Table1:TabularformofFHSS(Θ1, Γ1) and (Θ2, Γ2) u1 u2 u3 k1 0, 400, 7 k2 00, 30, 6 k3 0, 700, 8 k4 00, 50, 7

Table2:TabularformofFHSS(Θ2, Γ2) Thenthe”AND”and”OR”operationsofthesesetsaregivenasbelow.

(Θ1, Γ1) ∧ (Θ2, Γ2) u1 u2 u3

η1 × k1 0, 300

η1 × k2 00, 30

η1 × k3 0, 300

η1 × k4 00, 40

η2 × k1 0, 200, 1

η2 × k2 00, 30, 1

η2 × k3 0, 200, 1

η2 × k4 00, 50, 1

η3 × k1 0, 400, 7

η3 × k2 000, 6

η3 × k3 0, 600, 7

η3 × k4 000, 7

η4 × k1 000, 5

η4 × k2 00, 30, 5

η4 × k3 000, 5

η4 × k4 00, 40, 5

Table3:TabularfornofFHSS(Θ1, Γ1) ∧ (Θ2, Γ2)

(Θ1, Γ1) ∨ (Θ2, Γ2) u1 u2 u3

η1 × k1 0, 40, 40, 7

η1 × k2 0, 30, 40, 6

η1 × k3 0, 70, 40, 8

η1 × k4 0, 30, 50, 7

η2 × k1 0, 40, 50, 7

η2 × k2 0, 20, 50, 6

η2 × k3 0, 70, 50, 8

η2 × k4 0, 20, 50, 7

η3 × k1 0, 600, 7

η3 × k2 0, 60, 30, 7

η3 × k3 0, 700, 8

η3 × k4 0, 60, 50, 7

η4 × k1 0, 40, 40, 7

η4 × k2 00, 40, 6

η4 × k3 0, 70, 40, 8

η4 × k4 00, 50, 7

Table4:TabularformofFHSS(Θ1, Γ1) ∨ (Θ2, Γ2)

4.

ApplicationinDecision-MakingProblem

Thecomparisontableoffuzzyhypersoftset(Θ, Γ)isasquaretableinwhichthenumber ofrowsandnumberofcolumnsareequal,rowsandcolumnsbotharelabelledbytheobjects names u1,u2,...,un oftheuniverse U andtheentriescij (i,j =1, 2,...,n)isthenumberof parametersforwhichthemembershipvalueof ui exceedorequaltothemembershipvalueof uj Clearly0 ≤ cij ≤ k where k isthenumberofparametersinafuzzyhypersoftset.Thus, cij indicatesanumericalmeasureandintegernumberwhichis ui dominates uj in cij number ofparametersoutof k parameters.

4.1. ColumnSum,RowSumandScorevalueofanObject

a: Therowsumofobject ui isdenotedby ri andcalculatedbyfollowingformula, ri = n j=1 cij

Itisclearthat ri indicatesthetotalnumberofparametersinwhich ui dominatesall themembersof U.

b: Thecolumnsumofobject uj isdenotedby ti andcalculatedbufollowingformula, ti = n i=1 cij

Itisclearthat tj indicatesthetotalnumberofparametersinwhich uj dominatesall themembersof U.

c: Thescorevalueof ui is Si andcalculatedbyfollowingformula, Si = ri ti

4.2. Algorithm

Theproblemhereistoselectanobjectfromthesetofobjectsgivenbasedonasetof parametersΓ.Wenowpresentanobjectrecognitionalgorithm,basedoninputdatafrom multiobserversdefinedbyage,foreignlanguageknowledgeandeducation.

(1) Inputthefuzzyhypersoftsets(Θ1, Γ1), (Θ2, Γ2), (Θ3, Γ3). (2) InputtheparametersetΣasobservedbythedecisionmakers.

(3) Computethecorrespondingresultantfuzzyhypersoftsetfromthefuzzyhypersoftsets (Θ1, Γ1), (Θ2, Γ2), (Θ3, Γ3)andplacethetabularform.

(4) ConstructtheComparisonTableofthefuzzyhypersoftsetandcomputerowsum ri andcolumnsum ti for ui

(5) ComputetheScorevalueof ui. (6) Thedecisionis Sk =max Si

(7) If k hasmorethanonevaluethenanyoneof uk maybechosen.

4.3. Application

Considertheproblemofselectingthemostsuitablestafffromthesetofstaffwithrespect toasetofchoiceparameters.Let U bethesetofstaffgivenas U = {u1,u2,u3,u4,u5} also considerthesetofattributesas E1 =Age, E2 =Foreignlanguageknowledge, E3 =Education, andtheirrespectiveattributesaregivenas

E1 =Age= {18-25(α1),25-30(α2),30-35(α3)}

E2 =Foreignlanguageknowledge= {English(β1),Turkish(β2),German(β3)}

E3 =Education= {BSc(γ1),MSc(γ2),PhD(γ3)}

Supposethat

A1 = {α3},A2 = {β1,β2},A3 = {γ2,γ3}

B1 = {α1,α2},B2 = {β2,β3},B3 = {γ3} C1 = {α2,α3},A2 = {β1,β3},A3 = {γ1} aresubsetof Ei foreach i =1, 2, 3. Thenthefuzzyhypersoftsets(Θ1, Γ1)and(Θ2, Γ2)defined asfollows; (Θ1, Γ1)= < (α3,β1,γ2), { u1 0,4 , u2 0,5 , u3 0,6 , u4 0,1 , u5 0,8 } >,< (α3,β1,γ3), { u1 0,4 , u2 0,2 , u3 0,6 , u4 0,1 , u5 0,3 } >, < (α3,β2,γ2), { u1 0,2 , u2 0,7 , u3 0,4 , u4 0,4 , u5 0,5 } >,< (α3,β2,γ3), { u1 0,7 , u2 0,1 , u3 0,4 , u4 0,2 , u5 0,6 } > , (Θ2, Γ2)= < (α1,β2,γ3), { u1 0,8 , u2 0,4 , u3 0,7 , u4 0,3 , u5 0,8 } >,< (α1,β3,γ3), { u1 0,4 , u2 0,5 , u3 0,6 , u4 0,1 , u5 0,8 } >, < (α2,β2,γ3), { u1 0,2 , u2 0,2 , u3 0,6 , u4 0,3 , u5 0,1 } >,< (α2,β3,γ3), { u1 0,5 , u2 0,5 , u3 0,1 , u4 0,1 , u5 0,7 } >

(Θ3, Γ3)= < (α2,β1,γ1), { u1 0,2 , u2 0,3 , u3 0,1 , u4 0,7 , u5 0,6 } >,< (α2,β3,γ1), { u1 0,6 , u2 0,2 , u3 0,3 , u4 0,3 , u5 0,1 } >, < (α3,β1,γ1), { u1 0,7 , u2 0,5 , u3 0,8 , u4 0,4 , u5 0,8 } >,< (α3,β3,γ1), { u1 0,3 , u2 0,4 , u3 0,1 , u4 0,6 , u5 0,5 } >

Thetabularrepresentionsof(Θ1, Γ1), (Θ2, Γ2), (Θ3, Γ3)areshowninTable5-7. (Θ1, Γ1) u1 u2 u3 u4 u5 (α3,β1,γ2)= η1 0, 40, 50, 60, 10, 8 (α3,β1,γ3)= η2 0, 40, 20, 60, 10, 3 (α3,β2,γ2)= η3 0, 20, 70, 40, 40, 5 (α3,β2,γ3)= η4 0, 70, 10, 40, 20, 6

Table-5:TabularformofFHSS(Θ1, Γ1)

(Θ2, Γ2) u1 u2 u3 u4 u5

(α1,β2,γ3)= k1 0, 80, 40, 70, 30, 8

(α1,β3,γ3)= k2 0, 40, 50, 60, 10, 8 (α2,β2,γ3)= k3 0, 20, 20, 60, 30, 1

(α2,β3,γ3)= k4 0, 50, 50, 10, 10, 7

Table-6:TabularformofFHSS(Θ2, Γ2)

(Θ3, Γ3) u1 u2 u3 u4 u5

(α2,β1,γ1)= m1 0, 20, 30, 10, 70, 6 (α2,β3,γ1)= m2 0, 60, 20, 30, 30, 1 (α3,β1,γ1)= m3 0, 70, 50, 80, 40, 8 (α3,β3,γ1)= m4 0, 30, 40, 10, 60, 5

Table-7:TabularformofFHSS(Θ3, Γ3)

Nowlet’sseehowtheoriginalproblemcanbesolvedusingthealgorithm.Considerthe fuzzyhypersoftsets(Θ1, Γ1), (Θ2, Γ2)asdefinedabove.Ifweperform(Θ1, Γ1) ∧ (Θ2, Γ2)then wewillhave4 × 4=16parameters.Ifwerequirefuzzyhypersoftsetfortheparameters R = {η1 × k1,η2 × k4,η3 × k2,η3 × k3,η4 × k1}, thenweobtaintheresultantfuzzyhypersoft forthefuzzyhypersoftsets(Θ1, Γ1), (Θ2, Γ2) So,afterperforming(Θ1, Γ1) ∧ (Θ2, Γ2)forsome parametersthetabularformoftheresultantfuzzyhypersoftset(K,R)willtaketheformas Table-8, (K,R) u1 u2 u3 u4 u5 η1 × k1 0, 40, 40, 60, 10, 8 η2 × k4 0, 40, 20, 10, 10, 3 η3 × k2 0, 20, 50, 40, 10, 5 η3 × k3 0, 20, 20, 40, 30, 1 η4 × k1 0, 70, 10, 40, 20, 6

Table-8:TabularformofFHSS(K,R)

Considerthefuzzyhypersoftsets(Θ1, Γ1), (Θ2, Γ2)and(Θ3, Γ3)asdefinedabove.Suppose that P besetofchoiceparametersofadecisionmaker.Onthebasisofthisparameterwe havetotakethedecisionfromtheavailabilityset U .Thetabularrepresentationofresultant fuzzyhypersoftset(S,P )willbeas.(Table19). (S,P ) u1 u2 u3 u4 u5 (η1 × k1) × m1 0, 20, 30, 10, 10, 6 (η2 × k4) × m3 0, 40, 20, 10, 10, 3 (η3 × k2) × m2 0, 20, 20, 30, 10, 1 (η3 × k3) × m4 0, 20, 20, 10, 30, 1 (η4 × k1) × m3 0, 70, 10, 40, 20, 6

Table-9:TabularformofFHSS(S,P )

Thecomparisontableoftheaboveresultantfuzzyhypersoftsetisasbelow, u1 u2 u3 u4 u5 u1 54444 u2 35332 u3 12542 u4 12352 u5 13445

Table-10:Thecomparisontableoftheaboveresultantfuzzyhypersoftset Nextwecomputethecolumnsum,rowsumandscorevalueforeach ui asshownfollowing, RowSum(ri) ColumnSum(ti) Score(Si) u1 211110 u2 1616 0 u3 1419 5 u4 1320 7 u5 17152

Table-11:Therowsum,columnsumandscoreof ui

Fromtheabovescoretable,itisclearthatthemaximumscoreis10,scoredby u1. Therefore thedecisionisinfavourofselecting u1

5. Conclusion

Inthepresentpaper,FuzzyHypersoftset’soperationssuchaslikesubset,union,intersection,equalset,complement,ANDandORareintroduced.Thevalidityandcomplementarity oftheproposedoperationsandmeaningsischeckedbypresentingthecorrectexample.Fuzzy hypersoftsetNHSSwouldbeanewmethodforcorrectselectionindecisiontakingproblems.wegivealsoanapplicationoffuzzyhypersofttheoryinobjectrecognitionproblem.The recognitionstrategyisbasedoncollectionofdataparametersformultiobserverinputs.ThealgorithminvolvesconstructingtheComparisonTablefromtheresultingfuzzyhypersoftsetand makingthefinaldecisionbasedonthemaximumscoredeterminedfromtheComparisonTable (Tables10and11).Inordertoexpandourwork,moreresearchshouldbeundertakentostudy theissuesofreducingfuzzyhypersoftsetsparameterization,andtoexplorethepossibilitiesof usingthefuzzyhypersoftsetsmethodtosolvereal-worldproblemssuchasdecision-making, forecasting,anddataanalysis.Matrices,similaritymeasure,singleandmulti-valued,interval valued,functions,distancemeasures,algorithms:scorefunction,VIKOR,TOPSIS,AHPof IntutionisticHypersoftsetwillbefuturework.Also,inthetopologicalfield,topicssuchas fuzzyhypersofttopologicalspaces,separationaxioms,compactness,connectedness,continuity canbestudiedinthefuture.

References

[1] Zadeh,L.A.(1965).Fuzzysets,InformationandControl,8(3),338–353.

[2] Pawlak,Z.(1982).Roughsets,InternationalJournalofComputerandInformationSciences,11(5),341–356.

[3] Molodtsov,D.,(1999).Softsettheory—firstresults,ComputersandMathematicswithApplications,37(45),19–31.

[4] Maji,P.K.,Biswas,R.,andRoy,A.R.,(2001).Fuzzysoftsets,TheJournalofFuzzyMathematics,9, 589–602.

[5] Feng,F.,Y.Jun,Y.B.andZhao,X.,(2008).Softsemirings,ComputersandMathematicswithApplications, 56(10),2621–2628.

[6] Jiang,Y.,Tang,Y.,Chen,Q.,Wang,J.andTang,S.,(2010).Extendingsoftsetswithdescriptionlogics, ComputersandMathematicswithApplications,59(6),2087–2096.

[7] Yang,X.,Lin,T.Y,Yang,J.,Li,Y.,andYu,D.,(2009).Combinationofinterval-valuedfuzzysetandsoft set,ComputersandMathematicswithApplications,58(3),521–527.

[8] Ahmad,B.,&Kharal,A.(2009).Onfuzzysoftsets.Advancesinfuzzysystems,2009.

[9] Maji,P.K.andRoy,A.R.(2002).Anapplicationofsoftsetsinadecisionmakingproblem,Computers &MathematicswithApplications,44(8-9),1077–1083.

[10] Chen,D.,(2005).Theparameterizationreductionofsoftsetsanditsapplications,ComputersandMathematicswithApplications,49(5-6),757–763.

[11] Cagman,N.andEnginoglu,S.,(2010).Softmatrixtheoryanditsdecisionmaking,ComputersandMathematicswithApplications,59(10),3308–3314.

[12] Cagman,N.andEnginoglu,S.,(2010).Softsettheoryanduni-intdecisionmaking,EuropeanJournalof OperationalResearch,207(2),848–855.

[13] Xie,N.,Wen,G.,&Li,Z.(2014).Amethodforfuzzysoftsetsindecisionmakingbasedongreyrelational analysisandDStheoryofevidence:applicationtomedicaldiagnosis.ComputationalandMathematical MethodsinMedicine,2014.

[14] C¸elik,Y.,&Yamak,S.(2013).Fuzzysoftsettheoryappliedtomedicaldiagnosisusingfuzzyarithmetic operations.Journalofinequalitiesandapplications,2013(1),82.

[15] Roy,A.R.,&Maji,P.K.(2007).Afuzzysoftsettheoreticapproachtodecisionmakingproblems.Journal ofcomputationalandAppliedMathematics,203(2),412-418.

[16] Kwun,Y.C.,Park,J.H.,Koo,J.H.,&Lee,Y.K.(2012).Anapplicationofgeneralizedinterval-valued intuitionisticfuzzysoftsetsinadecisionmakingproblem.InMechanicalEngineeringandTechnology(pp. 193-198).Springer,Berlin,Heidelberg.

[17] Guleria,A.,&Bajaj,R.K.(2018).PythagoreanFuzzy-NormInformationMeasureforMulticriteria Decision-MakingProblem.AdvancesinFuzzySystems,2018,1-11.

[18] Guleria,A.,&Bajaj,R.K.(2019).Techniqueforreducingdimensionalityofdataindecision-making utilizingneutrosophicsoftmatrices.NeutrosophicSetsandSystems,29(1),129–141.

[19] Guleria,A.,Srivastava,S.,&Bajaj,R.K.(2019).Onparametricdivergencemeasureofneutrosophicsets withitsapplicationindecision-makingmodels.NeutrosophicSetsandSystems,29(1),101–120.

[20] Guleria,A.,&Bajaj,R.K.(2019).EigenSphericalFuzzySetanditsApplicationinDecisionMaking Problem.ScientiaIranica.1–29.

[21] Chakraborty,A.,Banik,B.,Mondal,S.P.,&Alam,S.(2020).ArithmeticandGeometricOperatorsof PentagonalNeutrosophicNumberanditsApplicationinMobileCommunicationServiceBasedMCGDM Problem.NeutrosophicSetsandSystems,32(1),6.

[22] Chakraborty,A.,Mondal,S.P.,Alam,S.,&Dey,A.(2020).Classificationoftrapezoidalbipolarneutrosophicnumber,de-bipolarizationtechniqueanditsexecutionincloudservice-basedMCGDMproblem. Complex&IntelligentSystems,1-18.

[23] Chakraborty,A.,Mondal,S.P.,Alam,S.,&Mahata,A.(2020).Cylindricalneutrosophicsingle-valued numberanditsapplicationinnetworkingproblem,multi-criteriongroupdecision-makingproblemand graphtheory.CAAITransactionsonIntelligenceTechnology.

[24] F.Smarandache,ExtensionofSoftSettoHypersoftSet,andthentoPlithogenicHypersoftSet,Neutrosophicsetsandsystem.22(2018),168-170.

[25] Saqlain,M.,Jafar,N.,Moin,S.,Saeed,M.,&Broumi,S.(2020).SingleandMulti-valuedNeutrosophic HypersoftsetandTangentSimilarityMeasureofSinglevaluedNeutrosophicHypersoftSets.Neutrosophic

Sets and Systems, 32, 317-329.

[26] Gayen,S.,Smarandache,F.,Jha,S.,Singh,M.K.,Broumi,S.,&Kumar,R.(2020).Introductionto plithogenichypersoftsubgroup.NeutrosophicSetsandSystems,33(1),14.

[27] Rana,S.,Qayyum,M.,Saeed,M.,Smarandache,F.,&Khan,B.A.(2019).PlithogenicFuzzyWhole HypersoftSet,ConstructionofOperatorsandtheirApplicationinFrequencyMatrixMultiAttribute

[28] Martin,N.,&Smarandache,F.(2020).ConcentricPlithogenicHypergraphbasedonPlithogenicHypersoft

[29] Saeed,M.,Ahsan,M.,Siddique,M.K.,&Ahmad,M.R.(2020).AStudyofTheFundamentalsofHypersoft SetTheory,InternationalJournalofScientific&EngineeringResearch,11(1).

[30] Saqlain,M.,Moin,S.,Jafar,M.N.,Saeed,M.,&Smarandache,F.(2020).AggregateOperatorsofNeu-

Decision Making Technique. sets–A Novel Outlook. Neutrosophic Sets and Systems, 33. trosophic Hypersoft Set. Neutrosophic Sets and Systems, 32, 294-306.

Matrix Theory for Intuitionistic Fuzzy Hypersoft Sets and its application in Multi-Attributive Decision-Making Problems

Muhammad Naveed Jafar1 , Muhammad Saeed1* , Muhammad Haseeb2, Ahtasham Habib3

1 Department of Mathematics University of Management and Technology, Johar Town C-Block 54000, Pakistan, E-Mail: 15009265004@umt.edu.pk

1* Department of Mathematics University of Management and Technology, Johar Town C-Block 54000, Pakistan, E-Mail: muhammad.saeed@umt.edu.pk

2-3 Department of Mathematics Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan, Correspondence: muhammad.saeed@umt.edu.pk

Abstract: The main aspect of this study is to broaden the concept of intuitionistic fuzzy hypersoft set (IFHSS) and apply it as intuitionistic fuzzy hypersoft matrix (IFHSM). Fundamental workings of intuitionistic hypersoft matrices have been elaborated through suitable examples. Analytical study of common operations of IFHSM has been devised. A new algorithm, based on score function, has been constituted to represent decision-making issues and these issues can now be defined as numerical values to elaborate the hiring of employees for a private firm.

Keywords: MCDM, Hypersoft Matrix, Score Function, Intuitionistic Fuzzy Hypersoft Matrix.

1. Introduction

Zadeh L.A introduced the concept ‘Fuzzy sets’ to deal with the problem of uncertainty in 1965 [1]. Many problems arising due to uncertainty in many applications have been successfully dealt by using fuzzy sets and fuzzy logic. In 1975, Zadeh [2] introduced the interval valued fuzzy sets. This concept emphasized on the degree of membership of an element on closed subinterval of [0,1] rather than an element of [0,1]. Fuzzy set theory can better handle the problem of uncertainty that is due to unclarity or incomplete belonging of an element in a set, but it has its limitations when dealing with various real physical problems or incomplete data.

Atanassov [12] in 1986 worked on the generalization of fuzzy set known as intuitionistic fuzzy set (IFS). For the first time, the concept of natural hesitation occurring in human beings’ brain was awarded a membership or non-membership value in closed interval, so that the impact and effect of uncertainty could be portrayed realistically as happening in real world during decision making processes. It has been seen that situations where insufficient data is present, membership values may not be possible up to the level of our satisfaction. The same happens with

Theory and Application of Hypersoft Set

the non-membership values as allocating them values is not sometimes possible and degree of hesitation remains. In such scenarios, fuzzy set theory cannot be used effectively and this is where intuitionistic fuzzy set theory plays its role. In fact, the problems for which fuzzy set theory is used, these problems can be solved by intuitionistic fuzzy set theory as well and it is well suited for solving complicated problems. Hence, IFS has been found quite effective in dealing and solving the issue of uncertainty. [12,13]

In 1999, Molodtsov introduced a new mathematical tool know as soft set to deal with the complexities arising due to uncertainties. He demonstrated the way in which soft sets could be used. Their application can be used in game theory, operation research, Peron integration, Riemman-integration, probability, theory of management etc. Presently, much progress is being made in the working and applications of the soft set theory. [25,27]

The parameters in soft set theory are fuzzy concepts taken from the fuzzy set theory. Hence, set soft is a specified classification of subsets of the universe. Results have been projected by the application of using fuzzy soft sets in decision making in the works of Roy and Maji [32]. Measures showing similarity in fuzzy soft sets were proposed by Majumdar and Samanta [24]. By minimization of fuzzy soft sets, Yang et al. [37] did an analysis of decision-making problems using fuzzy soft sets. The idea of soft number, soft integral and soft derivative was developed through an analysis relying on the theory of soft sets. The problems of soft optimization were solved through this manner by Kovkov et al. [21]. For the forecasting of the volume of export and import in international trade, using fuzzy soft sets of soft set theory were introduced by Xiao et al. [34]. For the studying of business competitive capacity evaluation, Xiao et al. [35] also introduced soft set theory.

Solutions regarding data analysis due to the conditions of incomplete information by using soft sets were introduced by Zou and Xiao [38]. These solutions, by using soft sets, reflect the accurate state of incomplete data. Problems arising with the classifications of natural textures were dealt by a new algorithm by Mushrif et al. [28]. Analysis of various operations of soft sets was introduced by Ali et al. [14].

The idea of intuitionistic fuzzy soft set theory made up of intuitionistic fuzzy set and soft set models was introduced by Maji et al. [41-42]. Interval-valued fuzzy soft sets containing the combination of interval-valued fuzzy set and soft set models was worked upon by Yang et al. [42]

Importance of matrices cannot be ignored in the fields of science and engineering. But due to uncertainties arising due to imprecision in the nature of our environment, classic matrix theory cannot meet the demand of solving the problem of uncertainty. However, fuzzy soft sets represented as matrices were successfully used in the process known as fuzzy soft matrix to deal with the problems of decision making by Yong Tang and Chenli [39]. Jafar et al [30] worked on IFSS in Selection of Laptop.

Theory and Application of Hypersoft Set

Recently in 2018, another tool was created by Smarandache [43] in diversifying the modes available in overcoming the problems related to uncertainty. He generalized soft set theory to hyper soft set theory.

The propose of this research work is to introduce a new tool of dealing with uncertainty by combining intuitionist fuzzy set theory with hyper soft set theory. This combination of the two theories will make a new tool by the name of ‘Intuitionistic fuzzy hyper soft set’. Later on, it would be converted in the form of intuitionistic fuzzy hyper soft matrix.

2. Preliminaries

Definition 2.1: Fuzzy set

A pair (��,��) is said to be fuzzy set if there exist a function ��:�� →[0,1] where �� is set. The function �� =���� is called membership function of the fuzzy set �� =(��,��)

Definition 2.2: Soft set

Let �� be the universal set and ƿ(��) be the power set of �� Let �� ̌ be the set of attributes then the pair (��, ��) is said to be soft set over �� if ��:�� ̌ →ƿ(��)

Definition 2.3: Hypersoft set

Let �� be the universal set and ��(��)be the power set of ��. Suppose ℎ1,ℎ2,ℎ3 …ℎ�� where �� ≥1 be n distinct attributes whose corresponding attributive values respectively the sets ��1,��2,��3 ���� with ���� ∩���� =∅,�� ≠�� and ��,����{0,1,2,3 ��} then the pair (��,��1 ×��2 × ��3 × ×����),where ��:��1 ×��2 × ��3 × ×���� →��(��) is called a hypersoft set over ��

Definition 2.4: Intuitionistic fuzzy soft set

Consider �� is a universal set and Ĕ be the set of parameters. Let ��(��) denote the set of all intuitionistic fuzzy sets of ��. Let �� ⊆Ĕ. A pair (��,Ĕ) is an intuitionistic fuzzy soft set over ��, where is mapping given by ��:��→��(��) .

Definition 2.4: Intuitionistic fuzzy Hypersoft set (IFHSS)

Let �� be the initial universe of discourse and P(��) is the set of all possibilities of ��. suppose ℎ1,ℎ2,ℎ3 ℎ�� where �� ≥1 be n distinct attributes whose corresponding attributive values respectively the sets ��1,��2,��3 …����with ���� ∩���� =∅,�� ≠�� and ��,����{0,1,2,3,…,��} then the relation ��1 ×��2 × ��3 ×…×���� =�� then the pair (��,��) is said to be Intuitionistic fuzzy hypersoft set (IFHSS) ��:��1 ×��2 × ��3 × ×���� →��(��) and ��(��1 ×��2 × ��3 × ×����)={< ��,��(��(��)),��(��(��))>,������} where �� is the value of membership and �� is the value of non-membership such that ��:�� →[0,1],��:�� →[0,1] and also 0<��(��(��))+��(��(��))<2

3. Calculations

Definition 3.1: Intuitionistic fuzzy Hyper Soft Matrix (IFHSM): Let Ƒ={ƒ1,ƒ2,…,ƒ��} be the universal set and P(Ƒ) be the power sets of Ƒ. Suppose Ʈ1,Ʈ2, ,Ʈ�� where α ≥ 1, α be the well-defined attributes , whose corresponding attributive values are Ʈ1 ��,Ʈ2 ��,…,Ʈ�� �� and their relation is Ʈ1 �� ×Ʈ2 �� × ×Ʈ�� �� where ��,��,��, ,�� = ��,��, ,�� then the pair, (��,Ʈ1 �� ×Ʈ2 �� × × Ʈ�� �� ) is said to be Intuitionistic fuzzy Hyper Soft set over Ƒ where ��:(Ʈ1 �� ×Ʈ2 �� × ×Ʈ�� �� )→ ��(Ƒ) and it is define as ��(Ʈ1 �� ×Ʈ2 �� ×…×Ʈ�� �� )={<ƒ,��£(ƒ),��£(ƒ)> ƒ∈Ƒ,£∈(Ʈ1 �� ×Ʈ2 �� ×…×Ʈ�� �� )} Let ��£ =Ʈ1 �� ×Ʈ2 �� × ×Ʈ�� �� be the relation and its characteristic function is ����£:(Ʈ1 �� ×Ʈ2 �� × × Ʈ�� �� )→��(Ƒ) and it is define as ����£ ={<ƒ,��£(ƒ),��£(ƒ)> ƒ∈Ƒ,£∈(Ʈ1 �� ×Ʈ2 �� ×…×Ʈ�� �� )} Then the Tabular form of ��£ is given as Table 1: Tabular form of ��£ Ʈ�� �� Ʈ�� �� ⋯ Ʈ�� �� ƒ�� ����£(ƒ1,Ʈ1 ��) ����£(ƒ1,Ʈ2 ��) ⋯ ����£(ƒ1,Ʈ�� �� ) ƒ�� ����£(ƒ2,Ʈ1 ��) ����£(ƒ2,Ʈ2 ��) ����£(ƒ2,Ʈ�� �� ) ⋮ ⋮ ⋮ ⋱ ⋮ ƒ�� ����£(ƒ��,Ʈ1 ��) ����£(ƒ��,Ʈ2 ��) ����£(ƒ��,Ʈ�� �� ) If ������ =����£(ƒ��,Ʈ�� ��) where i=1,2,3…n , j=1,2,3…m , k=a,b,c, ,z Then a matrix is defining as

Thus, we can represent any Intuitionistic Fuzzy Hyper Soft Set in term of Intuitionistic Fuzzy Hyper Soft matrix.

Example 3.1.1: To illustrate the working of this theory, a need arises for a company to hire an employee to fill one of its vacant spaces. A total of five promising applicants apply to fill up the vacant space. From

Hypersoft Set

the human resources department of the company, a decision maker (DM) has been tasked for selection.

Let �� ={ƒ1,ƒ2,ƒ3,ƒ4,ƒ5} be the set of five applicants also consider the set of attributes as Ʈ1 = Qualification, Ʈ2 =Exprince, Ʈ3 = Age, Ʈ4 =Gender And their respective attributes are Ʈ1 �� =Qualification = {BS Hons., MS, Ph.D., Post Doctorate} Ʈ2 �� =Exprince= {5yr, 7yr, 10yr, 15yr} Ʈ3 �� =Age= {Less than thirty, Greater than thirty} Ʈ4 �� =Gender= {Male, Female} Let the function be F: Ʈ1 �� ×Ʈ2 �� ×Ʈ3 �� ×Ʈ4 �� → ��(��) Blew are the Intuitionistic values table from different dissuasion maker

Table 2: Decision maker Intuitionistic values for Qualification Ʈ�� ��(Qualification) ƒ�� ƒ�� ƒ�� ƒ�� ƒ��

BS Hons. (0.5,0.7) (0.3,0.5) (0.5,1.0) (0,0.9) (0.6,0.5) MS (0.3,0.8) (0.1,0.6) (0.1,0.3) (0.2,0.5 ) (0.3,0.9) Ph.D. (0.9,1) (0.4,0.6) (0.6,0.8) (0.1,0.9) (0.1,0.7) Post Doctorate (0.2,0.3) (0.3,0.4) (0.4,0.5) (0.5,0.6) (0.6,0.7)

Table 3: Decision maker Intuitionistic values for Experience Ʈ�� ��(Experience) ƒ�� ƒ�� ƒ�� ƒ�� ƒ��

5yr (0.4,0.7) (0.1,0.3) (0.2,0.4) (0.3,0.5) (0.4,0.6)

7yr (0.5,0.7) (0.6,0.8) (0.7,0.9) (0.8,1) (0.9,0)

10yr (0.4,0.7) (0.5,0.8) (0.6,0.9) (0.7,1) (0.8,0.3)

15yr (0.1,0.5) (0.2,0.6) (0.3,0.7) (0.4,0.8) (0.5,0.9)

Table 4: Decision maker Intuitionistic values for Age

(Age)

Less than thirty (0.6,0.8) (0.6,0.9) (0.3,0.5) (0.4,0.8) (0.5,0.9)

Greater than thirty (0.7,1) (0.5,0.7) (0.3,0.7) (0.5,0.6) (0.9,1)

Table 5: Decision maker Intuitionistic values for Gender Ʈ�� ��(Gender) ƒ�� ƒ�� ƒ�� ƒ�� ƒ��

Male (0.5,0.7) (0.1,0.5) (0.6,0.9) (0.4,0.8) (0.5,0.9) Female (0.3,0.7) (0.2,0.6) (0.2,0.4) (0.7,1) (0.9,0)

Intuitionistic fuzzy Hyper Soft set is defining as F:(Ʈ1 �� ×Ʈ2 �� ×Ʈ3 �� ×Ʈ4 ��)→ ��(��)

Let’s assume that F(Ʈ1 �� ×Ʈ2 �� ×Ʈ3 �� ×Ʈ4 ��)=F(MS,7yr,Greaterthanthirty,Male)=(ƒ1,ƒ2,ƒ3,ƒ5)

Then Intuitionistic Fuzzy Hyper Soft set of above relation F(Ʈ1 �� ×Ʈ2 �� ×Ʈ3 �� ×Ʈ4 ��)= {<ƒ1,((����(03,08),7����(05,07),Greaterthanthirty(07,1),Male(05,07))> <ƒ2,(����(0.1,0.6),7����(0.6,0.8),Greaterthanthirty(0.5,0.7),Male(0.1,0.5))> <ƒ3,(����(01,03),7����(07,09),Greaterthanthirty(03,07),Male(06,09))> <ƒ5,(����(03,09),7����(09,0 ),Greaterthanthirty(09,1),Male(05,09))>} The above relation can be written in the form of

Table 6: Tabular Representation of above relation

(����(0.3,0.8)) (7����(0.5,0.7)) (Greaterthanthirty(0.7,1)) (Male(0.5,0.7))

(����(01,06)) (7����(06,08)) (Greaterthanthirty(05,07)) (Male(01,05)

(����(01,03)) (7����(07,09)) (Greaterthanthirty(03,07)) (Male(06,09))

(����(0.3,0.9)) (7����(0.9,0)) (Greaterthanthirty(0.9,1)) (Male(0.5,0.9)) And its matrix form is defined as [��]4×4

[

(����,(0.3,0.8)) (7����,(0.5,0.7)) (Greaterthanthirty,(0.7,1)) (Male,(0.5,0.7)) (����,(01,06)) (7����,(06,08)) (Greaterthanthirty,(05,07)) (Male,(01,05)) (����,(01,03)) (7����,(07,09)) (Greaterthanthirty,(07,1)) (Male,(06,09)) (����,(03,09)) (7����,(09,0)) (Greaterthanthirty,(09,1)) (Male,(05,09))]

Definition 3.2: Square Intuitionistic Fuzzy Hyper Soft Matrix

Let �� =[������] be the IFHSM of order ��×�� where ������ =(�������� �� ,�������� �� ), then �� is said to be square IFHSM if n=�� . It means that if a IFHSM have same number of columns (alternatives) and rows (attributes) then it’s called square IFHSM.

Example 3.2.2: [��]4×4 = [

(����,(03,08)) (7����,(05,07)) (Greaterthanthirty,(07,1)) (Male,(05,07)) (����,(0.1,0.6)) (7����,(0.6,0.8)) (Greaterthanthirty,(0.5,0.7)) (Male,(0.1,0.5)) (����,(0.1,0.3)) (7����,(0.7,0.9)) (Greaterthanthirty,(0.7,1)) (Male,(0.6,0.9)) (����,(03,09)) (7����,(09,0)) (Greaterthanthirty,(09,1)) (Male,(05,09))]

Definition 3.3:

Transpose of square IFHSM

Let �� =[������] be the IFHSM of order ���� where ������ =(�������� �� ,�������� �� ) then ���� is said to be transpose of square IFHSM if rows interchange with column (column interchange with rows) of ��. It is denoted as ���� =[������]�� =(�������� �� ,�������� �� )�� =(�������� �� ,�������� �� )=[������]

Example 3.3.1 [��]4×4 = [

(����,(03,08)) (7����,(05,07)) (Greaterthanthirty,(07,1)) (Male,(05,07)) (����,(0.1,0.6)) (7����,(0.6,0.8)) (Greaterthanthirty,(0.5,0.7)) (Male,(0.1,0.5)) (����,(01,03)) (7����,(07,09)) (Greaterthanthirty,(07,1)) (Male,(06,09)) (����,(03,09)) (7����,(09,0)) (Greaterthanthirty,(09,1)) (Male,(05,09))]

Transpose of the upper Metrix is defining as [��]�� 4×4 = [

(����,(03,08)) (����,(01,06)) (����,(01,03)) (����,(03,09)) (7����,(05,07)) (7����,(06,08)) (7����,(07,09)) (7����,(09,0)) (Greaterthanthirty,(07,1)) (Greaterthanthirty,(05,07)) (Greaterthanthirty,(07,1)) (Greaterthanthirty,(09,1)) (Male,(05,07)) (Male,(01,05) (Male,(06,09)) (Male,(05,09)) ]

Definition 3.4:

Symmetric IFHSM

Let �� =[������] be the IFHSM of order ���� where ������ =(�������� �� ,�������� �� ) then Q is said to be symmetric IFHSM if ���� =Q i.e (�������� �� ,�������� �� )=(�������� �� ,�������� �� )

Definition 3.5: Scalar multiplication of IFHSM

Let �� =[������] be the IFHSM of order ��×�� where ������ =(�������� �� ,�������� �� ) and µ be any scalar. Then the product of scalar µ and matrix �� is defined by multiplying each element �� by µ. It is denoted as µ�� =[µ������] where0≤µ≤1

Example 3.5.1: Consider a IFHSM [��]4×4 and 0.3 is a scalar. [��]4×4 = [

(����,(03,08)) (7����,(05,07)) (Greaterthanthirty,(07,1)) (Male,(05,07)) (����,(0.1,0.6)) (7����,(0.6,0.8)) (Greaterthanthirty,(0.5,0.7)) (Male,(0.1,0.5) (����,(0.1,0.3)) (7����,(0.7,0.9)) (Greaterthanthirty,(0.7,1)) (Male,(0.6,0.9)) (����,(03,09)) (7����,(09,0)) (Greaterthanthirty,(09,1)) (Male,(05,09))]

Then scalar multiplication of IFHSM [��]4×4 is shown as [(0.3)��]4×4 = [

(����,(0.09,0.24)) (7����,(0.15,0.21)) (Greaterthanthirty,(0.21,0.3)) (Male,(0.15,0.21)) (����,(003,018)) (7����,(018,024)) (Greaterthanthirty,(015,021)) (Male,(003,015)) (����,(0.03,0.09)) (7����,(0.21,0.27)) (Greaterthanthirty,(0.21,0.3)) (Male,(0.18,0.27)) (����,(0.09,0.27)) (7����,(0.27,0)) (Greaterthanthirty,(0.27,0.3)) (Male,(0.15,0.27))]

Proposition 3.6

⇒��[������]⊆ ��[������] ⇒���� ⊆ ����

Theorem 3.7

Let �� =[������] be the IFHSM of order n×m , where ������ =(�������� �� ,�������� �� ), then

I. (����)�� =������ where α∈[0,1].

II. (����)��=��

Proof

I. Here (����)��, (α ��)�� ∈IFHSMn×m so (����)�� =[(α�������� �� ,���������� �� )]�� =[(α�������� �� ,���������� �� )] =��[(�������� �� ,�������� �� )] =��[(�������� �� ,�������� �� )]�� =������

II. Since ���� ∈������������×�� so (����)�� ∈IFHSMn×m Now (����)�� =([(�������� �� ,�������� �� )]��)�� =([(�������� �� ,�������� �� )])�� =[(�������� �� ,�������� �� )]=��

Definition 3.8: Trace of IFHSM

Let �� =[������] be the square IFHSM of order n×m , where ������ =(�������� �� ,�������� �� ) and �� =�� then trace of IFHSM is written as tr(��) and define as ����(��)=∑ [�������� �� �������� �� ] ��,�� ��=1,��=��

Example 3.8.1: Let us consider a IFHSM [��]4×4

[��]4×4 = [

(����,(03,08)) (7����,(05,07)) (Greaterthanthirty,(07,1)) (Male,(05,07)) (����,(0.1,0.6)) (7����,(0.6,0.8)) (Greaterthanthirty,(0.5,0.7)) (Male,(0.1,0.5)) (����,(0.1,0.3)) (7����,(0.7,0.9)) (Greaterthanthirty,(0.7,1)) (Male,(0.6,0.9)) (����,(03,09)) (7����,(09,0)) (Greaterthanthirty,(09,1)) (Male,(05,09))]

Then tr(��) =(0.3 0.8)+(0.6 0.8)+(0.7 1)+(0.5 0.9)= 1.4

Proposition 3.9

Let �� =[������] be the square IFHSM of order ����, where ������ =(�������� �� ,�������� �� ) and �� =�� �� be any scalar then ����(����) =������(��)

Proof: ����(����)=∑ [���������� �� ���������� �� ] ��,�� ��=1,��=�� =��∑ [�������� �� �������� �� ]=������(��) ��,�� ��=1,��=��

Let �� =[������] and �� =[������] be two IFHSM, where ������ =(�������� �� ,�������� �� ) and ������ =(�������� �� ,�������� �� )

Definition 3.10: Max-Min Product of IFHSM

Let �� =[������] and �� =[������] be two IFHSM, where������ =(�������� �� ,�������� �� ) and ������ =(�������� �� ,�������� �� ). Then, �� and �� are said to be conformable if their dimensions are equal to each other (number of columns of �� is equal to number of rows of ��). If �� =[������]��×�� and �� =[������]��×�� then ��⨂ �� = [������]��×�� where [������]=(���������� min(�������� �� ,�������� �� ),���������� max(�������� �� ,�������� �� ))

Theorem 3.11

Let �� =[������] and �� =[������] be two IFHSM, where ������ =(�������� �� ,�������� �� ) and ������ =(�������� �� ,�������� �� ). Then, (��⨂��)�� =���� ⨂ ����

Proof: Let ��⨂ �� =[������]��×��, then (��⨂��)�� =[������]��×��,���� =[������]��×�� ,���� =[������]��×�� Now (��⨂��)�� =(�������� �� ,�������� �� )��×�� = (���������� min(�������� �� ,�������� �� ),���������� max(�������� �� ,�������� �� ))��×�� =(�������� �� ,�������� �� )��×�� ⊗ (�������� �� ,�������� �� )��×�� =��⨂ ����

4. Operators of IFHSMs

Let �� =[������] and �� =[������] be the two IFHSM, where������ =(�������� �� ,�������� �� ) and ������ =(�������� �� ,�������� �� ). Then, i. Union: ��∪�� =�� where �������� �� =max(�������� �� ,�������� �� ), �������� �� =min(�������� �� ,�������� �� )

ii. Intersection: ��∩�� =�� where �������� �� =min(�������� �� ,�������� �� ), �������� �� =max(�������� �� ,�������� �� ).

iii. Arithmetic Mean: ��⨁�� =�� where �������� �� = (�������� �� +�������� �� ) 2 ,�������� �� = (�������� �� +�������� �� ) 2 .

iv. Weighted Arithmetic Mean: ��⨁ ���� =�� where �������� �� = (��1�������� �� +��2�������� �� ) ��1+��2 ,�������� �� = (��1�������� �� +��2�������� �� ) ��1+��2 ��1,��2 >0

v. Geometric Mean: ��⨀�� =�� where �������� �� =√�������� �� .�������� �� , �������� �� =√�������� �� .�������� �� .

vi. Weighted Geometric Mean: ��⨀ ���� =�� where �������� �� = √(�������� �� )��1 (�������� �� )��2 ��1+��2 , �������� �� = √(�������� �� )��1.(�������� �� )��2 ��1+��2 and ��1,��2 >0 vii. Harmonic Mean: ��⊘�� =�� where �������� �� = 2�������� �� �������� �� �������� �� +�������� �� , �������� �� = 2�������� �� �������� �� �������� �� +�������� �� . viii. Weighted Harmonic Mean: ��⊘�� �� =�� where �������� �� = ��1+��2 ��1 �������� �� + ��2 �������� �� , �������� �� = ��1+��2 ��1 �������� �� + ��2 �������� �� Proposition 4.1 Let �� =[������] and �� =[������] be two IFHSM, where ������ =(�������� �� ,�������� �� ) and ������ =(�������� �� ,�������� �� ). Then, i. (��∪��)�� =���� ∪���� ii. (��∩��)�� =���� ∩���� iii. (��⨁��)�� =����⨁���� iv. (��⨁ ���� )�� =����⨁������ v. (��⨀��)�� =����⨀���� vi. (��⨀ ���� )�� =���� ⨀ ���� �� vii. (��⊘��)�� =���� ⊘���� viii. (��⊘�� ��)�� =���� ⊘�� ���� Proof: i. (��∪��)�� =[(max(�������� �� ,�������� �� ),min(�������� �� ,�������� �� ))]��

Theory and Application of Hypersoft Set

=[(max(�������� �� ,�������� �� ),min(�������� �� ,�������� �� ))] =[(�������� �� , ℱ������ �� )]∪[(�������� �� ,�������� �� )] =[(�������� �� ,�������� �� )]�� ∪[(�������� �� ,�������� �� )]�� =���� ∪����

Proposition 4.2

Let �� =[������] and �� =[������] be the two-upper triangular IFHSM, where ������ =(�������� �� ,�������� �� ) and ������ =(�������� �� ,�������� �� ). Then, (��∪��), (��∩��), (��⨁��), (��⨁ ��)�� , (��⨀��) and (��⨀ ���� ) are all upper triangular IFHSM and vice versa.

Theorem 4.3:

Let �� =[������] and �� =[������] be the two IFHSM, where������ =(�������� �� ,�������� �� ) and ������ =(�������� �� ,�������� �� ). Then, i. (��∪��)⋄ =��⋄ ∩��⋄ ii. (��∩��)⋄ =��⋄ ∪��⋄ iii. (��⨁��)⋄ =��⋄⨁��⋄ iv. (��⨁ ���� )⋄ =��⋄⨁����⋄ v. (��⨀��)⋄ =��⋄⨀��⋄ vi. (��⨀ ���� )⋄ =��⋄ ⨀ ���� ⋄ vii. (��⊘��)⋄ =��⋄ ⊘��⋄ viii. (��⊘�� ��)⋄ =��⋄ ⊘�� ��⋄

Proof:

(��∪��)⋄ =[(max(�������� �� ,�������� �� ),,min(�������� �� ,�������� ��

Theorem 4.4 Let �� =[������] and �� =[������] be the two IFHSM, where������ =(�������� �� ,�������� �� ) and ������ =(�������� �� ,�������� �� ). Then, i. (��∪��)=(��∪��) ii. (��∩��)=(��∩��) iii. (��⨁��)= (��⨁��)

Theory and Application of Hypersoft Set

iv. (��⨁ ��)�� =(��⨁ ��)��

v. (��⨀��)=(��⨀��)

vi. (��⨀ ���� )=(��⨀ ���� ) vii. (��⊘��) =(��⊘��)

viii. (��⊘�� ��)=(��⊘�� ��)

Proof:

i. (��∪��)=[(max(�������� �� ,�������� �� ),min(�������� �� ,�������� �� ))] =[(max(�������� �� ,�������� �� ),min(�������� �� ,�������� �� ))] =[(�������� �� , �������� �� )]∪[(�������� �� ,�������� �� )] =(��∪��)

Theorem 4.5

Let =[������] , �� =[������] and ��=[������] be the two IFHSM, where ������ =(�������� �� ,�������� �� ), ������ =(�������� �� ,�������� �� ) and ������ =(�������� �� ,�������� �� ), then

i. (��∪��)∪�� =��∪(��∪��)

ii. (��∩��)∩�� =��∩(��∩��)

iii. (��⨁��)⨁�� ≠��⨁(��⨁��)

iv. (��⨀��)⨀�� ≠��⨀(��⨀��)

v. (��⊘��)⊘�� ≠��⊘(��⊘��)

Proof:

i. (��∪��)∪�� =[(max(�������� �� ,�������� �� ),min(�������� �� ,�������� �� ))]∪[(�������� �� ,�������� �� )] =[(max(�������� �� ,�������� �� ,�������� �� ),min(�������� �� ,�������� �� ,�������� �� ))] = [(�������� �� ,�������� �� )] ∪[(max(�������� �� ,�������� �� ),min(�������� �� ,�������� �� ))] =[(�������� �� ,�������� �� )] ∪[((�������� �� ,�������� �� )∪(�������� �� ,�������� �� ))] =��∪(��∪��)

Theorem 4.6 Let =[������] , �� =[������] and ��=[������] be the two IFHSM, where ������ =(�������� �� ,�������� �� ) , ������ =(�������� �� ,�������� �� ) and ������ =(�������� �� ,�������� �� ) Then, i. ��∩(��⨁��)=(��∩��)⨁(��∩��) ii. (��⨁��)∩�� =(��∩��)⨁(��∩��) iii. ��∪(��⨁��)=(��∪��)⨁(��∪��)

5. Applications

Intuitionistic fuzzy Hypersoft Matrix (IFHSM) in Decision Making Using Score Function

For example, a task of selection befalls a group of decision makers who have been tasked to select from �� number of objects. The objects are further presented on the basis of their �� number of attributes. IFHSM can be used to show their relation with each other. These attributes are allocated Intuitionistic values by the decision-makers and are portrayed by IFHSM of order ����. We can get score matrix by using IFHSM to calculate value matrices. In this way, total score of each object from score matrix can be determined. Value matrices are considered to be real matrices because they abide by the properties of real matrices. Score function also serves as a real matrix because we acquire it from two or more value matrices.

Definition 5.1 Let �� =[������] be the IFHSM of order ��×��, where ������ =(�������� �� ,�������� �� ), then the value of matrix �� is denoted as ��(��) and it is defined as ��(��)=[������ ��] of order ��×��, where ������ �� =�������� �� �������� �� . The Score of two IFHSM, �� = [������] and �� =[������] of order ��×�� is given as ��(��,��)=��(��)+ ��(��) and ��(��,��)=[������] where ������ = ������ �� +������ ��. The total score of each object in universal set is |∑ ������ �� ��=1 |.

5.2 Algorithm

Step 1: Construct a IFHSM as define in 3.1.

Step 2: Calculate the value matrix from IFHSM. Let �� =[������] be the IFHSM of order ����, where ������ =(�������� �� ,�������� �� ) then the value of matrix �� is denoted as ��(��) and it is defined as ��(��) = [������ ��] of order ����, where������ �� =�������� �� �������� ��

Step 3: Compute score matrix with the help of value matrices. The Score of two IFHSM �� =[������] and �� =[������] of order ���� is given as ��(��,��)=��(��)+��(��) and ��(��,��)= [������] where ������ = ������ �� +������ ��

Step 4: Compute total score from score matrix. The total score of each object in universal set is |∑ ������ �� ��=1 |.

Step 5: Find optimal solution by selecting an object of maximum score from total score matrix. Figure 1: Algorithm design for the proposed technique

5.3 Numerical Example

To illustrate the working of this theory, a need arises for a company to hire an employee to fill one of its vacant spaces. A total of fifteen promising applicants apply to fill up the vacant space. From the human resources department of the company, a decision maker (DM) has been tasked for selection.

Theory and Application of Hypersoft Set

The decision maker finds it quite hard and a time-consuming task to interview all of them to fill up its vacant seat in the company. But by the help of the theory Intuitionistic fuzzy hypersoft matrix he would be able to narrow down the criteria for selection of the best possible candidate to just one candidate. Let us assume that Ƒ be the set of all candidates Ƒ= {ƒ1,ƒ2,ƒ3,ƒ4,ƒ5,ƒ6,ƒ7,ƒ8,ƒ9,ƒ10,ƒ11,ƒ12,ƒ13,ƒ14,ƒ15 } And the selection criteria set by the company is given in the form of attributes as Ʈ1 = Qualification, Ʈ2 =Exprince, Ʈ3 = Age, Ʈ4 =Gender Also, these attributes are further classified as Ʈ1 �� =Qualification ={BS Hons. , MS , Ph.D. , Post Doctorate} Ʈ2 �� =Exprince= {5yr, 7yr, 10yr, 15yr} Ʈ3 �� =Age= {Less than thirty, Greater than thirty} Ʈ4 �� =Gender= {Male, Female}

The function F:Ʈ1 �� ×Ʈ2 �� ×Ʈ3 �� ×Ʈ4 �� →��(Ƒ) is Let’s assume the relation F((Ʈ1 �� ×Ʈ2 �� ×Ʈ3 �� ×Ʈ4 ��)=F(MS,7yr,Greaterthanthirty,Male) is the actual requirement of company for the selection of candidates? Four candidates {ƒ4,ƒ8,ƒ10,ƒ13} are shortlisted on the basis of assumed relation i.e. (MS,7yr,Greaterthanthirty,Male) Decision makers {��,��} are set for the selection of short-listed candidates. These decision makers give their valuable opinion in the form of IFHSSs separately as ��=F(MS,7yr,Greaterthanthirty,Male)={<ƒ2,(MS{05,06},7����{03,07},

Greaterthanthirty{05,09},Male{06,05})>,<ƒ6,(MS{03,01},7����{06,03}, Greaterthanthirty{07,03},Male{07,03})>,<ƒ8(MS{07,06},7����{06,08},

Greaterthanthirty{0.8,0.4},Male{0.6,0.1})>,<ƒ14,(MS{0.5,0.5},7����{0.3,0.7}, Greaterthanthirty{0.9,0.1},Male{0.4,0.3})>}

��=ℱ(MS,7yr,Greaterthanthirty,Male)={<��2,(MS{0.8,0.2},7����{0.7,0.3},

Greaterthanthirty{04,03},Male{05,05})>,<��6,(MS{08,01},7����{07,03},

Greaterthanthirty{08,01},Male{09,02})>,<��8(MS{05,04},7����{07,02},

Greaterthanthirty{0.9,0.1},Male{0.4,0.7})>,<��14,(MS{0.7,0.2},7����{0.2,0.7},

Greaterthanthirty{07,01},Male{06,04})>,}

Let’s apply the above define algorithm for the calculation of total score

Step I: The above two NHSSs are given in the form of IFHSMs as

[��]= [

(MS,(05,06)) (7����,(03,07)) (Greaterthanthirty,(05,09)) (Male,(06,05)) (MS,(0.3,0.1)) (7����,(0.6,0.3)) (Greaterthanthirty,(0.7,0.3)) (Male,(0.7,0.3)) (MS,(0.7,0.6)) (7����,(0.6,0.8)) (Greaterthanthirty,(0.8,0.4)) (MS,(05,05)) (7����,(03,07)) (Greaterthanthirty,(09,01)) (Male,(0.6,0.1)) (Male,(04,03))]

[��]= [

(MS,(04,05)) (7����,(02,05)) (Greaterthanthirty,(02,04)) (Male,(05,05)) (MS,(0.4,0.2)) (7����,(0.7,0.4)) (Greaterthanthirty,(0.8,0.9)) (Male,(0.8,0.9)) (MS,(05,02)) (7����,(02,03)) (Greaterthanthirty,(05,05)) (MS,(04,03)) (7����,(02,07)) (Greaterthanthirty,(01,01)) (Male,(07,02)) (Male,(04,04))]

Step II: Now calculate the value matrices of IFHSMs define in Step I.

[��(��)]= [

(MS,( 0.1)) (7����,( 0.4)) (Greaterthanthirty,( 0.4)) (Male,(0.1)) (MS,(0.2)) (7����,(0.3)) (Greaterthanthirty,(0.4)) (Male,(0.4)) (MS,(01)) (7����,( 02)) (Greaterthanthirty,(04)) (MS,(0)) (7����,( 04)) (Greaterthanthirty,( 08)) (Male,(05)) (Male,( 01))]

[��(��)]= [

(MS,( 01)) (7����,( 03)) (Greaterthanthirty,( 02)) (Male,(0)) (MS,(02)) (7����,(03)) (Greaterthanthirty,( 01)) (Male,( 01)) (MS,(0.3)) (7����,( 0.1)) (Greaterthanthirty,(0)) (MS,(0.1)) (7����,( 0.5)) (Greaterthanthirty,(0)) (Male,(0.5)) (Male,(0)) ]

Step III: Now compute score matrix by adding value matrices obtained in Step II.

[��(��,��)]= [

(MS,( 02)) (7����,( 07)) (Greaterthanthirty,( 06)) (Male,(01)) (MS,(04)) (7����,(06)) (Greaterthanthirty,(03)) (Male,(03)) (MS,(0.4)) (7����,( 0.3)) (Greaterthanthirty,(0.4)) (MS,(01)) (7����,( 09)) (Greaterthanthirty,(08)) (Male,(1)) (Male,( 01))]

Step IV: Now total score of score matrix is given as;

�������������������� =[1.4 1.6 1.5 01]

Step V: The candidate ƒ8 will be selected for vacant spaces as the total score of ƒ8 is highest among the rest of the total score of candidates.

6. Conclusions

In this study, new operations and definitions of IFHSM have been put forward and their workings have been illustrated by using a numerical example. Utility of IFHSM, concerning decision-making troubles, has been portrayed using a score matrix. Its usefulness is effective and helps us get accurate results. In future, this study will further help researchers in decision-making related calculations like TOPSIS, AHP and VIKOR

Acknowledgments:

The authors are highly thankful to the Editor-in-chief and the referees for their valuable comments and suggestions for improving the quality of this chapter.

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

References

[1] Zadeh L.A. (1965). Fuzzy sets, Information and Control, 8, 338-353.

[2] Zadeh L.A. (1975). The concept of a linguistic and application to approximate reasoning I, Information Sciences, 8,199-249.

[3] Pawlak, Z. (1982). Rough sets. International Journal of Computer Information and Science. 11, 341-356

[4] Pawlak, Z. (1991). Rough sets, Theoretical Aspects of Reasoning About Data. Kluwer, Netherlands,

[5] Skowron, A., and Grzymala-Busse, J. W. (1994). Rough set theory to evidence theory. In Advances in the Dempster-Shafer Theory of Evidence. R. R. Yaeger, M. Fedrizzi, and J. Kacprzyk, Eds. Wiley, New York, 193-236.

[6] Pawlak, Z., and Skowron. (1994). A. Rough membership functions, Advances in the Dempster-Shafer Theory of Evidence. Wiley, New York, 251-271.

[7] Krusinska, E., Slowinski, R., and Stefanowski, J. (1992). Discriminant versus rough set approach to vague data analysis. Appl. Stochastic Models and Data Anal. 8, 43 56.

[8] Skowron, A., and Rauszer, C. (1992). The discernibility matrices and functions in information systems. Intelligent Decision Support. Handbook of Advances and Applications of the Rough Set Theory, 8,331 362.

Theory and Application of Hypersoft Set 83

[9] Pawlak, Z., and Slowinski, R. (1994). Rough set approach to multiattribute decision analysis, European Journal of Operational Research, 72, 443 459.

[10] Grzymala-Busse, J. W. (1988). Knowledge acquisition under uncertainty A rough set approach. Journal of Intellectual Rob. System.1,3 16.

[11] Atanassov K. (1986). Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87–96.

[12] Biswas R. (1997). On fuzzy sets and intuitionistic fuzzy sets, International Journal of Fuzzy System. 3 ,3–11.

[13] Ali M.I, Feng F, Liu X, Min W.K, Shabir M. (2009). On some new operations in soft set theory, Computers and Mathematics with Applications, 57, 1547–1553.

[14] Aktas H, Çag˘man N. (2007). Soft sets and soft groups, Information Sciences 177, 2726–2735.

[15] Chen D, Tsang E.C.C, Yeung D.S, Wang X. (2005). The parameterization reduction of soft sets and its applications, Computers and Mathematics with Applications, 49, 757–763.

[16] Feng F, Jun Y. B, Zhao X. (2008). Soft semi rings, Computers and Mathematics with Applications 56(10), 2621–2628.

[17] Jun Y.B. (2008). Soft BCK/BCI-algebras, Computers and Mathematics with Applications 56, 1408–1413.

[18] Jun Y.B, Park C.H. (2008). Applications of soft sets in ideal theory of BCK/BCI-algebras, Information Sciences, 178, 2466–2475.

[19] Kong. Z, Gao.L, Wang.L, Li S. (2008). The normal parameter reduction of soft sets and its algorithm, Computers and Mathematics with Applications 56 ,3029–3037.

[20] Kovkov D.V, Kolbanov V.M, Molodtsov D.A. (2007). Soft sets theory-based optimization, Journal of Computer and Systems Sciences International 46 (6), 872–880.

[21] Maji P.K, Bismas R, A.R. Roy. (2003). Soft set theory, Computers and Mathematics with Applications, 45, 555–562.

[22] Maji P.K, Roy A.R, Biswas R. (2002). An application of soft sets in a decision making problem, Computers and Mathematics with Applications, 44, 1077–1083.

[23] Majumdar P, Samanta S.K. (2008). Similarity measure of soft sets, New Mathematics and Natural Computation, 4 (1), 1–12.

[24] Molodtsov D. (2004). The Theory of Soft Sets, URSS Publishers, Moscow

[25] Molodtsov D.A, Leonov V. Yu, Kovkov D.V. (2006). Soft sets technique and its application, Nechetkie Sistemy i Myagkie Vychisleniya 1 (1), 8–39.

[26] Molodtsov D. (1999). Soft set theory – first results, Computers and Mathematics with Applications 37, 19–31.

[27] Mushrif M.M, Sengupta S, Ray A.K. (2006). Texture classification using a novel, soft-set theory-based classification algorithm, Lecture Notes in Computer Science, 3851, 246–254.

[28] Pawlak Z. (1982). Rough sets, International Journal of Information and Computer Sciences 11, 341–356.

[29] Park C.H, Jun Y.B, Öztürk, (2008), M.A, Soft WS-algebras, Communications of Korean Mathematical Society, 23 (3) ,313–324.

Theory and Application of Hypersoft Set 84

[30] Jafar. M. N, Saeed. A, Waheed. M, Shafique. A. (2020). A comprehensive study of Intuitionistic Fuzzy Hypersoft Matrices and its Application in selection of Laptop by using Score Function, International Journal of Computer Applications, 177 (38), 8-17.

[31] Roy A.R, Maji P.K (2007). A fuzzy soft set theoretic approach to decision making problems, Journal of Computational and Applied Mathematics 203, 412–418.

[32] Sun Q-M, Zhang Z-L, Liu J. (2008). Soft sets and soft modules, Proceedings of Rough Sets and Knowledge Technology, RSKT-2008, Springer,403–409.

[33] Xiao Z, Gong K, Zou Y, (2009). A combined forecasting approach based on fuzzy soft sets, Journal of Computational and Applied Mathematics 228 (1), 326–333.

[34] Xiao Z, Li Y, Zhong B, Yang X. (2003) Research on synthetically evaluating method for business competitive capacity based on soft set, Statistical Research, 52–54.

[35] Xiao Z, Chen L, Zhong B, Ye S. (2005). Recognition for soft information based on the theory of soft sets, Proceedings of ICSSSM-05, 2, 1104–1106.

[36] Yang X, Yu D, Yang J, Wu C. (2007). Generalization of soft set theory: From crisp to fuzzy case, in: Bing-Yuan, Fuzzy Information and Engineering: Proceedings of ICFIE-2007, Advances in Soft Computing, 40, 345–355.

[37] Zou Y, Xiao Z. (2008). Data analysis approaches of soft sets under incomplete, Information, Knowledge-Based Systems, 21, 941–945

[38] Yang Yong and Ji.Chenli (2011). Fuzzy Soft Matrices and Their Applications” AICI 2011, Part I, LNAI 7002, 618–627,

[39] Maji P.K. (2009). More on intuitionistic fuzzy soft sets. Proceedings of the 12th International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, RSFDGrC 2009, in: Lecture Notes in Computer Science,5908, Springer, 231–240.

[40] Maji P.K, Biswas R, A.R. Roy. (2001). Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics 9(3), 677–692.

[41] Maji P.K, Roy A.R, Biswas R. (2004). On intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics, 12(3), 669–683.

[42] Yang X.B, Lin T.Y, Yang J.Y, Li Y. Yu D. (2009). Combination of interval-valued fuzzy set and soft set, Computers & Mathematics with Applications, 58 (3), 521–527.

[43] Smarandache.F. (2018). Extension of soft set to Hypersoft Set, and then Plithogenic Hypersoft Set, Neutrosophic sets and systems, 22, 168-170.

A Development of Pythagorean fuzzy hypersoft set with basic operations and decision-making approach based on the

correlation coefficient

1 School of Mathematics, Northwest University Xi’an, China. E-mail: ranazulqarnain7777@gmail.com

2* School of Science, Xi'an Polytechnic University, Xi'an 710048, China; xlxin@nwu.edu.cn

3 School of Science, Department of Mathematics, University of Management and Technology, Lahore, Pakistan. E-mail: muhammad.saeed@umt.edu.pk

2* Correspondence: E-mail: xlxin@nwu.edu.cn

Abstract: The idea of the Pythagorean fuzzy hypersoft set is a generalization of the intuitionistic fuzzy hypersoft set, which is used to express insufficient evaluation, uncertainty, and anxiety in decision-making. Compared with the intuitionistic fuzzy hypersoft set, the Pythagorean fuzzy hypersoft set can accommodate more uncertainty, which is the most important strategy for analyzing fuzzy information in the decision-making process. The most important determination of the present research is to perform basic operations under the Pythagorean fuzzy hypersoft set (PFHSS) with their mandatory properties. In it, we establish logical operators and propose the idea of necessity and possibility operations under PFHSS. In the following research under PFHSS, some desirable properties are proposed by using the proposed operations. We also introduce the correlation coefficient under the PFHSS structure and develop an algorithm for decision-making by using the developed correlation coefficient. Furthermore, a case study on decision-making difficulties proves the application of the proposed algorithm. Finally, a comparative analysis with the advantages, effectiveness, flexibility, and numerous existing studies demonstrates this method's effectiveness.

Keywords: Hypersoft set, intuitionistic fuzzy set, Pythagorean fuzzy soft set, Pythagorean fuzzy hypersoft set, correlation coefficient.

1. Introduction

Imprecision performs a dynamic part in many facets of life (such as modeling, medicine, engineering, etc.). However, people have raised a general question, that is, how can we express and use the concept of uncertainty in mathematical modeling. Many researchers in the world have proposed and recommended different methods of using uncertainty theory. First, Zadeh stepped forward the theory of fuzzy set (FS) [1] to resolve the problem of uncertainty and ambiguity.

Rana Muhammad Zulqarnain1, Xiao Long Xin1, 2 *, Muhammad Saeed3

Theory and Application of Hypersoft Set

In some cases, we need to investigate membership as a non-membership value to properly interpret objects that FS cannot handle. To overcome the above-mentioned issues, Atanasov proposed the idea of intuitionistic fuzzy sets (IFS) [2]. Researchers have also used several other theories, such as cubic intuitionistic fuzzy sets [3], interval value IFS [4], linguistic interval-valued IFS [5], etc. After carefully considering the above theories, the experts considered the essence, and the sum of its two membership values and non-membership values cannot exceed one.

Atanassov's intuitionistic fuzzy sets only deal with insufficient data due to membership and nonmembership values, but IFS cannot deal with incompatible and imprecise information. Molodtsov [6] proposed a general mathematical tool to deal with uncertain, ambiguous, and uncertain matters, called soft set (SS). Maji et al. [7] extended the concept of SS and developed some operations with properties and used the established concepts for decision-making [8]. Maji et al. [9] proposed the concept of a fuzzy soft set (FSS) by combining FS and SS. They also proposed an Intuitionistic Fuzzy Soft Set (IFSS) with basic operations and properties [10]. Yang et al. [11] proposed the concept of interval-valued fuzzy soft sets with operations (IVFSS) and proved some important results by combining IVFS and SS, and they also used the developed concepts for decision-making. Jiang et al. [12] proposed the concept of interval-valued intuitionistic fuzzy soft sets (IVIFSS) by extending IVIFS. They also introduced the necessity and possibility operators of IVIFSS and their properties.

Garg and Arora [13] progressed the generalized version of the IFSS with weighted averaging and geometric aggregation operators and built a decision-making technique to resolve complications beneath an intuitionistic fuzzy environment. Garg [14] developed some improved score functions to analyze the ranking of the normal intuitionistic and interval-valued intuitionistic sets and established the new methodologies to solve multi-attribute decision making (MADM) problems. The idea of entropy measure and TOPSIS under the correlation coefficient (CC) has been developed by using complex q-rung orthopair fuzzy information and used the established strategies for decision making [15]. The authors [16] developed the aggregate operators by using dual hesitant fuzzy soft numbers and utilized the proposed operators to solve multi-criteria decision making (MCDM) problems. To measure the relationship among dual hesitant fuzzy soft set Arora and Garg [17] introduced the CC and developed a decision-making approach under the presented environment to solve the MCDM approach, they also used the proposed methodology for decision making, medical diagnoses, and pattern recognition. They also developed the operational laws and presented some prioritized aggregation operators under linguistic IFS environment [18] and extended the Maclaurin symmetric mean (MSM) operators to IFSS based on Archimedean Tconorm and T-norm [19].

As the above work is considered an environment where linear inequalities have been examined between membership degree (MD) as well as non-membership degree (NMD). However, if the decision-maker goes steady with object MD = 0.7 and NDM = 0.6, then ��.��+��.�� ≰ ��. We can see that; it cannot be handled by the above studied IFS theories. To overcome the above-mentioned limitations, Yager [20, 21] prolonged the IFS to Pythagorean fuzzy sets (PFSs) by modifying the condition ��+ ��≤�� to ���� + ���� ≤��. Zhang and Xu [22] defined some operational laws and extended the TOPSIS technique to solve MCDM problems under PFSs environment. Many researchers used the TOPSIS method for medical diagnoses, pattern recognition, and decisionmaking, etc. to find the positive ideal alternative in different structures [23-31]. Wei and Lu [32] presented several Pythagorean fuzzy power aggregation operators with their properties and proposed the decision-making approaches to solving MADM problems based on developed operators. Wang and Li [33] proposed the Pythagorean fuzzy interaction operational laws and power Bonferroni mean operators. Then, they discussed some specific cases of established operators

Theory and Application of Hypersoft Set 87 and considered their properties. Zhang [34] established a novel decision-making technique based on Pythagorean fuzzy numbers (PFNs) to solve multiple criteria group decision making (MCGDM) problems. He also developed the accuracy function for the ranking of PFNs and similarity measures under a PFSs environment with some desirable properties. Guleria and Bajaj [35] introduced a Pythagorean fuzzy soft matrix and its various possible types and binary operations with their properties. Further, they used the proposed Pythagorean fuzzy soft matrices for decision making by developing a new algorithm by using a choice matrix and weighted choice matrix. They also presented some noel information measures to solve MCDM problems [36]. Bajaj and Guleria [37] proposed the notion of object-oriented Pythagorean fuzzy soft matrix and the parameter-oriented Pythagorean fuzzy soft matrix has been utilized to outline an algorithm for the dimensionality reduction in the process of decision making. The authors developed the new (R, S)-norm discriminant measure of PFSs has been proposed along with its various properties and proposed a decision-making approach to solving MCDM problems [38]. Zulqarnain et al. [39] proposed the aggregation operators for Pythagorean fuzzy soft sets and established the decision-making approach using their developed technique. The authors of [40] extended the TOPSIS technique under Pythagorean fuzzy soft sets and used their proposed method for supplier selection in green supply chain management.

Recently, Smarandache [41] extended the concept of soft sets to hypersoft sets (HSS) by replacing the one-parameter function F with a multi-parameter (sub-attribute) function defined on the Cartesian product of n different attributes. The established HSS is more flexible than soft sets and is more suitable for the decision-making environment. He also introduced the further extension of HSS, such as crisp HSS, fuzzy HSS, intuitionistic fuzzy HSS, neutrosophic HSS, and plithogenic HSS. Nowadays, HSS theory and its extensions are developing rapidly. Many researchers have developed different operators and properties based on HSS and its extensions [42-46]. Zulqarnain et al. [47] introduced the TOPSIS technique and aggregation operators for PFHSS. They also utilized their developed technique for the selection of anti-virus face mask. Abdel-Basset [48] uses plithogenic set theory to resolve uncertain information and evaluate the financial performance of manufacturing. Then, they use VIKOR and TOPSIS methods to find the weight vector of financial ratios using the AHP method to achieve this goal. Abdel basset, etc. [49] proposed an effective combination of plithogenic aggregation operations and quality feature deployment methods. The advantage of this combination is that it can improve accuracy and thus evaluate decision-makers. The following research is organized as follows: In Section 2, we review some basic definitions used in the following sequels, such as SS, FSS, IFS, IFSS, and IFHSS, etc. In Section 3, we propose some operations with their necessary properties such as union, intersection, restricted union, and extended intersection, etc. under PFHSS. We develop the AND operator, OR operator, necessity operation, and possibility operation with their several desirable properties in section 4. In section 5, the idea of correlation coefficient in PFHSS structure is introduced, and develop the decisionmaking technique based on the presented CC. We also used the developed approach to solve decision making problems in an uncertain environment. Furthermore, we use some existing techniques to present comparative studies between our proposed methods. Likewise, present the advantages, naivety, flexibility as well as effectiveness of the planned algorithms. We organized a brief discussion and a comparative analysis of the recommended approach and the existing techniques in section 6.

2. Preliminaries

Theory and Application of Hypersoft Set 88

In this section, we recollect some basic definitions which are helpful to build the structure of the following manuscript such as soft set, hypersoft set, fuzzy hypersoft set, and intuitionistic fuzzy hypersoft set.

Definition 2.1 [6]

Let �� be the universal set and ℰ be the set of attributes concerning ��. Let ��(��) be the power set of �� and �� ⊆ℰ. A pair (ℱ,��) is called a soft set over �� and its mapping is given as ℱ:�� →��(��)

It is also defined as: (ℱ,��)={ℱ(ℯ) ∈��(��):ℯ ∈ ℰ,ℱ(ℯ) = ∅����ℯ ∉��}

Definition 2.2 [9] ℱ(��) be a collection of all fuzzy subsets over �� and ℰ be a set of attributes. Let �� ⊆ ℰ, then a pair (ℱ,��) is called FSS over ��, where ℱ is a mapping such as ℱ: �� → ��(��).

Definition 2.3 [41]

Let �� be a universe of discourse and ��(��) be a power set of �� and �� = {��1, ��2, ��3,..., ����},(n ≥ 1) be a set of attributes and set ���� a set of corresponding sub-attributes of ���� respectively with ���� ∩ ���� = φ for �� ≥ 1 for each ��, �� �� {1,2,3 … ��} and �� ≠ �� Assume ��1 × ��2 × ��3× … × ���� = �� ⃛ = {��1ℎ ×��2�� ×⋯×������} be a collection of multi-attributes, where 1 ≤ ℎ ≤ ��, 1 ≤ �� ≤ ��, and 1 ≤ �� ≤ ��, and ��, ��, and �� ∈ ℕ Then the pair (ℱ, ��1 × ��2 × ��3× … × ���� = �� ⃛ ) is said to be HSS over �� and its mapping is defined as ℱ: ��1 × ��2 × ��3× … × ���� = �� → ��(��). It is also defined as (ℱ, ��) = {�� ̌,ℱ��(��̌):�� ̌ ∈ ��, ℱ��(��̌) ∈ ��(��)}

Definition 2.4 [2] An IFS is an object of the form �� = {〈(����,����(����),����(����))⎸���� ∈ ��〉} on a universe ��, where ���� and ����: �� → [0,1] represents the degree of membership and non-membership respectively of any element ���� ∈ ��, to set �� with the following condition 0 ≤����(����)+����(����) ≤ 1.

Definition 2.5 [10] A mapping ℱ: �� → ��(��) is known as an IFSS and defined as ℱ����(��) = {(����,����(����),����(����))⎸���� ∈ ��}, where ����(����) and ����(����) are the degree of acceptance and rejection respectively for all ���� ∈ �� and 0 ≤ ����(����),����(����), ����(����)+����(����) ≤ 1.

Definition 2.6 [41]

Let �� be a universe of discourse and ��(��) be a power set of �� and �� = {��1, ��2, ��3,..., ����},(n ≥ 1) be a set of attributes and set ���� a set of corresponding sub-attributes of ���� respectively with ���� ∩ ���� = φ for �� ≥ 1 for each ��, �� �� {1,2,3 … ��} and �� ≠ �� “Assume ��1 × ��2 × ��3× … × ���� = �� ⃛ = {��1ℎ ×��2�� ×⋯×������} be a collection of sub-attributes, where 1 ≤ ℎ ≤ ��, 1 ≤ �� ≤ ��, and 1 ≤ �� ≤ ��, and ��, ��, and �� ∈ ℕ and ���� be a collection of all fuzzy subsets over �� Then the pair (ℱ, ��1 × ��2 × ��3× … × ���� = �� ⃛ ) is said to be FHSS over �� and its mapping is defined as ℱ: ��1

2.7 Consider the universe of discourse �� = {��1,��2} and �� = {ℓ1 =��������ℎ������������ℎ������������,ℓ2 = ����������������,ℓ3 =��������������} be a collection of attributes with following their corresponding attribute

Theory and Application of Hypersoft Set 89 values are given as teaching methodology = ��1 = {��11 =����������������������,��12 = ������������������������������}, Subjects = ��2 = {��21 =������ℎ��������������,��22 = ������������������������������,��23 = ��������������������}, and Classes = ��3 = {��31 =��������������,��32 = ����������������}. Let �� = ��1 × ��2 × ��3 be a set of attributes �� = ��1 × ��2 × ��3 = {��11,��12} × {��21,��22,��23} × {��31,��32} = {(��11,��21,��31),(��11,��21,��32),(��11,��22,��31),(��11,��22,��32),(��11,��23,��31),(��11,��23,��32), (��12,��21,��31),(��12,��21,��32),(��12,��22,��31),(��12,��22,��32),(��12,��23,��31),(��12,��23,��32),} �� = {�� ̌ 1,�� ̌ 2,�� ̌ 3,�� ̌ 4,�� ̌ 5,�� ̌ 6,�� ̌ 7,�� ̌ 8,�� ̌ 9,�� ̌ 10,�� ̌ 11,�� ̌ 12}

Then the FHSS over �� is given as follows (ℱ, �� ⃛ ) =

{(�� ̌ 1,(��1,.6),(��2,.3)),(�� ̌ 2,(��1,.7),(��2,.5)),(�� ̌ 3,(��1,.8),(��2,.3)),(�� ̌ 4,(��1,.2),(��2,.8)), (�� ̌ 5,(��1, 4),(��2, 3)),(�� ̌ 6,(��1, 2),(��2, 5)),(�� ̌ 7,(��1, 6),(��2, 9)),(�� ̌ 8,(��1, 2),(��2, 3)), (�� ̌ 9,(��1, 4),(��2, 7)),(�� ̌ 10,(��1, 1),(��2, 7)),(�� ̌ 11,(��1, 4),(��2, 6)),(�� ̌ 5,(��1, 2),(��2,

7))}

Definition 2.8 [46] Let �� be a universe of discourse and ��(��) be a power set of �� and �� = {��1, ��2, ��3,..., ����},(n ≥ 1) be a set of attributes and set ���� a set of corresponding sub-attributes of ���� respectively with ���� ∩ ���� = φ for �� ≥ 1 for each ��, �� �� {1,2,3 … ��} and �� ≠ ��. Assume ��1 × ��2 × ��3× … × ���� = �� = {��1ℎ ×��2�� ×⋯×������} be a collection of sub-attributes, where 1 ≤ ℎ ≤ ��, 1 ≤ �� ≤ ��, and 1 ≤ �� ≤ ��, and ��, ��, and �� ∈ ℕ and �������� be a collection of all intuitionistic fuzzy subsets over �� Then the pair (ℱ, ��1 × ��2 × ��3× … × ���� = ��) is said to be IFHSS over �� and its mapping is defined as ℱ: ��1 × ��2 × ��3× … × ���� = �� ⃛ → �������� It is also defined as (ℱ, �� ⃛ ) = {(�� ̌,ℱ��(��̌)):�� ̌ ∈�� ⃛ , ℱ��(��̌) ∈ �������� ∈ [0,1]}, where ℱ��(��̌) = {〈��,��ℱ(��̌)(��),��ℱ(��̌)(��)〉:�� ∈ ��}, where ��ℱ(��)(��) and ��ℱ(��)(��) represents the membership and non-membership values of the attributes such as ��ℱ(��̌)(��), ��ℱ(��̌)(��) ∈ [0,1], and 0 ≤ ��ℱ(��̌)(��) + ��ℱ(��̌)(��) ≤ 1. Simply an intuitionistic fuzzy hypersoft number (IFHSN) can be expressed as ℱ = {(��ℱ(��̌)(��),��ℱ(��̌)(��))}, where 0 ≤��ℱ(��̌)(��)+��ℱ(��̌)(��) ≤ 1.

3. Basic Operations and properties on a Pythagorean fuzzy hypersoft set

In this section, we introduce PFHSS and some basic operations with their properties under the Pythagorean fuzzy hypersoft environment. Definition 3.1 Let �� be a universe of discourse and ��(��) be a power set of �� and �� = {��1, ��2, ��3,..., ����},(n ≥ 1) be a set of attributes and set ���� a set of corresponding sub-attributes of ���� respectively with ���� ∩ ���� = φ for �� ≥ 1 for each ��, �� �� {1,2,3 … ��} and �� ≠ ��. Assume ��1 × ��2 × ��3× … × ���� = �� = {��1ℎ ×��2�� ×⋯×������} be a collection of sub-attributes, where 1 ≤ ℎ ≤ ��, 1 ≤ �� ≤ ��, and 1 ≤ �� ≤ ��, and ��, ��, and �� ∈ ℕ and �������� be a collection of all Pythagorean fuzzy subsets over �� Then the pair (ℱ, ��1 × ��2 × ��3× … × ���� = ��) is said to be PFHSS over �� and its mapping is defined as ℱ: ��1 × ��2 × ��3× … × ���� = �� ⃛ → �������� It is also defined as (ℱ, �� ⃛ ) = {(�� ̌,ℱ�� ⃛(��̌)):�� ̌ ∈�� ⃛ , ℱ�� ⃛(��̌) ∈ �������� ∈ [0,1]}, where ℱ�� ⃛(��̌) = {〈��,��ℱ(��̌)(��),��ℱ(��̌)(��)〉:�� ∈ ��}, where ��ℱ(��̌)(��) and ��ℱ(��̌)(��) represents the membership and non-membership values of the attributes such as ��ℱ(��̌)(��), ��ℱ(��̌)(��) ∈ [0,1], and 0 ≤ (��ℱ(��̌)(��))2 + (��ℱ(��̌)(��))2 ≤ 1.

Theory and Application of Hypersoft Set 90 Simply a Pythagorean fuzzy hypersoft number (PFHSN) can be expressed as ℱ = {(��ℱ(��̌)(��),��ℱ(��̌)(��))}, where 0 ≤(��ℱ(��̌)(��))2 + (��ℱ(��̌)(��))2 ≤ 1.

Example 3.2 Consider the universe of discourse �� = {��1,��2} and �� = {ℓ1 =��������ℎ������������ℎ������������,ℓ2 = ����������������,ℓ3 =��������������} be a collection of attributes with following their corresponding attribute values are given as teaching methodology = ��1 = {��11 =����������������������,��12 = ������������������������������}, Subjects = ��2 = {��21 =������ℎ��������������,��22 = ������������������������������,��23 = ��������������������}”, and Classes = ��3 = {��31 =��������������,��32 = ����������������}. Let �� = ��1 × ��2 × ��3 be a set of attributes

= ��1

2

21,��22,��23} × {��31,��32} = {(��11,��21,��31),(��11,��21,��32),(��11,��22,��31),(��11,��22,��32),(��11,��23,��31),(��11,��23,��32), (��12,��21,��31),(��12,��21,��32),(��12,��22,��31),(��12,��22,��32),(��12,��23,��31),(��12,��23,��32),} �� = {�� ̌ 1,�� ̌ 2,�� ̌ 3,�� ̌ 4,�� ̌ 5,�� ̌ 6,�� ̌ 7,�� ̌ 8,�� ̌ 9,�� ̌ 10,�� ̌ 11,�� ̌ 12} Then the PFHSS over �� is given as follows (ℱ, ��) = {

(�� ̌ 1,(��1,(6, 3)),(��2,(5, 7))),(�� ̌ 2,(��1,(6, 7)),(��2,(7, 5))),(�� ̌ 3,(��1,(4, 8)),(��2,(3, 7))), (�� ̌ 4,(��1,( 6, 5)),(��2,( 5, 6))),(�� ̌ 5,(��1,( 7, 3)),(��2,( 4, 8))),(�� ̌ 6,(��1,( 5, 4)),(��2,( 6, 5))), (�� ̌ 7,(��1,(5, 6)),(��2,(4, 5))),(�� ̌ 8,(��1,(2, 5)),(��2,(3, 9))),(�� ̌ 9,(��1,(4, 6)),(��2,(8, 5))), (�� ̌ 10,(��1,(.7,.4)),(��2,(.7,.2))),(�� ̌ 11,(��1,(.4,.5)),(��2,(.5,.3))),(�� ̌ 12,(��1,(.5,.7)),(��2,(.4,.7))) }

Definition 3.3

Let (ℱ, �� ⃛ ) and (��, ��) be two PFHSS over ��, then (ℱ, �� ⃛ ) is said to be a Pythagorean fuzzy hypersoft subset of (��, �� ⃛ ), if

1. �� ⊆ �� ⃛ 2. ℱ��(��)(��) ⊆ ����(��̌)(��) for all �� ∈ �� Where ��ℱ(��̌)(��) ≤ ����(��̌)(��), and ��ℱ(��̌)(��) ≥ ����(��̌)(��): �� ∈ ��

Definition 3.4

Let (ℱ, ��) and (��, �� ⃛ ) be two PFHSS over ��, then (ℱ, ��) = (��, �� ⃛ ), if (ℱ, ��) ⊆ (��, �� ⃛ ) and (��, �� ⃛ ) ⊆ (ℱ, �� ⃛ ).

Definition 3.5

Let �� be a universe of discourse and �� ⃛ be a set of attributes, then a pair (∅,��⃛) is said to be empty PFHSS, if ��ℱ(��)(��) = 0, and ��ℱ(��)(��) = 1 for all �� ̌ ∈ �� and �� ∈ ��. It can be represented by ∅ℱ(��̌)(��) and defined as follows (∅,��) = {�� ̌,(��,(0,1)):�� ∈ ��,�� ̌ ∈ ��}

Definition 3.6

Let �� be a universe of discourse and �� ⃛ be a set of attributes, then a pair (∅,��⃛) is said to be universal PFHSS, if ��ℱ(��)(��) = 1, and ��ℱ(��)(��) = 0 for all �� ̌ ∈ �� and �� ∈ ��. It can be represented by ��ℱ(��̌)(��) and defined as follows (��,��) = {�� ̌,(��,(1,0)):�� ∈ ��,�� ̌ ∈ ��}.

Definition 3.7

Let (ℱ, �� ⃛ ) = {(�� ̌,〈��,��ℱ(��̌)(��),��ℱ(��̌)(��)〉:�� ∈��):�� ̌ ∈��⃛} be a PFHSS over ��, then its complement is denoted by (ℱ,��)��, and is defined as follows (ℱ,��)�� = {(�� ̌,〈��,��ℱ(��)(��),��ℱ(��)(��)〉:�� ∈��):�� ̌ ∈��}.

Proposition 3.8

If (ℱ, �� ⃛ ) be a PFHSS, then

1. (ℱ��,��)�� = (ℱ, ��)

2. (∅��,��⃛)= (��,��⃛)

3. (����,��) = (∅,��)

Proof 1

Let ( ℱ , �� ⃛ ) = {(�� ̌,〈��,��ℱ(��̌)(��),��ℱ(��̌)(��)〉:�� ∈��):�� ̌ ∈��⃛} be a PFHSS over �� , then by using definition 3.7. we have (ℱ��,�� ⃛ ) = {(�� ̌,〈��,��ℱ(��̌)(��),��ℱ(��̌)(��)〉:�� ∈��):�� ̌ ∈��⃛}. Again, “by using definition 3.7 (ℱ��,��)�� = {(�� ̌,〈��,��ℱ(��̌)(��),��ℱ(��̌)(��)〉:�� ∈��):�� ̌ ∈��}

Hence, (ℱ��,�� ⃛ )�� = (ℱ, �� ⃛ ). Similarly, we can prove 2 and 3.

Definition 3.9

Let (ℱ, �� ⃛ ) and (��, �� ⃛ ) be two PFHSS over ��, then their union is defined as (ℱ, �� ⃛ ) ∪ (��, ��) = {��,(������{��ℱ(��̌)(��),����(��̌)(��)},������{��ℱ(��̌)(��),����(��̌)(��)}):�� ∈ ��,�� ̌ ∈ ��⃛}

Proposition 3.10

Let (ℱ, ��), (��, �� ⃛ ), and (ℋ, C) be three PFHSS over ��. Then

1. (ℱ, �� ⃛ ) ∪ (ℱ, �� ⃛ ) = (ℱ, �� ⃛ )

2. (ℱ, ��) ∪ (∅,��) = (ℱ, ��)

3. (ℱ, �� ⃛ ) ∪ (��,��⃛) = (��,��⃛)

4. (ℱ, �� ⃛ ) ∪ (��, �� ⃛ ) = (��, �� ⃛ ) ∪ (ℱ, �� ⃛ )

5. ((ℱ, ��) ∪ (��, �� ⃛ )) ∪ (ℋ, C ⃛ ) = (ℱ, ��) ∪ ((��,�� ⃛ ) ∪ (ℋ, C ⃛ ))

Proof 1 As we know that (ℱ,��) = {(��,��ℱ(��̌)(��),��ℱ(��̌)(��)) ⎸�� ∈ ��} be an PFHSS, then (ℱ, ��) ∪ (ℱ, ��) = {��,(������{��ℱ(��)(��),��ℱ(��)(��)},������{��ℱ(��)(��),��ℱ(��)(��)}):�� ∈ ��,�� ̌ ∈ ��} (ℱ, ��) ∪ (ℱ, ��) = {(��,��ℱ(��)(��),��ℱ(��)(��)) ⎸�� ∈ ��}

Hence (ℱ, �� ⃛ ) ∪ (ℱ, �� ⃛ ) = (ℱ, �� ⃛ ).

Proof 2 As we know that (ℱ,��) = {(��,��ℱ(��)(��),��ℱ(��)(��)) ⎸�� ∈ ��} be a PFHSS, and (∅,��) = {�� ̌,(��,(0,1)):�� ∈ ��,�� ̌ ∈ ��} be an empty PFHSS. Then, (ℱ, �� ⃛ ) ∪ (∅,��⃛) = {��,(������{��ℱ(��̌)(��),0},������{��ℱ(��̌)(��),1}):�� ∈ ��,�� ̌ ∈ ��⃛} (ℱ, �� ⃛ ) ∪ (∅,��⃛) = {(��,��ℱ(��̌)(��),��ℱ(��̌)(��)) ⎸�� ∈ ��} (ℱ, ��) ∪ (∅,��) = (ℱ, ��).

Proof 3 As we know that

Theory and Application of Hypersoft Set

(ℱ,��) = {(��,��ℱ(��̌)(��),��ℱ(��̌)(��)) ⎸�� ∈ ��} be a PFHSS, and (��,��) = {�� ̌,(��,(1,0)):�� ∈ ��,�� ̌ ∈ ��} be an empty PFHSS. Then, (ℱ, �� ⃛ ) ∪ (��,��⃛) = {��,(������{��ℱ(��̌)(��),1},������{��ℱ(��̌)(��),0}):�� ∈ ��,�� ̌ ∈ ��⃛} (ℱ, ��) ∪ (��,��) = {�� ̌,(��,(1,0)):�� ∈ ��,�� ̌ ∈ ��}. (ℱ, �� ⃛ ) ∪ (��,��⃛) = (��,��⃛) Proof 4 As (ℱ,��) = {(��,��ℱ(��̌)(��),��ℱ(��̌)(��)) ⎸�� ∈ ��} and (��,�� ⃛ ) = {(��,����(��̌)(��),����(��̌)(��)) ⎸�� ∈ ��} be two

PFHSS, then (ℱ, �� ⃛ ) ∪ (��, ��) = {��,(������{��ℱ(��̌)(��),����(��̌)(��)},������{��ℱ(��̌)(��),����(��̌)(��)}):�� ∈ ��,�� ̌ ∈ �� ⃛∪��} (ℱ, ��) ∪ (��, �� ⃛ ) = {��,(������{����(��)(��),��ℱ(��)(��)},������{����(��)(��),��ℱ(��)(��)}):�� ∈ ��,�� ̌ ∈ ��∪��⃛} (ℱ, ��) ∪ (��, �� ⃛ ) = (��, �� ⃛ ) ∪ (ℱ, ��).

Similarly, we can prove 5.

Definition 3.11

Let (ℱ, �� ⃛ ) and (��, ��) be two PFHSS over ��, then their intersection is defined as follows: (ℱ, ��) ∩ (��, �� ⃛ ) = {��,(������{��ℱ(��)(��),����(��)(��)},������{��ℱ(��)(��),����(��)(��)}):�� ∈ ��,�� ̌ ∈ ��}.

Proposition 3.12

Let (ℱ, �� ⃛ ), (��, ��), and (ℋ, C ⃛ ) be three PFHSS over ��. Then

1. (ℱ, ��) ∩ (ℱ, ��) = (ℱ, ��)

2. (ℱ, �� ⃛ ) ∩ (∅,��⃛) = (ℱ, �� ⃛ )

3. (ℱ, ��) ∩ (��,��) = (��,��)

4. (ℱ, �� ⃛ ) ∩ (��, ��) = (��, ��) ∩ (ℱ, �� ⃛ )

5. ((ℱ, �� ⃛ ) ∩ (��, ��)) ∩ (ℋ, C) = (ℱ, �� ⃛ ) ∩ ((��,��) ∩ (ℋ, C))

Proof By using Definition 3.11 we can prove easily.

Proposition 3.13

Let (ℱ,��) and (��,�� ⃛ ) be three PFHSS, then

1. ((ℱ,��) ∪ (��,�� ⃛))�� = (ℱ,��)�� ∩ (��,��⃛)��

2. ((ℱ,��⃛) ∩ (��,�� ⃛))�� = (ℱ,��⃛)�� ∪ (��,��⃛)��

Proof 1

As we know that (ℱ,�� ⃛ ) = {(��,��ℱ(��̌)(��),��ℱ(��̌)(��)) ⎸�� ∈ ��} and (��,�� ⃛ ) = {(��,����(��̌)(��),����(��̌)(��)) ⎸�� ∈ ��} be two PFHSS, then by using Definition 3.9 (ℱ, ��) ∪ (��, �� ⃛ ) = {��,(������{��ℱ(��)(��),����(��)(��)}",������{��ℱ(��)(��),����(��)(��)}):�� ∈ ��,�� ̌ ∈ ��}.

By using definition 3.7, we have ((ℱ,��) ∪ (��,�� ⃛))�� = {��,(������{��ℱ(��)(��),����(��)(��)},������{��ℱ(��)(��),����(��)(��)}):�� ∈ ��,�� ̌ ∈ ��}

Now (ℱ,��)�� = {〈(��, ��ℱ(��)(��), ��ℱ(��)(��)) ⎸�� ∈ ��〉} and (��,��⃛)�� = {〈(��,����(��)(��),����(��)(��)) ⎸�� ∈ ��〉}.

Theory and Application of Hypersoft Set 93

By using Definition 3.11 (ℱ,��)�� ∩ (��,��⃛)�� = {��,(������{��ℱ(��)(��),����(��)(��)},������{��ℱ(��)(��),����(��)(��)}):�� ∈ ��,�� ̌ ∈ ��} So ((ℱ,��⃛) ∪ (��,�� ⃛))�� = (ℱ,��⃛)�� ∩ (��,��⃛)�� Proof 2

As we know that (ℱ,�� ⃛ ) = {〈(��,��ℱ(��̌)(��),��ℱ(��̌)(��)) ⎸�� ∈ ��〉} and (��,��) = {〈(��,����(��̌)(��),����(��̌)(��)) ⎸�� ∈ ��〉} be two PFHSS, then by using Definition 3.11 (ℱ, ��) ∩ (��, �� ⃛ ) = {��,(������{��ℱ(��)(��),����(��)(��)},������{��ℱ(��)(��),����(��)(��)}):�� ∈ ��,�� ̌ ∈ ��}.

By using Definition 3.7, we have ((ℱ,��) ∩ (��,�� ⃛))�� = {��,(������{��ℱ(��)(��),����(��)(��)},������{��ℱ(��)(��),����(��)(��)}):�� ∈ ��,�� ̌ ∈ ��} Now (ℱ,��⃛)�� = {(��, ��ℱ(��̌)(��), ��ℱ(��̌)(��)) ⎸�� ∈ ��} and (��,��⃛)�� = {(��,����(��̌)(��),����(��̌)(��)) ⎸�� ∈ ��}

By using Definition 3.9 (ℱ,��⃛)�� ∪ (��,��⃛)�� = {��,(������{��ℱ(��̌)(��),����(��̌)(��)},������{��ℱ(��̌)(��),����(��̌)(��)}):�� ∈ ��,�� ̌ ∈ ��⃛} So ((ℱ,��) ∩ (��,�� ⃛))�� = (ℱ,��)�� ∪ (��,��⃛)�� Definition 3.14 Let (ℱ, �� ⃛ ) and (��, �� ⃛ ) be two PFHSS over ��, then their restricted union is defined as ��(ℱ, �� ⃛ ) ∪��(��, ��) = {��ℱ(��̌)(��) ������ ̌ ∈�� ⃛ �� ⃛ ,�� ∈ �� ����(��̌)(��) ������ ̌ ∈�� �� ⃛ ,�� ∈ �� ������{��ℱ(��̌)(��),����(��̌)(��)} ������ ̌ ∈ ��∩�� ⃛ ,�� ∈ �� ��(ℱ, �� ⃛ ) ∪��(��, ��) = {��ℱ(��̌)(��) ������ ̌ ∈�� ⃛ ��,�� ∈ �� ����(��̌)(��) ������ ̌ ∈�� ⃛ ��,�� ∈ �� ������{��ℱ(��)(��),����(��)(��)} ������ ̌ ∈��∩�� ⃛ ,�� ∈ �� Definition 3.15 Let (ℱ, �� ⃛ ) and (��, �� ⃛ ) be two PFHSS over ��, then their extended intersection is defined as ��(ℱ, �� ⃛ ) ∩�� (��, ��) =

Proposition 3.16

Theory and Application of Hypersoft Set 94 4. (ℱ,��) ∩�� ((ℱ,��) ∪�� (��,�� ⃛)) = (ℱ,��)

Proof 1 Consider ( ℱ , �� ⃛ ) = {(�� ̌,〈��,��ℱ(��̌)(��),��ℱ(��̌)(��)〉:�� ∈��):�� ̌ ∈�� ⃛,} , and (��,��) = {(�� ̌,〈��,����(��̌)(��),����(��̌)(��)〉:�� ∈ ��):�� ̌ ∈�� ⃛,} are two PFHSS over the universe of discourse �� (ℱ,�� ⃛ ) ∩ (��,��) = {��,(������{��ℱ(��̌)(��),����(��̌)(��)},������{��ℱ(��̌)(��),����(��̌)(��)}):�� ∈ ��} (ℱ,�� ⃛ ) ∪ ((ℱ,��⃛) ∩ (��,��)) = {��,(������{��ℱ(��̌)(��),������{��ℱ(��̌)(��),����(��̌)(��)}},������{��ℱ(��̌)(��),������{��ℱ(��̌)(��),����(��̌)(��)}}):�� ∈ ��} (ℱ,��) ∪ ((ℱ,��) ∩ (��,�� ⃛)) = {(�� ̌,〈��,��ℱ(��̌)(��),��ℱ(��̌)(��)〉:�� ∈��):�� ̌ ∈��}. Therefore, (ℱ,�� ⃛ ) ∪ ((ℱ,��⃛) ∩ (��,��)) = (ℱ,�� ⃛ ) Proof 2 Consider ( ℱ , �� ⃛ ) = {(�� ̌,〈��,��ℱ(��̌)(��),��ℱ(��̌)(��)〉:�� ∈��):�� ̌ ∈��⃛} , and (��,��) = {(�� ̌,〈��,����(��̌)(��),����(��̌)(��)〉:�� ∈ ��):�� ̌ ∈��⃛} are two PFHSS over the universe of discourse �� (ℱ,�� ⃛ ) ∪ (��,��) = {��,(������{��ℱ(��̌)(��),����(��̌)(��)},������{��ℱ(��̌)(��),����(��̌)(��)}):�� ∈ ��} (ℱ,�� ⃛ ) ∩ ((ℱ,��⃛) ∪ (��,�� ⃛)) = {��,(������{��ℱ(��̌)(��),������{��ℱ(��̌)(��),����(��̌)(��)}},������{��ℱ(��̌)(��),������{��ℱ(��̌)(��),����(��̌)(��)}}):�� ∈ ��} (ℱ,��) ∩ ((ℱ,��) ∪ (��,�� ⃛)) = {(�� ̌,〈��,��ℱ(��)(��),��ℱ(��)(��)〉:�� ∈��):�� ̌ ∈��}. Therefore, (ℱ,��) ∩ ((ℱ,��) ∪ (��,�� ⃛)) = (ℱ,��). Similarly, we can prove 3 and 4. 4. Logical operators and Necessity and Possibility operators under the Pythagorean fuzzy hypersoft set. In this section, we propose the idea of AND-operator, OR-operator, Necessity operator, and possibility operator under the PFHSS with their several desirable properties. We also introduce the correlation coefficient under the PFHSS environment.

Definition 4.1

Let (ℱ, ��) and (��, �� ⃛ ) be two PFHSS over ��, then their OR-operator is represented by (ℱ, ��) ˅ (��, ��) and defined as follows (ℱ, ��) ˅ (��, �� ⃛ ) = (⅄,��×��⃛), where ⅄(�� ̌ 1 ×�� ̌ 2) = ℱ��(�� ̌ 1) ∪ ����(�� ̌ 2) for all (�� ̌ 1 ×�� ̌ 2) ∈ ��×�� ⃛ . ⅄(�� ̌ 1 ×�� ̌ 2) = {������{��ℱ(�� ̌ 1)(��),����(�� ̌ 2)(��)},������{��

Theory and Application of Hypersoft Set 95

Let (ℱ, ��), (��, �� ⃛ ), and (ℋ, C ⃛ ) be three PFHSS over ��. Then

1. (ℱ,�� ⃛ )˅(��,��) = (��,��)˅ (ℱ, �� ⃛ )

2. (ℱ,��)˄(��,�� ⃛ ) = (��,�� ⃛ )˄ (ℱ, ��)

3. (ℱ,�� ⃛ )˅((��,��)˅ (ℋ, C ⃛ )) = ((ℱ,�� ⃛ )˅(��,��))˅ (ℋ, C ⃛ )

4. (ℱ,��)˄((��,�� ⃛ )˄ (ℋ, C)) = ((ℱ,��)˄(��,�� ⃛ ))˄ (ℋ, C)

5. ((ℱ,��⃛)˅(��,��))�� = ℱ��(�� ⃛ ) ˄ ℘��(��)

6. ((ℱ,��)˄(��,�� ⃛))�� = ℱ��(��) ˅ ℘��(�� ⃛ )

Proof 1 By using Definitions 4.1, 4.2 we can prove easily.

Definition 4.4

Let (ℱ, �� ⃛ ) be a PFHSS, then necessity operation on PFHSS represented by ⊕ (ℱ, �� ⃛ ) and defined as follows

⊕ (ℱ, �� ⃛ ) = {(�� ̌,〈��,��ℱ(��̌)(��),1 ��ℱ(��̌)(��)〉:�� ∈��):�� ̌ ∈�� ⃛,}

Definition 4.5

Let (ℱ, �� ⃛ ) be a PFHSS, then possibility operation on PFHSS represented by (ℱ, �� ⃛ ) and defined as follows

⊗ (ℱ, ��) = {(�� ̌,〈��,1 ��ℱ(��)(��),��ℱ(��)(��)〉:�� ∈��):�� ̌ ∈��,}

Proposition 4.6

Let (ℱ, ��) and (��, �� ⃛ ) be two PFHSS, then

1. ⊕((ℱ,��) ∪�� (��,�� ⃛)) = ⊕ (��, �� ⃛ ) ∪�� ⊕ (ℱ, ��)

2. ⊕((ℱ,��) ∩�� (��,�� ⃛)) = ⊕ (��, �� ⃛ ) ∩�� ⊕ (ℱ, ��)

Proof 1

As we know that (ℱ,�� ⃛ ) = {(��,��ℱ(��̌)(��),��ℱ(��̌)(��)) ⎸�� ∈ ��} and (��,��) = {(��,����(��̌)(��),����(��̌)(��)) ⎸�� ∈ ��} are two PFHSS. Let ((ℱ,��⃛) ∪�� (��,��)) = (ℋ, C ⃛ )

,�� ∈ ��

By using Definition 4.4 ⊕��(ℋ, C ⃛ ) = {��ℱ(��̌)(��) ������ ̌ ∈�� ⃛ �� ⃛ ,�� ∈ �� ����(��̌)(��) ������ ̌ ∈�� ⃛ �� ⃛ ,�� ∈ �� ������{��ℱ(��)(��),����(��)(��)} ������ ̌ ∈��∩�� ⃛ ,�� ∈ �� ⊕��(ℋ, C ⃛ ) = {1 ��ℱ(��̌)(��) ������ ̌ ∈�� ⃛ ��,�� ∈ �� 1 ����(��̌)(��) ������ ̌ ∈�� ⃛ ��,�� ∈ �� ������{1 ��ℱ(��)(��),1 ����(��)(��)} ������ ̌ ∈��∩�� ⃛ ,�� ∈ �� Assume ⊕ (��, ��) ∩�� ⊕ (ℱ, �� ⃛ ) = ℵ, where ⊕(ℱ,�� ⃛ ) and ⊕(��,��) are given as follows by using the definition of necessity operation. ⊕(ℱ,�� ⃛ ) = {(��,��ℱ(��̌)(��),1 ��ℱ(��̌)(��)) ⎸�� ∈ ��} and ⊕(��,��) = {(��,����(��̌)(��),1 ����(��̌)(��)) ⎸�� ∈ ��}. By using Definition 3.15 ��ℵ = {��ℱ(��)(��) ������ ̌ ∈�� �� ⃛ ,�� ∈ �� ����(��̌)

Theory and Application of Hypersoft Set

∈�� �� ⃛ ,�� ∈ ��

��ℱ(��̌)(��),1 ����(��̌)(��)} ������ ̌ ∈�� ⃛ ∩��,�� ∈ �� ⊗��(ℋ, C ⃛ ) =

ℱ(��)(��) ������ ̌ ∈�� �� ⃛ ,�� ∈ �� ����(��̌)(��) ������ ̌ ∈�� �� ⃛ ,�� ∈ �� ������{��ℱ(��̌)(��),����(��̌)(��)} ������ ̌ ∈�� ⃛ ∩��,�� ∈ �� Assume ⊗(ℱ,��) ∪�� ⊗(��,�� ⃛ )= ℵ, where ⊗(ℱ,��) and ⊗(��,�� ⃛ ) are given as follows by using the definition of necessity operation. ⊗(ℱ,�� ⃛ ) = {(��,1 ��ℱ(��̌)(��), ��ℱ(��̌)(��)) ⎸�� ∈ ��} and ⊗(��,�� ⃛ ) = {(��,1 ����(��̌)(��),����(��̌)(��)) ⎸�� ∈ ��}. By using Definition 3.14 ��ℵ = {1 ��ℱ(��̌)(��) ������ ̌ ∈�� �� ⃛ ,�� ∈ �� 1 ����(��̌)(��) ������ ̌ ∈�� �� ⃛ ,�� ∈ �� ������{1 ��ℱ(��̌)(��),1 ����(��̌)(��)} ������ ̌ ∈�� ⃛ ∩��,�� ∈ �� ��ℵ = {��ℱ(��̌)(��) ������ ̌ ∈�� �� ⃛ ,�� ∈ �� ����(��̌)(��) ������ ̌ ∈�� �� ⃛ ,�� ∈ �� ������{��ℱ(��̌)(��),����(��̌)(��)} ������ ̌ ∈�� ⃛ ∩��,�� ∈ �� Consequently ⊗ (ℋ, C) and ℵ are the same, so ⊗((ℱ,��) ∪�� (��,�� ⃛)) = ⊗ (��, �� ⃛ ) ∪�� ⊗ (ℱ, ��). Proof 2 As we know that (ℱ,��) = {(��,��ℱ(��̌)(��),��ℱ(��̌)(��)) ⎸�� ∈ ��} and (��,�� ⃛ ) = {(��,��

are the same, so ⊗ ((ℱ,��⃛) ∩�� (��,��)) = ⊗ (��, ��) ∩�� ⊗ (ℱ, �� ⃛ ).

Proposition 4.8

Let (ℱ, �� ⃛ ) and (��, �� ⃛ ) be two PFHSS, then

1. ⊕((ℱ,��⃛)˄(��,��)) = ⊕ (ℱ, �� ⃛ ) ˄ ⊕ (��, ��)

2. ⊕((ℱ,��⃛)˅(��,��)) = ⊕ (ℱ, �� ⃛ ) ˅ ⊕ (��, ��)

3. ⊗((ℱ,��)˄(��,�� ⃛)) = ⊗(ℱ,��) ˄ ⊗(��,�� ⃛ )

4. ⊗((ℱ,��⃛)˅(��,�� ⃛)) = ⊗(ℱ,�� ⃛ ) ˅ ⊗(��,�� ⃛ )

Proof 1 Proof is straight forward.

5. Application of Correlation Coefficient for Decision Making Under PFHSS Environment

In this section, we present the correlation coefficient under the PFHSS environment and establish an algorithm based on the proposed CC under PFHSS and utilize the proposed approach for decision making in real-life problems.

Definition 5.1

Let (ℱ,�� ⃛ ) = {(����,��ℱ(�� ̌ ��)(����),��ℱ(�� ̌ ��)(����)) ⎸���� ∈ ��} and (��,�� ⃛ ) = {(����,����(�� ̌ ��)(����),����(�� ̌ ��)(����)) ⎸���� ∈ ��} be two PFHSSs defined over a universe of discourse ��. Then, their informational energies of (ℱ,�� ⃛ ) and (��,��) can be described as follows: Ϛ����������(ℱ,�� ⃛ ) = ∑ ∑ ((��ℱ(�� ̌ ��)(����))4 +(��ℱ(�� ̌ ��)(����))4) �� ��=1 �� ��=1 Ϛ����������(��,��) = ∑ ∑ ((����(�� ̌ ��)(����))�� +(����(�� ̌ ��)(����))��)

Definition 5.2 Let (ℱ,�� ⃛ ) = {(����,��ℱ(�� ̌ ��)(����),��ℱ(�� ̌ ��)(����)) ⎸���� ∈��} and (��,��) = {(����,����(�� ̌ ��)(����),����(�� ̌ ��)(����)) ⎸���� ∈ ��} be two PFHSSs defined over a universe of discourse ��. Then, their correlation measure between (ℱ,�� ⃛ ) and (��,��) can be described as follows: ������������((ℱ,��⃛),(��,��)) = ∑ ∑ ((��ℱ(�� ̌ ��)(����))2 ∗(����(�� ̌ ��)(����))2 + (��ℱ(�� ̌ ��)(����))2 ∗(����(�� ̌ ��)(����))2) �� ��=1 �� ��=1 Definition 5.3

5.1 Algorithm for Correlation Coefficient under PFHSS

Step 1. Pick out the set containing sub-attributes of parameters.

Step 2. Construct the PFHSS according to experts in form of PFHSNs. Step 3. Find the informational energies of PFHSS.

Step 4. Calculate the correlation between PFHSSs by using the following formula ������������((ℱ,��⃛),(��,�� ⃛ )) = ∑ ∑ ((��ℱ(�� ̌ ��)(����))2 ∗(����(�� ̌ ��)(����))2 + (��ℱ(�� ̌ ��)(����))2 ∗(����(�� ̌ ��)(����))2) �� ��=1 �� ��=1

Step 5. Calculate the CC between PFHSSs by using the following formula ������������((ℱ,��⃛),(��,��)) = ������������((ℱ,��),(��,��)) √Ϛ����������(ℱ,��)∗√Ϛ����������(��,��)

Step 6. Choose the alternative with a maximum value of CC.

Step 7. Analyze the ranking of the alternatives. A flowchart of the above-presented algorithm can be seen in Figure 1.

Step 1 •Input Alternatives and sub-attributes of parameters

Step 2 •Expert's evaluation for each alternative in terms of PFHSNs according to sub-attributes values of the parameters

Step 3 •Compute the informational energies for each alternative Step 4 •Measure the correlation between alternatives sub-attributes and department requirenment

Step 5 •Compute the correlation coefficient

Step 6 •Choose the alternative with a maximum value of CC Step 7 •Analyze the ranking of the alternatives

Figure 1: Flowchart for correlation coefficient under PFHSS

5.2 Problem Formulation and Application of PFHSS For Decision Making

Department of the scientific discipline of some university �� will have one scholarship for the position of the doctoral degree. Several students apply to get a scholarship but referable probabilistic along with CGPA (cumulative grade points average), simply four students call for enrolled for undervaluation such as �� = {��1,��2,��3,��4} be a set of selected students call for the interview. The president of the university hires a committee of four decision-makers (DM) �� = {��1,��2,��3,��4} for the selection of doctoral degree student. The team of DM decides the criteria (attributes) for the selection of doctoral degree such as �� = {ℓ1 =Publications,ℓ2 =Subjects,ℓ3 =IF} be a collection of attributes and their corresponding sub-attribute are given as Publications = ℓ1 = {��11 =����������ℎ����10,��12 = ����������ℎ����10} , Subjects = ℓ2 = {��21 =������ℎ��������������,��22 = ������������������������������}, and IF = ℓ3 = {��31 =45,��32 = 47}. Let ��′ = ℓ1 × ℓ2 × ℓ3 be a set of subattributes ��′ = ℓ1 × ℓ2 × ℓ3 = {��11,��12} × {��21,��22} × {��31,��32} = {(��11,��21,��31),(��11,��21,��32),(��11,��22,��31),(��11,��22,��32), (��12,��21,��31),(��12,��21,��32),(��12,��22,��31),(��12,��22,��32)}, ��′ = {�� ̌ 1,�� ̌ 2,�� ̌ 3,�� ̌ 4,�� ̌ 5,�� ̌ 6,�� ̌ 7,�� ̌ 8} be a set of all multi sub-attributes. Each DM will evaluate the ratings of each alternative in the form of PFHSNs under the considered multi sub-attributes. The developed method to find the best alternative is as follows.

5.3 Application of PFHSS For Decision Making

Assume �� = {��1,��2,��3,��4} be a set of alternatives who are shortlisted for interview and �� = {ℓ1 =Publications,ℓ2 =Subjects,ℓ3 =Qualification} be a set of parameters for the selection of scholarship positions. Let the corresponding sub-attribute are given as Publications = ℓ1 = {��11 =����������ℎ����10,��12 = ����������ℎ����10} , Subjects = ℓ2 = {��21 =������ℎ��������������,��22 = ������������������������������}, and IF = ℓ3 = {��31 =45,��32 = 47}. Let ��′ = ℓ1 × ℓ2 × ℓ3 be a set of subattributes. The development of the decision matrix according to the requirement of the scientific discipline department in terms of PFHSNs is given in Table 1.

Table

1. Decision Matrix for Concerning Department

���� ( 3, 8) ( 7 3) ( 6, 7) ( 5, 4) ( 2, 4) ( 4, 6) ( 5, 8) ( 9, 3) ���� (.6,.7) (.4,.6) (.3,.4) (.9,.2) (.3,.8) (.2,.4) (.7,.5) (.4,.5) ���� ( 7, 3) ( 2, 5) ( 1, 6) ( 3, 4) ( 4 6) ( 8, 4) ( 6, 7) ( 2, 5) ���� (.8,.4) (.2,.9) (.2,.4) (.4,.6) (.6,.5) (.5,.6) (.4,.5) (.8,.3)

Develop the decision matrices for each alternative in terms of PFHSNs by considering their subattributes values of given attributes can be seen in Table 2- Table 5.

Table 2. Decision Matrix for Alternative ��(1)

(��)

��

��

�� ��

��

���� ( 7, 6) ( 3, 4) ( 6, 5) ( 3, 9) ( 5, 4) ( 4, 6) ( 7, 5) ( 4, 8)

Theory and Application of Hypersoft Set 101

( 8, 5) ( 7, 4) ( 9, 2) ( 7, 4) ( 4, 5) ( 9, 3) ( 2, 7) ( 3, 8)

(.3,.7) (.4,.5) (.4,.8) (.3,.4) (.6,.7) (.3,.4) (.9,.2) (.7,.2)

( 5, 4) ( 7, 6) ( 9, 3) ( 8, 5) ( 9, 2) ( 2, 4) ( 4, 6) ( 6, 5)

Table 3. Decision Matrix for Alternative ��(2)

(.6,.5) (.3,.8) (.4,.5) (.7,.4) (.6,.4) (.4,.5) (.3,.4) (.7,.5)

( 8, 4) ( 9, 3) ( 1, 8) ( 1, 2) ( 4, 6) ( 3, 7) ( 6, 8) ( 8, 4)

( 6, 7) ( 7, 4) ( 7, 5) ( 3, 4) ( 9, 2) ( 6, 5) ( 3, 5) ( 6, 7)

( 5, 4) ( 4, 8) ( 5, 6) ( 3, 4) ( 7, 6) ( 7, 5) ( 4, 9) ( 5, 2)

Table 4. Decision Matrix for Alternative ��(3)

( 5, 7) ( 8, 5) ( 7, 4) ( 4, 3) ( 4, 9) ( 2, 4) ( 8, 4) ( 7, 5)

(.8,.5) (.7,.4) (.8,.5) (.5,.2) (.5,.7) (.7,.5) (.7,.6) (.6,.4)

( 6, 8) ( 4, 5) ( 6, 5) ( 6, 4) ( 7, 5) ( 8, 4) ( 5, 8) ( 4 5)

(.5,.7) (.9,.3) (.3,.5) (.5,.7) (.3,.5) (.8,.5) (.7,.5) (.2,.5)

Table 5. Decision Matrix for Alternative ��(5)

���� ( 6, 5) ( 3, 8) ( 4, 5) ( 7, 4) ( 6, 4) ( 4, 5) ( 3, 4) ( 7, 5) ���� ( 8, 4) ( 9, 3) ( 1, 8) ( 1, 2) ( 4, 6) ( 3, 7) ( 6, 8) ( 8, 4) ���� (.6,.7) (.7,.4) (.7,.5) (.3,.4) (.9,.2) (.6,.5) (.3,.5) (.6,.7) ���� ( 5, 4) ( 4, 8) ( 5, 6) ( 3, 4) ( 7, 6) ( 7, 5) ( 4, 9) ( 5, 2)

By using Tables 1-5, compute the correlation coefficient between ������������(℘,��(1)), ������������(℘,��(2)), ������������(℘,��(3)), ������������(℘,��(4)) by using Definition 5.3 given as follows: ������������(℘,��(1)) = .99658, ������������(℘,��(2)) = .99732, ������������(℘,��(3)) = .99894, and ������������(℘,��(4)) = .99669. This shows that ������������(℘,��(3))> ������������(℘,��(2))>������������(℘,��(4)) >������������(℘,��(1)). It can be seen from this ranking alternative ��(3) is the most suitable alternative. Therefore ��(3) is the best alternative, the ranking of other alternatives given as ��(3) >��(2) >��(4) >��(1). Graphical results of alternatives ratings can be seen in Figure 2.

0.9995

0.999

0.9985

0.998

0.9975

0.997

0.9965

0.996

0.9955

0.995

Correlation Coefficient for PFHSS

T^(3), 0.99894

T^(1), 0.99658

T^(2), 0.99732

T^(4), 0.99669

T^(1) T^(2) T^(3) T^(4)

T^(1) T^(2) T^(3) T^(4)

Figure 2: Alternatives rating based on correlation coefficient under PFHSS

6. Discussion and Comparative Analysis

In the following section, we are going to debate the effectivity, naivety, flexibility as well as favorable position of the suggested method along with the algorithm. We also organized a brief comparative analysis of the following: The suggested method along with the prevailing approaches.

6.1 Superiority of the Proposed Method

Through this research and comparative analysis, we have concluded that the proposed methods' results are more general than prevailing techniques. However, in the decision-making process, compared with the existing decision-making methods, it contains more information to deal with the data's uncertainty. Moreover, many of FS's mixed structure has become a special case of PFHSS, add some suitable conditions. In it, the information related to the object can be expressed more accurately and empirically, so it is a convenient tool for combining inaccurate and uncertain information in the decision-making process. Therefore, our proposed method is effective, flexible, simple, and superior to other hybrid structures of fuzzy sets.

Table 6: Comparative analysis between some existing techniques and the proposed approach

Set Truthine ss Falsit y Attribute s Multi subattributes

Advantages

Zadeh [1] FS ✓ × ✓ × Deals uncertainty by using fuzzy interval Atanassov [2] IFS ✓ ✓ ✓ × Deals uncertainty by using MD and NMD

Theory and Application of Hypersoft Set 103

Yager [21] PFS ✓ ✓ ✓ × Deals more uncertainty by using MD and NMD comparative to IFS

Zulqarnain et al. [46] IFHSS ✓ ✓ ✓ ✓ Deals uncertainty of multi subattributes

Proposed approach PFHS S ✓ ✓ ✓ ✓ Deals more uncertainty comparative to IFHSS

6.2 Comparative Analysis

By using the technique of Zadeh [1], we deal with the true information of the alternatives, but this method cannot deal with falsity objects and multi sub-attributes of the alternatives. By utilizing the Atanassov [2], and Yager [21] methodologies, we cannot deal with the alternatives' multi-subattribute information. But our proposed method can easily solve these obstacles and provide more effective results for the MCDM problem. Zulqarnain et al. [46] presented IFHSS deals with the uncertainty by using the MD and NMD of the sub-attributes of a set of parameters, but the sum of MD and NMD of sub-attributes cannot exceed 1. In some cases the sum of MD and NMD exceeds 1, then existing IFHSS fails to deal with such situations. Instead of this, our developed method is an advanced technique that can handle alternatives with multi-sub-attributes information when the sum of MD and NMD exceeds 1. A comparison can be seen in the above-listed Table 6. However, on the other hand, the methodology we have established deals with the truthiness and falsity of alternatives with multi sub-attributes. Therefore, the technique we developed is more efficient and can provide better results for decision-makers through a variety of information comparative to existing techniques.

7. Conclusion

The Pythagorean fuzzy hypersoft set is a novel concept that is an extension of the intuitionistic fuzzy soft set and generalization of the intuitionistic fuzzy hypersoft set. In this work, we studied some basic concepts and developed some basic operations for PFHSS with their properties. We proposed the AND-operation and OR-operation under the PFHSS environment with their desirable properties in the following research. The idea of necessity and possibility operations with numerous properties under the Pythagorean fuzzy hypersoft set is also presented in it. Furthermore, the concept of CC is also established in this research with its decision-making methodology. We used the developed methodology to solve decision-making problems to ensure the validity and applicability of the developed decision-making methodology. Furthermore, A comparative analysis is presented to verify the validity and demonstration of the proposed method. Finally, based on the results obtained, it is concluded that the suggested techniques showed higher stability and practicality for decision-makers in the decision-making process. In the future, anyone can extend the PFHSS to interval-valued PFHSS, aggregation operators, and TOPSIS technique under PFHSS.

8. Acknowledgment

This research is partially supported by a grant of National Natural Science Foundation of China (11971384).

References

1. Zadeh, L. A (1965). Fuzzy Sets Information and Control, 8, 338–353.

2. Atanassov, K. (1986) Intuitionistic Fuzzy Sets Fuzzy Sets and Systems, 20, 87–96.

Theory and Application of Hypersoft Set 104

3. Garg, H., & Kaur, G. (2019). Cubic intuitionistic fuzzy sets and its fundamental properties. J. Mult-valued Log. S, 33, 507–537.

4. Atanassov, K., & Gargov, G. (1989). Interval valued intuitionistic fuzzy sets Fuzzy Sets and Systems, 31, 343-349.

5. Garg, H., & Kumar, K. (2019). Linguistic interval-valued Atanassov intuitionistic fuzzy sets and their applications to group decision-making problems IEEE T. Fuzzy Syst, 27, 2302–2311.

6. Molodtsov, D. (1999) Soft Set Theory First Results Computers & Mathematics with Applications, 37, 19–31.

7. Maji, P. K., Biswas, R., & Roy, A. R. (2003) Soft set theory Computers and Mathematics with Applications, 45(4–5), 555–562.

8. Maji, P. K., Roy, A. R., & Biswas, R. (2002). An Application of Soft Sets in A Decision Making Problem Computers and Mathematics with Applications, 44, 1077–1083.

9. Maji, P. K., Biswas, R., & Roy, A. R. (2001). Fuzzy soft sets Journal of Fuzzy Mathematics, 9, 589–602.

10. Maji, P. K., Biswas, R., & Roy, A. R. (2001). Intuitionistic fuzzy soft sets Journal of Fuzzy Mathematics, 9, 677–692.

11. Yang, X., Lin, T. Y., Yang, J., Li, Y., & Yu, D. (2009). Combination of interval-valued fuzzy set and soft set Computers and Mathematics with Applications, 58, 521–527.

12. Jiang, Y., Tang, Y., Chen, Q., Liu, H., & Tang, J. (2010). Interval-valued intuitionistic fuzzy soft sets and their properties. Computers and Mathematics with Applications, 60, 906–918.

13. Garg, H., & Arora, R. (2018). Generalized and group-based generalized intuitionistic fuzzy soft sets with applications in decision-making. Applied intelligence, 48, 343–356.

14. Garg, H. (2020) New ranking method for normal intuitionistic sets under crisp, interval environments and its applications to multiple attribute decision making process. Complex & Intelligent Systems, 6(3), 559-571.

15. Mahmood, T., & Ali, Z. (2020). Entropy measure and TOPSIS method based on correlation coefficient using complex q-rung orthopair fuzzy information and its application to multiattribute decision making. Soft Computing, 11, 1-27.

16. Garg, H., & Arora, R. (2020). Maclaurin symmetric mean aggregation operators based on tnorm operations for the dual hesitant fuzzy soft set. Journal of Ambient Intelligence and Humanized Computing, 11, 375–410.

17. Arora, R., & Garg, H. (2018). A robust correlation coefficient measure of dual hesitant fuzzy soft sets and their application in decision making Eng Appl Artif Intell, 72, 80–92.

18. Arora, R., & Garg, H. (2019). Group decision-making method based on prioritized linguistic intuitionistic fuzzy aggregation operators and its fundamental properties Comput Appl Math, 38(2), 1–36.

19. Garg, H., & Arora, R. (2020) Generalized Maclaurin symmetric mean aggregation operators based on Archimedean t-norm of the intuitionistic fuzzy soft set information Artifcial Intelligence Review, 1-41.

20. Yager, R. R. (2013) Pythagorean fuzzy subsets. Proceedings joint IFSA world congress and NAFIPS annual meeting. Edmonton, Canada, 57–61.

21. Yager, R. R. (2014). Pythagorean membership grades in multicriteria decision making IEEE Trans Fuzzy Syst, 22(4) 958–965.

22. Zhang, X., & Xu, Z. (2014). Extension of topsis to multiple criteria decision making with Pythagorean fuzzy sets. International Journal of Intelligent Systems, 29(12), 1061–1078.

23. Zulqarnain, R. M., Abdal, S., Maalik, A., Ali, B., Zafar, Z., Ahamad, M. I., Younas, S., Mariam, A., Dayan, F. (2020). Application of TOPSIS Method in Decision Making Via Soft Set. BIOMEDICAL Journal of Scientific & Technical Research, 24(3), 18208-18215.

Theory and Application of Hypersoft Set 105

24. Zulqarnain, M., & Dayan, F. (2017). Choose Best Criteria for Decision Making Via Fuzzy TOPSIS Method. Mathematics and Computer Science, 2(6), 113-119.

25. Zulqarnain, R. M., Abdal, S., Ali, B., Ali, L., Dayan, F., Ahamad, M. I., & Zafar, Z. (2020). Selection of Medical Clinic for Disease Diagnosis by Using TOPSIS Method International Journal of Pharmaceutical Sciences Review and Research, 61(1), 22-27.

26. Zulqarnain, M., & Dayan, F. (2017). Selection of Best Alternative For An Automotive Company

By Intuitionistic Fuzzy TOPSIS Method International journal of scientific & technology research, 6(10), 126-132.

27. Zulqarnain, M., Dayan, D., & Saeed, M. (2018). TOPSIS analysis for the prediction of diabetes based on general characteristics of humans International Journal of Pharmaceutical Sciences and Research, 9(7), 2932-2939.

28. Zulqarnain, R. M., Xin, X.L., Saeed, M., Ahmad, N., Dayan, F., & Ahmad, N. (2020). Recruitment of Medical Staff in Health Department by Using TOPSIS Method International Journal of Pharmaceutical Sciences Review and Research, 62(1), 1-7.

29. Zulqarnain, R. M., Saeed, M., Ahmad, N., Dayan, F., & Ahmad, N. (2020). Application of TOPSIS Method for Decision Making International Journal of Scientific Research in Mathematical and Statistical Sciences, 7(2), 76-81.

30. Zulqarnain, R. M., Saeed, M., Ali, B., Abdal, S., Saqlain, M., Ahamad, M. I., & Zafar, Z. (2020). Generalized Fuzzy TOPSIS to Solve Multi-Criteria Decision-Making Problems. Journal of New Theory, 32, 40-50.

31. Zulqarnain, R. M., Xin, X. L., Saeed, M., Smarandache, F., & Ahmad, N. (2020). Generalized Neutrosophic TOPSIS to Solve Multi-Criteria Decision-Making Problems. Neutrosophic Sets and Systems, 38, 276-292.

32. Wei, G., & Lu, M. (2018). Pythagorean fuzzy power aggregation operators in multiple attribute decision making. International Journal of Intelligent Systems, 33(1), 169-186.

33. Wang, L., & Li, N. (2020). Pythagorean fuzzy interaction power Bonferroni mean aggregation operators in multiple attribute decision making. International Journal of Intelligent Systems, 35(1), 150-183.

34. Zhang, X. L. (2016). A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making. Int J Intell Syst, 31, 593–611.

35. Guleria, A., & Bajaj, R. K. (2019). On Pythagorean fuzzy soft matrices, operations and their applications in decision making and medical diagnosis Soft Computing, 23(17), 7889-7900.

36. Guleria, A., & Bajaj, R. K. (2018). Pythagorean Fuzzy-Norm Information Measure for Multicriteria Decision-Making Problem Advances in Fuzzy Systems, Jan 1;2018.

37. Bajaj, R. K., & Guleria, A. (2020). Dimensionality reduction technique in decision making using Pythagorean fuzzy soft matrices Recent Advances in Computer Science and Communications (Formerly: Recent Patents on Computer Science), 13(3), 406-413.

38. Guleria, A., & Bajaj, R. K. (2020). Pythagorean fuzzy (R, S)-norm discriminant measure in various decision making processes Journal of Intelligent & Fuzzy Systems, 38(1), 761-777.

39. Zulqarnain, R. M., Xin, X. L., Garg, H., & Asghar Khan, W. (2021). Aggregation operators of pythagorean fuzzy soft sets with their application for green supplier chain management Journal of Intelligent and Fuzzy Systems, DOI: 10.3233/JIFS-202781.

40. Zulqarnain, R. M., Xin, X. L., Siddique, I., Asghar Khan, W., & Yousif, M. A. (2021). TOPSIS Method Based on Correlation Coefficient under Pythagorean Fuzzy Soft Environment and Its Application towards Green Supply Chain Management. Sustainability, 13, 1642. https://doi.org/10.3390/ su13041642.

41. Smarandache, F. (2018). Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set. Neutrosophic Sets and Systems, 22, 168-170.

Theory and Application of Hypersoft Set 106

42. Gayen, S., Smarandache, F., Jha, S., Singh, K., Broumi, S., & Kumar, R. (2020). Introduction to Plithogenic Subgroup. In Neutrosophic Graph Theory and Algorithms IGI Global, 213-259.

43. Zulqarnain, R. M., Xin, X. L., Saqlain, M., & Smarandache, F. (2020). Generalized Aggregate Operators on Neutrosophic Hypersoft Set Neutrosophic Sets and Systems, 36, 271-281.

44. Saqlain, M., Jafar, N., Moin, S., Saeed, M., & Broumi, S. (2020). Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure of Single valued Neutrosophic Hypersoft Sets Neutrosophic Sets and Systems, 32(1), 317-329

45. Saqlain, M., Moin, S., Jafar, M N., Saeed, M., & Smarandache, F. (2020). Aggregate Operators of Neutrosophic Hypersoft Set. Neutrosophic Sets and Systems, 32(1), 18.

46. Zulqarnain, R. M., Xin, X. L., & Saeed, M. (2021). Extension of TOPSIS method under intuitionistic fuzzy hypersoft environment based on correlation coefficient and aggregation operators to solve decision making problem AIMS Mathematics, 6 (3), 2732-2755.

47. Zulqarnain, R. M., Siddique, I., Jarad, F., Ali, R., & Abdeljawad, T. (2021). Development of TOPSIS Technique Under Pythagorean Fuzzy Hypersoft Environment Based on Correlation Coefficient and Its Application Towards the Selection of Antivirus Mask in COVID-19 Pandemic Complexity, Accepted.

48. Abdel-Basset, M., Ding, W., Mohamed, R., & Metawa, N. (2020). An integrated plithogenic MCDM approach for financial performance evaluation of manufacturing industries Risk Manag, 22, 192–218.

49. Abdel-Basset, M., Mohamed, R., Zaied, A. E. N. H., & Smarandache, F. (2019). A Hybrid Plithogenic Decision-Making Approach with Quality Function Deployment for Selecting Supply Chain Sustainability Metrics. Symmetry, 11(7), 90.

Development of TOPSIS using Similarity Measures and Generalized weighted distances for Interval Valued Neutrosophic Hypersoft Matrices along with Application in MAGDM Problems

Adeel Saleem1, Muhammad Saqlain1* , Sana Moin2

2Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: adeelsaleem1992@hotmail.com

1*Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: msaqlain@lgu.edu.pk

2Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: moinsana64@gmail.com

Abstract: In this chapter, we stretch out the TOPSIS technique to illuminate MAGDM issues in the interval valued neutrosophic hypersoft set environment To build up the proposed TOPSIS procedures, distance measures, similarity measures and the concept of interval valued neutrosophic hypersoft matrices (IVNHSM) is introduced along with definitions, theorems, propositions and examples. To exhibit dependability and suitability of the proposed TOPSIS technique, we solve a numerical example for the selection of the best marriage proposal TOPSIS is a fascinating tool for dealing with MCDM. The Future directions and the limitations of the proposed algorithm are also presented.

Keywords: Interval Valued Neutrosophic Hypersoft Set (IVNHSS), MCDM, Neutrosophic Hypersoft Set (NHSS), Neutrosophic Hypersoft Matrix (NHSM), TOPSIS, MAGDM, Similarity Measures, Distance Measures.

1.Introduction

In our daily life every being has to decide something and it is very common that most of us faces problem in decision making. To make decision making easy different researchers and mathematicians invent many techniques. Multi criteria decision making is also one of the decisionmaking techniques. MADM is a procedure of finding an ideal alternative that is most suitable to fulfill the need from a lot of options associated with various different attributes. A huge range of strategies has been adapted for managing MADM issues, such as, VIKOR [1], AHP [2], TOPSIS [3,4] and etc. In MADM problems, evaluation of attributes can't be constantly given in crisp values due to uncertain and unpredictable nature of the characteristics in real life. Fuzzy set [5] introduced by Zadeh is equipped for managing uncertainty in scientific structure. Thus MADM [6-9] can be demonstrated very well by utilizing the fuzzy set theory into the field of developing decisionmaking techniques. Fuzzy set are progressing rapidly but the mathematicians faces problem in handling uncertainty because fuzzy set can just concentrate on the participation degree and it

Theory and Application of Hypersoft Set 108 neglects to consider non-participation degree and indeterminacy level of imprecise information. Atanassov [10] presented Intuitionistic Fuzzy set (IFS) for dealing with uncertainty and unpredictability in the more accurate manner by considering the truth membership and falsity membership values. Working on intuitionistic fuzzy set it is noticed that some of the incomplete or indeterminate information do exist. Hence, IFS can't deal with uncertainty in the appropriate manner in MADM [11-13] and MAGDM [14-16] issues in which some problems arise due to indeterminate information. Smarandache [17] supports the idea of Neutrosophic set that is a scientific gadget to overcome issued like indeterminant, uncertain and contradictory information. Neutrosophic set displays real membership value, indeterminacy membership value and falsity membership value. Such an idea is of great significance in numberless applications because indeterminacy is checked extraordinarily and truth membership values, indeterminacy membership values and falsity values are independent. Molodtsov [18] highlights the idea of soft set to manage issues Indefinite circumstances. It was said that soft set was parameterized family of subsets of universal sets. Soft sets have their own paramount importance in the fields of artificial insight, game hypothesis and fundamental decision-making issues., it also helps us to find out various functions for different parameters and benefit values against the accepted and established parameters. We come to know that the fundamentals of soft set theories have been mediated and pondered over by different learned people for the last two years. For example, Maji et al [19,20] presented a hypothetical analysis of soft sets and upper set of soft sets, equality of soft sets and operation on soft sets, for example, union, intersection AND, and OR operation between different sets. Ali at el [21] new operations in soft theory that covers restricted union, intersection and difference. Cagman and Egniloglus [22,23] support soft set theory that empowers itself a very important measurement while looking after issues to make different choices. Onyeozili [24] presented research which claims that soft set is equal to corresponding soft matrices. From Molodtsov [25] to present different beneficial applications identified with soft set theory have been presented and annexed to numberless fields of science and data innovation. Maji [26] presented Neutrosophic soft set exhibited by truth, indeterminacy and falsity membership values that are independent in nature. Neutrosophic soft set can manage inadequate, uncertain and inconstant data while intuitionistic fuzzy soft set can only deal with partial data. Smarandache [27] came up with new plans to cope with uncertainly. Soft sets were generalized to hypersoft set by altering the function into multi decision function. When features and attributes are more than one and further diverge, Neutrosophic soft set environment can't help to cope with such kind of issues owing to soft sets as soft sets deal with sole argument function. In order to overcome such problem, Neutrosophic hypersoft set [28] was introduced. There are various MADM methodologies are accessible in the literature. Among them TOPSIS is extremely mainstream to manage MADM.TOPSIS strategy offered by Hwang and Yoon [3] can be called one of the strategies which can provide suitable solutions. The central idea of TOPSIS is that the best alternative must keep the shortest distance from the positive ideal solution (PIS)and the farther distance from the negative

Theory and Application of Hypersoft Set 109

ideal solution (NIS) simultaneously. TOPSIS is very renowned and well known to deal with MADM. Behzaian et al. [4] offered a detailed survey on TOPSIS applications in various fields. Many researchers proposed TOPSIS technique to solve MCDM and MAGDM problems. [29-35]. MultiCriteria decision problems (MCDM) consist of several attributes and indeterminacy, to deal with such types neutrosophic sets (N's) and interval valued neutrosophic sets (IVN’s) are used because (N's) fully deal with indeterminacy whereas to deal with vagueness and uncertainty neutrosophic soft sets (NS's) are used. But when attributes are more than one and further bifurcated the concept of neutrosophic soft set (NS's) cannot be used to tackle such type of issues so, there was a dire need to define the new environment, for this purpose the concept of neutrosophic hypersoft set (NHSS) was proposed by [28][36] Matrix notations on neutrosophic hypersoft set is proposed in []. This concept was extended to IVNHSS [37] with the generalization of definition and operators. To solve MCDM and MAGDM problems in IVNHSS environment it is necessary to propose any technique. So, in this chapter the basic operation like; interval valued neutrosophic hypersoft Matrix (IVNHSM), Generalized weighted distance for IVNHSM and IVNHSS the similarity measure for IVNHSM and IVNHSS are proposed. By using these operations, the algorithm of TOPSIS is extended to IVNHSS TOPSIS.

1.1 The Chapter Presentation:

After introduction rest of the chapter is contain section 2 of some basic preliminaries. Then we have Section 3, in which definition of IVNHSM along with some operations are defined. Section 4 consists of generalized weighted similarity measure for both IVNHSS and IVNHSM. Section 6, have distance measures for the both IVNHSS and IVNHSM. Section 7, consist of the TOPSIS algorithm which is proposed using the similarity measure, distance measure and IVNHSM. In section 7, a case study is presented. Finally, Sect. 8 presents some concluding remarks and future scope of research. Figure: 1 represents the chapter presentation graphically.

2.Preliminaries

This section consists of some basic definitions which leads us to neutrosophic hypersoft sets and will be helpful in rest of the article.

Definition 2.1: Neutrosophic Soft Set

Let �� be the universal set and �� be the set of attributes with respect to ��. Let ℙ(��) be the set of Neutrosophic values of �� and ��⊆�� . A pair (₣, ��) is called a Neutrosophic soft set over �� and its mapping is given as ₣:�� →ℙ(��)

Definition 2.2: Hyper Soft Set: Let �� be the universal set and ℙ(��) be the power set of ��. Consider l1,l2,l3 ln for n≥1, be n well-defined attributes, whose corresponding attributive values are respectively the set L1,L2,L3 …Ln with Li ∩Lj =∅, for i≠j and i,jϵ{1,2,3…n} , then the pair (₣,L1 ×L2 ×L3 …Ln) is said to be Hypersoft set over �� where ₣:L1 ×L2 ×L3 Ln →ℙ(��)

Hypersoft Set

Definition 2.3: Neutrosophic Hypersoft Set [28]

Let �� be the universal set and ℙ(��)be the power set of ��. Consider l1,l2,l3 ln for n≥1, be n well-defined attributes, whose corresponding attributive values are respectively the set L1,L2,L3 Ln with Li ∩Lj =∅, for i≠j and i,jϵ{1,2,3 n} and their relation L1 ×L2 ×L3 Ln =$, then the pair (₣,$) is said to be Neutrosophic Hypersoft set (NHSS) over �� where ₣:L1 ×L2 ×L3 Ln →ℙ(��)and ₣(L1 ×L2 ×L3 Ln)={<x,T(₣($)),I(₣($)),F(₣($))>,x∈��} where T is the membership value of truthiness, I is the membership value of indeterminacy and F is the membership value of falsity such that T,I,F:��→[0,1] also 0≤T(₣($))+ I(₣($))+ F(₣($))≤3

Definition 2.4: Interval Valued Neutrosophic Hypersoft Set (IVNHSS) [37] Let �� be the universal set and ℙ(��)be the power set of ��. Consider l1,l2,l3 ln for n≥1, be n well-defined attributes, whose corresponding attributive values are respectively the set L1,L2,L3 …Ln with Li ∩Lj =∅, for i≠j and i,jϵ{1,2,3…n} and their relation L1 ×L2 ×L3 …Ln =$, then the pair (₣,$) is said to be Interval Valued Neutrosophic Hypersoft set (IVNHSS) over �� where

₣:L1 ×L2 ×L3 Ln →ℙ(��)and ₣(L1 ×L2 ×L3 …Ln)={<x,[TL(₣($)),TU(₣($))],[IL(₣($)),IU(₣($))],[FL(₣($)),FU(₣($))]>,x∈ ��} where [TL(₣($))] is the lower membership value of truthiness, IL(₣($)) is the lower membership value of indeterminacy and FL(₣($)) is the lower membership value of falsity. Similarly, [TU(₣($))] is the upper membership value of truthiness, IU(₣($)) is the upper membership value of indeterminacy and FU(₣($)) is the upper membership value of falsity. Such that [TL(₣($)),TU(₣($))],[IL(₣($)),IU(₣($))],[FL(₣($)),FU(₣($))]:��→[0,1] Also 0≤supT(₣($))+ supI(₣($))+ supF(₣($))≤3

Definition 2.5: Neutrosophic Hypersoft Matrix (NHSM) [38] Let ��={��1,��2 , ����} and ℙ(��) be the universal set and power set of universal set respectively, also consider ��1,��2, ���� for �� ≥1, �� well defined attributes, whose corresponding attributive values are respectively the set ��1 ��,��2 �� ,… ���� �� and their relation ��1 �� ×��2 �� × … ���� �� where ��,��,��,…�� = 1,2, �� then the pair (��,��1 �� ×��2 �� × ���� ��) is said to be Neutrosophic Hypersoft set over �� where ��:(��1 �� ×��2 �� × … ���� ��)→ℙ(��) and it is define as ��:(��1 �� ×��2 �� × ���� ��)={<��,��ℓ(��),��ℓ(��),��ℓ(��)>�� ∈ ��,ℓ∈(��1 �� ×��2 �� × ���� �� )}

Figure: 1 The chapter presentation

3. Interval Valued Neutrosophic Hypersoft Matrix (IVNHSM)

In this Section IVNHSM, theorems and propositions with examples are defined.

Definition 3.1: Interval Valued Neutrosophic Hypersoft matrix (IVNHSM) Let ��={��1,��2 ,… ����} and ℙ(��) be the universal set and power set of universal set respectively, also consider ��1,��2, ���� for �� ≥1, �� well defined attributes, whose corresponding attributive

Theory and Application of Hypersoft Set 112 values are respectively the set ��1 ��,��2 �� ,… ���� �� and their relation ��1 �� ×��2 �� × … ���� �� where ��,��,��,…�� = 1,2, �� then the pair (��,��1 �� ×��2 �� × ���� ��) is said to be Neutrosophic Hypersoft set over �� where ��:(��1 �� ×��2 �� × ���� ��)→ℙ(��) and it is define as ��:(��1 �� ×��2 �� × ���� ��)={<��,([��ℓ ��(��),��ℓ ��(��)],[��ℓ ��(��),��ℓ ��(��)],[��ℓ ��(��),��ℓ ��(��)])>�� ∈ ��,ℓ∈(��1 �� × ��2 �� × … ���� ��)} Let ℝℓ =(��1 �� ×��2 �� × ���� ��) be the relation and its characteristic function is ��ℝℓ:(��1 �� ×��2 �� × … ���� ��)→ℙ(��) and it is define as ��ℝℓ ={<��,([��ℓ ��(��),��ℓ ��(��)],[��ℓ ��(��),��ℓ ��(��)],[��ℓ ��(��),��ℓ ��(��)])> �� ∈ ��,ℓ∈(��1 �� ×��2 �� × … ���� ��)}. The tabular representation of ℝℓ is given as ���� �� ���� �� ���� ��

Theory and Application of Hypersoft Set 113

Definition 3.3: Column IVNHSM Let ��=[������] be the IVNHSM of order ��×��, where ������ = (�� ̿ ������ �� ,�������� �� ,�� ̿ ������ �� ) then �� is said to be Column IVNHSM if �� =1 . It means that if an IVNHSM contain only one alternative i.e. ℓ={��1 ��} then it is a Column IVNHSM.

Definition 3.4: Zero IVNHSM Let �� =[������] be the IVNHSM of order ��×�� , where ������ = (�� ̿ ������ �� ,�������� �� ,�� ̿ ������ �� ) then �� is said to be Zero IVNHSM if ������ =([0,0],[1,1],[1,1]) .

Definition 3.5: Universal IVNHSM Let �� =[������] be the IVNHSM of order ��×��, where ������ = (�� ̿ ������ �� ,�������� �� ,�� ̿ ������ �� ) then �� is said to be Universal IVNHSM if ������ =([1,1],[0,0],[0,0])

Definition 3.6: Interval Valued Neutrosophic Hypersoft Submatrix (IVNHSSM)

Let �� =[������]and �� ̿ =[�� ̿ ����] be the two IVNHSM of order ��×��, where ������ =(�� ̿ ������ �� ,�������� �� ,�� ̿ ������ �� ) and �� ̿ ���� =(�� ̿ ������ �� ,�������� �� ,�� ̿ ������ �� ) then[������]is said to be IVNHSSM of [�� ̿ ����] if [��L ������ �� ≤��L ������ �� ,��U ������ �� ≤ ��U ������ �� ],[��L ������ �� ≤��L ������ �� ,��U ������ �� ≤��U ������ �� ],[��L ������ �� ≥��L ������ �� , ��U ������ �� ≥��U ������ �� ].

Definition 3.7: Interval Valued Neutrosophic Hypersoft Equal Matrix (IVNHSEM) Let �� =[������]and �� ̿ =[�� ̿ ����] be the two IVNHSM of order ��×��, where ������ =(�� ̿ ������ �� ,�������� �� ,�� ̿ ������ �� ) and �� ̿ ���� =(�� ̿ ������ �� ,�������� �� ,�� ̿ ������ �� ) then[������]is said to be IVNHSEM of [�� ̿ ����] if [��L ������ �� =��L ������ �� ,��U ������ �� = ��U ������ �� ],[��L ������ �� =��L ������ �� ,��U ������ �� =��U ������ �� ],[��L ������ �� =��L ������ �� , ��U ������ �� =��U ������ �� ].

Definition 3.8: AND Operation

of order

, where

Definition 3.9: OR Operation

Hypersoft

Definition 3.10: Sum of IVNHSM

4. Similarity Measures for Interval Valued Neutrosophic Hypersoft Matrix (IVNHSM)

In this section similarity measures for Interval Valued Neutrosophic Hypersoft Matrices (IVNHSM) and Generalized weighted Similarity measure for NHSS’s are defined.

Proposition 4.1: Similarity Axioms Let �� =���� , ℬ =ℬ�� and �� =���� be the three IVNHSs where ���� =([���� ��,���� ��,���� ��) , ℬ�� = (���� ℬ,���� ℬ,���� ℬ) and ���� =(���� ��,���� ��,���� ��) and �� ={1,2,3…n} then it satisfies the following axioms 1. 0≤��(��,ℬ)≤1 2. ��(��,ℬ)=1������������������������ =ℬ 3. ��(��,ℬ)=��(ℬ,��) 4. If �� ⊂�� ⊂C then ��(��,��)≤��(��,ℬ) and ��(��,��)≤��(ℬ,��) Definition 4.2: Generalized weighted similarity measure for IVNHSS. Let �� =���� and , ℬ =ℬ�� be the two IVNHSs where ���� =([���� ��,���� ��,���� ��) and ℬ�� = (���� ℬ,���� ℬ,���� ℬ), and �� ={1,2,3…n} then the generalized weighted similarity measure is given as ����(��,ℬ)=1 [ 1 6��∑ ���� (|���� ���� ���� ����|�� +|���� ���� ���� ����|�� +|���� ���� ���� ����|�� +|���� ���� ���� ����|�� + �� ��

Definition

Example: Let ��= [��������������{[06,07],[04,05],[05,06]} 6����{[06,07],[01,02],[02,03]} ��������{[07,08],[01,02],[005,01]}] and �� ̿ = [��������������{[07,08],[005,01],[01,02]} 6����{[05,06],[005,01],[01,02]} ��������{[02,03],[05,06],[03,04]}] be the two IVNHSM then the similarity measure is given as; ��(��,��̿) = 1 1 6(1)(3)(|06 07|+|07 08|+|04 005|+|05 01|+|05 01| +|0.6 0.2|+|0.6 0.5|+|0.7 0.6|+|0.1 0.05|+|0.2 0.1|+|0.2 0.1| +|03 02|+|07 02|+|08 03|+|01 05|+|02 06|+|005 03| +|0.1 0.4|) ��(��,��̿) = 07417 Definition 4.4:

5. Distance for Interval Valued Neutrosophic Hypersoft Matrix (IVNHSM) In this section distance measures for Interval Valued Neutrosophic Hypersoft Matrices (IVNHSM)

and Application of Hypersoft Set

Proposition 5.1: Distance Axioms

Let �� =���� , ℬ =ℬ�� and �� =���� be the three IVNHSs where ���� =([���� ��,���� ��,���� ��) , ℬ�� = (���� ℬ,���� ℬ,���� ℬ) and ���� =(���� ��,���� ��,���� ��) and �� ={1,2,3 n} then it satisfies the following axioms

1. ��(��,ℬ) ≥ 0

2. ��(��,ℬ) = ��(ℬ,��)

3. ��(��,ℬ) = 0�������� =ℬ

4. ��(��,ℬ)+��(ℬ,��) ≥ ��(��,��)

Definition 5.2: Generalized weighted distance for IVNHSS

Let �� =���� and , ℬ =ℬ�� be the two IVNHSs where ���� =([���� ��,���� ��,���� ��) and ℬ�� = (���� ℬ,���� ℬ,���� ℬ), and �� ={1,2,3…n} then the generalized weighted distance is given as ����(��,ℬ)=[16��∑���� (|���� ���� ���� ����|�� +|���� ���� ���� ����|�� +|���� ���� ���� ����|�� +|���� ���� ���� ����|�� �� �� +|���� ���� − ���� ����|��

��(��,��̿) = 02583

for

��

6���� Example: Let �� = [��������������{[06,07],[04,05],[05,06]} 6����{[06,07],[01,02],[02,03]} ��������{[07,08],[01,02],[005,01]}] and �� ̿ = [��������������{[07,08],[005,01],[01,02]} 6����{[05,06],[005,01],[01,02]} ��������{[02,03],[05,06],[03,04]}] be the two IVNHSM then the normalized Euclidean distance is given as; ��(A , B ) =√ (((|0.6-0.7|^2+|0.7-0.8|^2+|0.4-0.05|^2+|0.5-0.1|^2+|0.5-0.1|^2+|0.6-0.2|^2+| 0.6-0.5|^2+|0.7-0.6|^2+|0.1-0.05|^2+|0.2-0.1|^2+|0.2-0.1|^2+|0.3-0.2|^2+|0.7-0.2|^2+|0.80.3|^2+|0.1-0.5|^2+|0.2-0.6|^2+|0.05-0.3|^2+|0.1-0.4|^2 ))/(6(1)(3)))

̿) = 0.3025

Definition 5.5:

3. TOPSIS for Multi Attribute Group Decision Making (MAGDM)

TOPSIS (The Technique for Order Preference by Similarly to Ideal Solution) is the suitable approach to deal with Multi Attribute Group Decision Making problems. TOPSIS technique (Shown in Figure:2) is consist of following steps.

2)

Theory and Application of Hypersoft Set

1. Compose a decision matrix.

2. Normalizing decision matrix.

3. Determine the weighted normalized decision matrix.

4. Calculate the positive and negative ideal solution.

5. Calculate the distance of each alternative to the positive and negative ideal solution.

6. Calculate the relative closeness coefficients.

7. Rank the alternatives.

Figure: 2 Proposed TOPSIS Algorithm for IVNHSS environment

4. TOPSIS Technique for MAGDM With Interval Valued Neutrosophic Hypersoft Set

Consider a Multi Attribute Group Decision Making (MAGDM) problem based on interval valued neutrosophic hypersoft set in which ��={��1,��2 , ����} be the set of alternatives and ��1,��2, ���� be the sets of attributes and their corresponding attributive values are respectively the set

��1 ��,��2 �� , ���� �� where ��,��,��, �� =1,2, ��. Let ���� be the weight of attributes ���� ��,�� =1,2 ��, where 0≤���� ≤1 and ∑ ���� =1 �� ��=1 . Suppose that ��=(��1,��2,… ����) be the set of �� decision makers and Δ�� be the weight of �� decision makers with 0≤Δ�� ≤1 and ∑ Δ�� =1 �� ��=1 . Let [������ �� ]��×�� be the decision matrix where ������ �� =(�� ̿ ������ �� ,�������� �� ,�� ̿ ������ �� )= ((�������� ���� ,�������� ���� ),(�������� ���� ,�������� ���� ),(�������� ���� ,�������� ���� )) , �� =1,2,3…��,�� =1,2,3,…��,�� =��,��,��,…�� and �� ̿ ������ �� = [�������� ���� ,�������� ���� ],�������� �� =[�������� ���� ,�������� ���� ],�� ̿ ������ �� =[��

the selection of alternatives is given as follow: Step 1: Determine the Weight of Decision Makers Let [������ �� ]

(��1),��

(��2),��

(��2)

(��2),��

(��2),��

2 ��

(��2) ��

�� �� �� (��2),������ �� �� (��2),�� ̿

�� �� �� (��2)

⋮ ⋱ ⋮ �� ̿ ��1 �� �� (����),����1 �� �� (����),�� ̿ ��1 �� �� (����) �� ̿ ��2 �� �� (����),����2 �� �� (����),�� ̿ ��2 �� �� (����) ⋯ �� ̿ ���� �� �� (����),������ �� �� (����),�� ̿ ���� �� �� (����)] To find the ideal matrix we will average all the individual decision matrix ������ �� ��ℎ�������� =1,2…�� as [������ ⋆ `]��⨯�� = [

where ������ ⋆

̿ ���� �� ⋆ (����),������ �� ⋆ (����),�� ̿ ���� �� ⋆ (����))

119 �� �� +|������ �� ����(����) �� ���� �� ⋆��(����)|+|������ �� ����(����) �� ���� �� ⋆��(����)|+|������ �� ����(����) �� ���� �� ⋆��(����)|) Now we calculate the weight Δ��(�� =1,2, ��) of �� decision makers using above equation Δ�� = ��(������ �� ,������ ⋆ ) ∑ ��(������ �� ,������ ⋆ ) �� ��=1 Where 0≤Δ�� ≤1 and ∑ Δ�� =1 �� ��=1 .

Step 2: Aggregate Neutrosophic Hypersoft Decision Matrices

By accumulating all the individual decision matrices, we construct an aggregated neutrosophic hypersoft decision matrix to obtain one group decision. Aggregated neutrosophic hypersoft decision matrix is denoted as ������ and it is given as

Step

3: Determine the weight of attributes

In decision making procedure, decision makers may perceive that all attribute are not similarly significant. In this manner, each decision maker may have their own one opinion regarding attribute weights. To acquire the gathering assessment of the picked attributes, all the decision makers’ opinions for the importance of each attribute need to be aggregated. For this purpose, weight �� ̿�� of attributes ���� ��,�� =1,2 ��is calculated as

Step 4: Calculate the weighted aggregated decision matrix After finding the weight of individual attributes, we apply these weights to each row of aggregated decision matrix (step 2) as

And then we get a weighted aggregated decision matrix. Step 5: Determine the ideal solution

In real life we deal with two type of attributes, one is benefit type attributes and other is cost type attribute. In our MAGDM problem we also deal with these two types of attributes. Let ℂ1 be the benefit type attributes and ℂ2 be the cost type attributes. Neutrosophic hypersoft positive ideal solution is given as ���� ��+ =(�� ̿ ���� �� �� +(����),������ �� �� +(����),�� ̿ ���� �� �� +(����)) ={[max �� {���� ���� �� �� (����)},max �� {���� ���� �� �� (����)}],[min �� {���� ���� �� �� (����)},min �� {���� ���� �� �� (����)}],[min �� {���� ���� �� �� (����)} ,min �� {���� ���� �� �� (����)}],�� ∈ℂ1 [min �� {���� ���� �� �� (����)},min �� {���� ���� �� �� (����)}],[max �� {���� ���� �� �� (����)},max �� {���� ���� �� �� (����)}],[max �� {���� ���� �� �� (����)} ,max �� {���� ���� �� �� (����)}],�� ∈ℂ2

Similarly, neutrosophic hypersoft negative ideal solution is given as ���� �� =(�� ̿ ���� �� �� (����),������ �� �� (����),�� ̿ ���� �� �� (����)) ={[min �� {���� ���� �� �� (����)},min �� {���� ���� �� �� (����)}],[max �� {���� ���� �� �� (����)},max �� {���� ���� �� �� (����)}],[max �� {���� ���� �� �� (����)} ,max �� {���� ���� �� �� (����)}],�� ∈ℂ1 [max �� {���� ���� �� �� (����)},max �� {���� ���� �� �� (����)}],[min �� {���� ���� �� �� (����)},min �� {���� ���� �� �� (����)}],[min �� {���� ���� �� �� (����)} ,min �� {���� ���� �� �� (����)}],�� ∈ℂ2

Step 6: Calculate the distance measure Now we will find the normalized hamming distance between the alternatives and positive ideal solution as; ����+(������ ��,���� ��+)= 1 6��∑ (|���� ���� �� �� (����) ���� ���� �� �� + (����)|+|���� ���� �� �� (����) ���� ���� �� �� + (����

Hypersoft Set

Step 7: Calculate the relative closeness co-efficient

Relative closeness index is used to rank the alternatives and it is calculated as ℝℂ�� = ���� (������ ��,���� �� ) max �� {���� (������ ��,���� �� )} ����+(������ ��,���� ��+) min �� {����+(������ ��,���� ��+)}

Where �� =1,2,3 �� The set of selected alternatives are ranked according to the descending order of relative closeness index.

5. Case study

Marriage is one of the most significant social organizations and today it is similarly as important as it was many years ago. With the time, the standards of adoration, steadfastness and responsibility has changes, which are the crucial segments of a solid society. We know that marriage brings steadiness and it ties us together. It helps make our families more grounded. A significant part of the quality of marriage lies in its capacity to change with the occasions. As society has changed, so marriage has changed, and become accessible to an undeniably expansive scope of people. This coordinate creation process has been messing up best choices.

Objective: The primary objective of this case study is to provide excellent matchmaking experience by exploring the opportunities and resources to meet true potential partner. Keeping the objective in mind, we can select the best proposal to fulfill the current demand in marriage for eligible bachelors. For this purpose, let ℝ={ℝ1,ℝ2,ℝ3,ℝ4,ℝ5,ℝ6,ℝ7,ℝ8} be the set of different bachelors for Marriage Proposal. Following are the attributes for respective bachelors. ��1 =�������������������� ��2 =�������������������� ��3 =������������

Theory and Application of Hypersoft Set

is the actual requirement of a family for marriage proposal. Four proposals {ℝ2,ℝ4,ℝ5,ℝ6} are selected on the basis of assumed relation i.e.(��������(��),��������������������(����),����������������������(��),��������������������������������������(������)) Four decision makers from the family members {��1,��2,��3,��4} are intended to select the most suitable proposal. These decision makers give their valuable opinion in the form of IVNHSM separately as.

1]4×4 = [

(��,([08,09], [0.1,0.2], [02,01])) (����,([02,03], [0.2,0.3], [06,07])) (��,([05,06], [0.3,0.4], [01,02])) (������,([06,07], [0.05,0.1], [02,03] )) (��,([07,08], [0.2,0.3], [0.1,0.2]) ) (����,([0.5,0.6], [0.1,0.2], [0.5,0.6])) (��,([0.7,0.8], [0.2,0.3], [0.05,0.1])) (������,([0.1,0.2], [0.5,0.6], [0.7,0.8])) (��,([0.5,0.6], [005,01], [0.2,0.3] )) (����,([0.7,0.8], [005,01], [0.1,0.2] )) (��,([0.7,0.8], [01,02], [0.05,0.1])) (��,([08,09], [005,01], [005,01])) (����,([08,09], [005,01], [005,01])) (��,([07,08], [005,01], [005,01])) (������,([0.5,0.6], [02,03], [0.3,0.4])) (������,([08,09], [005,01], [01,02] ))]

= [

(��,([02,03], [02,03], [06,07])) (����,([08,09], [01,02], [005,01])) (��,([05,06], [005,01], [02,03] )) (������,([02,03], [05,06], [01,02])) (��,([0.7,0.8], [01,02], [005,01]) ) (����,([07,08], [02,03], [01,02])) (��,([08,09], [005,01], [005,01])) (������,([07,08], [02,03], [005,01])) (��,([05,06], [0.2,0.3], [0.3,0.4])) (����,([07,08], [0.05,0.1], [0.1,0.2] )) (��,([08,09], [0.05,0.1], [0.05,0.1])) (��,([0.8,0.9], [0.05,0.1], [0.1,0.2] )) (����,([0.7,0.8], [0.05,0.1], [0.05,0.1])) (��,([0.6,0.7], [0.05,0.1], [0.2,0.3] )) (������,([01,02], [0.2,0.3], [0.7,0.8])) (������,([0.5,0.6], [0.2,0.3], [0.3,0.4]))]

(��,([0.5,0.6], [02,03], [0.3,0.4])) (����,([0.1,0.2], [02,03], [0.7,0.8])) (��,([0.2,0.3], [05,06], [0.1,0.2])) (������,([0.2,0.3], [05,06], [0.1,0.2])) (��,([08,09], [005,01], [0.05,0.1])) (����,([08,09], [005,01], [0.05,0.1])) (��,([08,09], [005,01], [0.05,0.1])) (������,([07,08], [02,03], [0.05,0.1])) (��,([07,08], [02,03], [01,02])) (����,([08,09], [01,02], [005,01])) (��,([08,09], [005,01], [005,01])) (��,([02,03], [0.2,0.3], [06,07])) (����,([08,09], [0.05,0.1], [01,02] )) (��,([06,07], [0.05,0.1], [02,03] )) (������,([01,02], [02,03], [07,08])) (������,([05,06], [0.2,0.3], [03,04]))

Importance of selected attributes by each decision maker is given as

��1 ⟶(��,([0.8,0.9], [0.1,0.2], [0.05,0.1])) (����,([0.6,0.7], [0.1,0.2], [0.1,0.2])) (��,([0.7,0.8], [0.05,0.1], [0.05,0.1])) (������,([0.5,0.6], [0.2,0.3], [0.4,0.5]))

��2 ⟶(��,([07,08], [005,01], [0.05,0.1])) (����,([05,06], [02,03], [0.3,0.4])) (��,([04,05], [01,02], [0.3,0.4])) (������,([06,07], [01,02], [0.1,0.2])) ��3 ⟶(��,([04,05], [01,02], [03,04])) (����,([07,08], [005,01], [005,01])) (��,([08,09], [005,01], [005,01])) (������,([08,09], [005,01], [01,02] )) ��4 ⟶(��,([08,09], [0.05,0.1], [005,01])) (����,([08,09], [0.1,0.2], [005,01])) (��,([08,09], [0.1,0.2], [005,01])) (������,([06,07], [0.05,0.1], [02,03] ))

Step 1: Determine the Weight of Decision Makers To find the ideal matrix we will average all the individual decision matrix ��1,��2,��3,��4using

]

(��

��=1 ]) �������� =1,2,3 ��,�� =1,2,3, ��,�� =��,��,��, ��and �� =1,�� =3,�� =2,�� = 2 By averaging all decision matrices, we get [��⋆] = [

(��,([0.6443,0.7700], [01189,02060], [0.2060,0.2300])) (����,([0.5880,0.7264], [01189,02060], [0.2141,0.3253])) (��,([0.5051,0.6131], [01655,02632], [0.1,0.1861] )) (������,([0.3273,0.4336], [02812,03834], [0.1189,0.2213])) (��,([0.7289,0.8318], [0.1,0.1861], [00595,01189])) (����,([0.6920,0.800], [0.1,0.1861], [01507,01861])) (��,([0.7217,0.8318], [0.1,0.1732], [00783,01414])) (������,([06052,0.7172], [0.2515,0.3568], [01150,0200] )) (��,([06127,07172], [0.1,0.1732], [01316,02213])) (����,([07289,,08318], [0.0595,0.1189], [01107,01682])) (��,([07787,08811], [0.0595,0.1189], [005,0100] )) (��,([07172,08373], [0.0707,0.1316], [01316,02300])) (����,([06870,08066], [0.0707,0.1316], [01107,01934])) (��,([06536,07551], [0.0707,0.1316], [01189,02060])) (������,([02455,03494], [0.2515,0.3568], [03482,04757])) (������,( [06838,0800], [0.1,0.1732], [01456,02378]))]

One calculation is provided for the convenience of reader for �� =1,�� =1,�� =�� =1

(1 (1 0.8) 1 4(1 0.2) 1 4(1 0.5) 1 4(1 0.8) 1 4 , 1 (1 0.9) 1 4(1 0.3) 1 4(1 0.6) 1 4(1 0.9) 1 4), ((01)1 4(02) 1 4(02) 1 4(005) 1 4 , (02) 1 4(03) 1 4(03) 1 4(01) 1 4 ) ((02)1 4(06) 1 4(03) 1 4(005) 1 4 , (01) 1 4(07) 1 4(04) 1 4(01) 1 4 ) ) ��11 ⋆ = ([06443,07700],[01189,02060],[02060,02300]) To determine the weights of the decision makers, first we will find the similarity measure between each decision matrix

1,��2,��3,��4and the ideal matrix ��⋆ using

1 (1 ����1 1 1��(��1)) 1 4 (1 ����1 1 2��(��1)) 1 4 (1 ����1 1 3��(��1)) 1 4 (1 ����1 1 4��(��1)) 1 4 , 1 (1 ����1 1 1��(��1)) 1 4 (1 ����1 1 2��(��1)) 1 4 (1 ����1 1 3��(��1)) 1 4 (1 ����1 1 4��(��1)) 1 4) , ( (����1 1 1��(��1)) 1 4 (����1 1 2��(��1)) 1 4 (����1 1 3��(��1)) 1 4 (����1 1 4��(��1)) 1 4 , (����1 1 1��(��1)) 1 4 (����1 1 2��(��1)) 1 4 (����1 1 3��(��1)) 1 4 (����1 1 4��(��1)) 1 4) , ( (����1 1 1��(��1)) 1 4 (����1 1 2��(��1)) 1 4 (����1 1 3��(��1)) 1 4 (����1 1 4��(��1)) 1 4 , (����1 1 1��(��1)) 1 4 (����1 1 2��(��1)) 1 4 (����1 1 3��(��1)) 1 4 (����1 1 4��(��1)) 1 4) ) ��11 ⋆ = ( �� �� +|������ �� ����(����) �� ���� �� ⋆��(����)|+|������ �� ����(����) �� ���� �� ⋆��(����)|+|������ �� ����(����) �� ���� �� ⋆��(����)|) So, ��(��1,��⋆)=0.8774, ��(��2,��⋆)=09133, ��(��3,��⋆)=08834, ��(��4,��⋆)=09101 Now we calculate the weight Δ�� ������(�� =1,2,3,4) of each decision makers using Δ�� = ��(������ �� ,������ ⋆ ) ∑ ��(������ �� ,������ ⋆ ) �� ��=1 Δ1 = 08774 (08774+09133+08834+09101) =02448

=0.2465

=0.2539

Step

2: Aggregate Neutrosophic Hypersoft Decision Matrices

Now we construct an aggregated neutrosophic hypersoft decision matrix to obtain one group decision. Aggregated neutrosophic hypersoft decision matrix is denoted as ������ and it is given as

∏(1 ��

��

(����))

,1 ∏(1 ��

(����))

],[∏(��

�� ��

(����))

��=1 ,∏(��

�� ��

(����))

],[∏(������ �� ����(����))Δ�� �� ��=1 ,∏(������ �� ����(����))Δ�� �� ��=1 ]) Where �� =1,2,3…��,�� =1,2,3,…���������� =1,2,…��. After calculations, the aggregated neutrosophic hypersoft decision matrices is [��] = [

(��,([06431,07691], [01187,02057], [0.2055,0.2309])) (����,([05932,07303], [01179,02051], [0.2099,0.3212])) (��,([05069,06146], [01631,02606], [0.1007,0.1862])) (������,([03249,04322], [02846,03854], [0.1185,0.2211])) (��,([0.7285,0.8313], [00999,01864], [0.0592,0.1187])) (����,([0.6924,0.7998], [01006,01870], [0.1048,0.1857])) (��,([0.7213,0.8316], [00998,01731], [0.0788,0.1419])) (������,([0.6074,0.7183], [02503,03560], [0.1138,0.1991])) (��,([0.6128,0.7168], [01002,01733], [0.1315,0.2215])) (����,([0.7285,0.8318], [00593,01186], [0.0843,0.1687])) (��,([0.7791,0.8813], [00592,01187], [0.0500,0.100] ))

(��,([0.7185,0.8386], [00704,01310], [01313,02293])) (����,([0.6847,0.8051], [00711,01319], [01115,01937])) (��,([0.6535,0.7550], [00711,01319], [01195,02065])) (������,([0.2436,0.3482], [02524,03573], [0.3471,0.4752])) (������,([0.6834,0.7998], [01002,01733], [01455,02376])) ]

One calculation is provided for the convenience of reader for �� =1,�� =1,�� =�� =1

11 = (

( (1 (1 ����1 1 1��(��1))Δ1 (1 ����1 1 2��(��1))

2 (1 ����1 1 3��(��1))Δ3 (1 ����1 1 4��(��1))Δ4), (1 (1 ����1 1 1��(��1))

1 (1 ����1 1 2��(��1))

2 (1 ����1 1 3��(��1))

3 (1 ����1 1 4��(��1))Δ4)), ( (����1 1 1��(��1))Δ1 (����1 1 2��(��1))Δ2 (����1 1 3��(��1))Δ3 (����1 1 4��(��1))Δ4 , (����1 1 1��(��1))Δ1 (����1 1 2��(��1))Δ2 (����1 1 3��(��1))Δ3 (����1 1 4��(��1))Δ4) , ( (����1 1 1��(��1))Δ1 (����1 1 2��(��1))Δ2 (����1 1 3��(��1))Δ3 (����1 1 4��(��1))Δ4 , (����1 1 1��(��1))Δ1 (����1 1 2��(��1))Δ2 (����1 1 3��(��1))Δ3 (����1 1 4��(��1))Δ4) ) ��11 = (

(1 (1 0.8)02448(1 0.2)02548(1 0.5)02465(1 0.8)02539 , 1 (1 0.9)02448(1 0.3)02548(1 0.6)02465(1 0.9)02539), ((01)02448(02)02548(02)02465(005)02539 , (0.2)02448(0.3)02548(0.3)02465(0.1)02539 ) ((0.2)02448(0.6)02548(0.3)02465(0.05)02539 , (01)02448(07)02548(04)02465(01)02539 ) )

Step 3: Determine the weight of attributes Weight

of attributes

,�� =1,2…�� is calculated using

��

,∏(������

)

],[∏(������

)

,∏(������

)

�� ��=1 ]) To calculate the weight of attributes, we use importance of selected attributes by each decision maker ��1 ⟶(��,([0.8,0.9], [01,02], [0.05,0.1])) (����,([0.6,0.7], [01,02], [0.1,0.2])) (��,([0.7,0.8], [005,01], [0.05,0.1])) (������,([0.5,0.6], [02,03], [0.4,0.5])) ��2 ⟶(��,([07,08], [005,01], [0.05,0.1])) (����,([05,06], [02,03], [0.3,0.4])) (��,([04,05], [01,02], [0.3,0.4])) (������,([06,07], [01,02], [0.1,0.2])) ��3 ⟶(��,([04,05], [0.1,0.2], [03,04])) (����,([07,08], [0.05,0.1], [005,01])) (��,([08,09], [0.05,0.1], [005,01])) (������,([08,09], [0.05,0.1], [01,02] ))

⟶(��,([08,09], [005,01], [005,01])) (����,([08,09], [01,02], [005,01])) (��,([08,09], [01,02], [005,01])) (������,([06,07], [005,01], [02,03] ))

Using this importance of attributes, we get ��1 =([0.7092,0.8228],[0.0703,0.1407],[0.0778,0.1406]) ��2 = ([06692,07785],[01006,01870],[00935,01424]) ��3 = ([0.7078,0.8212],[0.0711,0.1421],[0.0789,0.1424]) ��4 = ([06439,07542],[00838,01565],[01674,02778]) One calculation is provided for the convenience of reader for �� =1 ��1 = (

( (1 (1 ����1 1��(��1))Δ1 (1 ����1 2��(��1))Δ2 (1 ����1 3��(��1))Δ3 (1 ����1 4��(��1))Δ4), (1 (1 ����1 1��(��1))Δ1 (1 ����1 2��(��1))Δ2 (1 ����1 3��(��1))Δ3 (1 ����1 4��(��1))Δ4)), ((����1 1��(��1))Δ1 (����1 2��(��1))Δ2 (����1 3��(��1))Δ3 (����1 4��(��1))Δ4 , (����1 1��(��1))Δ1 (����1 2��(��1))Δ2 (����1 3��(��1))Δ3 (����1 4��(��1))Δ4), ((����1 1��(��1))Δ1 (����1 2��(��1))Δ2 (����1 3��(��1))Δ3 (����1 4��(��1))Δ4 , (����1 1��(��1))Δ1 (����1 2��(��1))Δ2 (����1 3��(��1))Δ3 (����1 4��(��1))Δ4) ) ��1 = (

(1 (1 0.8)02448(1 0.7)02548(1 0.4)02465(1 0.8)02539 , 1 (1 09)02448(1 08)02548(1 05)02465(1 09)02539), ((0.1)02448(0.05)02548(0.1)02465(0.05)02539 , (0.2)02448(0.1)02548(0.2)02465(0.1)02539 ) ((005)02448(005)02548(03)02465(005)02539 , (01)02448(01)02548(04)02465(01)02539 ) ) ��1 =([07092,08228],[00703,01407],[00778,01406])

Step 4: Calculate the weighted aggregated decision matrix After finding the weight of individual attributes, we apply these weights to each row of aggregated decision matrix

(��,([04562,06328], [01807,03175], [02675,03390])) (����,([03960,05685], [02066,03537], [02838,04179])) (��,([03588,05047], [02226,03657], [01703,03021])) (������,( [02092,0326], [03446,04816], [02661,04375]))

(��,([0.5167,0.6840], [01632,03009], [01324,02426])) (����,([0.4634,0.6226], [01911,03390], [01885,03017])) (��,([0.5106,0.6829], [01638,02906], [01507,02641])) (������,([0.3911,0.5417], [03131,04568], [02622,04216])) (��,([0.4347,0.5898], [01635,02896], [01991,0331] )) (����,([0.4875,0.6476], [01539,02834], [01699,02871])) (��,([0.5515,0.7237], [01261,02439], [01241,02282])) (��,( [0.5096,0.69], [01358,02533], [01989,03377])) (����,([0.4582,0.6268], [01645,02942], [01946,03085])) (��,( [0.4626,0.62], [01371,02551], [01882,03195])) (������,([0.1569,0.2626], [03150,04579], [04564,0621] )) (������,([0.4400,0.6032], [01756,03027], [02885,04494]))] One calculation is provided for the convenience of reader for �� =1,�� =1,�� =�� =1 ��11 �� =([���� ��1 1(��1) ����

1,����

1 1(��1) ���� ��1],[(���� ��1 1(��1)+���� ��1 ���� ��1 1(��1) ���� ��1),(���� ��1 1(��1)+���� ��1 ���� ��1 1(��1) ���� ��1)],[(���� ��1 1(��1)+���� ��1 ���� ��1 1(��1) ���� ��1),(���� ��1 1(��1)+���� ��1 ���� ��1 1(��1).���� ��1)]) ��11 �� =(((0.6431)(0 .7092),(0.7691)(0.8228)),((0.1187+0.0703 (0.1187)(0.0703)),(0.2057 +0.1407 (0.2057)(0.1407))) ,((0.2057+0.0778 (0.2057)(0.0778)),(0.2309 +01406 (02309)(01406)))) ��11 �� =([0.4562,0.6328],[0.1807,0.3175],[0.2675,0.3390])

Step 5: Determine the ideal solution Since we are dealing with benefit type (ℂ1) attributes so Neutrosophic hypersoft positive ideal solution is calculated using

=[(��,( [0.5167,0.69], [0.1358,0.2533], [01324,02426])) (����,([0.4875,0.6476], [0.1539,0.2834], [01699,02871])) (��,([0.5515,0.7237], [0.1261,0.3439], [01507,02282])) (������,([0.4400,0.6032], [0.1756,0.3027], [02622,04216]))] One calculation is provided for the convenience of reader

�� (����)},max �� {���� ���� �� �� (����)}],[max �� {���� ���� �� �� (����)} ,max �� {���� ���� �� �� (����)}],�� ∈ℂ1 ��

=[(��,([04347,05898], [01807,03175], [0.2675,0.3390])) (����,([03960,05685], [02066,03537], [0.2838,0.4179])) (��,([03588,05047], [02226,03657], [0.1882,0.3195])) (������,([01569,02626], [03446,04816], [0.4564,0.621]))] One calculation is provided for the convenience of reader for �� =1,2,3,4, �� =1,�� = �� =1 ��1 �� =( (������{04562,05167,04347,05096},������{06328,06840,05898,069}), (������{01807,01632,01635,01358},������{03175,03009,02896,02533}), (������{0.2675,0.1324,0.1991,0.1989},������{0.3390,0.2426,0.331,0.3377})) ��1 �� =(��,([0.4347,0.5898],[0.1807,0.3175],[0.2675,0.3390]))

Step 6: Calculate the distance measure Now we will find the normalized hamming distance between the alternatives and positive ideal solution using

(��

)= 1 24((|04562 01358|+|01807 01358|+|02675 01324|) +(|0.6328 0.69|+|0.69 0.2533|+|0.3390 0.2426|) +(|03960 04875|+|02066 01539|+|02838 01699|) +(|0.5685 0.6476|+|0.3537 0.2834|+|0.4179 0.2871|) +(|03588 05515|+|02226 01261|+|01703 01507|) +(|0.5047 0.7237|+|0.3657 0.2439|+|0.3021 0.2282|) +(|02092 04400|+|03446 01756|+|02661 02622|) +(|0.326 0.6032|+|0.4816 0.3027|+|0.4375 0.4216|)) ��1+(��1 ��,����+)=0.1081

Similarly, we will find the normalized hamming distance between the alternatives and negative ideal solution using

We get,

One calculation is provided for the convenience of reader

0.1807|+|0.2675 0.2675|) +(|06328 05898|+|03175 03175|+|03390 03390|) +(|0.3960 0.3960|+|0.2066 0.2066|+|0.2838 0.2838|) +(|05685 05685|+|03537 03537|+|04179 04179|) +(|0.3588 0.3588|+|0.2226 0.2226|+|0.1703 0.1882|) +(|05047 05047|+|03657 03657|+|03021 03195|) +(|0.2092 0.1569|+|0.3446 0.3446|+|0.2661 0.4564|) +(|0326 02626|+|04816 04816|+|04375 0621|))

ℝℂ4 = 00968 01023 00283 00283 = 005

Since we know that our set of four selected proposal is {ℝ2,ℝ4,ℝ5,ℝ6} for �� =1,2,3,4 respectively. So, we rank the selected alternatives (shown in Figure: 3) according to the descending order of relative closeness index as; ℝ6 >ℝ4 >ℝ5 >ℝ2

ALTERNATIVES RELATIVES CLOSENESS COEFFICIENT MEASUREMENT

0 R2 R4 R5 R6

-0.5

-1

-1.5

-2

-2.5

-3

-3.5

-0.05 -4

-3.64

-0.27 -1.93

Figure: 3 Alternatives relative closeness coefficient measurement and ranking This show that ℝ6 is the best proposal.

Result Comparison

The proposed Algorithm is compared with the existing interval valued techniques which are widely used in decision-making problems and results are presented in Table: 1 listing the ranking of top four alternatives and optimal alternative.

Methods

Ranking of alternatives Best alternative

I. Deli [39] ℝ6 > ℝ4 > ℝ5 > ℝ2 ℝ6 S. Alkhazaleh [40] ℝ6 > ℝ5 > ℝ4 > ℝ2 ℝ6 Riaz et al. (Proposed TOPSIS) ℝ6 > ℝ4 > ℝ5 > ℝ2 ℝ6

Table 1: Comparison analysis of final ranking with existing methods

6. Conclusion

In this chapter, TOPSIS technique has been proposed for interval valued neutrosophic hypersoft set IVNHSS by using generalized weighted similarity measure and distance measures for both IVNHSS and IVNHSM and the implementation is shown by letting a case study of life partner section. Also, chapter consists of the definitions of IVNHSM, generalized weighted similarity measures for interval valued neutrosophic hypersoft set and interval valued neutrosophic hypersoft matrix. In this process, decision makers’ opinions are consolidated initially. Further, we construct ideal matrix to find the weight of decision makers using the concept of interval valued

Theory and Application of Hypersoft Set 135

similarity measure. Afterwards, we compute a group opinion by accumulating all discrete opinions. We also gauge the weights of selected attributes to find the interval valued neutrosophic hypersoft ‘positive ideal’ and interval valued neutrosophic hypersoft ‘negative ideal’. Interval valued having distance measure is employed to compute the distance of alternatives from positive ideal and negative ideal. Thereafter, we rank the alternatives on the basis of relative closeness index.

o This proposed TOPSIS technique with interval valued neutrosophic hypersoft set IVNHSS cannot be compared because no work is present in this direction.

o It has immense chances for multi criteria decision making issues in several fields like HR selection, supplier determination, manufacturing frameworks, and various other areas of management frameworks.

o In enlargement, the proposed TOPSIS approach can be augmented in various directions to involve wide range of decision-making issues in several interval valued neutrosophic hypersoft conditions. In future, we propose matrix operations, theorems and propositions for interval valued neutrosophic hypersoft set along with algorithms and also the aggregate operators for IVNHSS.

References

[1] Opricovic, S., Tzeng, G.H. (2004). Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156, 445–455

[2] Saaty, T.L. (1980) The Analytic Hierarchy Process. McGraw-Hill, New York

[3] Hwang, C.L., Yoon, K. (1981). Multiple attribute decision making, methods and applications. Springer, New York.

[4] Behzadian, M., Khanmohammadi Otaghsara, S., Yazdani, M., Ignatius, J. (2012). A state-of the-art survey of TOPSIS applications. Expert System and Applications, 39, 13051–13069

[5] Zadeh, L.A. (1965) Fuzzy sets. Information Control, 8, 338–353

[6] Chen, S.J., Hwang, C.L. (1992). Fuzzy multiple attribute decision making. Springer.

[7] Kahraman, C., Onar, S.C., Oztaysi, B. (2015). Fuzzy multicriteria decision-making: a literature review. International Journal of Computer and Intelligent System, 8(4), 637–666

[8] Pramanik, S., Mondal, K. (2015) Weighted fuzzy similarity measure based on tangent function and its application to medical diagnosis. International Journal of Innovation Research in Scientific Engineering and Technology, 4(2), 158–164.

[9] Kahraman, C., Suder, A., Bekar, E. T. (2015). Fuzzy multi-attribute consumer choice among health insurance options. Technol. Econ. Dev. Econ. https://doi.org/10.3846/20294913.2014. 984252

[10]Atanassov, K. (1986) Intuitionistic fuzzy sets. Fuzzy Sets System, 20, 87–96

[11]Wei, G.W., Cao, H.B. (2008). Generalizing TOPSIS method for multiple attribute decision making in intuitionistic fuzzy setting. Information Computing and Automation, 1427–1430.

Theory and Application of Hypersoft Set

[12]Xu, Z., Zhao, N. (2016). Information fusion for intuitionistic fuzzy decision making: An overview. Information Fusion, 28, 10–23.

[13]Mondal, K., Pramanik, S. (2015) Intuitionistic fuzzy similarity measure based on tangent function and its application to multi-attribute decision making. Global Journal of Advance Research, 2(2), 464–471

[14]Pramanik, S., Mukhopadhyaya, D. (2011). Grey relational analysis based intuitionistic fuzzy multi criteria group decision-making approach for teacher selection in higher education. International Journal of Computer and Applications, 34(10), 21–29

[15]Mondal, K., Pramanik, S. (2014) Intuitionistic fuzzy multi-criteria group decision making approach to quality-brick selection problem. Journal of Applied Quantitative Methods, 9(2), 35–50.

[16]Dey, P.P., Pramanik, S., Giri, B.C. (2015). Multi-criteria group decision making in intuitionistic fuzzy environment based on grey relational analysis for weaver selection in Khadi institution, Journal of Applied Quantitative Methods, 10(4), 1–14

[17] F. Smarandache, (2005). Neutrosophic set, a generalization of the intuitionistic fuzzy sets, International Journal of Pure and Applied Mathematics, 24, 287–297.

[18] D. Molodtsov, (1999). Soft set theory - First results, Computers and mathematics with applications, 37, 19-31.

[19] P.K. Maji, A.R. Roy, R. Biswas, (2002). An application of soft sets in a decision-making problem, Computers and mathematics with applications, 44, 1077-1083.

[20] P.K. Maji, A.R. Roy, R. Biswas, (2003) Soft set theory, Computers and mathematics with applications, 45, 555-562.

[21] M.I. Ali, F. Feng, X. Liu, W.K. Min, (2009) On some new operations in soft set theory, Computers and mathematics with applications, 57(9), 1547-1553.

[22] N. Cagman, S. Enginoglu, (2010) Soft matrix theory and its decision making, Computers and mathematics with applications, 59, 3308-3314.

[23] N. Cagman, S. Enginoglu, (2010). Soft set theory and un-int decision making, European Journal of Operational Research, 207, 848-855.

[24] Z. Pawlak, (1982). Rough sets, International Journal of Computer Science and Information Technology, 11, 341-356.

[25] Tripathy B.K., Arun K.R. (2015) A new approach to soft sets, soft multisets and their properties, International Journal of Reasoning-based Intelligent Systems, 7(3/4), 244-253.

[26] P. K. Maji, (2013) Neutrosophic soft set. Annals of Fuzzy Mathematics and Informatics, 5 (1), 157-168.

[27] F. Smarandache, (2018) Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set, Neutrosophic sets and system, 22, 168-170.

[28] Saqlain M, Sana M, Jafar N, Saeed. M, Smarandache, F. (2020). Aggregate Operators of Neutrosophic Hypersoft Sets, Neutrosophic Sets and Systems, 32, 294–306

Theory and Application of Hypersoft Set 137

[29] Chi, P., Liu, P. (2013). An extended TOPSIS method for the multi-attribute decision making problems on interval neutrosophic set. Neutrosophic Sets System, 1, 63–70

[30] Zhang, Z., Wu, C. (2014). A novel method for single-valued neutrosophic multi-criteria decision making with incomplete weight information, Neutrosophic Sets System, 4, 35–49

[31] Biswas, P., Pramanik, S., Giri, B.C. (2016). TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Computing & Applications, 27, 727–737

[32] Sahin, R., Yigider, M. (2016). A multi-criteria neutrosophic group decision making method based TOPSIS for supplier selection Applied Mathematics & Information Sciences, 10(5), 1843–1852.

[33] Broumi, S., Ye, J., Smarandache, F. (2015) An Extended TOPSIS method for multiple attribute decision making based on interval neutrosophic uncertain linguistic variables. Neutrosophic Sets System, 8, 22–31.

[34]Saqlain M, Saeed M, Ahmad M. R, Smarandache F, (2019) Generalization of TOPSIS for Neutrosophic Hypersoft set using Accuracy Function and its Application, Neutrosophic Sets and Systems (NSS), 27, 131-137.

[35]S. Eraslan and F. Karaasalan, (2015), A group decision making method based on TOPSIS under fuzzy soft environment, Journal of New Theory, 3,30–40.

[36]Saqlain M, Sana M, Jafar N, Saeed. M, Said. B, (2020). Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure of Single valued Neutrosophic Hypersoft Sets, Neutrosophic Sets and Systems, 32, 317-329.

[37]Saqlain M, Xin L X., (2020), Interval Valued, m-Polar and m-Polar Interval Valued Neutrosophic Hypersoft Sets, Neutrosophic Sets and Systems (NSS), 36, 389-399.

[38]Saqlain M, Saeed. M, Zulqarnain M R, Sana M, (2021) Neutrosophic Hyper soft Matrix Theory: Its Definition, Operators and Application in Decision-Making of Personnel Selection Problem, Neutrosophic Operational Research, Springer, 10.1007/978-3-030-57197-9

[39]I. Deli, Interval-valued neutrosophic soft sets and its decision making, International Journal of Machine Learning and Cybernetics, DOI: 10.1007/s13042-015-0461-3.

[40]S. Alkhazaleh, (2016) N-valued refined neutrosophic soft set theory, Journal of Intelligent and Fuzzy System, 32(6), 4311–4318.

[41]Saqlain M, Saeed. M, Zulqarnain M R, Sana M, (2021) Neutrosophic Hyper soft Matrix Theory: Its Definition, Operators and Application in Decision-Making of Personnel Selection Problem, Neutrosophic Operational Research, Springer, 10.1007/978-3-030-57197-9

The Application of the Score Function of Neutrosophic Hypersoft Set in the Selection of SiC as Gate Dielectric For MOSFET

1 Department of Physics, Govt. Degree College Bhera, Sargodha, Pakistan. E-mail: umerfarooq900@gmail.com

1* School of Mathematics, Northwest University, Xian, China. E-mail: msgondal0@gmail.com

2 Department of Physics, University of Lahore, Sargodha Campus, Pakistan. E-mail: zakaurrehman52@gmail.com

Abstract: Neutrosophic hypersoft set are useful when attributes are more than one and are further bi-furcated. These sets are more accurate, precise and suitable for the ranking in decision making problems. Keeping in view; in this chapter, we present the study of various high k gate dielectrics for the metal oxide semiconductor filed effect transistor (MOSFET). The objective of this work is to choose the most suitable high k gate dielectric for the MOSFETs which can be used as gate dielectric.

By applying the score function of neutrosophic hypersoft set for the selection of best MOSFETS since these are the most important part of all the electronic communication systems today. In the last results conclude that ��4 =����2��3 is the best alternative that can be used as an optimal choice in digital communication devices.

Keywords: SiC, MOSFET, Gate Dielectric, Score Function, Neutrosophic Hypersoft Set (NHSS), MCDM

1. Introduction

Si based semiconductor devices have been used in power electronics applications for years. But such devices have certain limitations in terms of their operation at higher voltages, temperatures and switching speeds [1]. The voltage range for Si based semiconductor devices is limited to 6.5 kV which the switching frequency is limited to several hundred hertz [2, 3]. Also, the maximum temperature limit for such devices is 200 ⁰C [1]. Due to these problems, the wide band gap semiconductors like GaN and SiC have caught the attention of the researchers since first realization of SiC based metal oxide semiconductor field effect transistor (MOSFET). These semiconductor devices operate at higher voltages, higher temperatures and have higher switching frequencies as compared to Si based semiconductor devices [4-6]. SiC based field effect transistors (FET) have a voltage range 650 V to 1.7 kV [7]. Semiconductors are the essential part of all the communication systems today. The semiconductor devices have significantly reduced both power losses and cost of the communication systems. These devices are used for high power applications in all type of environments [8-17]. SiC is very suitable candidate for high power and high temperature applications due to their larger band gap and larger conductivity. Moreover, these materials offer

Muhammad Umer Farooq1 , Muhammad Saqlain1*, Zaka-ur-Rehman2

Theory and Application of Hypersoft Set 139 larger carrier saturation velocity. Such exceptional properties of SiC makes it ideal for its applications in high power communication devices [17-27]. It is used largely in devices like Schottky Diode [10], junction field effect transistors [15, 16], bipolar junction transistors [11-14], metal oxide semiconductor field effect transistors [18-22] and insulated gate field effect transistors [23-27]. In spite of such novel properties, SiC has some drawbacks too. It has low dielectric constant and poor interface properties at ������/ ������2 junction in MOSFETs. This causes an increase in electric field at junction interface. So, the search for new gate dielectrics as an alternate of SiC is required. The new gate dielectric may have dielectric constant similar to SiC but smaller interface densities than SiC for its use as gate dielectric in MOSFETs. The high k gate dielectrics like ������2, ������, ����

,

5 and ����2��3 are the potential candidates for gate dielectric as an alternate of SiC in MOSFETs [28-45]. With the development of fuzzy sets [46] decision making becomes easier but later on this theory was extended by [47] named as Intuitionistic fuzzy number theory. To deal with more precision, accuracy and indeterminacy this idea was extended by [48] called as neutrosophy theory. To, discuss the applications of these theory number of developments were made but the most important one is the theory of soft set [49]. Later on, fuzzy, intuitionist and neutrosophy theories were extended to fuzzy softset [50], intuitionistic soft set [51] and neutrosophic soft set [52]. In different fields the applications of these theories are presented by many researchers [53-60], but with the development of TOPSIS, WSM and WPM techniques [61-66] it becomes more powerful tool to solve the MCDM problems [67-72]. Samarandache [73] came up with strategy to handle uncertainty. Soft sets were generalized to hyper soft set (HSS’s) by changing the function into multi decision function. When attributes are more than one and further diverge, neutrosophic soft set environment cannot help to handle such type of issue because of soft sets as soft sets deal with single argument function. For this purpose, neutrosophic hypersoft set (NHSS) [74] was introduced. Later, aggregate operators, similarity measures and their applications are presented by [75-78]

Now the question arises why we are using these techniques in this case study? To get the answer of this question, firstly you need to know the attribute and alternatives; since dielectrics are of many types having different properties which makes it a perfect problem to apply the abovementioned MCDM techniques. Semiconductors are the essential part of all the communication systems today. The novel characteristics of SiC make it very suitable for high power communication. But due to problem of its low dielectric constant and poor interface properties, the search for new gate dielectric is essential. These new dielectrics may have dielectric constant similar to SiC but have smaller interface densities. In this research, ten such high k-gate dielectrics are being analyzed. The purpose of this research is to find which dielectric is most suitable alternate of SiC for high power communication with the help of MOSFETs. For this purpose, we apply neutrosophic hypersoft sets. The layout of this chapter is shown in Figure:1. Introduction, Literature review, motivation and contribution is presented in section 1. Next section includes the basic definitions. Section 3, comprises of score function of NHSS. The case study is presented in section 4. Result discussion is made in section 5. Finally, in section 6, the chapter is concluded with limitation and future directions.

•Introduction

• Literature Review Section 1

Section 2

•Preliminaries

•Score Function of NHSS

Section 3

Section 4 •Result Discussion Section 5

• Case Study

•Conclusion Section 6

Figure 1: The layout of the chapter 2.Preliminaries

Definition 2.1: Semiconductors [79]

These are the materials which are poorer conductors than metals but better than insulators. For example, Silicon (Si), Germanium (Ge) and Gallium Arsenide.

Definition 2.2: Transistors [79]

It is an electrical device which consist of two PN junctions fabricated on a same single crystal. It has three main parts i.e. emitter, base and collector. Transistors are used to amplify and switch the electronic signals and electrical power.

Definition 2.3: Field Effect Transistors (FET) [79]

It is a type of the transistor in which flow of majority charge carrier (current) is controlled by the signal voltage applied to a reverse biased pn junction This reverse biased pn junction is called gate. FET has three terminals i.e. source, gate and drain.

Definition 2.4: Metal Oxide Semiconductor FET (MOSFET) [79]

The MOSFET is a four terminal device i.e. source, gate, drain and substrate. The substrate is usually grounded. The charge carriers enter into conducting channel through source and exit through drain. The width of the channel is controlled by applying the voltage at the gate. The gate is present between source and drain. It is insulated from the channel near an extremely thin layer of metal oxide. The MOS capacity that exists in the device is a very important part as the entire operation occur across this. The metal contact over the insulator is known as gate electrode which is separated by a dielectric like Silicon dioxide or high k dielectric from the substrate.

Definition 2.4: Dielectrics

These are the materials which are poor conductors of electricity. It is an insulator but an effective supporter to electric filed.

Theory and Application of Hypersoft Set

Definition 2.5: Dielectrics Constant [80]

It can be defined as the ratio of the charge stored in the presence of an insulating material placed between two metallic plates to the charge that can be stored when the insulating material is replaced by vacuum or air.

Definition 2.6: High k Dielectrics Materials [81]

Dielectric materials which have high value of dielectric constant are called high k dielectric materials. Metal oxides have usually have high dielectric constant. They are usually used in MOSFETs as gate dielectric.

Definition 2.7: Band Gap [82]

The energy difference between the top of the valence band and the bottom of the conduction band is called band gap.

Definition 2.7: Wide Band Gap Materials [83]

These are the materials which have relatively larger band gap as compared to conventional semiconductors. Conventional semiconductors like silicon have a bandgap in the range of 1 - 1.5 eV, whereas wide-bandgap materials have bandgaps in the range of 2 - 4 eV

Definition 2.8: Neutrosophic Hypersoft Set (NHSS) [74]

Let �� be the universal set and ℙ(��)be the power set of ��. Consider l1,l2,l3 ln for n≥1, be n well-defined attributes, whose corresponding attributive values are respectively the set L1,L2,L3 Ln with Li ∩Lj =∅, for i≠j and i,jϵ{1,2,3 n} and their relation L1 ×L2 ×L3 Ln =$, then the pair (₣,$) is said to be Neutrosophic Hypersoft set (NHSS) over �� where; ₣:L1 ×L2 ×L3 Ln →ℙ(��)and ₣(L1 ×L2 ×L3 …Ln)={<x,T(₣($)),I(₣($)),F(₣($))>,x∈��} where T is the membership value of truthiness, I is the membership value of indeterminacy and F is the membership value of falsity such that T,I,F:��→[0,1] also 0≤T(₣($))+ I(₣($))+ F(₣($))≤3

3. Algorithm of Score Function of Neutrosophic Hypersoft set (NHSS)

In this section an algorithm is presented to solve MCDM problem under neutrosophic hypersoft set environment

Suppose that there are some decision makers who wish to select from �� number of objects. Each object is further characterized by �� number of attributes, whose respective attributes form a relation just like Neutrosophic Hypersoft Matrix (NHSM). Each decision makes gives different Neutrosophic values to these respective attributes. Corresponding to these Neutrosophic values for the required relation we get a NHSM of order ����. From these NHSM we calculate values matrices which helps to obtain a score matrix. And finally, we calculate the total score of each object from score matrix. Algorithm is presented in Figure 2.

Theory and Application of Hypersoft Set 142

Let �� =[������] be the NHSM of order ��×��, where ������ =(�������� �� ,ℐ������ �� ,ℱ������ �� ) then the value of matrix �� is denoted as ��(��) and it is defined as ��(��) =[������ ��] of order ��×��, where������ �� =�������� �� ℐ������ �� , ℱ������ �� . The Score of two NHSM �� =[������] and ℳ =[ℳ����] of order ��×�� is given as ��(��,ℳ)=��(��)+��(ℳ) and ��(��,ℳ)=[������] where ������ = ������ �� +������ ℳ . The total score of each object in universal set is |∑ ������ �� ��=1 |

Step 1: Construct a NHSM.

Step 2: Calculate the value matrix from NHSM. Let �� =[������] be the NHSM of order ��×��, where ������ =(�������� �� ,ℐ������ �� ,ℱ������ �� ) then the value of matrix �� is denoted as ��(��) and it is defined as ��(��) = [������ ��] of order ��×��, where������ �� =�������� �� ℐ������ �� , ℱ������ �� .

Step 3: Compute score matrix with the help of value matrices. . The Score of two NHSM �� =[������] and ℳ =[ℳ����] of order ��×�� is given as ��(��,ℳ)=��(��)+��(ℳ) and ��(��,ℳ)=[������] where ������ = ������ �� +������ ℳ .

Step 4: Compute total score from score matrix. The total score of each object in universal set is |∑ ������ �� ��=1 |

Step 5: Find optimal solution by selecting an object of maximum score from total score matrix.

Figure 2: Graphical representation of Score Function Algorithm of NHSS

4: Case Study

In this section a case study of SiC selection as gate dielectric for high power communication is considered and the selection is made by applying all the above-mentioned algorithm.

4.1 Problem Formulation

SiC based MOSFETs are widely used in high power communication devices. But due to its poor interface properties, the search for new dielectrics which have similar dielectric constant as SiC but have good interface properties is required. So, the more efficient high-power communication devices can be constructed with the help of these high k-gate dielectrics. These devices can operate at higher temperatures and higher voltages.

4.3 Assumptions

2.7

Theory and Application of Hypersoft Set 144 ��1 �� = Dielectricconstant ={39,70,40 70,90,25,25,26,15,30,914} ��2 �� = BandGap���� (����)={8.9,5.1,5 9,8.7,5.7,7.8,4.5,5.6,4.3,6.2} ��3 �� = ConductionbandwithrespecttoSi������ (����) ={3.2,2.0,2.8,2.8,1.5 1.7,1.4,1 1.5,2.3,2.3,2.2} ��4 �� =Conductionbandwithrespectto4H SiC������ (����)={2.2 2.7, , ,1.7,0.54,1.6, . , ,1.7} Let’s assume the relation for the function ℱ:��1 �� ×��2 �� ×��3 �� ×��4 �� →��(��) as; ℱ(��1 �� ×��2 �� ×��3 �� ×��4 �� ×��5 ��)=(39,51,14,16) is the actual requirement for the selection of material for making dielectric. Ten dielectric materials {��1 ,��2,��3,��4,��5,��6,��7,��8,��9,��10} are selected on the basis of assumed relation i.e.( 3.9,5.1,1.4,1.6).

Two decision makers {��1,��2} are intended to select the most suitable material for the formation of respective dielectric. These decision makers give their opinion in the form of NHSM separately as; Step 1: Construction of NHSM [��1]10×4 = [

(39,(08,01,005)) (51,(02,02,06)) (14,(05,03,01)) (16,(06,005,02)) (3.9,(0.7,0.2,0.1)) (5.1,(0.5,0.1,0.5)) (1.4,(0.7,0.2,0.05)) (1.6,(0.1,0.5,0.7)) (3.9,(0.5,0.05,0.2)) (5.1,(0.7,0.05,0.1)) (1.4,(0.7,0.1,0.05)) (1.6,(0.5,0.2,0.3)) (39,(08,005,005)) (51,(08,005,005)) (14,(07,005,005)) (16,(08,005,01)) (39,(02,02,06)) (51,(08,01,005)) (14,(05,005,02)) (16,(06,005,02)) (3.9,(0.7,0.2,0.1)) (5.1,(0.7,0.2,0.1)) (1.4,(0.8,0.05,0.05)) (1.6,(0.2,0.5,0.1)) (39,(07,01,005)) (51,(07,005,01)) (14,(08,005,005)) (16,(07,02,005)) (39,(05,02,03)) (51,(07,005,005)) (14,(07,005,005)) (16,(01,02,07)) (39,(08,005,01)) (51,(07,02,01)) (14,(06,005,02)) (16,(05,02,03)) (3.9,(0.2,0.2,0.6)) (5.1,(0.7,0.05,0.1)) (1.4,(0.6,0.05,0.2)) (1.6,(0.2,0.5,0.1)) ] [��2]

= [

(39,(05,02,03)) (51,(01,02,07)) (14,(02,05,01)) (16,(02,05,01)) (39,(08,005,005)) (51,(08,005,005)) (14,(08,005,005)) (16,(07,02,005)) (3.9,(0.7,0.2,0.1)) (5.1,(0.8,0.1,0.05)) (1.4,(0.8,0.05,0.05)) (1.6,(0.1,0.2,0.7)) (3.9,(0.2,0.2,0.6)) (5.1,(0.8,0.05,0.1)) (1.4,(0.6,0.05,0.2)) (1.6,(0.5,0.2,0.3)) (39,(02,02,06)) (51,(08,01,005)) (14,(05,005,02)) (16,(06,005,02)) (3.9,(0.7,0.2,0.1)) (5.1,(0.7,0.2,0.1)) (1.4,(0.8,0.05,0.05)) (1.6,(0.2,0.5,0.1)) (3.9,(0.8,0.05,0.1)) (5.1,(0.2,0.2,0.6)) (1.4,(0.7,0.05,0.5)) (1.6,(0.2,0.1,0.05)) (39,(07,005,005)) (51,(07,005,01)) (14,(05,02,03)) (16,(07,02,01)) (39,(07,01,005)) (51,(07,01,005)) (14,(08,005,02)) (16,(02,05,01)) (3.9,(0.8,0.05,0.05)) (5.1,(0.8,0.05,0.1)) (1.4,(0.7,0.5,0.2)) (1.6,(0.8,0.05,0.05))]

Step 2: Calculate the value matrices of NHSMs defined above using ������ �� =�������� �� ℐ������ �� ℱ������ ��

[��1]10×4 = [

(3.9,(0.65)) (5.1,( 0.6)) (1.4,(0.1)) (1.6,(0.35)) (39,(04)) (51,( 01)) (14,(045)) (16,( 11)) (39,(025)) (51,(055)) (14,(055)) (16,(00)) (3.9,(0.7)) (5.1,(0.7)) (1.4,(0.6)) (1.6,(0.65)) (39,( 06)) (51,(065)) (14,(025)) (16,(035)) (39,(04)) (51,(04)) (14,(07)) (16,( 04)) (3.9,(0.55)) (5.1,(0.55)) (1.4,(0.7)) (1.6,(0.45)) (3.9,(0.0)) (5.1,(0.6)) (1.4,(0.6)) (1.6,( 0.8)) (39,(065)) (51,(04)) (14,(035)) (16,(00)) (39,( 06)) (51,(055)) (14,(035)) (16,( 04))]

(39,(00)) (51,( 08)) (14,( 04)) (16,( 04)) (3.9,(0.7)) (5.1,(0.7)) (1.4,(0.7)) (1.6,(0.45)) (3.9,( 0.6)) (5.1,(0.65)) (1.4,(0.7)) (1.6,( 0.8)) (39,(06)) (51,(065)) (14,(035)) (16,(00)) (39,(06)) (51,(065)) (14,(025)) (16,(035)) (3.9,(0.4)) (5.1,(0.4)) (1.4,(0.7)) (1.6,( 0.4)) (39,(065)) (51,( 06)) (14,(015)) (16,(005)) (39,(06)) (51,(055)) (14,(00)) (16,(04)) (39,(055)) (51,(055)) (14,(055)) (16,( 04)) (3.9,(0.7)) (5.1,(0.65)) (1.4,(0.0)) (1.6,(0.7)) ]

Step 3: Now compute score matrix by adding value matrices obtained in Step 2.

[��(��1��2)]10×4 = [

(39,(065)) (51,( 14)) (14,( 03)) (16,( 005)) (3.9,(1.1)) (5.1,(0.6)) (1.4,(1.15)) (1.6,( 0.65)) (3.9,( 0.35)) (5.1,(1.2)) (1.4,(1.25)) (1.6,( 0.8)) (39,(13)) (51,(135)) (14,(095)) (16,(065)) (39,(00)) (51,(13)) (14,(05)) (16,(07)) (3.9,(0.8)) (5.1,(0.8)) (1.4,(1.4)) (1.6,( 0.8)) (39,(12)) (51,( 005)) (14,(085)) (16,(05)) (39,(06)) (51,(115)) (14,(06)) (16,( 04)) (3.9,(1.2)) (5.1,(0.95)) (1.4,(0.9)) (1.6,( 0.4)) (3.9,(0.1)) (5.1,(1.2)) (1.4,(0.35)) (1.6,(0.3)) ]

Step 4: Now total score of score matrix is given as;

�������������������� = [

�� �� �� �� ��.�� ��.���� �� �� �� �� �� �� ��.���� ��.���� ��.����]

Step 5: The material of dielectric ��4 will be selected as the total score of ��4 is highest among the rest of the total score of dielectrics.

Alternative Total Score Value D1 1.1 D2 2.2 D3 1.3 D4 4.25 D5 2.5 D6 2.2 D7 2.5 D8 1.95 D9 2.65 D10 1.95

Table 1: The total score function of each dielectric taken as alternative Figure 3: Total score value comparison of alternatives

6. Result Discussion

Everyday many researchers are trying to find the best operational research technique that can be useful in decision making problems. In very recent with the development of NHSS and its score function; it seems to be the success of the researchers who are working on it. Since NHSS is the tool which is most suitable in decision making problems and can be applied when attributes are more than one and are further bi-furcated. For the validity of the algorithm of score function in the applied mathematical issues and MCDM environment the case study of SiC gate dielectric for the MOSFET in communication devices is used.

In these calculations, the ranking of each dielectric with respect to each criterion is calculated which are shown in Table 1 and Figure 4 Result shows that the above-mentioned techniques can be used to find the optimal SiC dielectric in communication devises.

Total Score Value

1.1

2.2 1.3

4.25 2.5 2.2 2.5 1.95

2.65 1.95 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10

Total Score Value 1.1 2.2 1.3 4.25 2.5 2.2 2.5 1.95 2.65 1.95

Total Score Value

Figure 4: Ranking comparison of alternatives

The dielectric ����2��3 can be used as best alternative dielectric material which is most suitable in communication devices. So, the more efficient high-power communication devices can be constructed with the help of this high k-gate dielectric.

5. Conclusions

Among the ten potential alternatives for gate dielectric, the calculations show that ����2��3 is the most suitable high k-gate dielectric for the MOSFETs. Also ����2��3 has excellent lattice matching with SiC. It has good thermal stability and high conduction band off-set between ����2��3 and 4H-SiC. Moreover, it has high k- value and relatively larger dielectric band gap [28]. Such properties of ����2��3 makes it a very suitable substitute of SiC as gate dielectric. These calculations are done by applying score function of NHSS.

Since this study has not yet been studied yet, comparative study cannot be done with the existing methods.

Theory and Application of Hypersoft Set 148

In our forthcoming work, this method can be used to find the best alternative having same physical and structural properties in manufacturing process of different materials with low lost and good impact on environment.

Conflicts of Interest

The authors declare no conflict of interest.

Reference:

[1] J. Millán, P. Godignon, X. Perpiñà, A. Pérez-Tomás, and J. Rebollo, (2014). A survey of wide bandgap power semiconductor devices, IEEE Trans. Power Electron., 29(5), 2155–2163.

[2] A. Marzoughi and H. Imaneini, (2014). Optimal selective harmonic elimination for cascaded H-bridge-based multilevel rectifiers, IET Power Electron, 7(2), 350–356

[3] A. Marzoughi, H. Imaneini, and A. Moeini, (2013). An optimal selective harmonic mitigation technique for high power converters, Int. J. Elect. Power & Energy Syst., 49, 34–39

[4] R. Singh and M. Pecht, (2008). Commercial impact of silicon carbide, In IEEE Ind. Electron. Mag., 2(3), 19–31.

[5] J. Rabkowski, D. Peftitsis, and H. P. Nee, (2012). Silicon carbide power transistors: A new era in power electronics is initiated, IEEE Ind. Electron. Mag., 6(2), 17–26

[6] J. W. Palmour, J. A. Edmond, H. S. Kong, and C. H. Carter, Jr., (1993). 6H-silicon carbide power devices for aerospace applications, In Proc. 28th Intersociety Energy Conversion Engineering Conf., 1249–1254

[7] A. Marzoughi, R. Burgos, and D. Boroyevich, (2016). Characterization and comparison of latest generation 900-V and 1.2-kV SiC MOSFETs, In Proc. IEEE Energy Conversion Congress and Exposition, 1–8.

[8] S. m Madhusoodhanan, K. Hatua, S. Bhattacharyaetal., (2012). Comparison study of 12kV n-type SiC IGBT with 10kV SiC MOSFET and 6.5kV Si IGBT based on 3L-NPC VSC applications, In Proc. IEEE Energy Conversion Congress and Exposition, 310–317

[9] J. A. Cooper, T. Tamaki, G. G. Walden, Y. Sui, S. R. Wang, and X. Wang, (2009). Power MOSFETs, IGBTs, and thyristors in SiC: Optimization, experimental results, and theoretical performance, In Proc. of the International Electron Devices Meeting, 149–152

[10]L. Zhuand T. P. Chow, (2008). Advanced high-voltage 4H-SiC Schottky rectifiers, IEEE Transactions on Electron Devices, 55(8), 1871–1874

[11]M. Nawaz, C. Zaring, J. Bourceetal., (2009) Assessment of high and low temperature performance of SiC BJTs, Materials Science Forum, 615–617, 825–828.

Theory and Application of Hypersoft Set

[12]M. Nawaz, (2010) On the assessment of few design proposals for 4H SiC BJTs, Microelectronics Journal, 41(12), 801–808.

[13]B. Buono, R .Ghandi, M .Domeij ,B. G. Malm ,C.-M .Zetterling, and M. Ostling, (2010 ). Modeling and characterization of current gain versus temperature in 4H-SiC power BJTs, IEEE Transactions on Electron Devices, 57(3), 704–711.

[14]M. Usman, M. Nawaz, and A. Hallen, (2013). Position-dependent bulk traps and carrier compensation in 4H-SiC bipolar junction transistors, IEEE Transactions on Electron Devices, 60(1), 178–185.

[15]Y. Li, P. Alexandrov, and J. H. Zhao, (2008). (1.88-mΩ-����2) 1650-V normally on 4H-SiC TI-V JFET, IEEE Transactions on Electron Devices, 55(8), 1880–1886.

[16]A. Ritenour, I. Sankin, N. Merret et al., (2008). High temperature electrical characteristics of 20 A, 800V enhancement mode SiC VJFETs, In Proceedings of the IMAPS High Temperature Electronics Conference (HiTEC’08), 103–108.

[17]L. Cheng, A. K. Agarwal, C. Capell et al., (2013) 15 kV, large area (1 ����2), 4H-SiC ptype gate turn-off thyristors, Materials Science Forum, 740–742, 978–981.

[18]H. ¨ O. ´Olafsson, G. Gudj ´ onsson, P.-˚A. Nilsson et al., (2004) High field effect mobility in Si face 4H-SiC MOSFET transistors, Electronics Letters, 40(8), 508–510.

[19]J. Wang, T. Zhao, J. Li et al., (2008). Characterization, modeling, and application of 10kV SiC MOSFET, IEEE Transactions on Electron Devices, 55(8), 1798–1806.

[20]R. S. Howell, S. Buchoff, S. van Campen et al., (2008) A 10-kV large-area 4H-SiC power DMOSFET with stable subthreshold behavior independent of temperature, IEEE Transactions on Electron Devices, 55(8), 1807–1815.

[21]V. Tilak, K. Matocha, G. Dunne, F. Allerstam, and E. ¨ O. Sveinbjornsson, (2009). Trap and inversion layer mobility characterization using Hall effect in silicon carbide-based MOSFETs with gate oxides grown by sodium enhanced oxidation, IEEE Transactions on Electron Devices, 56(2), 162–169

[22]D. Okamoto, H. Yano, K. Hirata, T. Hatayama, and T. Fuyuki, (2010). Improved inversion channel mobility in 4H-SiC MOSFETs on Si face utilizing phosphorus-doped gate oxide, IEEE Electron Device Letters, 31(7), 710–713

[23]Q. Zhang, M. Das, J. Sumakeris, R. Callanan, and A. Agarwal, (2008). 12-kV p-channel IGBTs with low on-resistance in 4H-SiC, IEEE Electron Device Letters, 29(9), 1027–1029

[24]X. Wang and J. A. Cooper, (2010). High-voltage n-channel IGBTs on free-standing 4HSiC epilayers, IEEE Transactions on Electron Devices, 57(2), 511–515.

Theory and Application of Hypersoft Set 150

[25]S.-H. Ryu, C. Capell, L. Chengetal., (2012) High performance, ultra-high voltage 4HSiC IGBTs, In Proceedings of the IEEE Energy Conversion Congress and Exposition (ECCE’12), 3603–3608.

[26]M. Nawaz and F. Chimento, (2013) On the assessment of temperature dependence of 10–20 kV 4H-SiC IGBTs using TCAD, Materials Science Forum, 740–742, 1085–1088.

[27]M. Usman and M. Nawaz, (2014). Device design assessment of 4H-SiC n-IGBT a simulation study, Solid-State Electronics, 92, 5–11

[28]M. Nawaz, (2014). On the Evaluation of Gate Dielectrics for 4H-SiC Based Power MOSFETs, 2015.

[29]Q. Shui, M. S. Mazzola, X. Gu, C. W. Myles, and M. A. Gundersen, (2004) Investigation of ����2��3 and ������2 as gate insulators for 4H-SiC pulsed power devices, In Proceedings of the 26th International Power Modulator Symposium, 501–504.

[30]T. Tamaki, G. G. Walden, Y. Sui, and J. A. Cooper, (2008) Numerical study of the turn off behavior of high-voltage 4H-SiC IGBTs, IEEE Transactions on Electron Devices, 55(8), 1928– 1933.

[31]R. Singh and A. R. Hefner, (2004) Reliability of SiC MOS devices, Solid-State Electronics, 48(10-11), 1717–1720.

[32]A. K. Agarwal, S. Seshadri, and L. B. Rowland, (1997). Temperature dependence of Fowler-Nordheim current in 6H- and 4H-SiC MOS capacitors, IEEE Electron Device Letters, 18(12), 592–594

[33]L.Yu, Simulation, (2010). Modeling and characterization of SiC devices, [Ph.D. thesis], Graduate School-New Brunswick Rutgers, The State University of New Jersey

[34]M. Gurfinkel, J. C. Horst, J. S. Suehle et al., (2008). Time-dependent dielectric breakdown of 4H-SiC/ ������2 MOS capacitors, IEEE Transactions on Device and Materials Reliability, 8(4), 635–640

[35]O. Biserica, P. Godignon, X. Jord`a, J. Montserrat, N. Mestres, and S. Hidalgo, (2000). Study of ������/������2 as dielectric layer for SiC MOS structures, In Proceedings of the International Semiconductor Conference, 1, 205–208

[36]A. Taube, S. Gierałtowska, T. Guttetal., (2011). Electronic properties of thin ������2 films fabricated by atomic layer deposition on 4HSiC, Acta Physica Polonica A, 119(5), 696–698

[37]K.Y. Cheong, J.M.Moon, T.-J.Parketal., (2007). Improved electronic performance of ������2/ ������2 stackingg at dielectric on 4H SiC, IEEE Transactions on Electron Devices, 54(12), 3409– 3413

Theory and Application of Hypersoft Set 151

[38]K. Y. Cheong, J. H. Moon, D. Eom, H. J. Kim, W. Bahng, and N.-K. Kim, (2007). Electronic properties of atomic-layer-deposited ����2��3 /thermal-nitrided ������2 stacking dielectric on 4H SiC, Electro chemical and Solid-State Letters, 10(2), 69– 71.

[39]H. Yano, F. Katafuchi, T. Kimoto, and H. Matsunami, (1999). Effects of wet oxidation/anneal on interface properties of thermally oxidized ������2 / SiC MOS system and MOSFET’s, IEEE Transactions on Electron Devices, 46(3), 504–510.

[40]S. Hino, T. Hatayama, J. Kato, E. Tokumitsu, N. Miura, and T. Oomori, (2008). High channel mobility 4H-SiC metal-oxide semiconductor field-effect transistor with low temperature metal-organic chemical-vapor deposition grown ����2��3 gate insulator, Applied Physics Letters, 92(18)

[41]A. P´erez-Tom´as, M. R. Jennings, P. M. Gammon et al., (2008). SiC MOSFETs with thermally oxidized ����2���� stacked on ������2 as high-k gate insulator, Microelectronic Engineering, 85(4), 704–709

[42]A. Shima, K. Watanabe, T. Mine, N. Tega, H. Hamamura, and Y. Shimamoto, (2014). Reliable 4H-SiC MOSFET with high threshold voltage by ����2��3-inserted gate insulator, In Proceedings of the European Conference on Silicon Carbide & Related Materials (ECSCRM’14).

[43] T. Hatayama, S. Hino, N. Miura, T. Oomori , and E. Tokumitsu, (2008). Remarkable increase in the channel mobility of SiC-MOSFETs by controlling the interfacial ������2 layer between ����2��3 and SiC, IEEE Transactions on Electron Devices, 55(8), 2041– 2045

[44]A. Perez Tomas, (2005). Novel materials and processes for gate dielectrics on silicon carbide [Ph.D. thesis], Facultat De Ciencies, Department de Fisica Universitat Autonoma De Barcelona, Barcelona, Spain.

[45]K. Y. Cheong, J. H. Moon, H. J. Kim, W. Bahng, and N.-K. Kim, (2008). Current conduction mechanisms in atomic-layer-deposited ������2 /nitrided ������2 stacked gate on 4H silicon carbide, Journal of Applied Physics, 103(8)

[46]Zadeh, L. A, (1965). Fuzzy sets, Inform and Control, 8(3), 338-353.

[47]Atanassov, K. T, (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87-96

[48]F. Smarandache, (1998). Neutrosophy. Neutrosophic probability, set, and logic. ProQuest Information & Learning, Ann Arbor, Michigan, USA.

[49]Molodtsov, D. (1999). Soft set theory-First results, Computers and mathematics with applications, 37, 19-31.

[50]Maji, Pabitra & Biswas, R. & Roy, A.R. (2001). Fuzzy soft sets. 9. 589-602.

[51]Maji, Pabitra & Biswas, Ranjit & Roy, Akhil. (2001). Intuitionistic fuzzy soft sets. The Journal of Fuzzy Mathematics. 9.

Theory and Application of Hypersoft Set 152

[52]P. K. Maji, (2013). Neutrosophic soft set, Annals of Fuzzy Mathematics and Informatics, 5(1) 157-168.

[53]Wielki, Janusz & Jurczyk, Magdalena & Madera, Dariusz. (2019). Application of TOPSIS Method for Evaluation of IT Application in the Hospital. 10.34190/KM.19.134.

[54]Abdel-Baset, M., Chang, V., & Gamal, A. (2019). Evaluation of the green supply chain management practices: A novel neutrosophic approach. Computers in Industry, 108, 210220.

[55]Abdel-Basset, M., Saleh, M., Gamal, A., & Smarandache, F. (2019). An approach of TOPSIS technique for developing supplier selection with group decision making under type-2 neutrosophic number. Applied Soft Computing, 77, 438-452.

[56]Abdel-Basset, M., Atef, A., & Smarandache, F. (2019). A hybrid Neutrosophic multiple criteria group decision making approach for project selection. Cognitive Systems Research.

[57]M. Abdel-Basset, M. Mohamed, Y. Zhou, I. Hezam, (2017). Multi-criteria group decision making based on the neutrosophic analytic hierarchy process. Journal of Intelligent & Fuzzy Systems, 33(6): 4055-4066.

[58]M. Riaz, K. Naeem and M. O. Ahmad, (2017). Novel Concepts of Soft Sets with Applications, Annals of Fuzzy Mathematics and Informatics, 13(2), 239-251.

[59]M. Saqlain, N. Jafar, M. Rashid and A. Shahzad, (2019) Prediction of Cricket World Cup 2019 by TOPSIS Technique of MCDM-A Mathematical Analysis, International Journal of Scientific & Engineering Research 10(2), 789-792.

[60]M. Saqlain, M. N. Jafar, M. Riaz, (2020). A New Approach of Neutrosophic Soft Set with Generalized Fuzzy TOPSIS in Application of Smart Phone Selection, Neutrosophic Sets and Systems, 32, 307-316. https://doi.org/10.5281/zenodo.3723161

[61]Adriyendi, (2015). Multi-attribute decision making using simple additive weighting and weighted product in food choice. I.J. Information Engineering and Electronic Business, 6, 8–14.

[62]Calvo-Flores, Miguel & Verdegay, Jose & Vila, M.. (1992). Linguistic decision‐making models. International Journal of Intelligent Systems. 7. 479 - 492. 10.1002/int.4550070507.

[63]Afshari, A. R., & Mojahed, M. (2010). Simple additive weighting approach to personnel selection problem. International Journal of Innovation and Technology Management, 1(5), 511–515.

[64]Adriyendi, 2015). Multi-attribute decision making using simple additive weighting and weighted product in food choice. I.J. Information Engineering and Electronic Business, 6, 8–14.

[65]Lazim, A., & Rabiatul, A. C. W., 2014). Simple additive weighting methods of multi criteria decision making and applications: A decade review. International Journal of Information Processing and Management (IJIPM), 1(5), 39–49.

Theory and Application of Hypersoft Set 153

[66]C. Hwang, K. Yoon, (1981). Multiple Attribute Decision Making: Methods and Applications, A State-of-the-Art Survey, Springer

[67]Hwang, C.L.; Yoon, K., (1981). Multiple Attribute Decision Making: Methods and Applications. New York: Springer-Verlag

[68]M. Saqlain, N. Jafar, A. Riffat, (2018). Smart Phone Selection by Consumers in Pakistan: FMCGDM Fuzzy Multiple Criteria Group Decision Making Approach, Gomal University Journal of Research, 34(1), 27-31.

[69]Chourabi, Zouhour & Khedher, Faouzi & Dhouib, Amel & Cheikhrouhou, Morched. (2018). Multi-criteria decision making in workforce choice using AHP, WSM and WPM. The Journal of The Textile Institute. 110. 1-10. 10.1080/00405000.2018.1541434.

[70]Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. (2019). A group decision making framework based on neutrosophic TOPSIS approach for smart medical device selection. Journal of medical systems, 43(2), 38.

[71]Calvo-Flores, Miguel & Verdegay, Jose & Vila, M. (1992). Linguistic decision‐making models. International Journal of Intelligent Systems, 7, 479 - 492. 10.1002/int.4550070507.

[72]Şahin, R. (2014). Multi-criteria neutrosophic decision making method based on score and accuracy functions under neutrosophic environment. arXiv preprint arXiv:1412.5202

[73]F. Smarandache, (2018) Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set, Neutrosophic Sets and System, 22, 168-170.

[74]Saqlain M, Sana M, Jafar N, Saeed. M, Smarandache, F., (2020). Aggregate Operators of Neutrosophic Hypersoft Sets, Neutrosophic Sets and Systems (NSS), 32, 294–306.

[75]Saqlain M, Xin L X., (2020). Interval Valued, m-Polar and m-Polar Interval Valued Neutrosophic Hypersoft Sets, Neutrosophic Sets and Systems (NSS), 36, 389-399.

[76]Zulqarnain M R, Saqlain M, Xin L X, Smarandache F. (2020). Generalized aggregate operators on Neutrosophic Hypersoft set, Neutrosophic Sets and Systems (NSS), 36, 271281.

[77]Saqlain M, Saeed M, Ahmad M. R, Smarandache F (2019) Generalization of TOPSIS for Neutrosophic Hypersoft set using Accuracy Function and its Application, Neutrosophic Sets and Systems (NSS), 27, 131-137.

[78]Saqlain M, Sana M, Jafar N, Saeed. M, Said. B. (2020). Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure of Single valued Neutrosophic Hypersoft Sets, Neutrosophic Sets and Systems (NSS), 32, 317-329.

[79]James j. Brophy, (1990). Basic Electronics for Scientist, 5th Edition, McGraw-Hill Company, 99, 108, 119, 123.

[80]IEEE Standard Definitions of Terms for Radio Wave Propagation, in IEEE Std 211, 1-44. doi: 10.1109/IEEESTD.1997.8638365.

Theory and Application of Hypersoft Set 154

[81]Winarski, T.Y., (2001). Dielectrics in MOS devices, DRAM capacitors, and inter-metal isolation. Electrical Insulation Magazine, IEEE 17, 34 - 47.

[82]C. Kittel, (2005). Introduction to Solid State Physics. 8th Edition, John Wiley & Sons, Inc, 199.

[83]Yoshikawa, A. (2007). Development and Applications of Wide Bandgap Semiconductors. In Yoshikawa, A.; Matsunami, H.; Nanishi, Y. (eds.). Wide Bandgap Semiconductors. Springer. p. 2

Tangent, Cosine, and Ye Similarity Measures of m-Polar Neutrosophic Hypersoft Sets

Mahrukh Irfan1, Maryam Rani2, Muhammad Saqlain3 , Muhammad Saeed 1∗

1 University of Management and Technology, Township, Lahore, 54000, Pakistan. Email: f2018265015@umt.edu.pk

2 Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. Email: Maryum.rani009@gmail.com

3 School of Mathematics, Northwest University, Xian, China. Email: msgondal0@gmail.com

*Correspondence: Muhammad.saeed@umt.edu.pk

Abstract: In our daily lives, most of the problems are created due to wrong decisions. Similarity measures are very useful to make good decisions. In this manuscript, different similarities like Tangent, Ye, and Cosine similarity of m polar neutrosophic hypersoft sets and their comparison is presented. These techniques are very useful to deal with the problems whose attributes are numerous. In m-polar neutrosophic hypersoft set, attributes are large in number and further classified, that is why making effective and fruitful decision is cumbersome. Decision making become simple, easy and accurate through comparison of different proposed similarities. Finally, the proposed similarity measures applied to diagnose the Covid-19 will be extremely useful to prevent the spread of this virus.

Keywords: m-Polar Neutrosophic Hypersoft Set (mPNHSS), Similarity Measure, Tangent Similarity Measure, Ye Similarity, Cosine Similarity.

1. Introduction

A great researcher Lotfi A. Zadeh [4] proposed a revolutionary theory, known as fuzzy set theory in 1965 was based on membership values. In some problems it is difficult to allot the membership value, to overcome this problem Smarandache [10] extended this concept with the addition of indeterminacy and non-membership values along with the membership value, it is introduced as a neutrosophic set. Neutrosophic set is a scientific tool for handling all the issues which including indeterminacy, uncertain and conflicting information [5]. In this idea membership, indeterminacy, and non-membership values are independent. Theory of soft set was proposed by Molodtsov [2], it resolves the issue of uncertain conditions. He introduced the soft set by considering a universal set

andtheparameterizedfamilyofitssubsets.Eachelementofthesoftsetisconsideredasan approximateelement[7].Softsetsplayanimportantroleingamepredictionandbasicdecisionmakingproblems[11].FromMolodtsovtillnowitisusedinmanyapplicationsandproblems toconnectthedifferentfieldsofscienceanddataresearch[2,8,9,11].Maji[15]introduced theconceptoftheneutrosophicsoftsetbycombiningtheconceptofneutrosophicsetandsoft set,itisusedindecision-makingproblems[14]andforresearchpurpose[13].Todealwith uncertaintySmarandache[6]introducedtheconceptofthehypersoftsetbyconvertingsoft setfunctionvaluesintomulti-decisionfunctionvalues.InTOPSISandMCDMSmarandache gavebriefexplanationsofnumerousextensionsofneutrosophicsets[24–29].Saqlain[18] introducedneutrosophichypersoftsetconvertingitfromhypersoftsettohandletheproblems ofuncertainty.Theneutrosophichypersoftsetcanbeusedtopredictgames[20,21].Saeed andSaqlainintroducedtheconceptofmpolarneutrosophicsoftset[3]andusedinmedical diagnosisanddecisionmaking.weconvertthem-polarneutrosophicsoftsetintothem-polar neutrosophichypersoftsetstohandlealargenumberofattributesinmedicaldiagnosis.

Manymethodsareusedtomeasurethesimilaritybetweentwoneutrosophichypersoftsets. Theobjectofthispaperistouseseveralsimilaritymethodsform-polarneutrosophichypersoft sets.ThetangentsimilaritymeasureisintroducedbyPramanikandMondal[6],itsproperties andapplicationalsoexplained.J.YegavetheideaofYesimilaritymeasureandexplainits propertiesandapplications[17].SaidBroumiandFlorentinSmarandache[1]introducedcosine similaritywhichisbasedonBhattacharya’sdistance[12],itspropertiesalsobrieflydescribed. AnjanMukherjee,AbhikMukherjee[33]introducedsimilaritymeasuresofinterval-valued fuzzysoftsetfordeterminingCOVID-19patients.

1.1. Structureofthismanuscript

Theproposedworkiscatalogedasfollows,InSection1therelateddefinitionsofconcepts forunderstandingsoftset,neutrosophicset,neutrosophichypersoftset,m-polarneutrosophic softsetarepresented.Insection2weproposedthebasicdefinitionofmpolarneutrosophic hypersoftsetanditsaggregateoperatorslikeequal,null,subset,union,intersection.Insection 3comparisonamongtheseveralsimilaritymeasuresofmpolarneutrosophichypersoftsetand graphicalrepresentationoftheresultshavebeendonewiththehelpofanillustratedexample ofcovid-19.Insection4finally,conclusionandfuturedirectionsarepresented.

1.2. Motivation

SaqlainetalpresentedtheTangentsimilaritymeasureofsingle-valuedNHSS[23].Many piecesofresearchinvolvemulti-attributes,multi-agent,multi-object,multi-Polar,andmultiindexinformation.Totacklethesekindofsituations,wetheorizedseveralsimilaritymethods

form-Polarneutrosophichypersoftsets.ThisapproachwillbeextremelyusefulinMCDM, personalselection,codingtheory,managementproblems,andmedicaldiagnosis.Incomparisonwithsingle-valuedNHSS,m-PolarNHSSgivemoreaccurateresults.Wealsoappliedthis theoryonmedicaldiagnosistoshowitsimportanceinreallifeproblems.

2. Preliminaries

Inthissection,softset,hypersoftset,neutrosophicset,neutrosophichypersoftset,m-polar neutrosophicsoftset,m-polarneutrosophichypersoftsetarebrieflyexplained.

2.1. SoftSet

LetΥbeasetofparametersorattributesconcerntheuniversalsetK.P(K)representsthe powersetoftheuniversalsetK.TodefinesoftsetletB ⊆Υ.Here isamappingwhichis definedas

:B → P(K)

Thenapair( ,B)overKiscalledasoftset.Foranya∈B, (a)maybewrittenasasetof a-elementofthesoftset( ,B).Then( ,B)isdefinedas

( ,B)= (a)∈ P(K)ifa ∈ B (a)=φ if a/ ∈ B

2.2. HyperSoftSet

Thepair( , k1×k2×k3×...kn)issaidtobeahypersoftsetoverKwhere isdefinedas : k1×k2×k3×...kn → P(K)

HereKistheuniversalsetandP(K)isitspowerset.Consider κ1 , κ2 , κ3 κn forn≥1 isthenwell-definedattributes,whosecorrespondingattributevaluesare k1,k2,k3 , ... kn respectivelywiththecondition ki∩kj =φ fori=jandi∈{1,2,3,...n},j ∈{1,2,3,...n}

2.3. NeutrosophicSet

LetKbeauniversalsetandtheneutrosophicsetBisanobjecthavingtheform

B={ x:TB (x),IB (x),FB (x) ,x∈K}

WherethefunctionT,I,F:K →]-0,1+[definerespectivelythetruthmembershipfunction, indeterminacyfunction,andfalsitymembershipfunction.Asetisaneutrosophicsetwiththe condition

0 ≤ TB (x)+IB (x)+FB (x)≤3+

Toselecttheintervalfortheneutrosophicsetfromthephilosophicalpointofviewthe neutrosophicsettakesthevaluefromrealstandardornon-standardsubsetsof]0 ,1+[itis difficulttoapplyinrealapplicationsfortechnicalapplicationswetakeinterval[0,1][1].

2.4. NeutrosophicHypersoftset

LetKbetheuniversalsetandP(K)bethepowersetofK.Considernwell-definedattributes κ1 , κ2 , κ3 ... κn forn≥1whosecorrespondingattributevaluesarerespectively k1,k2,k3 ... kn withthecondition ki ∩ kj =φ fori=jandi∈{1,2,3,...n} j∈{1,2,3,...n} considertheirrelation k1 × k2 × k3× kn=Ψ defineamapping λ:Ψ → P(K) λ(Ψ)={ x,T(λ(Ψ)),I(λ(Ψ)),F(λ(Ψ)) ,x∈K} Thenthepair(λ,Ψ)issaidtobeaneutrosophichypersoftsetoveruniversalsetK.Where T(λ(Ψ))istruthmembershipfunction,I(λ(Ψ))istheindeterminacyfunctionandF(λ(Ψ))is thefalsitymembershipfunctionsuchthat

T(λ(Ψ)),I(λ(Ψ)),F(λ(Ψ)) → [0,1] withthecondition

0≤T(λ(Ψ))+I(λ(Ψ))+F(λ(Ψ))≤3

2.5.

m-PolarNeutrosophicsoftset

Anm-polarneutrosophicsoftsetisdefinedasifKisauniversalsetandP(K)isitspower setΥbethesetofattributesconcerningtoK.Bissaidtobem-polarneutrosophicsoftset overKifB⊆Υ.

Itsmappingisdefinedas

λ:B → P(K)

(λ,B)={ Ti λ(B),Ii λ(B),Fi λ(B) |k,k∈K}∀ i=1,2,3,...n

HereTiλ(B)denotesthedegreeofith truthmembershipfunction,Iiλ(B)denotesthedegree ofith indeterminacyfunctionandFiλ(B)denotesthedegreeofith falsitymembershipfunction foreachelementofk∈KtothesetBandholdthecondition

0≤ Tiλ(B)+Iiλ(B)+Fiλ(B)≤3 ∀ i=1,2,3,...n

3. m-PolarNeutrosophicHypersoftSet

LetK={k1,k2,...kn} betheuniversalsetandP(K)bethepowersetofK.Consider κ1, κ2,...κm form≥1bemwell-definedattributeswhosecorrespondingattribute valuesarerespectivelyΥ1 1,Υ2 2,...Υm n andtheirrelationisΥ1 1×Υ2 2×...Υm n thenthe pair(λ,Υ1 1×Υ2 2×...Υm n)issaidtobem-polarneutrosophichypersoftsetoverK.Let Υ1 1×Υ2 2×...Υm n=Ω where λ isdefinedas λ:Ω → P(K) λ(Ω)={ TΥ i(k),IΥ i(k),FΥ i(k) k∈K,Υ∈ Ω} i=1,2,...,n also 0≤ i=1 nTΥ i(k)≤10≤ i=1 nIΥ i(k)≤10≤ i=1 nFΥ i(k)≤1 where TΥ i(k)⊆[0,1]IΥ i(k)⊆[0,1]FΥ i(k)⊆[0,1] andholdsthecondition 0≤ i=1 nTΥ i(k)+ i=1 nIΥ i(k)+ i=1 nFΥ i(k)≤3

3.1. Example

LetKbethesetofdifferentteachersthosearenominatedforthebestteacheroftheyear. consider K{T1,T2,T3,T4,T5} Assumptions: • Everyteacherhasthesameprobabilitytoselect. • Independentattributesareconsidered. • HesitantEnvironmentisnotyetconsidered. Problemformulation: Assumethesetofattributesas

λ(masters,yes,good,uptodate)={ T2,(Q1 a[0.9,0.4,0.1],Q2 b[0.7,0.3,0.5],Q3 c[0.4,0.9, 0.7],Q4 d[0.6,0.3,0.5] , T4 (Q1 a[0.5,0.3,0.7],Q2 b[0.5,0.1,0.3],Q3 c[0.7,0.5,0.8],Q4 d[0.4, 0.6,0.3] }

Thenm-PolarNeutrosophicHypersoftSetofaboverelationis

λ1(masters,yes,good,uptodate)={ T 2 Q1 a[0.01,0.003,0.1,0.023,0.07],[0.092,0.073, 0.08,0.2,0.4],[0.2,0.17,0.06,0.13,0.3] , Q2 b[0.2,0.1,0.5,0.019,0.051],[0.21,0.14,0.27, 0.009,0.1],[0.113,0.35,0.25,0.12,0.03] , Q3 c[0.12,0.13,0.14,0.15,0.39],[0.17,0.20,0.24, 0.15,0.1],[0.2,0.1,0.5,0.019,0.051] , Q4 d[0.2,0.17,0.06,0.13,0.3],[0.12,0.025,0.07,0.22, 0.074],[0.01,0.003,0.1,0.023,0.07] }

λ2(masters,yes,good,uptodate)={T4 Q1 a[0.09,0.08,0.7,0.026,0.05],[0.04,0.03,0.02, 0.1,0.09],[0.32,0.51,0.06,0.03,0.02] , Q2 b[0.12,0.13,0.14,0.15,0.39],[0.17,0.20,0.041, 0.05,0.1],[0.2,0.1,0.5,0.019,0.051] , Q3 c[0.09,0.08,0.7,0.026,0.05],[0.04,0.03,0.02,0.1, 0.09],[0.02,0.51,0.06,0.03,0.12] , Q4 d[0.12,0.13,0.14,0.15,0.39],[0.12,0.025,0.07,0.22, 0.074],[0.12,0.025,0.07,0.22,0.4] }

3.2. AggregateOperatorsofm-PolarNeutrosophicHypersoftSet

Inthismanuscript,aggregateoperatorsofmpolarneutrosophichypersoftsetliketheequal set,nullset,subset,union,andintersectionareexplained.Thevalidityandrestrictionsof proposedoperatorsarebrieflydiscussed.WiththeHelpofaggregateoperators,anysetcan beusedtoobtaineddesiredresultstheseoperatorsaretoolsthatarebeneficialtousedifferent setsinproblems.

3.3. m-PolarNeutrosophicEqualHypersoftSet

ifitsatisfiedthefollowingconditions

TΥ(A) i(k)=TΥ(B) i(k)IΥ(A) i(k)=IΥ(B) i(k)FΥ(A) i(k)=FΥ(B) i(k)

3.4. m-PolarNeutrosophicNullHypersoftSet

letK={k1,k2,...kn} betheuniversalsetandP(K)bethepowersetofK.Consider κ1, κ2, ...κm form≥1bemwell-definedattributeswhosecorrespondingattributevaluesarerespectivelyΥ1 1,Υ2 2,...Υm n andtheirrelationareΥ1 1×Υ2 2×...Υm n

LetΥ1 1×Υ2 2×...Υm n=Ω

A(Ω)={ TΥ(A) i(k),IΥ(A) i(k),FΥ(A) i(k) k∈K,Υ∈ Ω } issaidtobeanm-polarneutrosophicNullhypersoftsetoverK.

ifitsatisfiedthefollowingconditions

TΥ(A) i(k)=0IΥ(A) i(k)=0FΥ(A) i(k)=0

3.5. m-PolarNeutrosophicSubsetHypersoftSet

letK={k1,k2,...kn} betheuniversalsetandP(K)bethepowersetofK.Consider κ1, κ2, κm form≥1bemwell-definedattributeswhosecorrespondingattributevaluesarerespectivelyΥ1 1,Υ2 2,...Υm n andtheirrelationareΥ1 1×Υ2 2×...Υm n . LetΥ1 1×Υ2 2×...Υm n=Ω

A(Ω)={ TΥ(A) i(k),IΥ(A) i(k),FΥ(A) i(k) k∈K,Υ∈ Ω} B(Ω)={ TΥ(B) i(k),IΥ(B) i(k),FΥ(B) i(k) k∈K,Υ∈ Ω } i=1,2,...,n aretwom-polarneutrosophichypersoftsetsoverK.ThenA(Υ1 1×Υ2 2×...Υm n)isthem polarneutrosophicsubsethypersoftset. ifitsatisfiedthefollowingconditions TΥ(A) i(k)≤TΥ(B) i(k)IΥ(A) i(k)≥IΥ(B) i(k)FΥ(A) i(k)≥FΥ(B) i(k)

3.6. UnionofTwom-PolarNeutrosophicHypersoftSets

letK={k1,k2,...kn} betheuniversalsetandP(K)bethepowersetofK.Consider κ1, κ2, κm form≥1bemwell-definedattributeswhosecorrespondingattributevaluesarerespectivelyΥ1 1,Υ2 2,...Υm n andtheirrelationareΥ1 1×Υ2 2×...Υm n

LetΥ1 1×Υ2 2×...Υm n=Ω

A(Ω)={ TΥ(A) i(k),IΥ(A) i(k),FΥ(A) i(k) k∈K,Υ∈ Ω} B(Ω)={ TΥ(B) i(k),IΥ(B) i(k),FΥ(B) i(k) k∈K,Υ∈ Ω } i=1,2,...,n aretwom-polarneutrosophichypersoftsetsoverK.

ThenA(Ω) ∪ B(Ω)isgivenas

 

 



 A(Ω) ifx ∈ A B(Ω) ifx ∈ B { (I(A(Ω))+I(B(Ω))) 2 } ifx ∈ A ∪ B)

  

F (A(Ω) ∪ B(Ω))=   

 A(Ω) ifx ∈ A B(Ω) ifx ∈ B min(F (A(Ω)),T (B(Ω))) ifx ∈ A ∪ B)

3.7. TheIntersectionofTwom-PolarNeutrosophicHypersoftSets

       



       



       

4. ComparisonamongTheSeveralSimilarityMeasureofm-PolarNeutrosophic Hypersoftsets

Intwoobjectssimilaritymeasureisanumericalvalueofthedegreeaccordingtotheobjects aresameornearlyequaltoeachother.Usually,thevaluesofsimilaritiesarepositiveand mostlyliesbetween0and1.Here0meansthereisnosimilarityand1meanscompletely similartoeachother,anysimilaritywhoanswernearto1ismoreaccurateandbeneficial forresearchproposes.Asimilaritymeasureisveryimportantindecision-makingproblems throughcalculatingsimilaritymeasuresamongtheidealandgivenoptions.LetAandBbe twom-polarneutrosophichypersoftsetintheuniversalsetK={ k1,k2,...,kn}.Explain severalsimilaritiestosolvem-polarneutrosophichypersoftset.

4.1. TangentSimilarityMeasureformPolarneutrosophicHypersoftSets

LetK={k1,k2,k3,...,kn} beauniversalsetP(K)beitspowerset.Considertwom-polar neutrosophichypersoftset

A={ Ta i(k),Ia i(k),Fa i(k) k∈K,a∈(a1 1×a2 2×...am l)}

B={ Tb i(k),Ib i(k),Fb i(k) k∈K,b∈(b1 1×b2 2×...bn l)}

Tomeasurethetangentsimilarityforthesetwom-polarneutrosophichypersoftsetusethe giventangentsimilarity

T(A,B)={ 1 n n i=1[1 tan π 12 (| T i a(k) T i b (k) | + | I i a(k) I i b(k) | + | F i a(k) F i b (k) |} here k∈K,a∈(a1 1×a2 2×...am l,b ∈ (b1 1× b2 2×...bn l)

0≤ i=1 nTA,B i(k)≤10≤ i=1 nIA,B i(k)≤10≤ i=1 nFA,B i(k)≤1 where TΥ i(k)⊆[0,1]IΥ i(k)⊆[0,1]FΥ i(k)⊆[0,1]

andholdstheconditions

0≤ i=1 nTa,b i(k)+ i=1 nIa,b i(k)+ i=1 nFa,b i(k)≤3

4.2. Proposition

Tangentsimilaritymeasurebetweentwom-polarneutrosophichypersoftsetssatisfiesthe followingproperties

(1) 0≤ TmPNHSS (A,B)≤1

(2) TmPNHSS (A,B)=1ifandonlyifA=B (3) TmPNHSS (A,B)=TmPNHSS (B,A)

(4) ifCisamPNHSSandA⊂B⊂CthenTmPNHSS (A,C) ≤ TmPNHSS (A,B)and TmPNHSS (A,C) ≤ TmPNHSS (B,C)

4.3. Proofs

(1).0≤ TmPNHSS (A,B)≤1

Asthetruthmembershipfunction,indeterminacyfunction,andfalsitymembershipfunction ofTmPNHSS liesbetween0and1,becausethevaluesofthetangentfunctionarewithin[0,1] andthesimilaritymeasurewhichisbasedontangentalsolieswithin[0,1].Henceproved 0≤ Tm PNHSS (A,B)≤1

(2)Tm PNHSS (A,B)=1ifandonlyifA=B Foranytwom-PNHSSAandB,letAandBareequalm-PNHSSasgiveninstatement(means A=B)thisimplies

TA(x)=TB (x)IA(x)=IB (x)FA(x)=FB (x) Hencetheredifferencealsoequaltozeromeans

| TA(x)-TB (x)|=0 | IA(x)-IB (x)|=0 | FA(x)-FB (x)|=0 afterputtingthesevaluesintangentsimilarityformula,sincetan(0)=0sowegettheanswer is1.HenceifA=BthenTmPNHSS (A,B)=1 Conversely

IfTmPNHSS (A,B)=1thenitisobviousthat

| TA(x)-TB (x)|=0 | IA(x)-IB (x)|=0 | FA(x)-FB (x)|=0 theseimplythat

TA(x)=TB (x)IA(x)=IB (x)FA(x)=FB (x) bydefinitionofequalmPNHssA=B

(3)TmPNHSS (A,B)=TmPNHSS (B,A) TocalculatethevalueofTmPNHSS (A,B)firstlyfind

| TA(x)-TB (x)|| IA(x)-IB (x)|| FA(x)-FB (x)| Asweknowthepropertyofmod

|-x|=x byusingthispropertyitisobviousthat

| TA(x)-TB (x)|=| TB (x)-TA(x)|| IA(x)-IB (x)|=| IB (x)-IA(x)| | FA(x)-FB (x)|=| FB (x)-FA(x)|

SothevaluesofTm PNHSS (A,B)=Tm PNHSS (A,B)

(4)ifCisam-PNHSSandA⊂B⊂CthenTm PNHSS (A,C) ≤ Tm PNHSS (A,B)and Tm PNHSS (A,C) ≤ Tm PNHSS (B,C)

IfA⊂B⊂Cthenbydefinitionifsubsetm-PNHSSTA(x)≤ TB (x) ≤ TC (x),IA(x) ≥ IB (x) ≥ IC (x),FA(x) ≥ FB (x) ≥ FC (x)forx∈X byusingdefinitionwegettheinequalities: | TA(x)-TB (x)|≤| TA(x)-TC (x)|| TB (x)-TC (x)|≤| TA(x)-TC (x)| | IA(x)-IB (x)|≤| IA(x)-IC (x)|| IB (x)-IC (x)|≤| IA(x)-IC (x)| | FA(x)-FB (x)|≤| FA(x)-FC (x)|| FB (x)-FC (x)|≤| FA(x)-FC (x)| Thusweconcludethat

Tm PNHSS (A,C) ≤ Tm PNHSS (A,B)andTm PNHSS (A,C) ≤ Tm PNHSS (B,C),because thetangentfunctionisincreasingintheinterval[0, π 4 ]

4.4.

Ye’ssimilarity

Ye’ssimilarityisdefinedas Sye(A,B)=1- 1 6 i=1 nwi[| TAj (ki)-TB j (ki)| +| IAj (ki)-IB j (ki)|+|FAj (k i )+FB j (k i )| where TA,B i(k)⊆[0,1]IA,B i(k)⊆[0,1]FA,B i(k)⊆[0,1] andholdsthecondition 0≤ i=1 nTA,B i(k)+ i=1 nIA,B i(k)+ i=1 nFA,B i(k)≤3

4.5.

Proposition

Yesimilaritymeasurebetweentwom-polarneutrosophichypersoftsetsatisfiesthefollowing properties

(1) 0≤ SyemPNHSS (A,B)≤1 (2) SyemPNHSS (A,B)=1ifandonlyifA=B (3) SyemPNHSS (A,B)=SyemPNHSS (B,A)

4.6. Proofs

(1).0≤ SyemPNHSS (A,B)≤1

Asthetruthmembershipfunction,indeterminacyfunction,andfalsitymembershipfunction ofSyemPNHSS liebetween0and1,thesimilaritymeasurewhichisbasedontheabovefunctions alsolieswithin[0,1].Henceproved

0≤ SyemPNHSS (A,B)≤1 (2).SyemPNHSS (A,B)=1ifandonlyifA=B

LetA=Bthenbydefinitionofequalmpolarneutrosophichypersoftset

TA(x)=TB (x)IA(x)=IB (x)FA(x)=FB (x) then SyemPNHSS (A,B)=1- 1 6 i=1 nwi[| TAj (ki)-TB j (ki)| +| IAj (ki)-IB j (ki)|+|FAj (k i )+FB j (k i )| SyemPNHSS (A,B)=1-0 SyemPNHSS (A,B)=1 Conversely, LetSyemPNHSS (A,B)=1thenonlyonepossibilityis TA(x)-TB (x)=0IA(x)-IB (x)=0FA(x)-FB (x)=0 thisimpliesthat TA(x)=TB (x)IA(x)=IB (x)FA(x)=FB (x) so,A=B (3)SyemPNHSS (A,B)=SyemPNHSS (B,A)

SyemPNHSS (A,B)=1- 1 6 i=1 nwi[| TAj (ki)-TB j (ki)| +| IAj (ki)-IB j (ki)|+|FAj (k i )+FB j (k i )| itcanbewriteas

SyemPNHSS (A,B)=1- 1 6 i=1 nwi[| TB j (ki)-TAj (ki)| +| IB j (ki)-IAj (ki)|+|FB j (k i )+FAj (k i )| SyemPNHSS (A,B)=SyemPNHSS (B,A)

4.7. CosineSimilarity

Candanandsapino[33]giveustheideaofcosinesimilaritywhichisafundamentalangle basedsimilarity.Ithelpstocalculatethesimilaritybetweentwon-dimensionalvectorsby usingthecosineoftheanglebetweenthem.ConsidertwovectorsA,B.Theirattributesare

A=((a1,a2,...,an)andB=((b1,b2,...,bn), θ istheanglebetweenAandB,similaritycanbe calculatedbyusingdotproductandmagnitudesofvectorsAandB cosθ = { n i=1 aibi √ n i=1 a2 i √ n i=1 b2 i } nowconsiderifthereisanm-polarneutrosophichypersoftsetsAandBinK={ k1,k2,..., kn } thenthecosinesimilaritybetweenAandBcanbecalculatedbythefollowingformula C(A,B)= 1 n n i=1 { ∆TA(ki)∆TB (ki)+∆IA(ki)∆IB (ki)+∆FA(ki)∆FB (ki) √(∆TA(ki))2+(∆IA(ki))2+(∆FA(ki))2√(∆TB (ki))2+(∆IB (ki))2+(∆FB (ki))2 } where ∆TA(ki)= j=1 nTΥj (k) ∆IA(ki)= j=1 nIΥj (k) ∆FA(ki)= j=1 nFΥj (k) andholdsthecondition 0≤ j=1 nTΥj (k)+ j=1 nIΥj (k)+ j=1 nFΥj (k)≤3

4.8. Proposition

Cosinesimilaritymeasurebetweentwom-polarneutrosophichypersoftsetssatisfiesthe followingproperties

(1) 0≤ Cm PNHSS (A,B)≤1 (2) Cm PNHSS (A,B)=1ifandonlyifA=B (3) Cm PNHSS (A,B)=Cm PNHSS (B,A)

4.9. Proofs

(1).0≤ Cm PNHSS (A,B)≤1

Asthetruthmembershipfunction,indeterminacyfunction,andfalsitymembershipfunction ofSyem PNHSS liebetween0and1,valuesofcosinefunctionalsoliesbetween0and1,the similaritymeasurewhichisbasedoncosinealsolieswithin[0,1].Henceproved 0≤ Cm PNHSS (A,B)≤1 (2).Cm PNHSS (A,B)=1ifandonlyifA=B

LetA=Bthen

TA(x)=TB (x)IA(x)=IB (x)FA(x)=FB (x) so,

Cm PNHSS (A,B)= 1 n

n i=1 { ∆TA(ki)∆TB (ki)+∆IA(ki)∆IB (ki)+∆FA(ki)∆FB (ki) (∆TA(ki))2 +(∆IA(ki))2 +(∆FA(ki))2 (∆TB (ki))2 +(∆IB (ki))2 +(∆FB (ki))2 }

afterusingthedefinitionofequalm-polarneutrosophichypersoftsetitbecame

Cm PNHSS (A,B)=1 Conversely, LetCm PNHSS (A,B)=1thenthereistheonlypossibility

TA(x)=TB (x)IA(x)=IB (x)FA(x)=FB (x) HenceA=B (3).Cm PNHSS (A,B)=Cm PNHSS (B,A)

Cm PNHSS (A,B)= 1 n

n i=1 { ∆TA(ki)∆TB (ki)+∆IA(ki)∆IB (ki)+∆FA(ki)∆FB (ki) (∆TA(ki))2 +(∆IA(ki))2 +(∆FA(ki))2 (∆TB (ki))2 +(∆IB (ki))2 +(∆FB (ki))2 } Itcanbewrittenas Cm PNHSS (A,B)= 1 n

n i=1 { ∆TB (ki)∆TA(ki)+∆IB (ki)∆IA(ki)+∆FB (ki)∆FA(ki) (∆TB (ki))2 +(∆IB (ki))2 +(∆FB (ki))2 (∆TA(ki))2 +(∆IA(ki))2 +(∆FA(ki))2 } Cm PNHSS (A,B)=Cm PNHSS (B,A)

4.10. ApplicationofTangentSimilarityMeasureform-PolrNeutrosophicHypersoftsetto DiagnosisCovid-19

Inmedicaldiagnosis,decisionmakingbecameverymuchcomplicatedbecausemedicaltest aretooexpansiveandtimetaking.Sometimeconditionofpatientisnotstable,thedoctor didnothaveenoughtimetowaitfortheresultofamedicaltest,theyshouldtakeurgent stepstosavethepatient’slife.Insuchconditionsthesekindofstudieshelpthemtotake effectivedecisionstosavepatient’slife.Inthedaysofcovid-19patientsarelargeinnumber andthemedicaltestisverymuchexpensiveandtime-taking.Somepatientshadfinancial problemsandtheconditionofsomepatientsiscritical.Tohandlethatpandemicsituation, wedemonstratetheapplicationoftheproposedtangentsimilaritymeasureforthem-polar neutrosophichypersoftsettomedicaldiagnosis.wediscussthecovid-19diagnosisasfollow: Forexample,thepatientsofcovid-19reportedthesymptomslike serioussymptoms=SS=difficultyinbreathing,chestpain,lossofspeech

mostcommonsymptoms=MCS=fever,drycough,tiredness lesscommonsymptoms=LCS=pain,sorethroat,headache weconstructamodel(patient)whoissufferingfromcovid-19.Notedownthesymptomsof covid-19accordingtothelistSS,MCS,LCSandfigureouttheirscalefordiagnosistakeunder observationandfigureoutthempolarneutrosophichypersoftset

PM ={SS (0.087,0.68,0.073,0.060),(0.087,0.073,0.79,0.05),(0.68,0.084,0.079,0.16) , MCS (0.53,0.067,0.071,0.068),(0.43,0.087,0.32,0.064),(0.411,0.080,0.076,0.4) , LCS (0.03,0.51,0.063,0.041),(0.063,0.072,0.54,0.010),(0.083,0.079,0.076,0.71) }

Fivepatientsaresufferingfromfeverandalsohavingsomesymptomsduetothesesymptoms theythoughttheyaresufferingfromcovid-19.Todiagnosisthatthosepatientsaresuffering fromcovid-19ornot.Forthispurposeconstructthempolarneutrosophichypersoftsetfor everypatient.Like

FirstPatient

P1={SS (0.078,0.77,0.068,0.05),(0.072,0.086,0.70,0.06),(0.58,0.085,0.075,0.13) , MCS (0.43,0.087,0.061,0.059),(0.53,0.057,0.12,0.034),(0.011,0.07,0.46,0.3) ,LCS (0.04, 0.61,0.053,0.051),(0.073,0.52,0.31,0.02),(0.073,0.069,0.080,0.72) }

SecondPatient

P2={SS (0.002,0.01,0.02,0.03),(0.032,0.056,0.02,0.05),(0.042,0.051,0.035,0.03) , MCS (0.04,0.05,0.041,0.029),(0.053,0.062,0.09,0.11),(0.001,0.04,0.16,0.05) ,LCS (0.063, 0.049,0.023,0.03),(0.053,0.42,0.21,0.01),(0.063,0.054,0.073,0.74) }

ThirdPatient

P3={SS (0.001,0.003,0.0012,0.004),(0.021,0.014,0.053,0.061),(0.012,0.003,0.0012, 0.021) ,MCS (0.002,0.013,0.053,0.060),(0.0017,0.023,0.04,0.0029),(0.0017,0.023,0.04, 0.0029) ,LCS (0.009,0.0014,0.0049,0.0021),(0.007,0.004,0.013,0.0029),(0.002,0.004, 0.0012,0.0014) }

ForthPatient

P4={SS (0.0011,0.0012,0.0002,0.0010),(0.0021,0.0041,0.0035,0.061),(0.0012,0.0001, 0.0021,0.0022) ,MCS (0.004,001,0.0051,0.006),(0.0017,0.0023,0.004,0.0019),(0.003, 0.00041,0.0003,0.005) ,LCS (0.0009,0.0013,0.0009,0.0012),(0.0005,0.0014,0.0002, 0.00091),(0.0003,0.0041,0.0012,0.0001) }

FifthPatient

P5={SS (0.077,0.66,0.073,0.060),(0.082,0.070,0.69,0.05),(0.66,0.083,0.070,0.11) , MCS (0.51,0.077,0.061,0.069),(0.53,0.077,0.22,0.074),(0.41,0.81,0.074,0.3) ,LCS (0.032, 0.49,0.061,0.042),(0.073,0.051,0.31,0.05),(0.083,0.078,0.075,0.70) }

Ouraimistodiagnosiswhichpatientissufferingfromcovid-19,forthispurposeuseseveral similaritiestoanalyzethisproblem.

Fromthetangentsimilarityformula,wecomputethesimilaritybetweenPi (i=1,2,3,4,5) andPM asfollows:

Tm PNHSS (P1,PM )=0.9148

Tm PNHSS (P2,PM )=0.6182

Tm PNHSS (P3,PM )=0.2499

Tm PNHSS (P4,PM )=0.2030

Tm PNHSS (P5,PM )=0.9077

Afterthatcalculationwediagnosisthatthefirstpatientissufferingfromcovid-19,andhis conditionissevere.Thefifthpatientisalsosufferingfromcovid-19.ThesecondPatientalso havingsomesymptomsofcovid-19heneedstoadoptthepreventingmeasures.Thirdand FourthPatientsaresufferingfromseasonaltemperature.

Fromtheyesimilarityformula,wecomputethesimilaritybetweenPi (i=1,2,3,4,5)and PM asfollows:

SyemPNHSS (P1,PM )=0.8377

SyemPNHSS (P2,PM )=0.3195

SyemPNHSS (P3,PM )=0

SyemPNHSS (P4,PM )=0

SyemPNHSS (P5,PM )=0.8248

Table1. SeveralSimilarityMeasures

PatientTangent Similarity YeSimilarity Cosine Similarity

P1 0.91480.83770.9961 P2 0.61820.31950.9599 P3 0.248900.7727 P4 0.20300,7342 P5 0.90770.82480.9864

Afterthatcalculation,wediagnosisthatFirstandFifthpatientissufferingfromcovid-19. ThesecondPatientalsohavingsomesymptomsodcovid-19heneedstoadopttheprevention measures.ThisSimilarityisnotusefultothediagnosisofcovid-19fortheThirdandFourth.

Fromthecosinesimilarityformula,wecomputethesimilaritybetweenPi (i=1,2,3,4,5)and PM asfollows:

CmPNHSS (P1,PM )=0.9961

CmPNHSS (P2,PM )=0.9599

CmPNHSS (P3,PM )=0.7727

CmPNHSS (P4,PM )=0.7342

CmPNHSS (P5,PM )=0.9864

Afterthatcalculation,wediagnosisthatthefirstpatientissufferingfromcovid-19,andhis conditionissevere.Thefifthandsecondpatientisalsosufferingfromcovid-19.Thethird andfourthpatientsalsohavingsomesymptomsodcovid-19heneedstoadopttheprevention measures.

4.11. TabularandGraphicalRepresentation

Acomparisonamongseveralsimilaritiesalsoexpressedthroughtabularandgraphicalrepresentationforquickanalysis.

5. Conclusion

SimilarityconceptisextremelybeneficialinMCDM,managementproblems,personalselection,codingtheory,medicaldiagnosis,andfeatureextraction,etc.Theaimofthispaperisto establishTangent,Cosine,Yesimilaritymeasuresofmulti-polarneutrosophichypersoftsets. Thisextensioncanbeappliedimmenselyindecision-makingproblems,managementproblems, timeseries,forecasting,andsupplychain,etc.Coronavirusdisease2019(Covid-19)isaninfectiousdiseasecausedofongoingpandemic.Intheend,weappliedthistechniqueinmedical diagnosisofcovid-19toshowthepracticalapplicationofthisidea.Infuture,wewouldlike togeneralizeentropymeasure,TOPSIS,VIKOR,etcofm-PolarandBipolarneutrosophic hypersoftsets.Theseconceptswillbeextremelyusefulforsolvingtheproblemshavinglarge numberofattributes.

References

[1] BroumiS.andSmarandacheF.,(2014)CosineSimilarityMeasureofInterval-valuedNeutrosophicSets, MathematicsandInformatics,5,15-20.

[2] MolodtsovD.,(1999)Softsettheory-firstresults, ComputersandMathematicswithApplications ,37(4/5), 19-31.

[3] SaeedM.,SaqlainM.,MahmoodA.,NaseerK.,andYaqoobS.,(2020)Multi-PolarNeutrosophicSoftSet withApplicationinMedicalDiagnosisandDecision-Making, NeutrosophicSetsandSystems(NSS),33, 183-207.

[4] ZadehL.A.,(1965)FuzzySets, Informationandcontrol,8,338-353.

[5] SaqlainM.,MoinS.,Naveed-JaferM.,SaeedM.andSmarandacheF.,(2020)AggregateOperatorsof NeutrosophicHypersoftSet, NeutrosophicSetsandSystems(NSS),32,294-306.

[6] SmarandacheF.,(2018)ExtensionofSoftSettoHypersoftSet,andthentoPlithogenicHypersoftSet, Neutrosophicsetsandsystem,22,168-170.

[7] Onyeozili,I.A.,Gwary,T.M.,(2014)AStudyoftheFundamentalsofSoftSetTheory, International JournalofScientificandTechnologyResearch,3,132-143.

[8] CagmanN.,EnginogluS.,(2010)SoftMatrixTheoryanditsDecisionMaking, ComputersandMathematics withApplications ,59,3308-3314.

[9] Tripathy,B.K.,ArumK.R.,(2015)ANewApproachtoSoftSet,SoftMultisetsandtheirProperties, InternationalJournalofReasoning-basedIntelligentSystems,7(3/4),244-253.

[10] SmarandacheF.,(1995)NeutrosophicSetAgeneralizationoftheIntuitionisticFuzzySet, Neutrosophic ProbebSet,LogicAmer.Res,1-15.

[11] MajiP.K.,RoyA.R.,BiswasR.,(2002)AnApplicationsofSoftSetsinaDecisionMakingProblem, ComputerandMathematicswithApplications,44,1077-1083.

[12] BhattacharyaA.,(1946)OnaMeasureofDivergenceoftwoMultinomialPopulation,SanakhayaSeaA, TheIndianJournalofStatistics,7,401-406.

[13] DeliI.,BroumiS.,andAliM.,(2016)NeutrosophicSoftMultisetTheory, Ital.j.Inf.Tech.Decis.Mak 15,5,1181-1210.

[14] DeliI.,(2017)Interval-valuedNeutrosophicSoftSetanditsDecisionMaking, InternationalJournalof MachineLearningandCybernetics,8,665-676.

[15] MajiP.K.,(2013)NeutrosophicSoftSet, AnnalsofFuzzyMathematicsandInformation,5(1),157-168.

[16] MondalK.andPramanikS.,(2015)IntuitionisticFuzzySimilarityMeasurebasedonTangentFunction anditsApplicationtoMulti-AttributesDecision, GlobalJournalofAdvancedResearch,2,464-471.

[17] YeJ.,(2014)SimilarityMeasurebetweenIntervalNeutrosophicSetsandtheirMulticriteriaDecisionMakingMethod, JournalofIntelligentandFuzzySystems,DOI:10.3233/IFS-120724,26,165-172.

[18] SaqlainM.,SaeedM.,AhmadM.R.,SmarandacheF.,(2019)GeneralizationofTOPSISforNeutrosophic HypersoftSetusingAccuracyFunctionanditsApplication, NeutrosophicSetsandSystems(NSS),27, 131-137.

[19] SaqlainM.,JafarN.,MoinS.,SaeedM.,andBroumiS.,(2020)SingleandMulti-valuedNeutrosophic HypersoftSetandTangentSimilarityMeasureofSinglevaluedNeutrosophicHypersoftSet, Neutrosophic SetsandSystems(NSS),32,317-329.

[20] RiazM.,SaqlainM.,andSaeedM.,(2019)ApplicationofGeneralizedFuzzyTOPSISinDecisionMakingforNeutrosophicSoftSettoPredicttheChampionofFIFA2018, AMathematicalAnalysis,Punjab UniversityJournalofMathematics,51(8),111-126.

[21] SaqlainM.,JaffarN.,HamidR.,ShahzadA.,(2019)PredictionofCricketWorldCup2019byTOPSISTechniqueofMCDM,AMathematicalAnalysis, InternationalJournalofScientificandEngineering Research,10(2),789-792.

[22] SaqlainM.,(2020)IntervalValued,mPolarandmPolarIntervalValuedNeutrosophicHypersoftSets, Neutrosophicsetsandsystem(NSS),36,389-399.

[23] SaeedM.,SaqlainM.,andMehmoodA.,(2020)ApplicationofSimilarityMeasureonm-PolarIntervalValuedNeutrosophicSetinDecisionMakinginSports, Neutrosophicsetsandsystem(NSS),38,317-334.

[24] Abdul-BassetM.,SalahM.,GamalA.,andSmarandacheF.,(2019)AnApproachoftheTOPSISTechnique forDevelopingSupplierSelectionwithGroupDecisionMakingundertype-2Neutrosophicnumber, Applied SoftComputingJournal,77,438-452.

[25] Abdul-BassetM.,Chang,V.,GanalA.,andSmarandacheF.,(2019)AnIntegratedNeutrosophicand VIKORMethodforachievingSustainableSupplierSelection,AcaseStudyinImportantField, Computers inIndustryJournal,106,94-110.

[26] Abdul-BassetM.,ManogaramG.,GamalA.,andSmarandacheF.,(2019)AGroupDecision-Making FrameWorkbasedontheNeutrosophicTOPSISApproachforSmartMedicalDeviceSelection, Journal ofMedicalSystems,43(2),38.

[27] NabeehN.A.,SmarandacheF.,Abdul-BassetM.,El-GhareebH.A.,andAboelfetouchA.,(2019)An IntegratedNeutrosophicTOPSISApproachanditsApplicationtoPersonnelSelection:ANewTrendin BrainProcessingandAnalysis, IEEEAccess,7,29734-29744.

[28] SmarandacheF.,(1998)Neutrosophy,NeutrosophicProbability,setandlogic,ProquestInformationand Learning,AnnArber,Michigan,USA,.

[29] SaqlainM.,Jaffar.N.M,Muniba.K,(2018)ChangeintheLayersofEarthinTermofFunctionalDerivative: AStudy, TheGomalUniversityJournalofResearch,34(2),12-31.

[30] SaqlainM.,SeeadM.,AhmadM.R,SmarandacheF.,(2019)GeneralizationofTOPSISforNeutrosophic HypersoftSetusingAccuracyFunctionanditsApplication, Neutrosophicsetsandsystem(NSS),27:131137.

[31] JafarN.,(2020)SimilarityMeasureofTangent,CotangentandCosinesinNeutrosophicEnvironmentand theirApplicationinSelectionofAcademicPrograms, InternationalJournalofComputerApplications,177, 17-24.

[32] CandanK.S.,andSapinoM.L.,(2010)Datamanagementformultimediaretrieval, CambridgeUniversity Press

[33] MukherjeeA.,MukherjeeA.,and(2020)IntervalvaluedFuzzysoftsetssimilaritymeasurefordetermining Covid-19patients, InternationalJournalofScientificandEngineeringResearch,11,1038-1047.

A Novel Approach to Mappings on Hypersoft Classes with Application

Muhammad Saeed 1, Muhammad Ahsan 1∗ , Atiqe Ur Rahman 2

1,2 Department of Mathematics, University of Management and Technology Lahore, Pakistan. E-mail: muhammad.saeed@umt.edu.pk , E-mail: ahsan1826@gmail.com , E-mail: aurkhb@gmail.com

Abstract: The soft set is inadequate to tackle the problems involving attribute-valued sets. Hypersoft set, an extension of the soft set, can tackle such issues efficiently. This work aims to adequate the existing concepts of mappings on fuzzy soft and soft classes for multi attribute-valued functions. First, mapping is characterized under a hypersoft set environment, then some of its essential properties like HS images, HS inverse images, etc., are developed with more generalized results. Moreover, practical application and comparative study is given to show the validity and predominance of the proposed technique.

Keywords: Fuzzy soft classes; Soft classes, Hypersoft set, Hypersoft classes, Hypersoft images, Hypersoft inverse images.

1. Introduction

For solving multifaceted problems in robotics, engineering, shortest-path selection, economics, and the environment, we cannot practice the conventional means successfully. Despite the variety of incomplete information, there are four theories specific to these problems Probability set theory (PST), Fuzzy set theory (FST) Zadeh [1], Rough set theory (RST) Pawlak [2] and Period mathematics (PM) can be assumed as a mathematical tool for dealing with lacking information. Every one of these apparatuses acquires the pre-determination of few parameters, to begin with, density function (DF) in PST, membership degree in FST, and a congruence relation in RST. Such a prerequisite, observed in the scrim of flawed or deficient information, escalate numerous issues. Simultaneously, fragmented information stays the most glaring attribute of humanitarian, organic, monetary, social, political, and large man-machine frameworks of different kinds. Heilpern [3], presented the idea of fuzzy mapping and demonstrated a fixed-point theory for fuzzy contraction mappings, which speculates the fixed-point hypothesis for multi-valued mappings of Nadler [4]. Estruch and Vidal give a fixed-point theory for fuzzy contraction mappings over a complete metric space, which is a generalization of the fundamental Heilpern’s fixed point hypothesis [5] In the pioneer research of Zhu and Xiao [6] they have examined the convexity, and quasiconvexity of fuzzy mappings

byconsideringtheideaoforderingduetoGoetschel-Voxman[7].Syau[8]demonstratedthe ideaofconvexandconcavefuzzymappings.Syau[9],presentedtheideaofdifferentiability, summedupconvexitye.g.,pseudoconvexityandinvexityforfuzzymappingsofafewfactors. HismethodologyisequaltoGoetschel-Voxmanapproachforfuzzymappingofasinglevariable inwhichthearrangementoffuzzynumbersisembeddedinatopologicalvectorspace. Molodtsov[10]successfullyimplementedsoftset(SS)theoryinseveraldirectionslikesthe easeoffunction,Riemannintegration(RI),Peranintegration(PI),Probabilitytheory(PT), Measurementtheory(MT)andsoon.HopefulresultshavebeenfoundbyKokovoetal.[11]. TheSStheoryisusedfortheprocessofoptimizationinOptimizationtheory(OPT),Game theory(GT),andOperationsresearch(OR).Majietal.[12]presentedS-setsapplications indecision-makingproblems.In[13],Yangetal.highlightedtherequirementsofS-setsin engineeringextendedapplications.Majietal.[14]presentedtheconceptoffuzzySSandits manyfeatures.TheyproposeditasanattractiveenlargementofS-sets,additionalfeatures touncertaintyandambiguityonthehighestlevelofincompleteness.Presentresearcheshave explained[15,16]howtocombinethetwoideasintoamoreflexible,highexpressionstructure formodelingandrefinedfoggydataintheinformationsystem.TheconceptofSSisapplied tosolvealotofproblemsin[17–23].

Karaaslan[24]presentedthesoftclassanditsrelevantoperations.Heapplieditsutilizations indecisionmakingsuccessfully.Atharetal.[25,26]presentedtheconceptofmappingson fuzzysoftclassesandmappingsonsoftclassesin2009and2011respectively.Theyconsidered S-images’properties,S-inverseimages,fuzzySS,fuzzyS-images,fuzzyS-inverseimagesof fuzzyS-sets,andillustratedtheseconceptswithexamplesandcounterexamples. Alkhazaleh[27]etal.introducedthenotionofamappingonclasseswheretheneutrosophicsoft classesarecollectionsoftheneutrosophicsoftsets.Additionally,theycharacterizedandstudy thepropertiesofneutrosophicsoftimagesandneutrosophicsoftinverseimagesofneutrosophic softsets.Sulaiman[28]et.alpresentedtheideaofmappingsonmulti-aspectfuzzysoftclasses. Theyexploredafewpropertiesrelatedtotheimageandpre-imageofmultiaspectfuzzysoft setsandfurtherillustratewithsomenumericalexamples.Maruah[29]et.alcharacterizedthe notationofmappingonintuitionisticfuzzysoftclasseswithsomepropertiesofintuitionistic fuzzysoftimagesandinverseimages.Manashetal.[30]gavetheideaofcompositemappings onhesitantfuzzysoftclassesin2016anddiscussedsomeinterestingpropertiesofthisidea. Samarandache[31]introducedtheconceptofHSsetasageneralizationofsoftsetin2018. Atthatpoint,hemadethedifferentiationbetweenthesortsofinitialuniverses,crisp,fuzzy, intuitionisticfuzzy,neutrosophic,andplithogenicrespectively.Thus,healsoshowedthat aHSsetcanbecrisp,fuzzy,intuitionisticfuzzy,neutrosophicandplithogenicrespectively. Saeedetal.[32]explainedsomebasicconceptslikeHSsubset,HScomplement,notHSset,

union,intersection,HSsetrelation,subrelation,complementrelation,HSrepresentationin matricesformanddifferentoperationsonmatrices.Inthisstudy,anextensionismadein existingliteratureregardingmappingonfuzzysoftandsoftclassesbydefiningmappingfor HSclasses.WedevelopsomepropertiesofmappingonHSclasseslikeHSimages,HSinverse images.Moreover,apracticalapplicationandcomparativestudyisgiventoshowthevalidity andpredominanceoftheproposedtechnique.

Theorderingofremainingarticleissortoutasfollows.InSection2,somebasicdefinition regardingsoftset,softclass,softimage,softinverseimage,HSset,HSrelationandHSfunction arere-imagined.InSection3,mappingonHSclasses,HSimage,HSinverseimageandits relevanttheoremswithproofsarecharacterized.InSection4,apracticalapplicationisgiven toshowthevalidityoftheproposedapproach.Inthelastsection,someconcludingremarks aredescribed.

1.1. Motivation

Inadiversityofreal-lifeapplications,theattributesshouldbefurthersub-partitionedinto attributevaluesforclearunderstanding.Samarandache[31]fulfilledthisneedanddeveloped theconceptoftheHSSasageneralizationoftheSS.Now,Itwillbeaquestionthathow andwhereittendstobeapplied?HSSsetissignificant?Howdowedefinemappingsfor HSSclasses?Toanswersthesequestionsandgettinginspirationfromtheabovewriting,it isrelevanttobroadentheideaofmappingsforthosesetsmanagingdisjointarrangementsof attributedvalues,i.e.,HSset.Inthisinvestigation,anextensionismadeinexistingtheories withrespecttomappingsonfuzzysoftandsoftclassesbycharacterizingmappingsonHS classes.ThestrikingcomponentofmappingsonHSclassesisthatitcanmirrortheinterrelationshipbetweenthemulti-attributefunction.Moreover,specificgeneralizedproperties ofmappingsonHSclasseslikeHSimagesandHSinverseimages,areestablishedsinceitis notyetcharacterized.Somerelatedresultsareprovedwiththehelpofillustrativeexamples. Moreover,apracticalapplicationandcomparativestudyisgiventoshowthevalidityand predominanceoftheproposedtechnique.

2. Preliminaries

Definition2.2. [24]Let ˘ U considerasinitialuniverse,suppose ˘ besetofparameters,and let ˘ = {˘ ςi : ˘ i =1, 2,...,n} beasetofdecisionmakers.IndexedclassofS-sets { ˘ ϑ ˘ ςi : ˘ ϑ ˘ ςi : ˘ → P ( ˘ U ), ˘ ςi ∈ ˘ } issaidtobesoftclassandisdenotedby ˘ ϑ ˘ .If,forany˘ ςi ∈ ˘ , ˘ ϑ ˘ ςi =Φ,theSS ˘ ϑ ˘ ςi / ∈ ˘ ϑ ˘ Definition2.3. [26]Let( ˘ , ˘ )and(˘℘, ˘ )betwoclassesofS-setsovertheuniversalset ˘ and˘ ℘ respectively.Let˘ µ : ˘ → ˘ ℘ and˘ γ : ˘ → ˘ bemappings.Thenamapping ˘ ϑ =(˘µ, ˘ γ):( ˘ , ˘ ) → (˘℘, ˘ )isdefinedas,forSS( ˘ , ˘ ℵ)in( ˘ , ˘ )and ˘ ϑ( ˘ , ˘ ℵ)isSSin(˘℘, ˘ ) obtainedasfollows,For ˘ β ∈ ˘ γ( ˘ ) ⊆ ˘ and˘ y ∈ ˘ ℘,then

1 (˘ y) ∪

∈γ 1 ( ˘ β)∩ ˘ ℵ

(˘ α) (˘ x), if˘ µ 1(˘ y) =Φ, ˘ γ 1( ˘ β) ∩ ˘ ℵ =Φ 0if otherwise

(1) ˘ ϑ( ˘ , ˘ ℵ)iscalledasoftimageofSS( ˘ , ˘ ℵ). Now,let( ˘ , ˘ )aSSin(˘℘, ˘ ),where ˘ ⊆ ˘ then ˘ ϑ 1( ˘ , ˘ )isaSSin( ˘ , ˘ )givenasfollows, ˘ ϑ 1( ˘ , ˘ )(˘ α)(˘ x)=    ˘ (˘ γ(˘ α)(˘ µ(˘ x)if˘ γ(˘ α) ∈ ˘ 0if otherwise (2) where˘ α ∈ ˘ γ 1( ˘ ) ⊂ ˘ ˘ ,then ˘ ϑ 1( ˘ , ˘ )saidtobethesoftinverseimageof( ˘ , ˘ ).

Definition2.4. [31]Let˘ o1, ˘ o2, ˘ o3, , ˘ on bethedistinctattributeswhosecorresponding attributevaluesbelongstothesets ˘ 1, ˘ 2, ˘ 3, ··· , ˘ n respectively,where ˘ i ∩ ˘ j = ˘ Φfor ˘ i = ˘ j.Apair( ˘ Υ, ˘ J )iscalledaHSsetovertheuniversalset ˘ U ,where ˘ Υisthemappinggiven by ˘ Υ: ˘ J −→ ˘ P ( ˘ U ),where ˘ J = ˘ 1 × ˘ 2 × ˘ 3 × ... × ˘ n.Formoredefinitionsee[32–35]. Definition2.5. [32]Let( ˘ , ˘ ℵ)and( ˘ , ˘ )bethetwoHSsetsoverthesameuniversalset ˘ U. Thentherelationfrom( ˘ , ˘ ℵ)to( ˘ , ˘ )iscalledaHSsetrelation( ˘ R, ˘ C)oritisinsimpleway ˘ R isaHSsubsetof( ˘ , ˘ ℵ) × ( ˘ , ˘ ),where ˘ C ⊆ ˘ ℵ× ˘ and ∀ (˘a, ˘ b) ∈ ˘ C ˘ R(˘a, ˘ b)= ˘ H(˘a, ˘ b),where ( ˘ H, ˘ ℵ× ˘ )=( ˘ , ˘ ℵ) × ( ˘ , ˘ ).AHSsetrelationon( ˘ , ˘ ℵ)isaHSsubsetof( ˘ , ˘ ℵ) × ( ˘ , ˘ ℵ).In similarway,theparameterizedformofrelation R ontheHSset( ˘ , ˘ ℵ)isdefinedasfollows. If( ˘ , ˘ ℵ)= { ˘ (˘ a), ˘ ( ˘ b),...},then ˘ (˘ a) ˘ R ˘ ( ˘ b) ⇔ ˘ (˘ a) × ˘ ( ˘ b)) ∈ ˘ R Definition2.6. [32]Let( ˘ , ˘ ℵ)and( ˘ , ˘ )bethetwoHSsetsoverthesameuniversalset ˘ U. ThentheHSrelationfrom( ˘ , ˘ ℵ)to( ˘ , ˘ )canbesymbolizeas ˘ ϑ :( ˘ , ˘ ℵ) −→ ( ˘ , ˘ )iscalleda HSsetfunction. (1) Ifeveryelementinthedomainof ˘ ϑ hasuniqueelementinrangeof ˘ ϑ

3. MappingsonHypersoftClasses

,...,

βn)= ˘ β Definition3.1. Suppose ˘ U beaninitialuniverse,let 1, 2, 3, , n bethedistinctattributes whoseattributevaluesbelongstothesets ˘ 1, ˘ 2, ˘ 3, ··· , ˘ n respectively,where ˘ i ∩ ˘ j = ˘ Φ for ˘ i = ˘ j,let ˘ = {˘ ςi : ˘ i =1, 2,...,n} beacollectionofdecisionmakers.IndexedclassofHSS { ˘ ϑ ˘ ςi : ˘ ϑ ˘ ςi : ˘ J → P ( ˘ U ), ˘ ςi ∈ ˘ },where ˘ J = ˘ 1 × ˘ 2 × ˘ 3 ×···× ˘ n issaidtobeHSclassandit canbesymbolizedinsuchaform ˘ ϑ .If,forany˘ ςi ∈ ˘ , ˘ ϑςi =Φ,theHSS ˘ ϑςi ∈ ˘ ϑ Example3.2. Let ˘ = {˘ a =LineInteractive, ˘ b =Standby-Ferro,˘ c =DeltaConversionOnLine} betypesofUPS(UninterruptiblePowerSupply)isconsideredasuniverseofdiscourse. Let 1 =efficiency, 2 =size, 3 =colour,distinctattributeswhoseattributevaluesbelongto thesets ˘ 1, ˘ 2, ˘ 3.Let ˘ 1 = {˘ τ1 =Good,˘ τ2 =VeryGood }, ˘ 2 = {˘ τ3 =medium,˘ τ4 = small}, ˘ 3 = {˘ τ5 =brown } andlet ˘ = {˘ ς1, ˘ ς1, ˘ ς1} beasetofdecisionmakers.IfweconsiderHSsets ˘ ϑ ˘ς1, ˘ ϑ ˘ς2, ˘ ϑ ˘ς3 givenas ˘ ϑ ˘ς1 = {((˘ τ1, ˘ τ3, ˘ τ5), {˘ a, ˘ b}), ((˘ τ1, ˘ τ4, ˘ τ5), {˘ c}), ((˘ τ2, ˘ τ3, ˘ τ5), { ˘ b, ˘ c}), ((˘ τ2, ˘ τ4, ˘ τ5), { ˘ b})}, ˘ ϑ ˘ς2 = {((˘ τ1, ˘ τ3, ˘ τ5), { ˘ b, ˘ c}), ((˘ τ1, ˘ τ4, ˘ τ5), {˘ a}), ((˘ τ2, ˘ τ3, ˘ τ5), {˘ a, ˘ c}), ((˘ τ2, ˘ τ4, ˘ τ5), { ˘ b})}, ˘ ϑ ˘ς3 = {(˘ τ1, ˘ τ3, ˘ τ5){˘ c, ˘ a}), ((˘ τ1, ˘ τ4, ˘ τ5), { ˘ b}), ((˘ τ2, ˘ τ3, ˘ τ5), { ˘ b, ˘ c}), ((˘ τ2, ˘ τ4, ˘ τ5), {˘ c})}, then ˘ ϑ ˘ = { ˘ ϑ ˘ς1, ˘ ϑ ˘ς2, ˘ ϑ ˘ς3} isaHSclass.Nowlet ˘ g ˘ς1 = {((˘ τ1, ˘ τ3, ˘ τ5), {˘ a}), ((˘ τ1, ˘ τ4, ˘ τ5), {˘ a}), ((˘ τ2, ˘ τ3, ˘ τ5), {˘ c, ˘ a}), ((˘ τ2, ˘ τ4, ˘ τ5), {˘ a})}, ˘ g ˘ς2 = {((˘ τ1, ˘ τ3, ˘ τ5), {˘ c}), ((˘ τ1, ˘ τ4, ˘ τ5), { ˘ b}), ((˘ τ2, ˘ τ3, ˘ τ5), { ˘ b, ˘ c}), ((˘ τ2, ˘ τ4, ˘ τ5), {˘ c})}, ˘ g ˘ς3 = {((˘ τ1, ˘ τ3, ˘ τ5), { ˘ b}), ((˘ τ1, ˘ τ4, ˘ τ5), { ˘ b}), ((˘ τ2, ˘ τ3, ˘ τ5), {˘ a, ˘ c}), ((˘ τ2, ˘ τ4, ˘ τ5), {˘ a})}, isalsoHSclass.ThenHSclassescanbewrittenas { ˘ ϑ ˘ς1, ˘ ϑ ˘ς2, ˘ ϑ ˘ς3},{ ˘ g ˘ς1, ˘ g ˘ς2, ˘ g ˘ς3}. Definition3.3. Let( ˘ , ˘ J )and(˘℘, ˘ K)betwoclassesofHSsetsovertheuniversalset ˘ and ˘ ℘ respectively.Let˘ µ : ˘ → ˘ ℘ and˘ γ : ˘ J → ˘ K bemappings.Thenamapping ˘ ϑ =(˘µ, ˘ γ): ( ˘ , ˘ J ) → (˘℘, ˘ K)isdefinedasforHSset( ˘ , ˘ ℵ)in( ˘ , ˘ J )and ˘ ϑ( ˘ , ˘ ℵ)isHSsetin(˘℘, ˘ K)obtained

 ∪˘

∈ ˘ µ 1(˘ y) ∪ ˘ α∈˘ γ 1( ˘ β)∩ ˘ ℵ ˘ (˘ α) (˘ x), if˘ µ 1(˘ y) =Φ, ˘ γ 1( ˘ β) ∩ ˘ ℵ =Φ 0if otherwise

(3) ˘ ϑ( ˘ , ˘ ℵ)iscalledaHSimageofHSset( ˘ , ˘ ℵ). Definition3.4. Let( ˘ , ˘ J )and(˘℘, ˘ K)betwoclassesofHSsetsovertheuniversalset ˘ and ˘ ℘ respectively.Let˘ µ : ˘ → ˘ ℘ and˘ γ : ˘ J → ˘ K bemappings.Now,let( ˘ , ˘ )beaHSsetin (˘℘, ˘ K),where ˘ ⊆ ˘ K then ˘ ϑ 1( ˘ , ˘ )isaHSsetin( ˘ , ˘ J )definedasfollows, ˘ ϑ 1( ˘ , ˘ )(˘ α)(˘ x)=    ˘ (˘ γ(˘ α)(˘ µ(˘ x)if˘ γ(˘ α) ∈ ˘ 0if otherwise (4) where˘ α ∈ ˘ γ 1( ˘ ) ⊂ ˘ J ,then ˘ ϑ 1( ˘ , ˘ )saidtobetheHSinverseimageofHSset( ˘ , ˘ ).

Example3.5. Let ˘ = {˘ a =LineInteractive, ˘ b =Standby-Ferro,˘ c =DeltaConversion On-Line} and˘ ℘ = {˘ x =MicrowavecumConvection,˘ y =Conventionaloven,˘ z =Convection oven } betypesofUPS(UninterruptiblePowerSupply)andOvensrespectively,consideredas universesofdiscourses.AMarketingmangerwantstoknowtherelationshipbetweenUPSand Ovenscharacteristicswhichwillbemoreeffectiveregardinghismarketing.Let 1 =efficiency, 2 =size, 3 =colour, 4 =priceand ˘ b1 =efficiency, ˘ b2 =colour, ˘ b3 =price, ˘ b4 =sizebe thetwotypesofdistinctattributeswhoseattributevaluesbelongtothesets ˘ 1, ˘ 2, ˘ 3, ˘ 4 and ˘ 1, ˘ 2, ˘ 3,

˘ 4)isaHSsetin(˘℘, ˘ 1 × ˘ 2 × ˘ 3 × ˘ 4),where ( ˘ 1 × ˘ 2 × ˘ 3 × ˘ 4)=˘ γ( ˘ ℵ1 × ˘ ℵ2 × ˘ ℵ3 × ˘ ℵ4)= {(˘ τ1 , ˘ τ3 , ˘ τ5 , ˘ τ7 ), (˘ τ2 , ˘ τ4 , ˘ τ5 , ˘ τ7 ))} obtainedas follows: =( ˘ ϑ( ˘ , ˘ ℵ1 × ˘ ℵ2 × ˘ ℵ3 × ˘ ℵ4), ˘ 1 × ˘ 2 × ˘ 3 × ˘ 4)(˘ τ1 , ˘ τ3 , ˘ τ5 , ˘ τ7 ) =˘ µ ∪ ˘ α∈˘ γ 1 (˘ τ2 , ˘ τ4 , ˘ τ5 , ˘ τ7 )∩( ˘ ℵ1 × ˘ ℵ2 × ˘ ℵ3 × ˘ ℵ4 ) ˘ (˘ α) (˘ s) =˘ µ ∪ ˘ α∈{(˘ τ1 , ˘ τ3 , ˘ τ5 , ˘ τ7 ),(˘ τ2 , ˘ τ4 , ˘ τ5 , ˘ τ7 )} ˘ (˘ α) (˘ s) =˘ µ ˘ (˘ τ1, ˘ τ3, ˘ τ5, ˘ τ7) ∪ ˘ (˘ τ2, ˘ τ4, ˘ τ5, ˘ τ7) (˘ s) =˘ µ Φ ∪{˘ a, ˘ b, ˘ c}) (˘ s) = {˘ y, ˘ z} ( ˘ ϑ( ˘ , ˘ ℵ1 × ˘ ℵ2 × ˘ ℵ3 × ˘ ℵ4), ˘ 1 × ˘ 2 × ˘ 3 × ˘ 4)(˘ τ1 , ˘ τ3 , ˘ τ5 , ˘ τ7 )= {˘ y, ˘ z} Similarly, ( ˘ ϑ( ˘ , ˘ ℵ1 × ˘ ℵ2 × ˘ ℵ3 × ˘ ℵ4), ˘ 1 × ˘ 2 × ˘ 3 × ˘ 4)(˘ τ2 , ˘ τ4 , ˘ τ5 , ˘ τ7 )= {˘ y, ˘ z} NowforHSinverseimage, ˘ ϑ 1(( ˘ , ˘ 1 × ˘ 2 × ˘ 3 × ˘ 4), ˘

aredefinedas; ( ˘ ϑ( ˘ , ˘ ℵ) ∪ ˘ ϑ( ˘ , ˘ ))( ˘ β)= ˘ ϑ( ˘ , ˘ ℵ)( ˘ β) ∪ ˘ ϑ( ˘ , ˘ )( ˘ β), ( ˘ ϑ( ˘ , ˘ ℵ) ∩ ˘ ϑ( ˘ , ˘ ))( ˘ β)= ˘ ϑ( ˘ , ˘ ℵ)( ˘ β) ∩ ˘ ϑ( ˘ , ˘ )( ˘ β) Definition3.7. ˘ ϑ :( ˘ , ˘ J ) → (˘℘, ˘ K)beamappingand( ˘ , ˘ ℵ)and( ˘ , ˘ )bethetwoHSsets in( ˘ , ˘ J ).Thenfor˘ α ∈ ˘ J ,HSunionandintersectionofHSinverseimagesof( ˘ , ˘ ℵ)and( ˘ , ˘ ) in( ˘ , ˘ J )aredefinedas; ( ˘ ϑ 1( ˘ , ˘ ℵ) ∪ ˘ ϑ 1( ˘ , ˘ ))(˘ α)= ˘ ϑ 1( ˘ , ˘ ℵ)(˘ α) ∪ ˘ ϑ 1( ˘ , ˘ )(˘ α), ( ˘ ϑ 1( ˘ , ˘ ℵ) ∩ ˘ ϑ 1( ˘ , ˘ ))(˘ α)= ˘ ϑ 1( ˘ , ˘ ℵ)(˘ α) ∩ ˘ ϑ 1( ˘ , ˘ )(˘ α) Theorem3.8. Let ˘ ϑ :( ˘ , ˘ J ) → (˘℘, ˘ K) beaHS-mapping,let ( ˘ , ˘ ℵ), ( ˘ , ˘ ) bethetwoHSsets in ( ˘ , ˘ J ) andwith ( ˘ i, ˘ ℵi) asthefamilyofaHSsetsin ( ˘ , ˘ J ) wehave, (1) ˘ ϑ( ˘ Φ)= ˘ Φ (2) ˘ ϑ( ˘ ) ˘ ⊂ ˘ ℘ (3) ˘ ϑ(( ˘ , ˘ ℵ) ∪ ( ˘ , ˘ ))= ˘ ϑ( ˘ , ˘ ℵ) ∪ ˘ ϑ( ˘ , ˘ ).Generally, ˘ ϑ(∪i( ˘ i, ˘ ℵi))= ∪i ˘ ϑ( ˘ i, ˘ ℵi) (4) ˘ ϑ(( ˘ , ˘ ℵ) ∩ ( ˘ , ˘ )) ˘ ⊇ ˘ ϑ( ˘ , ˘ ℵ) ∩ ˘ ϑ( ˘ , ˘ ).Generally, ˘ ϑ(∩i( ˘ i, ˘ ℵi)) ˘ ⊆∩i ˘ ϑ( ˘ i, ˘ ℵi). (5) If ( ˘ , ˘ ℵ) ˘ ⊆( ˘ , ˘ ) then ˘ ϑ( ˘ , ˘ ℵ) ˘ ⊆ ˘ ϑ( ˘ , ˘ ). Proof. Here(1), (2)aretrivialcase, (3):for ˘ β ∈ ˘ K,wehavetoprove ˘ ϑ(( ˘ , ˘ ℵ) ∪ ( ˘ , ˘ ))( ˘ β)=( ˘ ϑ( ˘ , ˘ ℵ) ∪ ˘ ϑ( ˘ , ˘ ))( ˘ β). Take ˘ ϑ(( ˘ , ˘ ℵ) ∪ ( ˘ , ˘ ))( ˘ β)= ˘ ϑ( ˘ S, ˘ ℵ∪ ˘ )( ˘ β)=    ˘ µ ∪ ˘ α∈˘ γ 1 ( ˘ β)∩( ˘ ℵ∪ ˘ )

˘ β) ( ˘ ϑ( ˘ , ˘ ℵ) ∩ ( ˘ , ˘ ))( ˘ β)= ˘ ϑ( ˘ S, ˘ ℵ∩ ˘ )( ˘ β) =    ˘ µ ∪ ˘ α∈˘ γ 1 ( ˘ β)∩( ˘ ℵ∩ ˘ ) ˘ S(˘ α) if˘ γ 1( ˘ β) ∩ ( ˘ ℵ∩ ˘ ) = ˘ Φ, ˘ Φ, Otherwise, (9) Where ˘ S(˘ α)= ˘ (˘ α) ∩ ˘ (˘ α). Forthenontrivialcase,˘ γ 1( ˘ β) ∩ ( ˘ ℵ∩ ˘ ) = ˘ Φ ˘ ϑ( ˘ S, ˘ ℵ∩ ˘ )( ˘ β)=˘ µ ∪ α∈˘ γ 1 ( ˘ β)∩( ˘ ℵ∩ ˘ ) ˘ S(˘ α) =˘ µ ∪ α∈˘ γ 1 ( ˘ β)∩( ˘ ℵ∩ ˘ ) ( ˘ (˘ α) ∩ ˘ (˘ α)) ( ˘ ϑ( ˘ , ˘ ℵ) ∩ ( ˘ , ˘ ))( ˘ β)=˘ µ ∪ α∈˘ γ 1 ( ˘ β)∩( ˘ ℵ∩ ˘ ) ( ˘ (˘ α) ∩ ˘ (˘ α)) ontheotherside,byusingdefinition3.6wehave, (

) =

(

,

)(

β). Thisyields5.Intheorem3.8,reversibleof(2)and(4)doesnothold.Itcanbeexplaininthe followingexample. Example3.9. Fromexample3.5, ˘ ℘

( ˘ )= {{(˘ τ1 , ˘

3 , ˘

5 , ˘

7 )= {˘ y, ˘ z}, (˘ τ2 , ˘ τ3 , ˘ τ5 , ˘ τ7 )= {˘ y, ˘ z}, (˘ τ2 , ˘ τ4 , ˘ τ5 , ˘ τ7 )= {˘ y, ˘ z}} Thisindicatethatreverseof(2)doesnothold.Nowwearegoingtoshowreverseof(4)also doesnothold.ForthispurposewechoosetwoHSsetsin( ˘ , ˘ 1 × ˘ 2 × ˘ 3 × ˘ 4)as ( ˘ , ˘ ℵ1 × ˘ ℵ2 × ˘ ℵ3 × ˘ ℵ4)= {(˘ τ1, ˘ τ3, ˘ τ5, ˘ τ7)=Φ, (˘ τ1, ˘ τ4, ˘ τ5, ˘ τ7)= {˘ a}, (˘ τ1, ˘ τ4, ˘ τ5, ˘ τ7)= {˘ a, ˘ b, ˘ c}, (˘ τ2, ˘ τ3, ˘ τ5, ˘ τ7)= { ˘ b}}, ( ˘ , ˘ 1 × ˘ 2 × ˘ 3 × ˘ 4)= {(˘ τ1, ˘ τ3, ˘ τ5, ˘ τ7)= {˘ a}, (˘ τ2, ˘ τ4, ˘ τ5, ˘ τ7)= { ˘ b, ˘ c}}

in ( ˘ , ˘ J ) andthefamilyofaHSsets ( ˘ i, ˘ ℵi) in ( ˘ , ˘ J ) wehave, (1) ˘

1( ˘ Φ)=

Φ (2) ˘ ϑ 1(˘ ℘)= ˘ (3) ˘ ϑ 1(( ˘ , ˘ ℵ) ∪ ( ˘ , ˘ ))= ˘ ϑ 1( ˘ , ˘ ℵ) ∪ ˘ ϑ 1( ˘ , ˘ ).Generally, ˘ ϑ 1( ˘ i, ˘ ℵi))= ∪i ˘ ϑ 1( ˘ i, ˘ ℵi). (4) ˘ ϑ 1(( ˘ , ˘ ℵ) ∩ ( ˘ , ˘ ))= ˘ ϑ 1( ˘ , ˘ ℵ) ∩ ˘ ϑ 1( ˘ , ˘ ).Generally, ˘ ϑ 1(∩i( ˘ i, ˘ ℵi))= ∩i ˘ ϑ 1( ˘ i, ˘ ℵi)

(5) If ( ˘ , ˘ ℵ) ˘ ⊆( ˘ , ˘ ) then ˘ ϑ 1(( ˘ , ˘ ℵ) ˘ ⊆ ˘ ϑ 1( ˘ , ˘ ) Proof. Here(1), (2)aretrivialcase, (3):for˘ α ∈ ˘ J ,wehavetoprove ˘ ϑ 1(( ˘ , ˘ ℵ) ∪ ( ˘ , ˘ ))(˘ α)=( ˘ ϑ 1( ˘ , ˘ ℵ) ∪ ˘ ϑ 1( ˘ , ˘ ))(˘ α). Take ˘ ϑ 1(( ˘ , ˘ ℵ) ∪ ( ˘ , ˘ ))(˘ α) = ˘ ϑ 1( ˘ S, ˘ ℵ∪ ˘ )(˘ α) = u 1( ˘ S(˘ γ(˘ α)), ˘ γ(˘ α) ∈ ˘ ℵ∪ ˘ = u 1( ˘ S( ˘ β)), ˘ γ(˘ α)= ˘ β where =˘ µ 1         ˘ ( ˘ β)if ˘ β ∈ ( ˘ ℵ− ˘ ) ˘ ( ˘ β)if ˘ β ∈ ( ˘ − ˘ ℵ) ˘ ( ˘ β) ∪ ˘ ( ˘ β)if ˘ β ∈ ( ˘ ∩ ˘ ℵ)

   (12)

Nowbyusingdefinition3.7,wehave ( ˘ ϑ 1( ˘ , ˘ ℵ) ∪ ˘ ϑ 1( ˘ , ˘ ))˘ α = ˘ ϑ 1( ˘ , ˘ ℵ)(˘ α) ∪ ˘ ϑ 1( ˘ , ˘ )(˘ α) = ˘ u 1( ˘ (˘ γ(˘ α))) ∪ ˘ u 1( ˘ (˘ γ(˘ α))), ˘ γ(˘ α) ∈ ˘ ℵ∩ ˘ =˘ µ 1         ˘ ( ˘ β)if ˘ β ∈ ( ˘ ℵ− ˘ ) ˘ ( ˘ β)if ˘ β ∈ ( ˘ − ˘ ℵ) ˘ ( ˘ β) ∪ ˘ ( ˘ β)if ˘ β ∈ ( ˘ ∩ ˘ ℵ)

   (13)

Where ˘ β =˘ γ(˘ α).From12and13weget(3).

(

,

(5):for˘ α ∈ ˘ J ,consider ˘ ϑ 1( ˘ , ˘ ℵ)(˘ α) =˘ µ 1( ˘ (˘ γ(˘ α))) =˘ µ 1( ˘ ( ˘ β)), ˘ γ(˘ α)= ˘ β ⊆ ˘ µ 1( ˘ ( ˘ β)) =˘ µ 1( ˘ (˘ γ(˘ α))) = ˘ ϑ 1( ˘ , ˘ )(˘ α)

Thisyields(5). 4. StatementofProblem 4.1. CasestudyandObjective

Beforestartinganewconstructionbusiness,itisessentialtounderstandtheinvestment andworkthatisinvolved.Establishinganewbusinessofanykindisnevereasy,andthere arealwaysthingstoconsiderthatmayormaynotbeattheforefrontofone’smind,evenan experiencedentrepreneur.Areyoureadytolaunchyourconstructionstartup?Hereareten thingsthatshouldbeatthefoundationofyourplans.

Apersondecidestosetaconstructioncompany.Totackleandplanforfuturehecreatean interrelationshipbetweentheConstructionCompaniesoftwomajoreconomicbodiesinthe world(ChinaandU.S.)byconsideringmulti-attributefunction.Forthis,let ˘ X = {˘ a =Beijing ConstructionEngineeringGroup, ˘ b =AnhuiConstructionEngineeringGroup,˘ c =ChinaState ConstructionEngineering } and ˘ Y = {˘ x =KiewitCorporation,˘ y =SkanskaConstruction,˘ z = Whiting-TurnerContractingCompany} bethetwouniversalsetsofconstructioncompanies ChinaandU.Srespectively.Let˘ x1 =Communication,˘ x2 =Leadership,˘ x3 =Standard,and ˘ y1 =Innovative,˘ y2 =Planning,˘ y3 =Budgetpotencybethetwodistinctattributeswhose attributevaluesbelongtothesets ˘ E1, ˘ E2, ˘ E3 and ˘ E 1, ˘ E 2, ˘ E 3 respectively,where ˘ E1 = {˘ e1 = Conciseness,˘ e2 =EffectiveCommunication,˘ e3 =LowlevelCommunication }, ˘ E2 = {˘ e4 = Learningagility,˘ e5 =Influence}, ˘ E3 = {˘ e6 =HighStandard,˘ e7 =Feasible} and ˘ E 1 = { ˘ e 1 =relativeadvantage, ˘ e 2 =compatibility}, ˘ E 2 = { ˘ e 3 =CorrectiveActionPlan, ˘ e 4 = Traditionalplanning}, ˘ E 3 = { ˘ e 5 =Highbudget} Let˘ u : ˘ X → ˘ Y ,˘ p : ˘ E1 × ˘ E2 × ˘ E3 → ˘ E 1 × ˘ E 2 × ˘ E 3 bemappingsasfollows˘ u(˘ a)=˘ x,˘ u( ˘ b)=˘ y, ˘ u(˘ c)=˘ x and ˘ p(˘ e1, ˘ e4, ˘ e6)=(˘ e1, ˘ e3, ˘ e5) ˘ p(˘ e1, ˘ e4, ˘ e7)=(˘ e1, ˘ e4, ˘ e5) ˘ p(˘ e1, ˘ e5, ˘ e6)=(˘ e1, ˘ e3, ˘ e5) ˘ p(˘ e1, ˘ e5, ˘ e7)=(˘ e1, ˘ e4, ˘ e5) ˘ p(˘ e2, ˘ e4, ˘ e6)=(˘ e2, ˘ e3, ˘ e5) ˘ p(˘ e2, ˘ e4, ˘ e7)=(˘ e1, ˘ e3, ˘ e5) ˘ p(˘ e2, ˘ e5, ˘ e6)=(˘ e1, ˘ e4, ˘ e5) ˘ p(˘ e2, ˘ e5, ˘ e7)=(˘ e1, ˘ e4, ˘ e5) ˘ p(˘ e3, ˘ e4, ˘ e6)=(˘ e1, ˘ e3, ˘ e5) ˘ p(˘ e3, ˘ e4, ˘ e6)=(˘ e1, ˘ e4, ˘ e5) ˘ p(˘ e3, ˘ e5, ˘ e6)=(˘ e1, ˘ e3, ˘ e5) ˘ p(˘ e3, ˘ e5, ˘ e7)=(˘ e1, ˘ e3, ˘ e5) respectively.ThisdatacanbeencodedintotwoHSSfromtwoclasses( ˘ X, ˘ E1 × ˘ E2 × ˘ E3)and ( ˘ Y, ˘ E 1 × ˘ E 2 × ˘ E 3)respectively,

( ˘ Λ, ˘ Σ1 × ˘ Σ2 × ˘ Σ3)= {(˘ e1, ˘ e4, ˘ e6)= {˘ a, ˘ b}, (˘ e1, ˘ e4, ˘ e7)= { ˘ b}}, ( ˘ ∆, ˘ Ω1 × ˘ Ω2 × ˘ Ω3)= {(˘ e1, ˘ e3, ˘ e5)= {˘ x}, (˘ e1, ˘ e4, ˘ e5)= {˘ x, ˘ y}, (˘ e2, ˘ e4, ˘ e5)= {˘ z}} ThentheHSimageof( ˘ Λ, ˘ Σ1 × ˘ Σ2 × ˘ Σ3)under ˘ f :( ˘ X, ˘ E1 × ˘ E2 × ˘ E3) → ( ˘ Y, ˘ E 1 × ˘ E 2 × ˘ E 3) isobtainedby ˘ f ( ˘ Λ, ˘ Σ1 × ˘ Σ2 × ˘ Σ3)(˘ e1, ˘ e3, ˘ e5)= {˘ x, ˘ y}, ˘ f ( ˘ Λ, ˘ Σ1 × ˘ Σ2 × ˘ Σ3)(˘ e1, ˘ e4, ˘ e5)= {˘ y} ˘ f ( ˘ Λ, ˘ Σ1 × ˘ Σ2 × ˘ Σ3)= {(˘ e1, ˘ e3, ˘ e5)= {˘ x, ˘ y}, (˘ e1, ˘ e4, ˘ e5)= {˘ y}},

whichmeansthat,withrespecttotheattributesvalues(Relativeadvantage,correctiveaction plan,Highbudget),KiewitCorporationandSkanskaConstructionisbestcompanies.Onthe otherhand,withrespecttotheattributesvalues(Relativeadvantage,Traditionalplanning, Highbudget),SkanskaConstructionisbest.Similarly, ˘ f 1( ˘ ∆, ˘ Ω1 × ˘ Ω2 × ˘ Ω3)(˘ e1, ˘ e4, ˘ e6)= {˘ a, ˘ c} ˘ f 1( ˘ ∆, ˘ Ω1 × ˘ Ω2 × ˘ Ω3)(˘ e1, ˘ e4, ˘ e7)= {˘ a, ˘ b, ˘ c} ˘ f 1( ˘ Λ, ˘ Σ1 × ˘ Σ2 × ˘ Σ3)= {(˘ e1, ˘ e4, ˘ e6)= {˘ a, ˘ c}, (˘ e1, ˘ e4, ˘ e7)= {˘ a, ˘ b, ˘ c}}, whichmeansthat,withrespecttotheattributesvalues(Conciseness,Learningagility,High standard),BeijingConstructionEngineeringGroupandAnhuiConstructionEngineeringis bestcompanies.Ontheotherside,withrespecttotheattributesvalues(Conciseness,Learning agility,Feasible),BeijingConstructionEngineeringGroup,AnhuiConstructionEngineering Group,ChinaStateConstructionEngineeringisbest.

4.2. Comparativestudies

Inthefollowing,fewcomparisonsoftheinitiatedtechniqueswithshortcomingsaretalked abouttoanalyzethevalidityandpredominanceoftheproposedtechnique.Additionally,we willcompareourproposedmappingsbasedhypersoftclasseswithnineotherexistingtheories, EstruchandVidal[5]giveafixedpointtheoryforfuzzycontractionmappingsoveracomplete metricspace,Syau[8]demonstratedtheideaofconvexandconcavefuzzymappings,Karaaslan [24]presentedthesoftclassanditsrelevantoperations,includingtheideapresentedbyAthar etal.[25,26]theconceptofmappingsonfuzzysoftclassesandmappingsonsoftclasses, Alkhazaleh[27]etal.introducedthenotionofamappingonclasseswheretheneutrosophic softclassesarecollectionsoftheneutrosophicsoftsets,Sulaiman[28]et.alpresentedtheidea

ofmappingsonmulti-aspectfuzzysoftclasses,Maruah[29]et.alcharacterizedthenotation ofmappingonintuitionisticfuzzysoftclasseswithsomepropertiesofintuitionisticfuzzysoft imagesandinverseimages,Manashetal.[30]gavetheideaofcompositemappingsonhesitant fuzzysoftclassesin2016anddiscussedsomeinterestingpropertiesofthisidea.However, whentheattributesarefurthersub-dividedintoattributevaluesthenallexistingtheoriesare failstomanage.Thisneedisfulfilledintheproposedmappingsbasedhypersoftclasses,see table1.

Table1. ComparisonoftheproposedmappingsbasedHypersoftclasseswith existingtheories

SN References Disadvantage Ranking

1 [5] lackofsub-partitionvaluesofparameter inapplicable

2 [8] lackofsub-partitionvaluesofparameter inapplicable

3 [24] lackofsub-partitionvaluesofparameter inapplicable

4 [25] lackofsub-partitionvaluesofparameter inapplicable

5 [26] lackofsub-partitionvaluesofparameter inapplicable

6 [27] lackofsub-partitionvaluesofparameter inapplicable

7 [28] lackofsub-partitionvaluesofparameter inapplicable

8 [29] lackofsub-partitionvaluesofparameter inapplicable 9 [30] lackofsub-partitionvaluesofparameter inapplicable

10 Proposed Methodin thispaper

5. Conclusions

Lengthyandheavycalculationsindecisionmaking applicable

Inthisstudy,anextensionismadeintheexistingmappingsoffuzzyandsoftclassestoHS classes.Thepresentedworkismoreeffectivethantheexistingone.Someimportantpropertiesandresultsarediscussedwiththesupportofillustratedexamples.Thefutureextension canbemadebydevelopingsuchmappingsonotherhybridstructuresofHSsets,topologicalstructures,algebraicstructures,graphtheory,computerscience,Electromagneticanalysis fordetection,Rocketlaunchtrajectoryanalysis,Materialsscience,Modelingofairflowover airplanebodiesandBehaviouralsciences.Moreover,apracticalapplicationandcomparative studyisgiventoshowthevalidityandpredominanceoftheproposedtechnique.Thisconcept mayhelptheresearchersindecision-makingproblemsbyconsideringdisjointssetsofattribute valuesratherthantakingonesetofparameters.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

References

1. Zadeh,L.A.(1965).Fuzzysets,InformationandControl,vol.8.GoogleScholarGoogleScholarDigital LibraryDigitalLibrary,338-353.

2. Pawlak,Z.(1982).Roughsets.Internationaljournalofcomputerandinformationsciences,11(5),341-356.

3. Heilpern,S.(1981).Fuzzymappingsandfixedpointtheorem.JournalofMathematicalAnalysisandApplications,83(2),566-569.

4. Nadler,S.B.(1969).Multi-valuedcontractionmappings.PacificJournalofMathematics,30(2),475-488.

5. Estruch,V.D.,andVidal,A.(2001).Anoteonfixedfuzzypointsforfuzzymappings.

6. Zou,Y.,andXiao,Z.(2008).Dataanalysisapproachesofsoftsetsunderincompleteinformation. Knowledge-BasedSystems,21(8),941-945.

7. GoetschelJr,R.,andVoxman,W.(1986).Elementaryfuzzycalculus.Fuzzysetsandsystems,18(1),31-43.

8. Syau,Y.R.(1999).Onconvexandconcavefuzzymappings.FuzzySetsandSystems,103(1),163-168.

9. Syau,Y.R.(2001).Differentiabilityandconvexityoffuzzymappings.ComputersandMathematicswith Applications,41(1-2),73-81.

10. Molodtsov,D.(1999).Softsettheoryfirstresults.ComputersandMathematicswithApplications,37(4-5), 19-31.

11. Kovkov,D.V.,Kolbanov,V.M.,andMolodtsov,D.A.(2007).Softsetstheory-basedoptimization.Journal ofComputerandSystemsSciencesInternational,46(6),872-880.

12. Maji,P.K.,Roy,A.R.,andBiswas,R.(2002).Anapplicationofsoftsetsinadecisionmakingproblem. ComputersandMathematicswithApplications,44(8-9),1077-1083.

13. Yang,X.,Yu,D.,Yang,J.,andWu,C.(2007).Generalizationofsoftsettheory:fromcrisptofuzzycase. InFuzzyInformationandEngineering(pp.345-354).Springer,Berlin,Heidelberg.

14. Maji,R.K.,andBiswas,R.(2001).ARRoy.Fuzzysoftsets,Jr.ofFuzzyMath,9(3),589-602.

15. Roy,A.R.,andMaji,P.K.(2007).Afuzzysoftsettheoreticapproachtodecisionmakingproblems.Journal ofcomputationalandAppliedMathematics,203(2),412-418.

16. Kong,Z.,Gao,L.,andWang,L.(2009).CommentonAfuzzysoftsettheoreticapproachtodecisionmaking problems.Journalofcomputationalandappliedmathematics,223(2),540-542.

17. Rauf,A.,Zeba,I.,andSaqlain,M.(2018).Modifieddust-lower-hybridwavesinquantumplasma.Scientific InquiryandReview,2(2),11-21.

18. Riaz,M.,Saeed,M.,Saqlain,M.,andJafar,N.(2019).Impactofwaterhardnessininstinctivelaundry

system based on fuzzy logic controller, Punjab University Journal of Mathematics, 51(2) 73-84.

19. Saeed,M.,Saqlain,M.,andJafar,N.(2018).ConsumerfsPerceptionandImpactofpHonDetergentin AutomaticWashingMachineBasedonFuzzyLogicController.ScientificInquiryandReview,2(3),40-52.

20. Saeed,M.,Mahmood,S.,andTaufeeq,H.HybridizationofSoftExpertSetwithAHPinDecisionMaking. InternationalJournalofComputerApplications,975,8887.

21. Saeed,M.,Anam,Z.,Kanwal,T.,Saba,I.,Memoona,F.,andTabassum,M.F.(2017).Generalizationof TOPSISfromsoftsettoFuzzysoftsetsindecisionmakingproblem.ScientificInquiryandReview,1(1), 11-18.

22. Saeed,M.,Sana,A.,andRubi,N.(2016).Comparativestudyofairportevaluationproblembyusingfuzzy softexpertsetandAHPtechnique.ScienceInternational,28(3).

23. Zulqarnain,M.,Dayan,F.,andSaeed,M.(2018).TOPSISanalysisforthepredictionofdiabetesbased ongeneralcharacteristicsofhumans.InternationalJournalofPharmaceuticalSciencesandResearch,9(7), 2932-2939.

24. Karaaslan,F.(2016).Softclassesandsoftroughclasseswithapplicationsindecisionmaking.Mathematical problemsinengineering,2016.

25. Kharal,A.,andAhmad,B.(2009).Mappingsonfuzzysoftclasses.AdvancesinFuzzySystems,2009.

26. Kharal,A.,andAhmad,B.(2011).Mappingsonsoftclasses.NewMathematicsandNaturalComputation, 7(03),471-481.

27. Alkhazaleh,S.,andMarei,E.(2014).Mappingsonneutrosophicsoftclasses.NeutrosophicSetsandSystems, 2,3-8.

28. Sulaiman,N.H.,andMohamad,D.(2014,June).Mappingsonmultiaspectfuzzysoftclasses.InAIP ConferenceProceedings(Vol.1602,No.1,pp.716-722).AmericanInstituteofPhysics.

29. Bashir,M.,andSalleh,A.R.(2013,April).Mappingsonintuitionisticfuzzysoftclasses.InAIPconference proceedings(Vol.1522,No.1,pp.1022-1032).AmericanInstituteofPhysics.

30. Borah,M.J.,andHazarika,B.(2016).CompositemappingonhesitantfuzzySoftClasses.arXivpreprint arXiv:1605.01304.

31. Smarandache,F.(2018).Extensionofsoftsettohypersoftset,andthentoplithogenichypersoftset. NeutrosophicSetsSyst,22,168-170.

32. Saeed,M.,Ahsan,M.,Siddique,M.K.,andAhmad,M.R.(2020).Astudyofthefundamentalsofhypersoft settheory.InternationalJournalofScientificandEngineeringResearch,11,2229-5518.

33. Rana,S.,Qayyum,M.,Saeed,M.,Smarandache,F.,andKhan,B.A.(2019).Plithogenicfuzzywhole hypersoftset,constructionofoperatorsandtheirapplicationinfrequencymatrixmultiattributedecision makingtechnique.NeutrosophicSetsandSystems(NSS).

34. Saqlain,M.,Jafar,N.,Moin,S.,Saeed,M.,andBroumi,S.(2020).Singleandmulti-valuedneutrosophic hypersoftsetandtangentsimilaritymeasureofsinglevaluedneutrosophichypersoftsets.NeutrosophicSets andSystems,32,317-329.

35. Saqlain,M.,Moin,S.,Jafar,M.N.,Saeed,M.,andSmarandache,F.(2020).Aggregateoperatorsof neutrosophic hypersoft set, Neutrosophic Sets and Systems, 32, 294-306.

Development of Rough Hypersoft Set with Application in Decision Making for the Best Choice of Chemical Material

1

1,2,3 Department of Mathematics, University of Management and Technology Lahore, Pakistan. E-mail: aurkhb@gmail.com , E-mail: f2019109046@umt.edu.pk, E-mail: muhammad.saeed@umt.edu.pk , E-mail: rayeesmalik.ravian@gmail.com , E-mail: f2019109047@umt.edu.pk

Abstract: Hypersoft set is the generalization of soft set as it replaces single set of distinct attributes with their corresponding disjoint attribute valued sets. This makes the hypersoft set more effective and useful parameterized tool to tackle vague information. Rough soft set, the combination of rough set and soft set, is studied by many researchers to undertake imperfect knowledge through lower and upper approximations. But it is inadequate to deal with disjoint attribute valued sets. In this study, this meagerness is addressed and rough hypersoft set is conceptualized with the development of its lower and upper approximations. Moreover, some of its elementary properties and results are discussed. A new algorithm is proposed to solve decision making problems and is demonstrated with illustrative examples.

Keywords: Soft set, Rough set, Rough soft set, Hypersoft set, Rough hypersoft set.

1. Introduction

Zadeh’s theory of fuzzy sets [1] is considered as mathematical tool to undertake many convoluted problems involving various uncertainties in different branches of mathematical sciences. But this theory is unable to solve these problems successfully due to the meagerness of the parameterization tool. This inadequacy is addressed by Molodtsov known as soft set theory [2] which is free from all such hindrances and appeared as a new parameterized family of subsets of the universe of discourse. Rough set theory [3-4] is assumed to be a novel mathematical device to tackle imperfect knowledge and vagueness through approximations. Rough soft set, a combination of soft set and rough set [5] was proposed to ensure the adequacy of rough set for parameterization purpose. This is achieved by developing lower and upper approximations for soft set. Rough soft set is studied by many researchers [6-11] through different approaches with applications in certain fields. The concept of hypersoft set as a generalization of soft set was proposed by [14]. Saeed

Atiqe Ur Rahman

∗, Abida Hafeez1, Muhammad Saeed2, Muhammad Rayees Ahmad3 , Ume-e-Farwa3

etal.[15]extendedtheconceptanddiscussedthefundamentalsofhypersoftsetsuchashypersoftsubset,complement,nothypersoftset,aggregationoperatorsalongwithhypersoft setrelation,subrelation,complementrelation,function,matricesandoperationsonhypersoft matrices.Mujahidetal.[16]discussedbasicingrediantsfortopologicalstructuresonthecollectionofhypersoftsets.Moreovertheyintroducedhypersoftpointsindifferentenvorinments likefuzzyhypersoftset,intuitionisticfuzzyhypersoftset,neutrosophichypersoft,plithogenic hypersoftset.Rahmanetal.[17]definedcomplexhypersoftsetanddevelopedthehybrids ofhypersoftsetwithcomplexfuzzyset,complexintuitionisticfuzzysetandcomplexneutrosophicsetrespectively.Theyalsodiscussedtheirfundamentalsi.e.subset,equalsets,nullset, absolutesetetc.andtheoreticoperationsi.e.complement,union,intersectionetc.Rahman etal.[18]conceptualizedconvexitycumconcavityonhypersoftsetandpresenteditspictorial versionswithillustrativeexamples.

Havingmotivationfromtheworkdescribedin[5],[14],[15]and[16],wedeveloproughhypersoftsetalongwithitssomefundamentalproperties.Furtherweproposeanewalgorithmto solvedecisionmakingproblemsandapplyitforthebestchoiceofchemicalmaterials.This workistheextensionofexistingconcept[5]inthesensethatitdealswithattributevalued setsinsteadoftakingjustattributes. Therestofthepaperisorganizedas:Section2recallsbasicdefinitionsfromliterature.Section 3presentshypersoftupperset,hypersoftlowersetandroughhypersoftset.Ittoopresents someimportantpropertiesandresultsofroughhypersoftset.Section4presentsapplication ofroughhypersoftsetindecisionmaking.Section5finallyconcludesthepaper.

2. Preliminaries

Herewerecallsomesupportingbasicdefinitionsandconceptsfromexistingliterature.

Definition2.1. [3]Arelation R on A = ∅ iscalledan equivalencerelation onAifitis reflexive,symmetricandtransitive.Wedenotetheequivalenceclassofanelement y ∈ S with respectto R as R[y]and R[y]= {z ∈ A : zRy}

Definition2.2. [3]Aset W ⊆ U issaidtobea Roughset w.r.t R on U,if βR(y)= R∗(y) R∗(y) isnonempty,where R∗(y)= {y : R[y] ⊆ W } and R∗(y)= {y : R[y] ∩ W = ∅} arelower andupperapproximationofWrespectively.Formoredefinitionsandtheoreticoperationson roughset,see[2].

Definition2.3. [2] Apair(ψ,A)iscalleda softset over U ,if A ⊂ E and ψ : A → P (U).Wewrite ψA for(ψ,A).

Definition2.4. [12]

Let ψA and ϕB besoftsetsoveracommonuniverseset U and A,B ⊂ E.Thenwesaythat

• ψA isa softsubset of ϕB ,denotedby ψA ⊂ ϕB ,if (1) A ⊂ E and (2) ψ(e) ⊂ ϕ(e)∀ e ∈ A

• ψA = ϕB ,denotedby ψA = ϕB if ψA ⊂ ϕB and ϕB ⊂ ψA Formoredefinitionsandoperationsofsoftset,see[2],[12]and[13].

Definition2.5. [5]Let ψE beasoftsetand R beanequivalencerelationon U thenwedefine twosoftsets → ψ : E → P (U)and ← ψ : E → P (U)asfollows. → ψ( e)= ∗ R(ψ(e)) and ← ψ( e)= R ∗ (ψ(e))

forevery e ∈ E. Wecallthesoftsets → ψ E and ← ψ E ,the uppersoftset andthe lowersoftset respectively.

Definition2.6. [5] Wesaythatasoftset ψE ,isa Roughsoftset ifthedifference → ψ E \ ← ψ E isanon-nullsoftset.

Definition2.7. [5] Bya Crispsoftset,wemeanasoftset ψE suchthat → ψ E ∼ = ← ψ E

Definition2.8. [14] Let Ai,i =1, 2,...,n,aredisjointsetshavingattributivevaluesofdistinctattributes ai,i = 1, 2,...,n.Thenthepair(ψ,G),where G = A1 × A2 × A3 × × An and ψ : G → P (U),is calleda hypersoftSet over U.Wesimplyuse ψG forhypersoftset. Moredefinitionsandexamplescanbeseenfrom[14]-[28].

3. RoughHypersoftSet

Inthissection,thelowerhypersoftset,upperhypersoftsetandroughhypersoftsetsare defined.Someimportantresultsareestablishedtoo.

Definition3.1. LetRbeanequivalencerelationon U and ψG isahypersoftsetthenwe definetwohypersoftsets → ψ : G → P (U)and ← ψ : G → P (U)asfollows. → ψ( e)= ∗ R( ψ(e))

and ← ψ( e)= R ∗ (ψ(e))

forevery e =(e1,e2,e3,...,en) ∈ G. Wecallthehypersoftsets → ψ G and ← ψ G ,the upperhypersoft set andthe lowerhypersoftset respectively.

Definition3.2. Wesaythatahypersoftset ψG ,a Roughhypersoftset ifthedifference → ψ G \ ← ψ G isanon-nullhypersoftset.

Definition3.3. Bya Crisphypersoftset,Wemeanahypersoftset ψG suchthat → ψ G ∼ = ← ψ G .

Theorem3.4. If ψG isahypersoftsetover U theuniversesetthen ← ψ G ⊆ ψG ⊆ → ψ G . Proof. Let ψG beahypersoftsetovertheuniverseset U andRbeanequivalencerelationon U.Let x ∈ R∗(ψ(e))then R[x] ⊆ ψ(e)for e ∈ G and R[x] ∩ ψ(e)= R[x] = ∅

so x ∈ R∗(ψ(e))whichimplies

R∗(ψ(e)) ⊆ R∗(ψ(e))(1) forevery e ∈ G

If x ∈ R∗(ψ(e))then R[x] ⊆ ψ(e)so x ∈ ψ(e) whichimplies R∗(ψ(e)) ⊆ ψ(e)(2) forevery e ∈ G. Let x ∈ ψ(e),then x ∈ R(x) ∩ ψ(e)so R[x] ∩ ψ(e) = ∅ whichimplies x ∈ R∗(ψ(e)) therefore, ψ(e) ⊆ R∗(ψ(e))(3) forevery e ∈ G. Fromequations(1),(2)and(3),wehave R∗(ψ(e)) ⊆ ψ(e) ⊆ R∗(ψ(e)) forevery e ∈ G Itfollowsthat ← ψ G ⊆ ψG ⊆ → ψ G .

Hencethetheorem.

Proposition3.5. If ΦG isthenullhypersoftsetand XG istheabsolutehypersoftset,defined by ΦG = ∅ and XG = U forevery e ∈ G,then (1) ← ΦG

Proposition3.6. If ψG and ϕG areanytwohypersoftsetssuchthat ψ(e) ∼ ⊆ ϕ(e) then (1) ← ψ(e)

Proposition3.7.

betwohypersoftsetsthen

4. ApplicationofRoughHypersoftsetinDecisionMaking

Hereanewdecisionmakingmethodisconstructedforroughhypersoftsetstofind ψ(e), theappropriatematerialon ψG w.r.tRon U andisillustratedbytwoexamples. Let ψG =(ψ,G)beanoriginaldescriptionhypersoftsetover U.Theproposedalgorithmis asunder:

Table1. tableforweightedhypersoftset ψG U e1 e2 e3 e4 w1 w2 w3 w4 m1 1100 m2 1010 m3 0101 m4 1110 m5 0010 m6 0001 m7 0001 m8 0101

Step1: Take ψG asinput. Step2: Compute → ψ G and ← ψ G on ψG,respectively. Step3: Computethedifferentvaluesof ψ(ei) ,where ψ(ei) = | → ψ G (ei)|−| ← ψ G (ei)| |ψ(ei| × wi where wi areassignedweights. Step4: Findtheminimumvalue ψ(el) of ψ(ei) ,where ψ(el) = minii ψ(ei) . Step5: Theoutputis ψ(el)

Example4.1. Let U = {m1,m2,m3,m4,m5,m6,m7,m8} bethematerials.Adrugmanufacturingcompanyintendstoselectthemostrelevantmaterialforitsspecificmoleculestructure. Therelevancyofthesematerialsonbasisofchemicalpropertiesisasunder: m1 ∼ m2, m3 ∼ m4 ∼ m5,and m6 ∼ m7 ∼ m8,where ∼ isforclosematching. Now,therearefourmaterialstobedetected,denotedby G = {e1 =(e11,e21),e2 = (e11,e22),e3 =(e12,e21),e4 =(e12,e22)}, where G = E1 × E2 with E1 = {e11,e12} and E2 = {e21,e22} Andtheiringredientsarerepresentedby ψ(e1)= {m1,m2,m4},ψ(e2)= {m1,m3,m4,m8},ψ(e3)= {m2,m4,m5},and ψ(e4)= {m3,m6,m7,m8},respectively. LetfollowingweightsareallocatedtoeachelementinG: w1 =0 7isfor e1, w2 =0 5isfor e2, w3 =0.6isfor e3,and w4 =0 3isfor e4.Thenwecanfindthetable(4)forweightedhypersoftset ψG.According

Table2. tableforhypersoftset ← ψ G .

U e1 e2 e3 e4 w1 w2 w3 w4 m1 1100 m2 1000 m3 0100 m4 0100 m5 0000 m6 0001 m7 0001 m8 0101

Table3. tableforhypersoftset → ψ G U e1 e2 e3 e4 w1 w2 w3 w4 m1 1110 m2 1110 m3 1111 m4 1111 m5 1111 m6 0101 m7 0101 m8 0101

totheproposedmethodandbydefinition(3.1),wehavetwohypersoftsets → ψ G and ← ψ G over U TheyarepresentedinTables(6)and(5). Nowbyapplyingproposedalgorithm,wehave, ψ(e1) =0 7, ψ(e2) =0 75, ψ(e3) =1.0,and ψ(e4) =0 215.

Thustheminimumvaluefor ψ(ei) is ψ(e4) =0.215. Whichimpliesthat ψ(e4)isthemostappropriateandsuitableone.

Example4.2. Acompanyisworkingoncancerdrugdiscovery.Somematerialscanbechecked againstatargetonthebasisofmolecularstructure.Thematerialsunderconsiderationare givenistheuniverse U = {M1,M2,M3,M4,M5,M6,M7,M8}. TheCompanywantstochoose thebestonematerialbyobservingfollowingattributesofcancer

Table4. Tableforweightedhypersoftset ψG U e1 e2 e3 e4 w1 w2 w3 w4 M1 0.70.6100 M2 0.800.40 M3 00.8500.73 M4 0.50.430.60 M5 000.510 M6 0000.49 M7 0000.8 M8 00.9200.67

a1 = Tumors,a2 = Fatigue.Theattributevaluesoftheattributesaregivenintheset correspondinglyas, E1 = {benign(e11), malignant(e12)}, E2 = {mildfatigue(e21), severefatigue(e22)}. Intermsoftheirchemicalproperties,weregardas M1 ∼ M2, M3 ∼ M4 ∼ M5, and M6 ∼ M7 ∼ M8, where ∼ representthechemicalpropertiesofthesematerialsareequivalent. Now,Setofmaterialstobeunderobservation,is G = {e1 =(e11,e21),e2 =(e11,e22),e3 =(e12,e21),e4 =(e12,e22)}, whereeach G = E1 × E2 with E1 = {e11,e12} and E2 = {e21,e22} Andeachkindofmaterial containing ψ(e1)= {M1,M2,M4},ψ(e2)= {M1,M3,M4,M8},ψ(e3)= {M2,M4,M5},and ψ(e4)= {M3,M6,M7,M8},respectively.

LetallocationofweightsbythecompanytoeachelementofG: w1 =0 81isassignedto e1, w2 =0 73isassignedto e2, w3 =0.66isassignedto e3,and w4 =0 45isassignedto e4 Thenwehavetables(4)forweighted ψG.And → ψ G and ← ψ G over U,arepresentedinTables(6) and(5),respectively.Thenwehave ψ(e1) =0 2872, ψ(e2) =0 1903, ψ(e3) =1.0372,and ψ(e4) =0 1422.

Table5. Tableforhypersoftset ← ψ G .

U e1 e2 e3 e4 w1 w2 w3 w4 M1 0.70.6100 M2 0.8000 M3 00.8500 M4 00.4300 M5 0000 M6 0000.49 M7 0000.8 M8 00.9200.67

Table6. Tableforhypersoftset → ψ G .

U e1 e2 e3 e4 w1 w2 w3 w4 M1 0.70.610.70 M2 0.80.50.40 M3 0.60.850.810.73 M4 0.50.430.60.63 M5 0.670.30.510.5 M6 00.700.49 M7 00.6500.8 M8 00.9200.67

Thustheminimumvaluefor ψ(ei) is ψ(e4) =0 1422. Thismeansthat ψ(e4)istheexpectedmaterialforthedrugdiscoveryofcancerdisease.

5. Conclusion

Inthisstudy,roughhypersoftsetisconceptualizedwiththedevelopmentofitsrelevant lowerandupperroughapproximations.Someinterestingresultsandpropertiesarediscussed. Moreover,anapplicationindecisionmakingisdiscussedwiththehelpofillustrativeexamples. Thisworkmayhelptheresearcherstoextendtheworkforotherfuzzy-likestructuresi.e. Interval-valuedfuzzy,Intuitionisticfuzzy,Pythagoreanfuzzy,sphericalfuzzy,neutrosophic fuzzyetc.sothatimperfectinformationmaybehandledproperlywithattributevalued setsthroughlowerandupperapproximations.Theproposeddecisionmakingalgorithmmay furtherbeappliedtootherdailylifeproblems.

References

[1] Zadeh,L.A.,Fuzzysets, Inform.Control. 1965,8,338-353.

[2] Molodtsov,D.,SoftSetTheory-FirstResults, ComputerandMathematicswithApplications 1999,37, 19-31.

[3] Pawlak,Z.,RoughSets, InternationalJournalofComputerandInformationSciences 1982,11,341-356.

[4] Pawlak,Z.,RoughSetsTheoreticalAspectsofReasoningaboutData, KluwerAcademicPublishers,Dordrecht 1991

[5] Moinuddin,K.,RoughSoftSets:AnovelApproach, InternationalJournalofComputationalandApplied MathematicsandSystems 2017,12(2),537-543.

[6] Mohana,K.,andMohanasundari,M.,OnSomeSimilarityMeasuresofSingleValuedNeutrosophicRough Sets, NeutrosophicSetsandSyst. 2019,24,10-22.

[7] Marei,E.,Singlevaluedneutrosophicsoftapproachtoroughsets,theoryandapplication, Neutrosophic SetsSyst 2018,20,76-85.

[8] Hashim,H.,Abdullah,L.,andAl-Quran,A.,Intervalneutrosophicvaguesets, NeutrosophicSetsandSyst. 2019,25,66-75.

[9] Samuel,A.E.,andNarmadhagnanam,R.,Pi-DistanceofRoughNeutrosophicSetsforMedicalDiagnosis, NeutrosophicSetsandSyst. 2019,28(1),6.

[10] Alias,S.,Mohamad,D.,andShuib,A.,Roughneutrosophicmultisetsrelationwithapplicationinmarketing strategy, NeutrosophicSetsSyst. 2018,21,36-55.

[11] Polkowski,L.,RoughSets, MathematicalFoundations,SpringerVerlag,Berlin 2002.

[12] Maji,P.K.,Biswas,R.,andRoy,A.R.,Softsettheory, ComputerandMathematicswithApplications 2008,45,3029-3037.

[13] Li,F.,NotesontheSoftOperations, ArpnJournalofSystemandSoftware 2011,66,205-208.

[14] Smarandache,F.,ExtensionofSoftSettoHypersoftSet,andthentoPlithogenicHypersoftSet, NeutrosophicSetsandSystems 2018,22,168-170.

[15] Saeed,M.,Ahsan,M.,Khubab,M.S.,Ahmad,M.R.,AStudyofTheFundamentalsofHypersoftSet Theory, InternationalJournalofScientificandEngineeringResearch 2020,11.

[16] Abbas,M.;Murtaza,G.;andSmarandache,F.,Basicoperationsonhypersoftsetsandhypersoftpoint, NeutrosophicSetsSyst. 2020,35,407-421.

[17] Rahman,A.U.,Saeed,M.,Smarandache,F.,andAhmad,M.R.,DevelopmentofHybridsofHypersoftSet withComplexFuzzySet,ComplexIntuitionisticFuzzysetandComplexNeutrosophicSet, Neutrosophic SetsandSyst. 2020,38,335-354.

[18] Rahman,A.U.,Saeed,M.,andSmarandache,F.,ConvexandConcaveHypersoftSetswithSomeProperties, NeutrosophicSetsandSyst. 2020,38,497-508. setandTangentSimilarityMeasureofSinglevaluedNeutrosophicHypersoftSets, Neutrosophicsetsand

[19] Saqlain, M., Jafar, N., Moin, S., Saeed, M., Broumi, S. (2020). Single and Multi-valued Neutrosophic Hypersoft Systems, 32, 317-329

[20] Saqlain, M.; Moin, S.; Jafar, M. N.; Saeed, M., and Smarandache, F. (2020). Aggregate Operators of Neutrosophic Hypersoft Set, Neutrosophic Sets Systems, 32, 294-306.

[21] Saqlain, M.; Saeed, M.; Ahmad, M. R., and Smarandache, F. (2019). Generalization of TOPSIS for Neutrosophic Hypersoft set using Accuracy Function and its Application, Neutrosophic Sets Systems, 27, 131-137.

[22] Martin,N.;Smarandache,F.,ConcentricPlithogenicHypergraphbasedonPlithogenicHypersoftsetsA NovelOutlook, NeutrosophicSetsSyst. 2020,33(1),5.

[23] Martin,N.;Smarandache,F.,IntroductiontoCombinedPlithogenicHypersoftSets, InfiniteStudy (2020)

[24] Rana,S.;Qayyum,M.;Saeed,M.;Smarandache,F.;Khan,B.A.,PlithogenicFuzzyWholeHypersoftSet, ConstructionofOperatorsandtheirApplicationinFrequencyMatrixMultiAttributeDecisionMaking Technique, InfiniteStudy (2019).

[25] Martin,N.;Smarandache,F.,ConcentricPlithogenicHypergraphbasedonPlithogenicHypersoftsetsA NovelOutlook, NeutrosophicSetsSyst. (2020),33(1),5.

[26] Saqlain,M.,;Xin,X.L.,IntervalValued,m-Polarandm-PolarIntervalValuedNeutrosophicHypersoft Sets, NeutrosophicSetsSyst. (2020),36,389-399.

[27] Gayen,S.;Smarandache,F.;Jha,S.;Singh,M.K.;Broumi,S.;Kumar,R.,Introductiontoplithogenic hypersoftsubgroup, NeutrosophicSetsSyst. (2020),33(1),14.

[28] Zulqarnain,R.M.;Xin,X.L.;Saqlain,M.;Smarandache,F.,GeneralizedAggregateOperatorsonNeutrosophicHypersoftSet, NeutrosophicSetsSyst. (2020),36(1),20.

A Novel Approach to the Rudiments of Hypersoft Graphs

1,2,3 Department of Mathematics, University of Management and Technology Lahore, Pakistan.

E-mail: muhammad.saeed@umt.edu.pk , E-mail: khubabsiddique@hotmail.com , E-mail: ahsan1826@gmail.com , E-mail: rayeesmalik.ravian@gmail.com , E-mail: aurkhb@gmail.com

Abstract: Graph theory, soft computing and machine learning are being used in our daily life problems in the field of science involving mathematics, optimization and decision sciences. Zadeh introduced the idea of fuzzy set. This idea helped Kauffman to present the concept of fuzzy graph. Molodtsov presented the idea of soft set. Using this concept, Thumbakara et al. discovered a novel idea of soft graphs and Akram et al. discussed the fundamentals of soft graphs. Smarandache conceptualized the hypersoft set (HS) which is the generalization of soft set. Hypersoft set transforms single attribute function to multi-attribute function. In this study, the existing concept of soft graph is extended to HS-graph and some of its rudiments like HS-subgraph, not HS-graph, HS-complete graph, HS-tree, etc., are conceptualized with the help of graphical representation and illustrative examples. Moreover, some theoretic operations are discussed with generalized results on hypersoft set. This study will help the researchers in multi-dimensional fields involving artificial intelligence (AI), soft computing, graph theory and networking, data sciences, etc.

Keywords: Soft set, Soft graph, Hypersoft set (HSS), Hypersoft graph (HS-graph), Hypersoft complete graph (HSC-graph), Hypersoft subgraph (HS-subgraph), Hypersoft tree (HS-tree).

1. Introduction

The world is full of uncertainties, complexities of incomplete network-based information with vague data. To deal with such kind of uncertain environment, there must be a tool which handles these kinds of problems very efficiently. To overcome this problem, Zadeh [1] introduced the idea of fuzzy set in 1965. This idea was motivated Kauffman, who presented the concept of fuzzy graphs using the notion of fuzzy set. Many researchers worked in this area and implement the new concept to solve real life problems. Poulik et al. [3–7] has many researches on bipolar fuzzy graphs in different environments and presented many applications in diverse fields of sciences.

Muhammad Saeed 1, Muhammad Khubab Siddique 2, Muhammad Ahsan 2 , Muhammad Rayees Ahmad 3∗, Atiqe Ur Rahman 3

Molodtsov[8]introducedanovelnotionofsoftsettheoryin1999.Heusedtheideaofsoft settheoryinseveralareasofmathematicsincludingRiemannintegration,theoryofprobability, Perronintegration,smoothnessoffunctionseffectively.Majietal.[9]presentedfundamental studyonsoftsetsincludingsubsets,superset,equationsandoperationsofsoftsetcontaining union,intersection,complementandmoreamongothersin2003.Peietal.[10]described therelationshipwithinformationsystemandsoftsetsin2005.Alietal.[11]presentedsome newoperationsincludingrestrictedunion,restrictedintersectionandtheirbasicscharacteristicsin2009.Babithaetal.[12]describedtheconceptofsoftsetrelation,softsetfunction, equivalencesoftset,orderingonsoftsets,softsetsdistribution,anti-symmetricrelationand transitiveclosurein2010.Nagarajanetal.[13]studiedthealgebraicstructureofsoftset theory.

Therearelotofnetworkapplicationsinwhichthebulkofdataisavailablebutnotool gavethesolutiontotheproblemproperly.Toovercomethissituation,Thumbakaraetal.[14] presentedtheconceptofsoftgraphasaparameterizedfamilyofgraphs.Havingmotivation fromthiswork,Thengeetal.[15]contributedfurthertosoftgraphsincludingtabularrepresentationofsoftgraph,center,degree,radiusanddiameterofsoftgraph.Akrametal.[16] explainedtheconceptofvertexandedgeinducedsoftgraph.Ratheesh[17]workedonsoft graphsandtheirapplications.

Smarandache[18],apioneerofhypersoftsettheory,generalizedthesoftsettohypersoft setin2018.Heintroducedthehybridsofhypersoftsetwithothersets.Fundamentalsof hypersoftsetsuchashypersoftsubset,complement,notstandardaggregationoperatorshave beendiscussedin[19].Rahmanetal.[20]definedcomplexhypersoftsetanddevelopedthe hybridsofhypersoftsetwithcomplexfuzzyset,complexintuitionisticfuzzysetandcomplex neutro-sophicsetrespectively.Theyalsodiscussedtheirfundamentalsi.e.subset,equalsets, nullset,absolutesetetc.andtheoreticoperationsi.e.complement,union,intersectionetc. Rahmanetal.[21]conceptualizedconvexitycumconcavityonhypersoftsetandpresentedits pictorialversionswithillustrativeexamples.

Havingmotivationfrom[17],[18]and[19],thenovelnotionsofhypersoftgraphs,hypersoft subgraph,nothypersoftgraph,hypersoftcompletegraphandhypersofttreeareproposed. Moreover,someoftheirpropertiesincludingintersectionofhypersoftgraphs,ANDoperation ofhypersoftgraphsarediscussedwiththesupportofgraphicalrepresentationandillustrative examples.

Theoverviewofthispaperispresentedasfollow.

Insection2,somefundamentaldefinitionsarerecalledfromliteraturetosupportmain results.Section3presentstheproposedconceptofhypersoftgraphs,theirpropertiesand someimportantresultswiththeirexamples.Attheend,theconclusionispresentedinsection 4.

2. Preliminaries

Heresomeimportantandmostrelevantdefinitionsarerecalledfromliteratureregarding softsetsandsoftgraphforclearunderstanding.

Definition2.1. [8]

Let χ bethenonemptyfiniteuniverse, T bethesetofattributesand S ⊆T .Apair(γ,S) representsa softset over χ suchthat γ : S → P (χ).

Definition2.2. [12] Let(γ1,J )and(γ2,K)betwosoftsetsoverthesameuniverse χ.

A softrelation R :(γ1,J ) → (γ2,K)is(R,L)with L ⊆ J × K, ∀ (x,y) ∈ L,thereexist R(x,y)= H(x,y)suchthat(H,J × K)=(γ1,J ) × (γ2,K).

Definition2.3. [12]

Asoftrelation(R,L)iscalleda softfunction,ifeveryelementofdomainof R,aunique elementinrangeof R.If γ1(x)Rγ2(y)for x ∈ J and y ∈ K thenitcanberepresented R(γ1(x))= γ2(y).

Formoredetail,see[8,9,11,12].

Definition2.4. [14]

Supposethat G =(V, T )representsthegraphsuchthat V and T representsthesetofvertices andedgesofthegraphrespectively.Consider J ⊆V suchthat J = φ and R isanarbitrary relationsuchthat R ⊆ J ×V.Letasoftsetis(γ,S),then softgraph of G isrepresentedby (F,J ),ifthesubgraphinducedin G by F (x),impliesthat F (x)isaconnectedsubgraphof G, ∀ x ∈ J.

Definition2.5. [14]

Let G1 =(F1,K1,J )and G2 =(F2,K2,K)betwosoftgraphsof G.Then G2 isa softsub graph of G1 if

(1) K ⊆ J.

(2) H2(x)issubgraphof H1(x), ∀ x ∈ K

Definition2.6. [14]

Let(F,J )beasoftgraphof G,then(F,J )representsthe softtree Ft(x), ∀ x ∈ J

Formoredefinitions,see[14–17].

Definition2.7. [18]

Let T1, T2, T3,..., Tn ben-distinctattributeswhoseattributevaluesbelongstodisjointsets J1,J2,J3,...,Jn respectively.Apair(γ,J )iscalleda hypersoftset overtheuniversalset χ, where γ isthemappinggivenby γ : J → P (U )suchthat J = J1 × J2 × J3 × × Jn

Definition2.8. [19]

Let(γ1,J )and(γ2,K)bethetwohypersoftsetsoverthesameuniversalsets χ,where J = J1 × J2 × J3 × ... × Jn and K = K1 × K2 × K3 × ... × Kn.Thentherelationfrom(γ1,J ) to(γ2,K)iscalleda hypersoftsetrelation ( ˇ R,W )oritisinsimpleway ˇ R isahypersoft subsetanditisdenotedby(γ1,J ) × (γ2,K),where W ⊆ J × K and ∀ (x,y) ∈ W ,implies that R(x,y)= H(x,y),where x =(x1,x2,x3,...,xn) ∈ J and y =(y1,y2,y3,...,yn) ∈ K and (H,J × K)=(γ1,J ) × (γ2,K).

Definition2.9. [19] Let(γ1,J )and(γ2,K)bethetwohypersoftsetsoverthesameuniversalsets χ. Thenthe hypersoftrelationfrom(γ1,J )to(γ2,K)definedas η :(γ1,J ) → (γ2,K)iscalleda hypersoft setfunction,ifeveryelementofdomainhasuniqueelementinrangeof γ.Ifitisclosed γ1(x)ηγ2(y)i.e γ1(x) × γ2(y) ∈ η for x ∈ J and y ∈ K,thenwecanrepresentitintheform η(γ1(x))= γ2(y).

FormoredetailregardingHS-set,see[18,19].

3. HypersoftGraph

Inthissection,wepresentstheHS-set,HS-graph,HS-intersection,ANDoperation,HSC graph,HS-subgraph,HS-tree,andtheirfundamentalsalongwiththeirrelatedimportantresults.

Forthis,let G =(V, T )beasimpleconnectedgraph.Let J = J1 × J2 × J3 × ... × Jn and Ji ⊆V with i =1, 2,...,n as Ji ∩ Jj = φ,i = j

Definition3.1. Let R beanyrelationwith J and V i.e. R ⊆ J ×V.Afunction F : J → P (V) representedas F (x)= {y ∈V| xRy}, ∀ x ∈ J. Thepair(F,J )ishypersoftsetover V

Figure1. HS-Graph

Definition3.2. AHS-set(F,J )of G over V,representsthe HS-graph,if F (x)isaconnected subgraphof G,inducedby F (x), ∀ x ∈ J.AsetcontainingalltheHS-graphsisdenotedby HsG(G).

Example3.3. Consider G =(V, T )asinfigure1.Wehave V = {v1,v2,v3,v4,v5}.Let J = {J1,J2},K = {K3,K4} and L = {L5} then J × K × L = {(J1,K3,L5), (J2,K3,L5), (J1,K4,L5), (J2,K4,L5)} = {x1,x2,x3,x4}. R = {(x1,v2), (x1,v4), (x2,v1), (x2,v4), (x3,v2), (x3,v3), (x4,v1), (x4,v3)} Then

F (ˇ x)= F (Ji,Jj ,Jk ))= {ˇ y ∈ V |ˇ xRˇ y ⇔ ˇ y/∈{vi,vj ,vk }} and

F (x1)= F ((J1,J3,J5))= {v2,v4}, F (x2)= F ((J2,J3,J5))= {v1,v4},

F (x3)= F ((J1,J4,J5))= {v2,v3}, F (x4)= F ((J2,J4,J5))= {v1,v3}.

Hence(F,A) ∈ HsG(G).

Example3.4. Suppose G =(V, T ),representedasfigure2.Wehave V = {v1,v2,v3,v4,v5}. Let J = {v1},K = {v2,v3} and L = {v4,v5} then M = J × K × L = {(v1,v2,v4), (v1,v2,v5), (v1,v3,v4), (v1,v3,v5)} = {x1,x2,x3,x4}, R = {(x1,v2), (x1,v4), (x2,v1), (x2,v4), (x3,v2), (x3,v3), (x4,v1), (x4,v3)}⊆ J × V Then F (x)= F ((vi,vj ,vk ))= {y ∈ V |xRy → y/∈{vi,vj ,vk }}

Figure2. NotHS-Graph

Figure3. IntersectionofHS-Graphs and

F (x1)= F ((v1,v2,v4))= {v3,v5},

F (x2)= F ((v1,v2,v5))= {v3,v4},

F (x3)= F ((v1,v3,v4))= {v2,v5},

F (x4)= F ((v1,v3,v5))= {v2,v4}

As F ((v1,v3,v4))= {v2,v5} isnotaconnectedsubgraphof G.Hence(F,J ) / ∈ HsG(G).

Proposition3.5. EverysimplegraphisHS-graph.

Proof. Let G =(V, T )beasimpleconnectedgraphand T = ∅.Let J = J1 × J2 × J3 × × Jn, where Ji = {vi} for i =1, 2,...,n andafunctionis F : J −→ P (V )as F ((v1,v2,...,vn))= {y ∈ V | y =endvertexofedge T} So,bydefinitionof F (x),wehave,forall x ∈ A, (F,J ) ∈ HsG(G).

Remark3.6. TheintersectionofHS-graphsisnotnecessarilyaHS-graph.

Example3.7. Let G =(V, T )asgiveninfigure2.Let J = J1 ×J2 ×J3,where J1 = {v1},J2 = {v2},J3 = {v3}.

⇒ F ((v1,v2,v3))= {y ∈ V |xRy ⇔ min{d(v1,y),d(v2,y),d(v3,y)}≤ 1}

Figure4. ANDoperationofHS-Graphs

⇒ F ((v1,v2,v3))= {v1,v2,v3,v4,v5}.

Supposethefunction H by H((v1,v2,v3))= {y ∈ V |xRy ⇔ min{d(v1,y),d(v2,y),d(v3,y)}≥ 1}

⇒ H((v1,v2,v3))= {v4,v5,v6}

⇒ F ((v1,v2,v3)) ∩ H((v1,v2,v3))= {v4,v5} thatisnotinducedconnectedsubgraphof G

Proposition3.8. Let (F,J ) and (H,K) ∈ ShG(G) with J ∩ K = ∅,then F (x) ∩H(x) ∀ x ∈ J ∩ K isaconnectedsubgraphof G suchthat

(F,J ) ∩ (H,K) ∈ HsG(G)

Proof. For(F,J )and(H,K) ∈ HsG(G), wehave

(F,J ) ∩ (H,K)=(χ,C) ,C = J ∩ K = ∅ and χ(T )= F (T ) ∩ H(T ), ∀T∈ C. Let x ∈ C = J ∩ K.Asconsideredby χ(x)= F (x) ∩H(x)isaconnectedsubgraphof G,so (χ,C)=(F,J ) ∩ (H,K) ∈ HsG(G)

Example3.9. Suppose G =(V, T )asinfigure4.Let J = J1 × J2,with J1 = {v1},J2 = {v2}, then J = {(v1,v2)} and K = K1 × K2,with K1 = {v3},K2 = {v4},then K = {(v3,v4)}. Defineafunction F by F ((x,y))= {z ∈V|(x,y)Rz ⇔ z/∈{x,y}}

Figure5. HS-Subgraphs

Then

F ((v1,v2))= {v3,v4,v5,v6}, F ((v3,v4))= {v1,v2,v5,v6}, F ((v1,v2)) ∧ F ((v3,v4))= {v5,v6} whichisnotaninducedconnectedsubgraphof G.Hence(F,J ) ∧ (F,K) / ∈ HsG(G).

Definition3.10. If(F,J )and(H,K)aretwoHS-graphsthen(H,K)betheHS-subgraphof (F,J ),if (1) K ⊆ J, (2) H(x)isaconnectedsubgraphof F (x), ∀ x ∈ K.

Example3.11. Consider G =(V, T )asinfigure5.Let V = {v1,v2,v3,v4} and J = J1 × J2 with J1 = {v1,v2,v3},J2 = {v4}, then J = {(v1,v4), (v2,v4), (v3,v4)} and K = K1 × K2, with K1 = {v1,v2},K2 = {v4},thenK = {(v1,v4), (v2,v4)}⊆ J. Nowdefinefunction F by, F ((x,y))= {z ∈V|(x,y)Rz ⇔ d(z, t) ≤ 1where t ∈{x,y}} Then F ((v1,v4))= {v1,v2,v4},F ((v2,v4))= {v1,v2,v3,v4} and F ((v3,v4))= {v1,v2,v3,v4}

Supposeafunction H : K −→ P (V)by H((x,y))= {z ∈V|(x,y)Rz ⇔ d(z, ˇ t) < 1, where ˇ t ∈{x,y}}

Figure6. HS-Tree

Then H((v1,v4))= {v1,v4} and H((v2,v4))= {v2,v4}.

Therefore(F,J ), (H,K) ∈ HsG(G).Here K ⊆ J and H(x)isaconnectedsubgraphof F (x), ∀ x ∈ K.Hence(H,K)isaHS-subgraphof(F,J ).

Definition3.12. Let(F,J )beaHS-graphof G,then(F,J )representstheHS-tree ∀ x ∈ J

Example3.13. Consider G =(V, T )asinfigure6.Here

V = {v1,v2,v3,v4,v5}.

Let J1 = {v1,v2},J2 = {v3},J3 = {v4,v5},then J = J1 × J2 × J3 = {(v1,v3,v4), (v1,v3,v5), (v2,v3,v4), (v2,v3,v5)}

Defineafunction F : J → P (V)by F (x)=(vi,vj ,vk )= {y ∈V|xRy ⇔ y ∈{vi,vj ,vk }}.

Then

F ((v1,v3,v4))= {v2,v5},F ((v1,v3,v5))= {v2,v4},

F ((v2,v3,v4))= {v1,v5}, F ((v2,v3,v5))= {v1,v4}.

Here F (x), ∀x ∈ J isatree.Hence(F,J )isaHS-tree.

Theorem3.14. EverymemberinsetofHS-graphoftreesisalsoaHS-tree.

Proof. Proofisstraightforward.

Remark3.15. AHS-graph(F,J )isaHS-treeiff F (x)isacyclic.

Definition3.16. Let(F,J )beaHS-graph,then(F,J )representsaHSC-graph,ifthegraph F (x)iscompleteforall x ∈ J

Figure7. HSCompleteGraph

Example3.17. Consider G =(V, T )asinfigure7 Here V = {v1,v2,v3,v4,v5}.Let J1 = {v1,v2},J2 = {v3},J3 = {v4},then J = J1 × J2 × J3 = {(v1,v3,v4), (v2,v3,v4)}

Thenwehave

F : J1 × J2 × J3 → P (V) by F (x)= F (vi,vj ,vk )= {y ∈ ˇ V |xRy ⇔ y ∈{vi,vj ,vk }}. Then F ((v1,v3,v4))= {v1,v3,v4},F ((v2,v3,v4))= {v2,v3,v4}. Since F (x)iscomplete ∀ x ∈ J.Therefore(F,J )isaHSC-graph.

Proposition3.18. Let G =(V, T ) isacompletegraph.TheneveryHS-graphin HsG(G) is aHSC-graph.

Proof. Let(F,J ) ∈ HsG(G).Sinceitisknownthateveryinducedsubgraphofacomplete graphiscomplete,therefore F (x)isacompletesubgraphforall x ∈ A.Hence(F,J )isa HSC-graph.

Remark3.19. AHS-graph(F,J )isaHSC-graphiff F (x)iscompleteforall x ∈ J

Example3.20. Forthesamefigure7,ifwetake V and J sameasexample3.17anddefine F : J → P (V)as

F (x)= F (vi,vj ,vk )= {y ∈V|xRy ⇔ y ∈{vi,vj ,vk }} (1) then

F ((v1,v3,v4))= {v2,v5} (2)

F ((v2,v3,v4))= {v1,v5} (3)

Since F ((v2,v3,v4))isnotcomplete,so(F,J ) ∈ HS C(G).Hencetheresultisverified.

4. Conclusions

Mathematicalmodeling,graphtheory,softcomputing,machinelearninginvolvesinour dailylifewhichcontainsuncertainsituationsinthefieldofscience.Inthisstudy,anovel conceptofhypersoftgraphisdevelopedbyextendingtheexistingconceptofsoftgraph.Some ofthefundamentals,propertiesandtheresultsofhypersoftgraphandhypersofttreeare discussedwiththehelpofillustratedexamplesandgraphs.Furtherextensioncanbesought byintroducingnewhybridstructureswithhypersoftgraph.Thisnovelconceptisveryusefulin developingnewalgorithmsinthefieldsofartificialintelligence,machinelearning,networking, softcomputing,etc.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

References

1. ZadehL.A.,Fuzzysets,Inform.Control,Vol.8,pp.338-353,1965.

2. Kauffman,A.,IntroductionalatheoriedesSous-emsemblesFlous,MassonetCie,Vol.1,1973.

3. Poulik,S.,Ghorai,G.,Detourg-interiornodesanddetourg-boundarynodesinbipolarfuzzygraphwith applications,HacettepeJournalofMathematicsandStatistics,Vol.49(1),pp.106-119,2020.

4. Poulik,S.,Ghorai,G.,Certainindicesofgraphsunderbipolarfuzzyenvironmentwithapplications,Soft Computing,Vol.24(7),pp.5119-5131,2020.

5. Poulik,S.,Ghorai,G.Noteon”Bipolarfuzzygraphswithapplications”,Knowledge-BasedSystems,Vol. 192,pp.105315,2020.

6. Poulik,S.,Ghorai,G.,Xin,Q.PragmaticresultsinTaiwaneducationsystembasedIVFGandIVNG.Soft Computing,2020.

7. Poulik,S.,andGhorai,G.Determinationofjourneysorderbasedongraph’sWienerabsoluteindexwith bipolarfuzzyinformation.InformationSciences,Vol.545,pp.608-619,2021.

8. Molodtsov,D.,Softsettheory-firstresults,Computersandmathematicswithapplications,Vol.37,pp. 19-31,1999.

9. Maji,P.K.,Biswas,R.,andRoy,A.R.,Softsettheory,Computersandmathematicswithapplications,Vol. 45,pp.555-562,2003.

10. Pei,D.,andMiao,D.,Fromsoftsettoinformationsystems,ProceedingsofGranularcomputingIEEE, Vol.2,pp.617-621,2005.

11. Ali,M.I.,FengF.,Lui,X.,Min,W.K.,andShabir,M.,Onsomenewoperationsinsoftsettheory, Computersandmathematicswithapplications,Vol.57,no.9,pp.1574-1553,2009.

12. BabithaK.V.,andSunil,J.J.,SoftsetRelationsandfunctions,Computersandmath-ematicswithapplications,Vol.60,no.7,pp.1840-1849,2010.

13. Nagarajan,E.K.R.,Meenambigai.G,Anapplicationofsoftsetstolattices,KragujevacJournalofMathematics,Vol.35,no.1,pp.75-87,2011.

14. Thumbakara,R.K.,George,B.,Softgraphs,GeneralMathematicsNotes,Vol.21,no.2,pp.75-86,2014.

15. Thenge,J.D.,SurendranathR.B.,andRupaliS.J.,ContributiontoSoftGraphandSoftTree,NewMathematicsandNaturalComputation(NMNC),WorldScientificPublishingCo.Pte.Ltd.,Vol.15,no.1,pp. 129-143,2019.

16. Akram,M.,andNawaz,S.,Operationsonsoftgraphs.FuzzyInformationandEngineering,Vol.7,no.4, pp.423-449,2015.

17. Ratheesh,K.P.,OnSoftGraphsandChainedSoftGraphs.InternationalJournalofFuzzySystemApplications,Vol.7,no.2,pp.85-102,2018.

18. Smarandache,F.,ExtensionofSoftSettoHypersoftSet,andthentoPlithogenicHypersoftSet,Neutrosophicsetsandsystems,Vol.22,pp.168-170,2018.

19. Saeed,M.,Ahsan,M.,Siddique,M.K.,andAhmad,M.R.,AStudyofTheFundamentalsofHypersoftSet Theory.InternationalJournalofScientificandEngineeringResearch,Vol.11,no.1,pp.320-329,2020.

20. Rahman,A.U.,Saeed,M.,Smarandache,F.,andAhmad,M.R.,DevelopmentofHybridsofHypersoftSet withComplexFuzzySet,ComplexIntuitionisticFuzzysetandComplexNeutrosophicSet,Neutrosophic SetsandSystems,Vol.38,pp.335-354,2020.

21. Rahman,A.U.,Saeed,M.,andSmarandache,F.,ConvexandConcaveHypersoftSetswithSomeProperties, NeutrosophicSetsandSystems,Vol.38,pp.497-508,2020.

On Neutrosophic Hypersoft Topological Spaces

Taha Yasin Ozturk 1 , Adem Yolcu 2

1 Department of Mathematics, Kafkas University, 36100, Kars, Turkey. E-mail: taha36100@gmail.com

2 Department of Mathematics, Kafkas University, 36100, Kars, Turkey. E-mail: yolcu.adem@gmail.com

Abstract: In this study, we redefine some operations on the neutrosophic hypersoft sets differently from the study [26]. Some basic properties of these operations have been characterized. Under the guidance of these redefined operations, we introduced the neutrosophic hypersoft topological spaces. Finally, on neutrosophic hypersoft topological spaces, we present simple concepts and theorems.

Keywords: Hypersoft sets; neutrosophic hypersoft sets; neutrosophic hypersoft topology; neutrosophic hypersoft interior; neutrosophic hypersoft closure.

1. Introduction

Complexity typically emerges out of confusion in the context of ambiguity in the real world. The challenges of modeling ambiguous data are discussed regularly by researchers in economics, history, medical science and many other areas. Classical approaches do not necessarily provide fruitful outcomes and there can be various forms of unknown presence in these domains. The theory of probability has become an age-old and powerful method to work with ambiguity, but it can only be extended to the random process. After that, to solve unknown problems, evidence theory, fuzzy set theory by Zadeh [32], and intuitionistic fuzzy set theory by Atanassov [4] were introduced. But each of these hypotheses, as Molodtsov [15] points out, has underlying difficulties. The underlying explanation for these problems is the inadequacy of the theory’s parametrization device.

The soft set theory was introduced by Molodtsov [15] as a new mathematical technique away from the parametrization inadequacy syndrome of multiple theories dealing with instability in 1999. This makes the theory, in practice, very convenient and easily applicable. Molodtsov [15] successfully applied many criteria for soft set theory applications, such as function smoothness, game theory, operational analysis, Riemann integration, integration and probability of Perron, soft set theory and its implementations are now evolving quickly in numerous fields. Shabir and Naz [27] introduced soft topological spaces and defined some notions of soft sets. In addition,

Theory and Application of Hypersoft Set

topological structure of the fuzzy, fuzzy soft, intuitionistic fuzzy and intuitionistic fuzzy soft sets have been defined by different researchers [2, 5, 7, 8, 12, 31].

The neutrosophic sets (NS) was first proposed by Smarandache [28, 29]. This is a classical set generalization, a fuzzy set, an intuitionist fuzzy set, etc. Later, a combined neutrosophic soft set (NSS) was proposed by Maji [13]. The topological structure on neutrosophic soft sets is defined by Bera [6]. Due to some structural deficiencies in the study [6], it was necessary to redefine some concepts on neutrosophic soft sets. Therefore, Ozturk et al. redefined operations on neutrosophic soft sets and studied neutrosophic soft topological spaces, neutrosophic soft separation axioms [3, 19]. Also several mathematicians have developed their research work in various mathematical systems using neutrosophic sets, neutrosophic soft sets etc. [3, 11, 14, 16-21, 24].

Smarandache [30] generalized the notion of a soft set to a hypersoft set by replacing the function with a multi-argument function specified in the cartesian product with a different set of parameters. This idea is more versatile than the soft set and more applicable in the sense of decisionmaking issues. Hypersoft set structure has attracted the attention of researchers because it is more suitable than soft set structure in decision making problems. Although it is a new concept, many studies have been done and the field of study continues to expand [1, 10, 22, 23, 25, 26, 33].

Due to some structural difficulties in the study [26], some problems were encountered in defining the neutrosophic hypersoft topological structure. For this reason, there was a need to redefine some concepts. In this study, firstly, basic operations on neutrosophic hypersoft sets such as complement, union, intersection, AND, OR are re-defined differently from [26] study and several properties have been characterized. After that, by using these redefined operations, the neutrosophic hypersoft topological spaces are introduced. On this topological structure, operations such as open (closed) set, interior and closure are defined and their properties are examined. The study is supported by appropriate examples.

2. Preliminaries

Definition 2.1 [15] Let  be an initial universe £ be a set of parameters and ()P  be the power set of  . A pair (,£) F is called a soft set over , where F is a mapping :£().FP→ In other words, the soft set is a parameterized family of subsets of the set 

Definition 2.2 [28,29] Let  be an initial universe and  be a neutrosophic set (NS) on  is defined as ={<,(),(),()>:}, xTxIxFxx   where T,I,F: ]0,1[ −+ → and 0()()()3. TxIxFx +  ++

Definition 2.3 [9] Let  be an initial universe £ be a set of parameters and ()NP  denote the set of all neutrosophic sets of . A pair (,£) F is called a neutrosophic soft set over  and its mapping is given as :£().FNP→

Theory and Application of Hypersoft Set 217

Definition 2.4 [30] Let  be the universal set and ()P  be the power set of  . Consider 123,,,..., k for 1 k  , be k well-defined attributes, whose corresponding attribute values are respectively the sets 12 £,£,...,£k with ££=, ij for ij  and ,{1,2,...}, ijk  then the pair 12 (,££...£) k  is said to be hypersoft set over  where 12 :££...£(). k P →

Definition 2.5 [26] Let  be the universal set and ()NP  be a family of all neutrosophic sets over  and 12 £,£,...,£k the pairwise disjoint sets of parameters Let iC be the nonempty subset of then the pair £i for each =1,2,...,. ik A neutrosophic hypersoft set (NHSS) over  defined as the pair 12 (,...) kCCC where 12 :...() k CCCNP → and 12()()() (...)={(,<,(),(),()>) k CCCxTxIxFx     1212 :,...££...£} kkxCCC   where T is the membership value of truthiness, I is the membership value of indeterminancy and F is the membership value of falsity such that ()()()(),(),()0,1]TxIxFx    also ()()() 0()()()3. TxIxFx   ++ For sake of simplicity, we write the symbols  for 12 ££...£, k   for 12 ... kCCC  and  for an element of the set . Throughout this paper, we denoted the family of all neutrosophic hypersoft sets over the universe set  with (,)NHSS 

3. Some Basic Operations on Neutrosophic Hypersoft Sets

In this section, operations of neutrosophic hypersoft subset, null neutrosophic hypersoft set, complement, union, intersection, AND, OR on neutrosophic hypersoft sets are defined differently from the study [26]. We also presented basic properties of these operations. Definition 3.1 Let  be the universal set and 1122 (,),(,)  be two neutrosophic hypersoft sets over  . Then 11 (,)  is the neutrosophic hypersoft subset of 22 (,)  if 1. 12,  2. For 1   and , x  ()()()() 1212()(),()(), TxTxIxIx 

Theory and Application of Hypersoft Set

where   and . x  It is clear that ( ) (,)=(,). c c 

Definition 3.3

1. A neutrosophic hypersoft set (,)  over the universe set  is said to be null neutrosophic hypersoft set if ()()=0,Tx   ()()=0,Ix   ()()=1Fx   where   and x  . It is denoted by (,)0 NH  .

2. A neutrosophic hypersoft set (,)  over the universe set  is said to be absolute neutrosophic hypersoft set if ()()=1,Tx   ()()=1,Ix   ()()=0Fx   where   and x  . It is denoted by (,)1 NH  Clearly, (,)(,)0=1 c NHNH  and (,)(,) 1=0 c NHNH  . Definition 3.4 Let  be the universal set and 11 (,)  and 22 (,)  be neutrosophic hypersoft sets over  . Extended union 11 (,)  22 (,) is defined as

Definition 3.7 Let  be the universal set and 11 (,)  and 22 (,)  be neutrosophic hypersoft sets over  . The "AND" operator 112212 (,)(,)=()  is defined as   1122()() 12 ((,)(,))=min(),(), TTxTx     1122()() 12 ((,)(,))=min(),(), IIxIx     1122()() 12 ((,)(,))=max(),(). FFxFx  

Definition 3.8 Let  be the universal set and 11 (,)  and 22 (,)  be neutrosophic hypersoft sets over  . The "OR" operator 112212 (,)(,)=()  is defined as   1122()() 12 ((,)(,))=max(),(), TTxTx  

2.

(,)1=(,).

   

x xx x xx x xx x

<(,,),{,,}>, 0.8,0.5,0.20.6,0.5,0.40.3,0.6,0.2 <(,,),{,,}>, 0.8,0.2,0.50.5,0.7,0.10.5,0.5,0.4 (,)= <(,,),{,,}>, 0.6,0.4,0.50.5,0.7,0.40.7,0.3,0.2 <(,,),{ 0.6,0.

3 12 111 3 12 112 11 3 12 121 1 122

 3 2 ,,}> 2,0.80.3,0.4,0.50.1,0.3,0.7 x x

             , 3 12 112 3 12 122 22 3 12 312 1 322

   

x xx x xx x xx x

 3 2 ,,}> 5,0.20.7,0.3,0.80.6,0.1,0.5 x x

<(,,),{,,}>, 0.9,0.2,0.10.2,0.2,0.40.2,0.1,0.7 <(,,),{,,}>, 0.8,0.4,0.30.7,0.4,0.20.4,0.6,0.8 (,)= <(,,),{,,}>, 0.5,0.2,0.60.6,0.5,0.30.8,0.3,0.7 <(,,),{ 0.4,0.

x xx x xx x xx

             The union, intersection, "AND", "OR" operation of these sets are follows: 3 12 111 3 12 112 3 12 121 1122 122

<(,,),{,,}>, 0.8,0.5,0.20.6,0.5,0.40.3,0.6,0.2 <(,,),{,,}>, 0.9,0.2,0.10.5,0.7,0.10.5,0.5,0.4 <(,,),{,,}>, 0.6,0.4,0.50.5,0.7,0.40.7,0.3,0.2 (,)(,)= <((,,),

     3 12 3 12 312 3 12 322

 

x xx x xx x xx

                   3 12 112 1122 3 12 122

, {,,}>, 0.8,0.4,0.30.7,0.4,0.20.4,0.6,0.7 <(,,),{,,}>, 0.5,0.2,0.60.6,0.5,0.30.8,0.3,0.7 <((,,),{,,}> 0.4,0.5,0.20.7,0.3,0.80.6,0.1,0.5

x xx x xx

<(,,),{,,}>, 0.8,0.2,0.50.2,0.2,0.40.2,0.1,0.7 (,)(,)=, <((,,),{,,}> 0.6,0.2,0.80.3,0.4,0.50.1,0.3,0.8

 

 

x xx ab x xx ab x xx ab xx ab

 

x x xx ab x xx ab x xx ab x ab

 

x x x xx ab x xx ab x xx ab x ab

x x x xx ab x xx ab x xx ab ab

   

                                                    

    3 2 ,,}), ,0.4,0.50.5,0.3,0.80.6,0.1,0.5 3 12 (,{,,}), 41 0.6,0.2,0.80.2,0.2,0.50.1,0.1,0.7 3 12 (,{,,}), 42 0.6,0.2,0.80.3,0.4,0.50.1,0.3,0.8 3 12 (,{,,}), 43 0.5,0.2,0.80.3,0.4,0.50.1,0.3,0.7 ( 4

, 3 12 4,{,,}) 0.4,0.2,0.80.3,0.3,0.80.1,0.1,0.7 x xx

3 12 11 3 12 12 3 12 13 12 14 1122

x xx ab x xx ab x xx ab xx ab

(,{,,}), 0.9,0.5,0.10.6,0.5,0.40.3,0.6,0.2 (,{,,}), 0.8,0.5,0.20.7,0.5,0.20.4,0.6,0.2 (,{,,}), 0.8,0.5,0.20.6,0.5,0.30.8,0.6,0.2 (,{,0.8,0.5,0.20.7, (,)(,)=

0.5,0.40.6,0.6,0.2,}), (,{,,}), 0.9,0.2,0.10.5,0.7,0.10.5,0.5,0.4 (,{,,}) 0.8,0.4,0.30.7,0.7,0.10.5,0.6,0.4 (,{,,}), 0.8,0.2,0.50.6,0.7,0.10.8,0.5,0.4 (,{0.8,0.5,0.2

x x xx ab x xx ab x xx ab x ab

3 2 3 12 31 3 12 32 3 12 33 1 34

,,}), 0.7,0.7,0.10.6,0.5,0.4 (,{,,}), 0.9,0.4,0.10.5,0.7,0.40.7,0.3,0.2 (,{,,}), 0.8,0.4,0.30.7,0.7,0.20.7,0.6,0.2 (,{,,}), 0.6,0.4,0.50.6,0.7,0.30.8,0.3,0.2 (,{0.6



x x x xx ab x xx ab x xx ab x ab

 3 2 3 12 41 3 12 42 3 12 43 4

,,}), ,0.5,0.20.7,0.7,0.40.7,0.3,0.2 (,{,,}), 0.9,0.2,0.10.3,0.4,0.40.2,0.3,0.7 (,{,,}), 0.8,0.4,0.30.7,0.4,0.20.4,0.6,0.7 (,{,,}), 0.6,0.2,0.60.6,0.5,0.30.8,0.3,0.7 (

x x x xx ab x xx ab x xx ab ab

                                                

    3 12 4,{,,}) 0.6,0.5,0.20.7,0.4,0.50.6,0.3,0.5 x xx

4. Neutrosophic Hypersoft Topological Spaces

Definition 4.1 Let (,)NHSS  be the family of all neutrosophic hypersoft sets over the universe set  and (,)NHSS   . Then  is said to be a neutrosophic hypersoft topology on  if

1. (,)0 NH  and (,)1 NH  belongs to 

2. the union of any number of neutrosophic hypersoft sets in  belongs to  3. the intersection of finite number of neutrosophic hypersoft sets in  belongs to .

Then (,,)   is said to be a neutrosophic hypersoft topological space over  . Each members of  is said to be neutrosophic hypersoft open set.

Definition 4.2 Let (,,)   be a neutrosophic hypersoft topological space over  and (,)  be a neutrosophic hypersoft set over  . Then (,)  is said to be neutrosophic hypersoft closed set iff its complement is a neutrosophic hypersoft open set .

Proposition 4.3 Let (,,)   be a neutrosophic hypersoft topological space over  . Then 1. (,)0 NH  and (,)1 NH  are neutrosophic hypersoft closed sets over 

Theory and Application of Hypersoft Set

2. The intersection of any number of neutrosophic hypersoft closed sets is a neutrosophic hypersoft closed set over 

3. The union of finite number of neutrosophic hypersoft closed sets is a neutrosophic hypersoft closed set over  .

Proof. It is clear that the definition of neutrosophic hypersoft topological space.

Definition 4.4 Let (,)NHSS  be the family of all neutrosophic hypersoft sets over the universe set  .

1. If  (,)(,) =0,1, NHNH   then  is said to be the neutrosophic hypersoft indiscrete topology and (,,)   is said to be a neutrosophic hypersoft indiscrete topological space over  .

2. If =(,), NHSS   then  is said to be the neutrosophic hypersoft discrete topology and (,,)   is said to be a neutrosophic hypersoft discrete topological space over  .

Proposition 4.5 Let 1 (,,)   and 2 (,,)   be two neutrosophic hypersoft topological spaces over the same universe set  . Then ( ) 12 ,,  is neutrosophic hypersoft topological space over  . Proof. 1. Since (,)(,)10,1 NHNH    and (,)(,)20,1 NHNH    , then (,)(,)120,1 NHNH    .

2. Suppose that   (,) ii iI be a family of neutrosophic hypersoft sets in 12  . Then 1 (,) ii  and 2 (,) ii  for all ,iI  so 1 (,) ii iI    and 2 (,) ii iI    . Thus 12 (,) ii iI   

3. Let   (,)=1, ii ik  be a family of the finite number of neutrosophic hypersoft sets in 12  . Then 1 (,) ii  and 2 (,) ii  for =1,,ik so 1 =1 (,) n ii i  and 2 =1 (,) n ii i  Thus 12 =1 (,) n ii i  . Remark 4.6 The union of two neutrosophic hypersoft topologies over  may not be a neutrosophic hypersoft topology on  . Example 4.7 We consider attributes of Example-3.12.   1(,)(,)1122 =0,1,(,),(,) NHNH    and   2(,)(,)3344 =0,1,(,),(,) NHNH   

Theory and Application of Hypersoft Set

be two neutrosophic hypersoft topologies over  . Here, the neutrosophic hypersoft sets 11 (,),  2233 (,),(,)  and 4 (,)  over  are defined as following: 3 12 111 3 12 112 11 3 12 121 1 122

   

x xx x xx x xx x

<(,,),{,,}>, 0.8,0.5,0.20.6,0.5,0.40.3,0.6,0.2 <(,,),{,,}>, 0.8,0.2,0.50.5,0.7,0.10.5,0.5,0.4 (,)= <(,,),{,,}>, 0.6,0.4,0.50.5,0.7,0.40.7,0.3,0.2 <(,,),{0.6,0.

 3 2

, 2,0.80.3,0.4,0.50.1,0.3,0.7,,}> x x

             3 12 111 3 12 112 22 3 12 121 1 122

   

x xx x xx x xx x

<(,,),{,,}>, 0.8,0.6,0.10.7,0.6,0.30.4,0.8,0.1 <(,,),{,,}>, 0.8,0.5,0.30.7,0.8,0.10.7,0.6,0.2 (,)= <(,,),{,,}>, 0.9,0.6,0.20.8,0.8,0.30.7,0.6,0.2 <(,,),{0.9,0.

 3 2

, 8,0.50.6,0.7,0.30.4,0.4,0.4,,}> x x

             3 12 212 3 12 222 33 3 12 312 1 322

   

x xx x xx x xx x

<(,,),{,,}>, 0.2,0.6,0.40.6,0.6,0.20.4,0.5,0.2 <(,,),{,,}>, 0.8,0.2,0.50.6,0.3,0.10.3,0.7,0.8 (,)= <(,,),{,,}>, 0.3,0.6,0.20.6,0.2,0.50.7,0.3,0.2 <(,,),{0.5,0.

 3 2

             3 12 212 3 12 222 44 3 12 312 1 322

, 3,0.40.5,0.1,0.30.4,0.5,0.5,,}> x x

x xx x xx x xx x

<(,,),{,,}>, 0.4,0.7,0.20.8,0.7,0.20.7,0.6,0.2 <(,,),{,,}>, 0.8,0.5,0.20.7,0.6,0.10.5,0.9,0.4 (,)= <(,,),{,,}>, 0.6,0.7,0.10.8,0.5,0.30.8,0.6,0.1 <(,,),{0.8,0.

   

 3 2 5,0.30.4,0.2,0.20.6,0.6,0.2,,}> x x

            

Theory and Application of Hypersoft Set

Since 1144 (,)(,) 

, then 1

2

is not a neutrosophic hypersoft topology over  .

Remark 4.8 In the (,)  hypersoft set,  consists of the cartesian product set 12 ... kCCC 

. If the parameters are selected from a single attribute set iC and the empty set from the other attribute sets while creating a neutrosophic hypersoft set, it is clear that the resulting structure will be a neutrosophic soft set structure. Neutrosophic hypersoft set structure is a generalized version of neutrosophic soft structure. Therefore, the neutrosophic hypersoft topological structure also provides all the conditions of the neutrosophic soft topological structure. (see Example-4.7)

Proposition 4.9 Let (,,)

be a neutrosophic hypersoft topological space over  and

  , 2 0,1 and 3 0,1 2. Suppose that   (,) ii iI be a family of neutrosophic hypersoft sets in  . Then   ()() i iI Tx

3. Suppose that   (,)=1, ii ik  be a family of finite neutrosophic hypersoft sets in  . Then 

This completes the proof.

Definition 4.10 Let (,,)   be a neutrosophic hypersoft topological space over  and (,)(,) NHSS  be a neutrosophic hypersoft set. Then, the neutrosophic hypersoft interior of (,)  , denoted (,)  , is defined as the neutrosophic hypersoft union of all neutrosophic hypersoft open subsets of (,)  Clearly, (,)  is the biggest neutrosophic hypersoft open set that is contained by (,). 

Example 4.11 Let us consider the neutrosophic hypersoft topology 1 given in Example 4.7 Suppose that an any (,)(,) NHSS  is defined as following:

Theory and Application of Hypersoft Set

Theorem 4.12 Let (,,)   be a neutrosophic hypersoft topological space over  and (,)(,) NHSS  . (,)  is a neutrosophic hypersoft open set iff (,)=(,)  .

Proof. Let (,)  be a neutrosophic hypersoft open set. Then the biggest neutrosophic hypersoft open set that is contained by (,)  is equal to (,)  . Hence, (,)=(,)  .

Conversely, it is known that (,)  is a neutrosophic hypersoft open set and if (,)=(,)  , then (,)  is a neutrosophic hypersoft open set.

Theorem 4.13 Let (,,)   be a neutrosophic hypersoft topological space over  and 1122 (,),(,)(,) NHSS  . Then,

11221122 (,)(,)=(,)(,),  5.   11221122 (,)(,)(,)(,).  Proof. 1. Let 1122 (,)=(,)  . Then 22 (,) iff 2222 (,)=(,)  . So, 1111 (,)=(,)   . 2. Straighforward. 3. It is known that 111122 (,)(,)(,)  and 2222 (,)(,)  . Since 22 (,)  is the biggest neutrosophic hypersoft open set contained in 22 (,)  and so, 1122 (,)(,)  4. Since 112211 (,)(,)(,)  and 112222 (,)(,)(,)  , then   112211 (,)(,)(,)  and   112222 (,)(,)(,)  and so,   11221122 (,)(,)(,)(,) 

Theory and Application of Hypersoft Set

Definition 4.14 Let (,,)   be a neutrosophic hypersoft topological space over  and (,)(,) NHSS  be a neutrosophic hypersoft set. Then, the neutrosophic hypersoft closure of (,)  , denoted (,)  , is defined as the neutrosophic hypersoft intersection of all neutrosophic hypersoft closed supersets of (,)  . Clearly, (,)  is the smallest neutrosophic hypersoft closed set that containing (,). 

  

Example 4.15 Let us consider the neutrosophic hypersoft topology 2 given in Example 4.7 Suppose that an any (,)(,) NHSS  is defined as following: 3 12 112 3 12 122 3 12 312 1 322

   

x xx x xx x xx x

 3 2 0.90.1,0.5,0.60.2,0.2,0.7,,}> x x

<(,,),{,,}>, 0.1,0.2,0.50.1,0.2,0.90.2,0.3,0.8 <(,,),{,,}>, 0.2,0.6,0.90.1,0.2,0.80.3,0.1,0.6 (,)= <(,,),{,,}>, 0.1,0.1,0.80.1,0.3,0.80.1,0.2,0.8 <(,,),{0.2,0.4,

Theorem 4.16 Let (,,)   be a neutrosophic hypersoft topological space over  and (,)(,) NHSS  . (,)  is neutrosophic hypersoft closed set iff (,)=(,)  .

Proof. Straightforward.

Proof. 1. Let 1122 (,)=(,)  . Then, 22 (,)  is a neutrosophic hypersoft closed set. Hence, 22 (,)  and 22 (,)  are equal. Therefore, 1111 (,)=(,)   .

2. Straightforward.

3. It is known that 1111 (,)(,)  and 2222 (,)(,)  and so, 112222 (,)(,)(,)  . Since 11 (,)  is the smallest neutrosophic hypersoft closed set containing 11 (,),  then 1122 (,)(,)  .

4. Since 111122 (,)(,)(,)  and 221122 (,)(,)(,)  , then

5. Conclusions

We re-defined neutrosophic hypersoft operations such as complement, null set, absolute set, union, intersection, "AND", "OR" differently from the study [26]. Providing de-morgan laws and other properties of these operations are demonstrated with various proofs and examples. Thus, operations on the neutrosophic hypersoft set structure are well defined. After that, by using these operations, we introduced neutrosophic hypersoft topological spaces, neutrosophic hypersoft open(closed) sets, neutrosophic hypersoft interior and neutrosophic hypersoft closure. Various properties of them have been studied. These are illustrated with appropriate examples. In the future, many topics such as compactness, continuity, connectedness and separation axioms can be studied on neutrosophic hypersoft topological spaces by considering this study. We hope this article will support future studies on neutrosophic hypersoft topological structure and many other general frameworks.

Conflicts of Interest

The authors declare no conflict of interest

References

[1] Abbas, M., Murtaza, G., & Smarandache, F. Basic operations on hypersoft sets and hypersoft point. Neutrosophic Sets & Systems, (2020), 35.

[2] Aras, C.G., Bayramov, S. Some results on fuzzy soft topological spaces. Math. Probl. Eng. 1–10 (2013)

[3] Aras, C.G., Ozturk, T. Y., & Bayramov, S. Separation axioms on neutrosophic soft topological spaces. Turkish Journal of Mathematics, (2019), 43(1), 498-510.

[4] Atanassov, K. Intuitionistic fuzzy sets, Fuzzy Sets and Systems (1986), 20, 87–96

Theory and Application of Hypersoft Set

[5] Bayramov, S., Gunduz, C. On intuitionistic fuzzy soft topological spaces. TWMS J. Pure Appl. Math. (2014), 5(1), 66–79.

[6] Bera T, Mahapatra NK. Introduction to neutrosophic soft topological space. Opsearch (2017), 54: 841-867.

[7] Chang, C.L. Fuzzy topological spaces. J. Math. Anal. Appl. (1968), 24(1), 182–190.

[8] Coker, D.: An introduction of intuitionistic fuzzy topological spaces. Fuzzy Sets Syst. (1997), 88, 81–89.

[9] Deli I., Broumi S., Neutrosophic soft relations and some properties, Ann. Fuzzy Math. Inform. (2015), 9(1), 169–182.

[10] Gayen, S., Smarandache, F., Jha, S., Singh, M.K., Broumi, S. & Kumar, R., Introduction to plithogenic hypersoft subgroup. Neutrosophic Sets and Systems, (2020), 33(1), 14.

[11] Karthika, M., Parimala, M., Jafari, S., Smarandache, F., , Alshumrani, M., Ozel, C., & Udhayakumar, R. Neutrosophic complex  -connectedness in neutrosophic complex topological spaces. Neutrosophic Sets and Systems, (2019), 29, 158-164.

[12] Li, Z., Cui, R. On the topological structure of intuitionistic fuzzy soft sets. Ann. Fuzzy Math. Inform. (2013), 5(1), 229–239.

[13] Maji, P.K. Neutrosophic soft set. Ann. Fuzzy Math. Inform. (2013), 5(1), 157–168.

[14] Mani, P., Muthusamy, K., Jafari, S., Smarandache, F., & Ramalingam, U. Decision-Making via Neutrosophic Support Soft Topological Spaces. Symmetry, (2018), 10(6), 217. doi:10.3390/sym10060217.

[15] Molodtsov D., “Soft set theory first results,” Computers and Mathematics with Applications, (1999), 37(4-5), 19–31.

[16] Ozturk, T. Y., Benek, A., & Ozkan, A. Neutrosophic soft compact spaces. Afrika Matematika, (2020) 1-16, Doi: 10.1007/s13370-020-00827-9.

[17] Ozturk, T. Y., & Dizman, T. H. A New Approach to Operations on Bipolar Neutrosophic Soft Sets and Bipolar Neutrosophic Soft Topological Spaces. Neutrosophic Sets and Systems, (2019), 30(1), 22–33.

[18] Ozturk, T. Y., & Ozkan, A. Neutrosophic Bitopological Spaces. Neutrosophic Sets and Systems, (2019), 30(1), 6.

[19] Ozturk, T. Y., Aras, C. G., & Bayramov, S. A New Approach to Operations on Neutrosophic Soft Sets and to Neutrosophic Soft Topological Spaces. Communication in Mathematics and Applications, (2019), 30(3), 481–493.

[20] Parimala, M., Jeevitha, R., Jafari, S., Smarandache, F., & Udhayakumar, R. Neutrosophic ����Homeomorphism in Neutrosophic Topological Spaces. Information, (2018), 9(8), 187.

[21] Parimala, M., Karthika, M., Jafari, S., Smarandache, F. and Udhayakumar, R. Neutrosophic Nano ideal Topological Spaces. Neutrosophic Sets and Systems, (2019), 24, 70-76.

Theory and Application of Hypersoft Set 234

[22] Rana, S., Qayyum, M., Saeed, M., Smarandache F. & Khan B. Plithogenic Fuzzy Whole Hypersoft Set, Construction of Operators and their Application in Frequency Matrix Multi Attribute Decision Making Technique. Neutrosophic Sets and Systems (2019), 28, 1.

[23] Saeed, M., Ahsan, M., Siddique, M.K., Ahmad M.R., A Study of The Fundamentals of Hypersoft Set Theory, International Journal of Scientific & Engineering Research, (2020), 11(1).

[24] Sahin, M., Alkhazaleh, S., Ulucay, V. Neutrosophic soft expert sets. Appl. Math. (2015), 6, 116–127, doi:10.4236/am.2015.61012.

[25] Saqlain, Jafar, M.N., Moin, S., Saeed M. and Broumi, S. (2020) Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure of Single valued Neutrosophic Hypersoft Sets. Neutrosophic Sets and Systems, 32, 317-329.

[26] Saqlain, M., Moin, S., Jafar, M.N., Saeed, M., & Smarandache, F. (2020). Aggregate Operators of Neutrosophic Hypersoft Set. Neutrosophic Sets and Systems, 32, 294–306

[27] Shabir, M., Naz, M. On soft topological spaces. Comput. Math. Appl. (2011), 61, 1786–1799.

[28] Smarandache, F. Neutrosophic set, a generalisation of the intuitionistic fuzzy sets. Int. J. Pure Appl. Math. (2005), 24, 287–297.

[29] Smarandache, F. Neutrosophy, Neutrosophic Probability, Set, and Logic. American Research Press, Rehoboth, USA, (1998). p. 105. http://fs.gallup.unm.edu/eBook-neutrosophics4 (fourth version)

[30] Smarandache, F., Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set, Neutrosophic sets and system. (2018), 22, 168-170.

[31] Tanay, B., Kandemir, M.B., Topological structure of fuzzy soft sets, Comput. Math. Appl. (2011), 61, 2952–2957.

[32] Zadeh L.A., “Fuzzy sets,” Information and Control, (1965), 8(3) 338–353.

[33] Zulqarnain, R. M., Xin, X. L., Saqlain, M., & Smarandache, F. (2020). Generalized Aggregate Operators on Neutrosophic Hypersoft Set. Neutrosophic Sets and Systems, 36, 271-281

Contributors

Prof. Dr. Florentin Smarandache

Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA. E-mail: smarand@unm.edu

Prof. Dr. Muhammad Saeed

Department of Mathematics, University of Management and Technology Township, 54000, Lahore, Pakistan E-mail: muhammad.saeed@umt.edu.pk

Prof. Dr. Irfan Deli

Muallim Rıfat Faculty of Education, 7 Aralık University, 79000 Kilis, Turkey. E-mail: irfandeli@kilis.edu.tr

Prof. Dr. Xiao Long Xin

School of Mathematics, Northwest University Xi’an, China. School of Science, Xi'an Polytechnic University, Xi'an 710048, China E-mail: xlxin@nwu.edu.cn

Associate Prof. Dr. Taha Yasin Ozturk

Department of Mathematic, Kafkas University, Kars, Turkey. E-mail: taha36100@hotmail.com

Assistant Prof. Dr. Adem Yolcu

Department of Mathematic, Kafkas University, Kars, Turkey. E-mail: yolcu.adem@gmail.com

Muhammad Saqlain

School of Mathematics, Northwest University, Xi’an, China. E-mail: msgondal0@gmail.com

Department of Mathematic, Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: msaqlain@lgu.edu.pk

Atiqe Ur Rahman

Department of Mathematics, University of Management and Technology Township, 54000, Lahore, Pakistan

E-mail: aurkhb@gmail.com

Rana Muhammad Zulqarnain

School of Mathematics, Northwest University, Xi’an, China.

E-mail: ranazulqarnain7777@gmail.com

Muhammad Naveed Jafar

Department of Mathematics, University of Management and Technology Township, 54000, Lahore, Pakistan

E-mail: 15009265004@umt.edu.pk

Department of Mathematic, Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan.

E-mail: naveedjafar@lgu.edu.pk

Muhammad Rayees Ahmad

Department of Mathematics, University of Management and Technology Township, 54000, Lahore, Pakistan

E-mail: rayeesmalik.ravian@gmail.com

Muhammad Khubab Siddique

Department of Mathematics, University of Management and Technology Township, 54000, Lahore, Pakistan

E-mail: khubabsiddique@hotmail.com

Muhammad Ahsan

Department of Mathematics, University of Management and Technology Township, 54000, Lahore, Pakistan

E-mail: ahsan1826@gmail.com

Mahrukh Irfan

Department of Mathematics, University of Management and Technology Township, 54000, Lahore, Pakistan

E-mail: f2018265015@umt.edu.pk

Adeel Saleem

Department of Mathematic, Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan.

E-mail: adeelsaleem1992@hotmail.com

Sana Moin

Department of Mathematic, Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan.

E-mail: moinsana64@gmail.com

Maryam Rani

Department of Mathematic, Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: Maryum.rani009@gmail.com

Abida Hafeez

Department of Mathematics, University of Management and Technology Township, 54000, Lahore, Pakistan E-mail: f2019109046@umt.edu.pk

Ume-e-Farwa

Department of Mathematics, University of Management and Technology Township, 54000, Lahore, Pakistan E-mail: f2019109047@umt.edu.pk

Muhammad Umer Farooq

Department of Physics, Govt. Degree College Bhera, Sargodha, Pakistan. E-mail: umerfarooq900@gmail.com

Zaka-ur-Rehman

Department of Physics, University of Lahore, Sargodha Campus, Pakistan. E-mail: zakaurrehman52@gmail.com

Muhammad Haseeb

Department of Mathematic, Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: hasich1122@gmail.com

Ahtasham Habib

Department of Mathematic, Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: ahtashamhabib1@gmail.com

Theory and Application of Hypersoft Set

The soft set was generalized in 2018 by Smarandache to the hypersoft set by transforming the function �� into a multiargument function. This extension reveals that the hypersoft set with neutrosophic, intuitionistic, and fuzzy set theory will be very helpful to construct a connection between alternatives and attributes. The Book “Theory and Application of Hypersoft Set” focuses on theories, methods, and algorithms for decision making problems. It also involves applications of neutrosophic, intuitionistic, and fuzzy information. Our goal is to develop a strong relationship with the MCDM solving techniques and to reduce the complexion in the methodologies.

Editors

Florentin Smarandache

University of New Mexico, Mathematics and Science Division, 705 Gurley Ave., Gallup, NM 87301, USA.

Muhammad Saqlain

School of Mathematics, Northwest University, Xi’an, China. Department of Mathematics, Lahore Garrison University, Sector-C, Phase VI, DHA, Lahore, Pakistan.

Muhammad Saeed

Department of Mathematics, University of Management and Technology Township, 54000, Lahore, Pakistan.

Mohamed Abdel-Baset

Department of Computer Science, Faculty of Computers and Informatics, Zagazig University, Egypt