Hypersoft set is an extension of soft set as it further partitions each attribute into its corresponding attribute-valued set. This structure is more flexible and useful as it addresses the limitation of soft set for dealing with the scenarios having disjoint attribute-valued sets corresponding to distinct attributes. The main purpose of this study is to make the existing literature regarding neutrosophic parameterized soft set in line with the need of multi-attribute approximate function. Firstly, we conceptualize the neutrosophic parameterized hypersoft sets under the settings of fuzzy set, intuitionistic fuzzy set and neutrosophic set along with some of their elementary properties and set theoretic operations. Secondly, we propose decision-making-based algorithms with the help of these theories. Moreover, illustrative examples are presented which depict the structural validity for successful application to the problems involving vagueness and uncertainties. Lastly, the generalization of the proposed structure is discussed.
Fuzzy sets theory (FST) [1] and intuitionistic fuzzy set theory (IFST) [2] are considered apt mathematical modes to tackle many intricate problems involving various uncertainties, in different mathematical disciplines. The former one emphasizes on the degree of true belongingness of a certain object from the initial sample space whereas the later one accentuates on degree of true membership and degree of non-membership with condition of their dependency on each other. These theories depict some kind of inadequacy regarding the provision of due status to degree of indeterminacy. Such impediment is addressed with the introduction of neutrosophic set theory (NST) [3,4] which not only considers the due status of degree of indeterminacy but also waives off the condition of dependency. This theory is more flexible and appropriate to deal with uncertainty and vagueness. NST has attracted the keen concentration of many researchers [5–19] to further utilization in statistics, topological spaces as well as in the development of certain neutrosophic-like blended structures with other existing models for useful applications in decision making. Edalatpanah [20] studied a system of neutrosophic linear equations (SNLE) based on the embedding approach. He used (α,β,γ)-cut for transformation of SNLE into a crisp linear system. Kumar et al. [21] exhibited a novel linear programming approach for finding the neutrosophic shortest path problem (NSSPP) considering Gaussian valued neutrosophic number.
FST, IFST and NST have some kind of complexities which restrain them to solve problems involving uncertainty professionally. The reason for these hurdles is, possibly, the inadequacy of the parametrization tool. It demands a mathematical tool free of all such impediments to tackle such issues. This scantiness is resolved with the development of soft set theory (SST) [22] which is a new parameterized family of subsets of the universe of discourse. The researchers [23–34] studied and investigated some elementary properties, operations, laws and hybrids of SST with applications in decision making. The gluing concept of NST and SST, is studied in [35,36] to make the NST adequate with parameterized tool. In many real life situations, distinct attributes are further partitioned in disjoint attribute-valued sets but existing SST is insufficient for dealing with such kind of attribute-valued sets. Hypersoft set theory (HST) [37] is developed to make the SST in line with attribute-valued sets to tackle real life scenarios. HST is an extension of SST as it transforms the single argument approximate function into a multi-argument approximate function. Certain elementary properties, aggregation operations, laws, relations and functions of HST, are investigated by [38–40] for proper understanding and further utilization in different fields. The applications of HST in decision making is studied by [41–44] and the intermingling study of HST with complex sets, convex and concave sets is studied by [45,46]. Deli [47] characterized hybrid set structures under uncertainly parameterized hypersoft sets with theory and applications. Gayen et al. [48] analyzed some essential aspects of plithogenic hypersoft algebraic structures. They also investigated the notions and basic properties of plithogenic hypersoft subgroups, i.e., plithogenic fuzzy hypersoft subgroup, plithogenic intuitionistic fuzzy hypersoft subgroup, plithogenic neutrosophic hypersoft subgroup.
Motivation
In miscellany of real-life applications, the attributes are required to be further partitioned into attribute values for more vivid understanding. Hypersoft set as a generalization of soft set, accomplishes this limitation and accentuates the disjoint attribute-valued sets for distinct attributes. This generalization reveals that the hypersoft set with neutrosophic, intuitionistic, and fuzzy set theory will be very helpful to construct a connection between alternatives and attributes. 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. This theory can successfully be applied to 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 many others. It is pertinent that the existing literature regarding soft set should be adequate with the existence and the consideration of attribute-valued sets, therefore, this study aims to develop novel theories of embedding structures of parameterized neutrosophic set and hypersoft set with the setting of fuzzy, intuitionistic fuzzy and neutrosophic sets through the extension of concept investigated in [49–54]. Moreover, decision-making based algorithms are proposed for each setting to solve a real life problem relating to the purchase of most suitable and appropriate product with the help of some essential operations of these presented theories.
Organization of Paper
The rest of the paper is systemized as:
Section 2
Some essential definitions and terminologies are recalled.
Section 3
Theory of neutrosophic parameterized fuzzy hypersoft set is developed with suitable examples.
Section 4
Theory of neutrosophic parameterized intuitionistic fuzzy hypersoft set is characterized with suitable examples.
Section 5
Theory of neutrosophic parameterized neutrosophic hypersoft set is investigated with suitable examples.
Section 6
Analysis of proposed structure is discussed.
Section 7
Paper is summarized with future directions.
Preliminaries
Here some basic terms are recalled from existing literature to support the proposed work. Throughout the paper, X, ℙ(X) and I will denote the universe of discourse, power set of X and closed unit interval respectively. In this work, algorithmic approaches are followed from decision making methods stated in [49–54].
Definition 2.1. [1]
A fuzzy setF defined as F={(â,AF(â))∣â∈X} such that AF:X→I where AF(â) denotes the belonging value of â∈F.
Definition 2.2. [2]
An intuitionistic fuzzy setY defined as Y={(b^,<AY(b^),BY(b^)>)∣b^∈X} such that AY:X→I and BY:X→I, where AY(b^) and BY(b^) denote the belonging value and not-belonging value of b^∈Y with condition of 0≤AY(b^)+BY(b^)≤1.
Definition 2.3. [3]
A neutrosophic setZ defined as Z={(ĉ,<AZ(ĉ),BZ(ĉ),CZ(ĉ)>)∣ĉ∈X} such that AZ(ĉ),BZ(ĉ),CZ(ĉ):X→(-0,1+), where AZ(ĉ),BZ(ĉ) and CZ(ĉ) denote the degrees of membership, indeterminacy and non-membership of ĉ∈Z with condition of -0≤AZ(ĉ)+BZ(ĉ)+CZ(ĉ)≤3+.
Definition 2.4. [22]
A pair (FS,Λ) is called a soft set over X, where FS:Λ→ℙ(X) and Λ be a subset of a set of attributes E.
For more detail on soft set, see [23–32].
Definition 2.5. [37]
The pair (W,G) is called a hypersoft set over X, where G is the cartesian product of n disjoint sets G1,G2,G3,…,Gn having attribute values of n distinct attributes ĝ1,ĝ2,ĝ3,…,ĝn respectively and W:G→ℙ(X).
For more definitions and operations of hypersoft set, see [38–40].
Neutrosophic Parameterized Fuzzy Hypersoft Set (npfhs-Set) with Application
In this section, npfhs-set theory is conceptualized and a decision making application is discussed.
Definition 3.1. Let X={X1,X2,X3,…,Xn} be a collection of disjoint attribute-valued sets corresponding to n distinct attributes α1,α2,α3,…,αn, respectively. A npfhs-set ΨA over X is defined as
ΨA={(<PA(g),QA(g),RA(g)>/g,ψA(g)):g∈G,ψA(g)∈F(X)}
where
F(X) is a collection of all fuzzy sets over X
G=X1×X2×X3×…×Xn
A is a neutrosophic set over G with PA,QA,RA:G→I as membership function, indeterminacy function and nonmembership function of npfhs-set.
ψA(g) is a fuzzy set for all g∈G with ψA:G→F(X) and is called approximate function of npfhs-set.
Note that collection of all npfhs-sets is represented by ΩNPFHS(X).
Definition 3.2. Let ΨA∈ΩNPFHS(X). If ψA(g)=ϕ,PA(g)=0,QA(g)=1,RA(g)=1 for all g∈G, then ΨA is called A-empty npfhs-set, denoted by ΨΦA. If A=ϕ, then A-empty npfhs-set is called an empty npfhs-set, denoted by ΨΦ.
Definition 3.3. Let ΨA∈ΩNPFHS(X). If ψA(g)=X,PA(g)=1,QA(g)=0,RA(g)=0 for all g∈G, then ΨA is called A-universal npfhs-set, denoted by ΨÃ. If A=G, then the A-universal npfhs-set is called universal npfhs-set, denoted by ΨG̃.
Example 3.1. Consider X={u1,u2,u3,u4,u5} and X={X1,X2,X3} with X1={x^11,x^12},
X2={x^21,x^22},X3={x^31}, then G=X1×X2×X3G={(x^11,x^21,x^31),(x^11,x^22,x^31),(x^12,x^21,x^31),(x^12,x^22,x^31)}={g1,g2,g3,g4}.
Case 1.
If A1={<0.2,0.3,0.4>/g2,<0,1,1>/g3,<1,0,0>/g4} and
ψA1(g2)={0.4/u2,0.6/u4},ψA1(g3)=∅,andψA1(g4)=X, thenΨA1={(<0.2,0.3,0.4>/g2,{0.4/u2,0.6/u4}),(<0,1,1>/g3,∅),(<1,0,0>/g4,X)}.
Case 2.
If A2={<0,1,1>/g2,<0,1,1>/g3},ψA2(g2)=∅ and ψA2(g3)=∅, then ΨA2=ΨΦA2.
Case 3.
If A3=∅ corresponding to all elements of G, then ΨA3=ΨΦ.
Case 4.
If A4={<1,0,0>/g1,<1,0,0>/g2},ψA4(g1)=X, and ψA4(g2)=X, then ΨA4=ΨA4̃.
Case 5.
If A5=X with respect to all elements of G, then ΨA5=ΨG̃.
Definition 3.4. Let ΨA1, ΨA2∈ΩNPFHS(X) then ΨA1 is an npfhs-subset of ΨA2, denoted by ΨA1⊆̃fΨA2 if
PA1(g)≤PA2(g),QA1(g)≥QA2(g),RA1(g)≥RA2(g) and ψA1(g)⊆fψA2(g)<transpanc/><d>:ltl:italic:gtl:for:ltl:/italic:gtl::ltl:italic:gtl:all:ltl:/italic:gtl:</d><transpanop/>g∈G.
Definition 3.5. Let ΨA1,ΨA2∈ΩNPFHS(X) then, ΨA1 and ΨA2 are npfhs-equal, represented as ΨA1=ΨA2, if and only if PA1(g)=PA2(g),QA1(g)=QA2(g),RA1(g)=RA2(g) and ψA1(g)=fψA2(g) for all g∈G.
Definition 3.6. Let ΨA∈ΩNPFHS(X) then, complement of ΨA (i.e., ΨAc̃) is an npfhs-set given as PAc̃(g)=1-PA(g),QAc̃(g)=1-QA(g),RAc̃(g)=1-RA(g) and ψAc̃(g)=X\fψA(g).
Proposition 3.1. Let ΨA∈ΩNPFHS(X) then,
(ΨAc̃)c̃=ΨA.
Ψϕc̃=ΨG̃.
Definition 3.7. Let ΨA1,ΨA2∈ΩNPFHS(X) then, union of ΨA1 and ΨA2, denoted by ΨA1∪̃fΨA2, is an npfhs-set defined by
PA1∪̃A2(g)=max{PA1(x),PA2(g)},
QA1∪̃A2(g)=min{QA1(x),QA2(g)},
RA1∪̃A2(g)=min{RA1(x),RA2(g)},
ψA1∪̃A2(g)=ψA1(g)∪̃fψA2(g), for all g∈G.
Definition 3.8. Let ΨA1,ΨA2∈ΩNPFHS(X) then intersection of ΨA1 and ΨA2, denoted by ΨA1∩̃fΨA2, is an npfhs-set defined by
PA1∩̃A2(g)=min{PA1(x),PA2(g)},
QA1∩̃A2(g)=max{QA1(x),QA2(g)},
RA1∩̃A2(g)=max{RA1(x),RA2(g)},
ψA1∩̃A2(g)=ψA1(g)∩̃fψA2(g), for all g∈G.
Remark 3.1. Let ΨA∈ΩNPFHS(X). If ΨA≠fΨG̃, then ΨA∪̃fΨAc̃≠fΨG̃ and ΨA∩̃fΨAc̃≠fΨΦ
Proposition 3.2. Let ΨA1,ΨA2∈ΩNPFHS(X) D. Morgan laws are valid
(ΨA1∪̃fΨA2)c̃=ΨA1c̃∩̃fΨA2c̃.
(ΨA1∩̃fΨA2)c̃=ΨA1c̃∪̃fΨA2c̃.
Proof. For all g∈G,
(1).Since(PA1∪̃A2)c̃(g)=1-PA1∪̃A2(g)=1-max{PA1(g),PA2(g)}=min{1-PA1(g),1-PA2(g)}=min{PA1c̃(g),PA2c̃(g)}=PA1∩̃A2c̃(g)
also
(QA1∪̃A2)c̃(g)=1-QA1∪̃A2(g)=1-min{QA1(g),QA2(g)}=max{1-QA1(g),1-QA2(g)}=max{QA1c̃(g),QA2c̃(g)}=QA1∩̃A2c̃(g)
and
(RA1∪̃A2)c̃(g)=1-RA1∪̃A2(g)=1-min{RA1(g),RA2(g)}=max{1-RA1(g),1-RA2(g)}=max{RA1c̃(g),RA2c̃(g)}=RA1∩̃A2c̃(g)
and
(ψA1∪̃A2)c̃(g)=X\fψA1∪̃A2(g)=X\f(ψA1(g)∪̃fψA2(g))=(X\fψA1(g))∩̃f(X\fψA2(g))=ψA1c̃(g)∩̃fψA2c̃(g)=ψA1∩̃A2c̃(g).
similarly (2) can be proved easily.
Proposition 3.3. Let ΨA1, ΨA2, ΨA3∈ΩNPFHS(X) then
ΨA1∪̃f(ΨA2∩̃fΨA3)=(ΨA1∪̃fΨA2)∩̃f(ΨA1∪̃fΨA3).
ΨA1∩̃f(ΨA2∪̃fΨA3)=(ΨA1∩̃fΨA2)∪̃f(ΨA1∩̃fΨA3).
Proof. For all g∈G,
(1).SincePA1∪̃(A2∩̃A3)(g)=max{PA1(g),PA2∩̃A3(g)}=max{PA1(g),min{PA2(g),PA3(g)}}=min{max{PA1(g),PA2(g)},max{PA1(g),PA3(g)}}=min{PA1∪̃A2(g),PA1∪̃A3(g)}=P(A1∪̃A2)∩̃(A1∪̃A3)(g)
and
QA1∪̃(A2∩̃A3)(g)=min{QA1(g),QA2∩̃A3(g)}=min{QA1(g),max{QA2(g),QA3(g)}}=max{min{QA1(g),QA2(g)},min{QA1(g),QA3(g)}}=max{QA1∪̃A2(g),QA1∪̃A3(g)}=Q(A1∪̃A2)∩̃(A1∪̃A3)(g)
and
RA1∪̃(A2∩̃A3)(g)=min{RA1(g),RA2∩̃A3(g)}=min{RA1(g),max{RA2(g),RA3(g)}}=max{min{RA1(g),RA2(g)},min{RA1(g),RA3(g)}}=max{RA1∪̃A2(g),RA1∪̃A3(g)}=R(A1∪̃A2)∩̃(A1∪̃A3)(g)
and
ψA1∪̃(A2∩̃A3)(g)=ψA1(g)∪fψA2∩̃A3(g)=ψA1(g)∪f(ψA2(g)∩fψA3(g))=(ψA1(g)∪fψA2(g))∩f(ψA1(g)∪fψA3(g))=ψA1∪̃A2(g)∩fψA1ŨA3(g)=ψ(A1∪̃A2)∩̃(A1∪̃A3)(g)
In the same way, (2) can be proved.
Definition 3.9. Let ΨA1,ΨA2∈ΩNPFHS(X) then OR-operation of ΨA1 and ΨA2, denoted by ΨA1⋓̃ΨA2, is an npfhs-set defined by
PA1⋓̃A2(g1,g2)=max{PA1(g1),PA2(g2)},
QA1⋓̃A2(g1,g2)=min{QA1(g1),QA2(g2)},
RA1⋓̃A2(g1,g2)=min{RA1(g1),RA2(g2)},
ψA1⋓̃A2(g1,g2)=ψA1(g1)∪fψA2(g2), for all (g1,g2)∈A1×A2.
Definition 3.10. Let ΨA1,ΨA2∈ΩNPFHS(X) then AND-operation of ΨA1 and ΨA2, denoted by ΨA1⋒̃ΨA2, is an npfhs-set defined by
PA1⋒̃A2(g1,g2)=min{PA1(g1),PA2(g2)},
QA1⋒̃A2(g1,g2)=max{QA1(g1),QA2(g2)},
RA1⋒̃A2(g1,g2)=max{RA1(g1),RA2(g2)},
ψA1⋒̃A2(g1,g2)=ψA1(g1)∩fψA2(g2), for all (g1,g2)∈A1×A2.
Proposition 3.4. Let ΨA1,ΨA2,ΨA3∈ΩNPFHS(X) then
ΨA1⋒̃ΨΦ=ΨΦ.
(ΨA1⋒̃ΨA2)⋒̃ΨA3=ΨA1⋒̃(ΨA2⋒̃ΨA3).
(ΨA1⋓̃ΨA2)⋓̃ΨA3=ΨA1⋓̃(ΨA2⋓̃ΨA3).
Neutrosophic Decision Set of npfhs-Set
An algorithm is presented with the help of characterization of neutrosophic decision set on npfhs-set which based on decision making technique and is explained with example.
Definition 3.11. Let ΨA∈ΩNPFHS(X) then a neutrosophic decision set of ΨA (i.e., ΨAD) is represented as
ΨAD={<TAD(u),IAD(u),FAD(u)>/u:u∈X}
where TAD, IAD, FAD:X→I and
TAD(u)=1|X|∑v∈S(A)TA(v)ΓψA(v)(u)IAD(u)=1|X|∑v∈S(A)IA(v)ΓψA(v)(u)FAD(u)=1|X|∑v∈S(A)FA(v)ΓψA(v)(u)
where |∙| denotes set cardinality with
ΓψA(v)(u)={ψA(v);u∈ΓψA(v)0;u∉ΓψA(v)
Definition 3.12. If ΨA∈ΩNPFHS(X) with neutrosophic decision set ΨAD then reduced fuzzy set of ΨAD is a fuzzy set represented as
ℝ(ΨAD)={ζΨAD(u)/u:u∈X}
where ζΨAD:X→I with ζΨAD(u)=TAD(u)+IAD(u)-FAD(u)
Algorithm 3.1. Once ΨAD has been established, it may be indispensable to select the best single substitute from the options. Therefore, decision can be set up with the help of following algorithm.
Example 3.2. Suppose that Mr. James Peter wants to buy a mobile tablet from a mobile market. There are eight kinds of tablets (options) which form the set of discourse X={T^1,T^2,T^3,T^4,T^5,T^6,T^7,T^8}. The best selection may be evaluated by observing the attributes, i.e., a1 = Storage (GB), a2 = Camera Resolution (mega pixels), a3 = Size (inches), a4 = RAM (GB), and a5 = Battery power (mAh). The attribute-valued sets corresponding to these attributes are:
A1={a11=64,a12=128}A2={a21=8,a22=16}A3={a31=10,a32=11}A4={a41=2,a42=4}A5={a51=5000}
then ℝ=A1×A2×A3×A4×A5ℝ={r1,r2,r3,r4,…,r16} where each ri,i=1,2,…,16, is a 5-tuples element.
Step 1:
From Tabs. 1–3, we can construct A as
A={<0.1,0.2,0.3>/r1,<0.2,0.3,0.4>/r2,<0.3,0.4,0.5>/r3,<0.4,0.5,0.6>/r4,<0.5,0.6,0.7>/r5,<0.6,0.7,0.8>/r6,<0.7,0.8,0.9>/r7,<0.8,0.9,0.1>/r8,<0.9,0.1,0.2>/r9,<0.16,0.27,0.37>/r10,<0.25,0.35,0.45>/r11,<0.45,0.55,0.65>/r12,<0.35,0.45,0.55>/r13,<0.75,0.85,0.95>/r14,<0.65,0.75,0.85>/r15,<0.85,0.95,0.96>/r16}.
Step 2:
Tab. 4 presents ψA(ri) corresponding to each element of G.
Degrees of membership TA(ri)
TA(ri)
Degree
TA(ri)
Degree
TA(r1)
0.1
TA(r9)
0.9
TA(r2)
0.2
TA(r10)
0.16
TA(r3)
0.3
TA(r11)
0.25
TA(r4)
0.4
TA(r12)
0.45
TA(r5)
0.5
TA(r13)
0.35
TA(r6)
0.6
TA(r14)
0.75
TA(r7)
0.7
TA(r15)
0.65
TA(r8)
0.8
TA(r16)
0.85
Degrees of indeterminacy IA(ri)
IA(r1)
0.2
IA(r9)
0.1
IA(r2)
0.3
IA(r10)
0.27
IA(r3)
0.4
IA(r11)
0.35
IA(r4)
0.5
IA(r12)
0.55
IA(r5)
0.6
IA(r13)
0.45
IA(r6)
0.7
IA(r14)
0.85
IA(r7)
0.8
IA(r15)
0.75
IA(r8)
0.9
IA(r16)
0.95
Degrees of non-membership FA(ri)
FA(ri)
Degree
FA(ri)
Degree
FA(r1)
0.3
FA(r9)
0.2
FA(r2)
0.4
FA(r10)
0.37
FA(r3)
0.5
FA(r11)
0.45
FA(r4)
0.6
FA(r12)
0.65
FA(r5)
0.7
FA(r13)
0.55
FA(r6)
0.8
FA(r14)
0.95
FA(r7)
0.9
FA(r15)
0.85
FA(r8)
0.1
FA(r16)
0.96
Approximate functions ψA(ri)
ri
ψA(ri)
ri
ψA(ri)
r1
{0.2/T^1,0.3/T^2}
r9
{0.4/T^2,0.6/T^7,0.5/T^8}
r2
{0.1/T^1,0.5/T^2,0.1/T^3}
r10
{0.2/T^6,0.6/T^7,0.4/T^8}
r3
{0.4/T^2,0.5/T^3,0.6/T^4}
r11
{0.5/T^2,0.6/T^4,0.7/T^6}
r4
{0.6/T^4,0.7/T^5,0.8/T^6}
r12
{0.7/T^2,0.8/T^3,0.9/T^6}
r5
{0.2/T^6,0.1/T^7,0.4/T^8}
r13
{0.2/T^3,0.4/T^5,0.6/T^7}
r6
{0.4/T^2,0.3/T^3,0.4/T^4}
r14
{0.2/T^1,0.5/T^3,0.6/T^5}
r7
{0.2/T^1,0.3/T^3,0.4/T^5}
r15
{0.6/T^5,0.4/T^7,0.2/T^8}
r8
{0.1/T^2,0.3/T^3,0.5/T^7}
r16
{0.3/T^4,0.5/T^5,0.7/T^6}
Step 3:
With the help of Step 1 and Step 2, we can construct ΨA as
From Tabs. 5–8, we can construct ℝ(ΨAD) as
ℝ(ΨAD)={0.0325/T^1,0.1412/T^2,0.1968/T^3,0.1052/T^4,0.2112/T^5,0.1675/T^6,0.2158/T^7,0.0867/T^8}
The graphical representation of this decision system is presented in Fig. 1.
Step 5:
Since maximum of ζΨAD(T^i) is 0.2158 so the tablet T^7 is selected.
Membership values TAD(T^i)
T^i
TAD(T^i)
T^i
TAD(T^i)
T^1
0.0413
T^5
0.2456
T^2
0.1700
T^6
0.2034
T^3
0.2006
T^7
0.1945
T^4
0.1331
T^8
0.1055
Indeterminacy values IAD(T^i)
T^i
IAD(T^i)
T^i
IAD(T^i)
T^1
0.0500
T^5
0.2856
T^2
0.1650
T^6
0.2474
T^3
0.2381
T^7
0.1628
T^4
0.1644
T^8
0.0685
Non-membership values FAD(T^i)
T^i
FAD(T^i)
T^i
FAD(T^i)
T^1
0.0588
T^5
0.3200
T^2
0.1938
T^6
0.2833
T^3
0.2419
T^7
0.1415
T^4
0.1923
T^8
0.0873
Reduced fuzzy membership ζΨAD(T^i)
T^i
ζΨAD(T^i)
T^i
ζΨAD(T^i)
T^1
0.0325
T^5
0.2112
T^2
0.1412
T^6
0.1675
T^3
0.1968
T^7
0.2158
T^4
0.1052
T^8
0.0867
Neutrosophic decision system on npfhs-set
Neutrosophic Parameterized Intuitionistic Fuzzy Hypersoft Set (npifhs-set) with Application
In this section, npifhs-set theory is developed and decision making based application is presented.
Definition 4.1. Let Y={Y1,Y2,Y3,…,Yn} be a collection of disjoint attribute-valued sets corresponding to n distinct attributes α1,α2,α3,…,αn, respectively. A npifhs-set ΨB over X is defined as
ΨB={(<LB(g),MB(g),NB(g)>/g,ψB(g)):g∈G,ψA(g)∈IF(X)}
where
IF(U) is a collection of all intuitionistic fuzzy sets over X
G=Y1×Y2×Y3×…×Yn
B is a neutrosophic set over G with LB,MB,NB:G→I as membership function, indeterminacy function and nonmembership function of npifhs-set.
ψB(g) is a fuzzy set for all g∈G with ψB:G→IF(X) and is called approximate function of npifhs-set.
Note that collection of all npifhs-sets is represented by ΩNPIFHS(X).
Definition 4.2. Let ΨB∈ΩNPIFHS(X). If ψB(g)=∅,LB(g)=0,MB(g)=1,NB(g)=1 for all g∈G, then ΨB is called B-empty npifhs-set, denoted by ΨΦB. If B=∅, then B-empty npifhs-set is called an empty npifhs-set, denoted by ΨΦ.
Definition 4.3. Let ΨB∈ΩNPIFHS(X). If ψB(g)=X,LB(g)=1,MB(g)=0,NB(g)=0 for all g∈G, then ΨB is called B-universal npifhs-set, denoted by ΨB̃. If B=G, then the B-universal npifhs-set is called universal npifhs-set, denoted by ΨG̃.
Example 4.1. Consider X={u1,u2,u3,u4,u5} and Y={Y1,Y2,Y3} with
Y1={ŷ11,ŷ12},Y2={ŷ21,ŷ22},Y3={ŷ31}, thenG=Y1×Y2×Y3G={(ŷ11,ŷ21,ŷ31),(ŷ11,ŷ22,ŷ31),(ŷ12,ŷ21,ŷ31),(ŷ12,ŷ22,ŷ31)}={g1,g2,g3,g4}.
Case 1.
If B1={<0.2,0.3,0.4>/g2,<0,1,1>/g3,<1,0,0>/g4} and
ψB1(g2)={<0.2,0.4>/u2,<0.3,0.5>/u4},ψB1(g3)=ϕ, and ψB1(g4)=X, thenΨB1={(<0.2,0.3,0.4>/g2,{<0.2,0.4>/u2,<0.3,0.5>/u4}),(<0,1,1>/g3,ϕ),(<1,0,0>/g4,X)}.
Case 2.
If B2={<0,1,1>/g2,<0,1,1>/g3},ψB2(g2)=ϕ and ψB2(g3)=ϕ, then ΨB2=ΨΦB2.
Case 3.
If B3=ϕ corresponding to all elements of G, then ΨB3=ΨΦ.
Case 4.
If B4={<1,0,0>/g1,<1,0,0>/g2},ψB4(g1)=X, and ψB4(g2)=X, then ΨB4=ΨB4̃.
Case 5.
If B5=X with respect to all elements of G, then ΨB5=ΨG̃.
Definition 4.4. Let ΨB1, ΨB2∈ΩNPIFHS(X) then ΨB1 is an npifhs-subset of ΨB2, denoted by ΨB1⊆̃ifΨB2 if
LB1(g)≤LB2(g),MB1(g)≥MB2(g),NB1(g)≥NB2(g) and ψB1(g)⊆ifψB2(g) for all g∈G.
Definition 4.5. Let ΨB1,ΨB2∈ΩNPIFHS(X) then, ΨB1 and ΨB2 are npifhs-equal, represented as ΨB1=ΨB2, if and only if LB1(g)=LB2(g),MB1(g)=MB2(g),NB1(g)=NB2(g) and ψB1(g)=ψB2(g) for all g∈G.
Definition 4.6. Let ΨB∈ΩNPIFHS(X) then, complement of ΨB (i.e., ΨBc̃) is an npifhs-set given as PBc̃(g)=1-LB(g),QBc̃(g)=1-MB(g),RBc̃(g)=1-NB(g) and ψBc̃(g)=X\ψB(g)
Proposition 4.1. Let ΨB∈ΩNPIFHS(X) then,
(ΨBc̃)c̃=ΨB.
Ψϕc̃=ΨG̃.
Definition 4.7. Let ΨB1,ΨB2∈ΩNPIFHS(X) then, union of ΨB1 and ΨB2, denoted by ΨB1∪̃ifΨB2, is an npifhs-set defined by
LB1∪̃B2(g)=max{LB1(x),LB2(g)},
MB1∪̃B2(g)=min{MB1(x),MB2(g)},
NB1∪̃B2(g)=min{NB1(x),NB2(g)},
ψB1∪̃B2(g)=ψB1(g)∪̃ifψB2(g), for all g∈G.
Definition 4.8. Let ΨB1,ΨB2∈ΩNPIFHS(X) then intersection of ΨB1 and ΨB2, denoted by ΨB1∩̃ifΨB2, is an npifhs-set defined by
LB1∩̃B2(g)=min{LB1(x),LB2(g)},
MB1∩̃B2(g)=max{MB1(x),MB2(g)},
NB1∩̃B2(g)=max{NB1(x),NB2(g)},
ψB1∩̃B2(g)=ψB1(g)∩̃ifψB2(g), for all g∈G.
Remark 4.1. Let ΨB∈ΩNPIFHS(X). If ΨB≠ifΨG̃, then ΨB∪̃ifΨBc̃≠ifΨG̃ and ΨB∩̃ifΨBc̃≠ifΨΦ
Proposition 4.2. Let ΨB1,ΨB2∈ΩNPIFHS(X) then following D. Morgan laws are valid:
(ΨB1∪̃ifΨB2)c̃=ΨB1c̃∩̃ifΨB2c̃.
(ΨB1∩̃ifΨB2)c̃=ΨB1c̃∪̃ifΨB2c̃.
Proof. For all g∈G,
(1).Since (LB1∪̃B2)c̃(g)=1-LB1∪̃B2(g)=1-max{LB1(g),LB2(g)}=min{1-LB1(g),1-LB2(g)}=min{PB1c̃(g),PB2c̃(g)}=PB1∩̃B2c̃(g)
also
(MB1∪̃B2)c̃(g)=1-MB1∪̃B2(g)=1-min{MB1(g),MB2(g)}=max{1-MB1(g),1-MB2(g)}=max{QB1c̃(g),QB2c̃(g)}=QB1∩̃B2c̃(g)
and
(NB1∪̃B2)c̃(g)=1-NB1∪̃B2(g)=1-min{NB1(g),NB2(g)}=max{1-NB1(g),1-NB2(g)}=max{RB1c̃(g),RB2c̃(g)}=RB1∩̃B2c̃(g)
and
(ψB1∪̃B2)c̃(g)=X\ifψB1∪̃B2(g)=X\if(ψB1(g)∪̃ifψB2(g))=(X\ifψB1(g))∩̃if(X\ifψB2(g))=ψB1c̃(g)∩̃ifψB2c̃(g)=ψB1∩̃B2c̃(g).
similarly (2) can be proved easily.
Proposition 4.3. Let ΨB1,ΨB2,ΨB3∈ΩNPIFHS(X) then
ΨB1∪̃if(ΨB2∩̃ifΨB3)=(ΨB1∪̃ifΨB2)∩̃if(ΨB1∪̃ifΨB3).
ΨB1∩̃if(ΨB2∪̃ifΨB3)=(ΨB1∩̃ifΨB2)∪̃if(ΨB1∩̃ifΨB3).
Proof. For all g∈G,
(1).Since LB1∪̃(B2∩̃B3)(g)=max{LB1(g),LB2∩̃B3(g)}=max{LB1(g),min{LB2(g),LB3(g)}}=min{max{LB1(g),LB2(g)},max{LB1(g),LB3(g)}}=min{LB1∪̃B2(g),LB1∪̃B3(g)}=L(B1∪̃B2)∩̃(B1∪̃B3)(g)
and
MB1∪̃(B2∩̃B3)(g)=min{MB1(g),MB2∩̃B3(g)}=min{MB1(g),max{MB2(g),MB3(g)}}=max{min{MB1(g),MB2(g)},min{MB1(g),MB3(g)}}=max{MB1∪̃B2(g),MB1∪̃B3(g)}=M(B1∪̃B2)∩̃(B1∪̃B3)(g)
and
NB1∪̃(B2∩̃B3)(g)=min{NB1(g),NB2∩̃B3(g)}=min{NB1(g),max{NB2(g),NB3(g)}}=max{min{NB1(g),NB2(g)},min{NB1(g),NB3(g)}}=max{NB1∪̃B2(g),NB1∪̃B3(g)}=N(B1∪̃B2)∩̃(B1∪̃B3)(g)
and
ψB1∪̃(B2∩̃B3)(g)=ψB1(g)∪̃ifψB2∩̃B3(g)=ψB1(g)∪̃if(ψB2(g)∩̃ifψB3(g))=(ψB1(g)∪̃ifψB2(g))∩̃if(ψB1(g)∪̃ifψB3(g))=ψB1∪̃B2(g)∩̃ifψB1ŨB3(g)=ψ(B1∪̃B2)∩̃(B1∪̃B3)(g)
In the same way, (2) can be proved.
Definition 4.9. Let ΨB1,ΨB2∈ΩNPIFHS(X) then OR-operation of ΨB1 and ΨB2, denoted by ΨB1⊻̃ΨB2, is an npifhs-set defined by
LB1⊻̃B2(g1,g2)=max{LB1(g1),LB2(g2)},
MB1⊻̃B2(g1,g2)=min{MB1(g1),MB2(g2)},
NB1⊻̃B2(g1,g2)=min{NB1(g1),NB2(g2)},
ψB1⊻̃B2(g1,g2)=ψB1(g1)∪ψB2(g2), for all (g1,g2)∈B1×B2.
Definition 4.10. Let ΨB1,ΨB2∈ΩNPIFHS(X) then AND-operation of ΨB1 and ΨB2, denoted by ΨB1⊼̃ΨB2, is an npifhs-set defined by
LB1⊼̃B2(g1,g2)=min{LB1(g1),LB2(g2)},
MB1⊼̃B2(g1,g2)=max{MB1(g1),MB2(g2)},
NB1⊼̃B2(g1,g2)=max{NB1(g1),NB2(g2)},
ψB1⊼̃B2(g1,g2)=ψB1(g1)∩ψB2(g2), for all (g1,g2)∈B1×B2.
Proposition 4.4. Let ΨB1,ΨB2,ΨB3∈ΩNPIFHS(X) then
ΨB1⊼̃ΨΦ=ΨΦ.
(ΨB1⊼̃ΨB2)⊼̃ΨB3=ΨB1⊼̃(ΨB2⊼̃ΨB3).
(ΨB1⊻̃ΨB2)⊻̃ΨB3=ΨB1⊻̃(ΨB2⊻̃ΨB3).
Neutrosophic Decision Set of npifhs-Set
Here an algorithm is presented with the help of characterization of neutrosophic decision set on npifhs-set which based on decision making technique and is explained with example.
Definition 4.11. Let ΨB∈ΩNPIFHS(X) then a neutrosophic decision set of ΨB (i.e., ΨBD) is represented as
ΨBD={<TBD(u),IBD(u),FBD(u)>/u:u∈X}
where TBD,IBD,FBD:X→I and
TBD(u)=1|X|∑v∈S(B)TB(v)ΓψB(v)(u)IBD(u)=1|X|∑v∈S(B)IB(v)ΓψB(v)(u)FBD(u)=1|X|∑v∈S(B)FB(v)ΓψB(v)(u)
where |∙| denotes set cardinality with
ΓψB(v)(u)={|TψB(u)-FψB(u)|;u∈ΓψB(v)0;u∉ΓψB(v)
Definition 4.12. If ΨB∈ΩNPIFHS(X) with neutrosophic decision set ΨBD then reduced fuzzy set of ΨBD is a fuzzy set represented as
ℝ(ΨBD)={ζΨBD(u)/u:u∈X}
where ζΨBD:X→I with ζΨBD(u)=TBD(u)+IBD(u)-FBD(u)
Proposed Algorithm
Once ΨBD has been established, it may be indispensable to select the best single substitute from the options. Therefore, decision can be set up with the help of following algorithm:
Example 4.2. Suppose that Mrs. Andrew wants to buy a washing machine from market. There are eight kinds of washing machines (options) which form the set of discourse X={Ŵ1,Ŵ2,Ŵ3,Ŵ4,Ŵ5,Ŵ6,Ŵ7,Ŵ8}. The best selection may be evaluated by observing the attributes i.e., b1 = Company, b2 = Power in Watts, b3 = Voltage, b4 = Capacity in kg, and b5 = Color. The attribute-valued sets corresponding to these attributes are:
B1={b11=National,b12=Hier}B2={b21=400,b22=500}B3={b31=220,b32=240}B4={b41=7,b42=10}B5={b51=White}
then ℚ=B1×B2×B3×B4×B5ℚ={q1,q2,q3,q4,…,q16} where each qi,i=1,2,…,16, is a 5-tuples element.
Step 1:
From Tabs. 9–11, we can construct B as
B={<0.1,0.2,0.3>/q1,<0.2,0.3,0.4>/q2,<0.3,0.4,0.5>/q3,<0.4,0.5,0.6>/q4,<0.5,0.6,0.7>/q5,<0.6,0.7,0.8>/q6,<0.7,0.8,0.9>/q7,<0.8,0.9,0.1>/q8,<0.9,0.1,0.2>/q9,<0.16,0.27,0.37>/q10,<0.25,0.35,0.45>/q11,<0.45,0.55,0.65>/q12,<0.35,0.45,0.55>/q13,<0.75,0.85,0.95>/q14,<0.65,0.75,0.85>/q15,<0.85,0.95,0.96>/q16}
Degrees of membership TB(qi)
TB(qi)
Degree
TB(qi)
Degree
TB(q1)
0.1
TB(q9)
0.9
TB(q2)
0.2
TB(q10)
0.16
TB(q3)
0.3
TB(q11)
0.25
TB(q4)
0.4
TB(q12)
0.45
TB(q5)
0.5
TB(q13)
0.35
TB(q6)
0.6
TB(q14)
0.75
TB(q7)
0.7
TB(q15)
0.65
TB(q8)
0.8
TB(q16)
0.85
Degrees of indeterminacy IB(qi)
IB(qi)
Degree
IB(qi)
Degree
IB(q1)
0.2
IB(q9)
0.1
IB(q2)
0.3
IB(q10)
0.27
IB(q3)
0.4
IB(q11)
0.35
IB(q4)
0.5
IB(q12)
0.55
IB(q5)
0.6
IB(q13)
0.45
IB(q6)
0.7
IB(q14)
0.85
IB(q7)
0.8
IB(q15)
0.75
IB(q8)
0.9
IB(q16)
0.95
Degrees of non-membership FB(qi)
FB(qi)
Degree
FB(qi)
Degree
FB(q1)
0.3
FB(q9)
0.2
FB(q2)
0.4
FB(q10)
0.37
FB(q3)
0.5
FB(q11)
0.45
FB(q4)
0.6
FB(q12)
0.65
FB(q5)
0.7
FB(q13)
0.55
FB(q6)
0.8
FB(q14)
0.95
FB(q7)
0.9
FB(q15)
0.85
FB(q8)
0.1
FB(q16)
0.96
Step 2:
Tab. 12 presents ψB(qi) corresponding to each element of G.
Approximate functions ψB(qi)
qi
ψB(qi)
qi
ψB(qi)
q1
{<0.2,0.1>/Ŵ1,<0.3,0.2>/Ŵ2}
q9
{<0.4,0.3>/Ŵ2,<0.6,0.4>/Ŵ7,<0.5,0.4>/Ŵ8}
q2
{<0.1,0.2>/Ŵ1,<0.5,0.4>/Ŵ2,<0.1,0.4>/Ŵ3}
q10
{<0.2,0.1>/Ŵ6,<0.6,0.4>/Ŵ7,<0.4,0.3>/Ŵ8}
q3
{<0.4,0.3>/Ŵ2,<0.5,0.4>/Ŵ3,<0.6,0.3>/Ŵ4}
q11
{<0.5,0.4>/Ŵ2,<0.6,0.3>/Ŵ4,<0.7,0.2>/Ŵ6}
q4
{<0.6,0.2>/Ŵ4,<0.7,0.3>/Ŵ5,<0.8,0.1>/Ŵ6}
q12
{<0.7,0.2>/Ŵ2,<0.8,0.1>/Ŵ3,<0.9,0.1>/Ŵ6}
q5
{<0.2,0.1>/Ŵ6,<0.1,0.2>/Ŵ7,<0.4,0.3>/Ŵ8}
q13
{<0.2,0.1>/Ŵ3,<0.4,0.3>/Ŵ5,<0.6,0.1>/Ŵ7}
q6
{<0.4,0.2>/Ŵ2,<0.3,0.4>/Ŵ3,<0.4,0.5>/Ŵ4}
q14
{<0.2,0.5>/Ŵ1,<0.5,0.4>/Ŵ3,<0.6,0.2>/Ŵ5}
q7
{<0.2,0.3>/Ŵ1,<0.3,0.4>/Ŵ3,<0.4,0.3>/Ŵ5}
q15
{<0.6,0.3>/Ŵ5,<0.4,0.3>/Ŵ7,<0.2,0.4>/Ŵ8}
q8
{<0.1,0.4>/Ŵ2,<0.3,0.5>/Ŵ3,<0.5,0.4>/Ŵ7}
q16
{<0.3,0.6>/Ŵ4,<0.5,0.4>/Ŵ5,<0.7,0.1>/Ŵ6}
Step 3: With the help of Step 1 and Step 2, we can construct ΨB as performed in step of Section 3.
Step 4:
From Tabs. 13–16, we can construct ℝ(ΨBD) as
ℝ(ΨBD)={0.0331/Ŵ1,0.1100/Ŵ2,0.1019/Ŵ3,0.0659/Ŵ4,0.0855/Ŵ5,0.1394/Ŵ6,0.0690/Ŵ7,0.0296/Ŵ8}
Membership values TBD(Ŵi)
Ŵi
TBD(Ŵi)
Ŵi
TBD(Ŵi)
Ŵ1
0.0406
Ŵ5
0.1006
Ŵ2
0.0950
Ŵ6
0.1676
Ŵ3
0.1006
Ŵ7
0.0728
Ŵ4
0.0800
Ŵ8
0.0358
Indeterminacy values IBD(Ŵi)
Ŵi
IBD(Ŵi)
Ŵi
IBD(Ŵi)
Ŵ1
0.0481
Ŵ5
0.1169
Ŵ2
0.1025
Ŵ6
0.2028
Ŵ3
0.1219
Ŵ7
0.0655
Ŵ4
0.0975
Ŵ8
0.0309
Non-membership values FBD(Ŵi)
Ŵi
FBD(Ŵi)
Ŵi
FBD(Ŵi)
Ŵ1
0.0556
Ŵ5
0.1320
Ŵ2
0.0875
Ŵ6
0.2310
Ŵ3
0.1206
Ŵ7
0.0693
Ŵ4
0.1116
Ŵ8
0.0371
Reduced fuzzy membership ζΨBD(Ŵi)
Ŵi
ζΨBD(Ŵi)
Ŵi
ζΨBD(Ŵi)
Ŵ1
0.0331
Ŵ5
0.0855
Ŵ2
0.1100
Ŵ6
0.1394
Ŵ3
0.1019
Ŵ7
0.0690
Ŵ4
0.0659
Ŵ8
0.0296
The graphical representation of this decision system is presented in Fig. 2.
Neutrosophic decision system on npifhs-set
Step 5:
Since maximum of ζΨBD(Ŵi) is 0.5313 so the washing machine Ŵ3 is selected.
Neutrosophic Parameterized Neutrosophic Hypersoft Set (npnhs-Set) with Application
In this section, neutrosophic parameterized hypersoft set is conceptualized and some of its fundamentals are discussed.
Definition 5.1. Let Z={Z1,Z2,Z3,…,Zn} be a collection of disjoint attribute-valued sets corresponding to n distinct attributes α1,α2,α3,…,αn, respectively. A npnhs-set ΨD over X is defined as
ΨD={(<AD(g),BD(g),CD(g)>/g,ψD(g)):g∈G,ψD(g)∈N(X)}
where
N(X) is a collection of all neutrosophic sets over X
G=Z1×Z2×Z3×…×Zn
D is a neutrosophic set over G with AD,BD,CD:G→I as membership function, indeterminacy function and nonmembership function of npnhs-set.
ψD(g) is a neutrosophic set for all g∈G with ψD:G→N(X) and is called approximate function of npnhs-set.
Note that collection of all npnhs-sets is represented by ΩNPNHS(X).
Definition 5.2. Let ΨD∈ΩNPNHS(X). If ψD(g)=∅,AD(g)=0,BD(g)=1,CD(g)=1 for all g∈G, then ΨD is called D-empty npnhs-set, denoted by ΨΦD. If D=∅, then D-empty npnhs-set is called an empty npnhs-set, denoted by ΨΦ.
Definition 5.3. Let ΨD∈ΩNPNHS(X). If ψD(g)=X,AD(g)=1,BD(g)=0,CD(g)=0 for all g∈G, then ΨD is called D-universal npnhs-set, denoted by ΨD̃. If D=G, then the D-universal npnhs-set is called universal npnhs-set, denoted by ΨG̃.
Example 5.1. Consider X={u1,u2,u3,u4,u5} and Z={Z1,Z2,Z3} with Z1={ẑ11,ẑ12},Z2={ẑ21,ẑ22},Z3={ẑ31}, then
G=Z1×Z2×Z3G={(ẑ11,ẑ21,ẑ31),(ẑ11,ẑ22,ẑ31),(ẑ12,ẑ21,ẑ31),(ẑ12,ẑ22,ẑ31)}={g1,g2,g3,g4}.
Case 1.
If D1={<0.2,0.3,0.4>/g2,<0,1,1>/g3,<1,0,0>/g4} and
ψD1(g2)={<0.2,0.4,0.6>/u2,<0.3,0.5,0.7>/u4},ψD1(g3)=∅, and ψD1(g4)=X, thenΨD1={(<0.2,0.3,0.4>/g2,{<0.2,0.4,0.6>/u2,0.3,0.5,0.7>/u4}),(<0,1,1>/g3,∅),(<1,0,0>/g4,X)}.
Case 2.
If D2={<0,1,1>/g2,<0,1,1>/g3},ψD2(g2)=∅ and ψD2(g3)=∅, then ΨD2=ΨΦD2.
Case 3.
If D3=∅ corresponding to all elements of G, then ΨD3=ΨΦ.
Case 4.
If D4={<1,0,0>/g1,<1,0,0>/g2},ψD4(g1)=X, and ψD4(g2)=X, then ΨD4=ΨD4̃.
Case 5.
If D5=X with respect to all elements of G, then ΨD5=ΨG̃.
Definition 5.4. Let ΨD1, ΨD2∈ΩNPNHS(X) then ΨD1 is an npnhs-subset of ΨD2, denoted by ΨD1⊆̃ΨD2 if AD1(g)≤AD2(g),BD1(g)≥BD2(g),CD1(g)≥CD2(g) and ψD1(g)⊆nψD2(g) for all g∈G.
Proposition 5.1. Let ΨD1,ΨD2,ΨD3∈ΩNPNHS(X) then
ΨD1⊆̃ΨG̃.
ΨΦ⊆̃ΨD1.
ΨD1⊆̃ΨD1.
if ΨD1⊆̃ΨD2 and ΨD2⊆̃ΨD3 then ΨD1⊆̃ΨD3.
Definition 5.5. Let ΨD1,ΨD2∈ΩNPNHS(X) then, ΨD1 and ΨD2 are npnhs-equal, represented as ΨD1=ΨD2, if and only if AD1(g)=AD2(g),BD1(g)=BD2(g),CD1(g)=CD2(g) and ψD1(g)=nψD2(g) for all g∈G.
Proposition 5.2. Let ΨD1,ΨD2,ΨD3∈ΩNPNHS(X) then,
if ΨD1=ΨD2 and ΨD2=ΨD3 then ΨD1=ΨD3.
if ΨD1⊆̃ΨD2 and ΨD2⊆̃ΨD1⇔ΨD1=ΨD2.
Definition 5.6. Let ΨD∈ΩNPNHS(X) then, complement of ΨD (i.e., ΨDc̃) is a npnhs-set given as PDc̃(g)=1-AD(g),QDc̃(g)=1-BD(g),RDc̃(g)=1-CD(g) and ψDc̃(g)=X\nψD(g).
Proposition 5.3. Let ΨD∈ΩNPNHS(X) then,
(ΨDc̃)c̃=ΨD.
Ψϕc̃=ΨG̃.
Definition 5.7. Let ΨD1,ΨD2∈ΩNPNHS(X) then, union of ΨD1 and ΨD2, denoted by ΨD1∪̃ΨD2, is an npnhs-set defined by
AD1∪̃D2(g)=max{AD1(x),AD2(g)},
BD1∪̃D2(g)=min{BD1(x),BD2(g)},
CD1∪̃D2(g)=min{CD1(x),CD2(g)},
ψD1∪̃D2(g)=ψD1(g)∪nψD2(g), for all g∈G.
Proposition 5.4. Let ΨD1,ΨD2,ΨD3∈ΩNPNHS(X) then,
ΨD1∪̃ΨD1=ΨD1,
ΨD1∪̃ΨΦ=ΨD1,
ΨD1∪̃ΨG̃=ΨG̃,
ΨD1∪̃ΨD2=ΨD2∪̃ΨD1,
(ΨD1∪̃ΨD2)∪̃ΨD3=ΨD1∪̃(ΨD2∪̃ΨD3).
Definition 5.8. Let ΨD1,ΨD2∈ΩNPNHS(X) then intersection of ΨD1 and ΨD2, denoted by ΨD1∩̃ΨD2, is an npnhs-set defined by
AD1∩̃D2(g)=min{AD1(x),AD2(g)},
BD1∩̃D2(g)=max{BD1(x),BD2(g)},
CD1∩̃D2(g)=max{CD1(x),CD2(g)},
ψD1∩̃D2(g)=ψD1(g)∩nψD2(g), for all g∈G.
Proposition 5.5. Let ΨD1,ΨD2,ΨD3∈ΩNPNHS(X) then
ΨD1∩̃ΨD1=ΨD1.
ΨD1∩̃ΨΦ=ΨΦ.
ΨD1∩̃ΨG̃=ΨD1̃.
ΨD1∩̃ΨD2=ΨD2∩̃ΨD1.
(ΨD1∩̃ΨD2)∩̃ΨΨD3=ΨD1∩̃(ΨD2∩̃ΨΨD3).
Note: It is pertinent to mention here that Propositions 5.1, 5.2, 5.4 and 5.5 are also valid for elements of ΩNPFHS(X) and ΩNPIFHS(X).
Remark 5.1. Let ΨD∈ΩNPNHS(X). If ΨD≠ΨG̃, then ΨD∪̃ΨDc̃≠ΨG̃ and ΨD∩̃ΨDc̃≠ΨΦ
Proposition 5.6. Let ΨD1,ΨD2∈ΩNPNHS(X) then following D. Morgan laws are valid:
(ΨD1∪̃ΨD2)c̃=ΨD1c̃∩̃ΨD2c̃.
(ΨD1∩̃ΨD2)c̃=ΨD1c̃∪̃ΨD2c̃.
Proof. For all g∈G,
(1).Since (AD1∪̃D2)c̃(g)=1-AD1∪̃D2(g)=1-max{AD1(g),AD2(g)}=min{1-AD1(g),1-AD2(g)}=min{PD1c̃(g),PD2c̃(g)}=PD1∩̃D2c̃(g)
also
(BD1∪̃D2)c̃(g)=1-BD1∪̃D2(g)=1-min{BD1(g),BD2(g)}=max{1-BD1(g),1-BD2(g)}=max{QD1c̃(g),QD2c̃(g)}=QD1∩̃D2c̃(g)
and
(CD1∪̃D2)c̃(g)=1-CD1∪̃D2(g)=1-min{CD1(g),CD2(g)}=max{1-CD1(g),1-CD2(g)}=max{RD1c̃(g),RD2c̃(g)}=RD1∩̃D2c̃(g)
and
(ψD1∪̃D2)c̃(g)=X\nψD1∪̃D2(g)=X\n(ψD1(g)∪nψD2(g))=(X\nψD1(g))∩n(X\nψD2(g))=ψD1c̃(g)∩ñψD2c̃(g)=ψD1∩̃D2c̃(g).
similarly (2) can be proved easily.
Proposition 5.7. Let ΨD1,ΨD2,ΨD3∈ΩNPNHS(X) then
ΨD1∪̃(ΨD2∩̃ΨD3)=(ΨD1∪̃ΨD2)∩̃(ΨD1∪̃ΨD3).
ΨD1∩̃(ΨD2∪̃ΨD3)=(ΨD1∩̃ΨD2)∪̃(ΨD1∩̃ΨD3).
Proof. For all g∈G,
(1).Since AD1∪̃(D2∩̃D3)(g)=max{AD1(g),AD2∩̃D3(g)}=max{AD1(g),min{AD2(g),AD3(g)}}=min{max{AD1(g),AD2(g)},max{AD1(g),AD3(g)}}=min{AD1∪̃D2(g),AD1∪̃D3(g)}=A(D1∪̃D2)∩̃(D1∪̃D3)(g)
and
BD1∪̃(D2∩̃D3)(g)=min{BD1(g),BD2∩̃D3(g)}=min{BD1(g),max{BD2(g),BD3(g)}}=max{min{BD1(g),BD2(g)},min{BD1(g),BD3(g)}}=max{BD1∪̃D2(g),BD1∪̃D3(g)}=B(D1∪̃D2)∩̃(D1∪̃D3)(g)
and
CD1∪̃(D2∩̃D3)(g)=min{CD1(g),CD2∩̃D3(g)}=min{CD1(g),max{CD2(g),CD3(g)}}=max{min{CD1(g),CD2(g)},min{CD1(g),CD3(g)}}=max{CD1∪̃D2(g),CD1∪̃D3(g)}=C(D1∪̃D2)∩̃(D1∪̃D3)(g)
and
ψD1∪̃n(D2∪̃nD3)(g)=ψD1(g)∪nψD2∩̃nD3(g)=ψD1(g)∪n(ψD2(g)∩nψD3(g))=(ψD1(g)∪nψD2(g))∩n(ψD1(g)∪nψD3(g))=ψD1∪̃D2(g)∩nψD1∪̃D3(g)=ψ(D1∪̃D2)∩̃(D1∪̃D3)(g)
In the same way, (2) can be proved.
Definition 5.9. Let ΨD1,ΨD2∈ΩNPNHS(X) then OR-operation of ΨD1 and ΨD2, denoted by ΨD1⊕̃ΨD2, is an npnhs-set defined by
AD1⊕̃D2(g1,g2)=max{AD1(g1),AD2(g2)},
BD1⊕̃D2(g1,g2)=min{BD1(g1),BD2(g2)},
CD1⊕̃D2(g1,g2)=min{CD1(g1),CD2(g2)},
ψD1⊕̃D2(g1,g2)=ψD1(g1)∪ψD2(g2), for all (g1,g2)∈D1×D2.
Definition 5.10. Let ΨD1,ΨD2∈ΩNPNHS(X) then AND-operation of ΨD1 and ΨD2, denoted by ΨD1⊗̃ΨD2, is an npnhs-set defined by
AD1⊗̃D2(g1,g2)=min{AD1(g1),AD2(g2)},
BD1⊗̃D2(g1,g2)=max{BD1(g1),BD2(g2)},
CD1⊗̃D2(g1,g2)=max{CD1(g1),CD2(g2)},
ψD1⊗̃D2(g1,g2)=ψD1(g1)∩ψD2(g2), for all (g1,g2)∈D1×D2.
Proposition 5.8. Let ΨD1,ΨD2,ΨD3∈ΩNPNHS(X) then
ΨD1⊗̃ΨΦ=ΨΦ.
(ΨD1⊗̃ΨD2)⊗̃ΨD3=ΨD1⊗̃(ΨD2⊗̃ΨD3).
(ΨD1⊕̃ΨD2)⊕̃ΨD3=ΨD1⊕̃(ΨD2⊕̃ΨD3).
Neutrosophic Decision Set of npnhs-Set
Here an algorithm is presented with the help of characterization of neutrosophic decision set on npnhs-set which based on decision making technique and is explained with example.
Definition 5.11. Let ΨD∈ΩNPNHS(X) then a neutrosophic decision set of ΨD (i.e., ΨDD) is represented as
ΨDD={<TDD(u),IDD(u),FDD(u)>/u:u∈X}
where TDD,IDD,FDD:X→I and
TDD(u)=1|X|∑v∈S(D)TD(v)ΓψD(v)(u)IDD(u)=1|X|∑v∈S(D)ID(v)ΓψD(v)(u)FDD(u)=1|X|∑v∈S(D)FD(v)ΓψD(v)(u)
where |∙| denotes set cardinality with
ΓψD(v)(u)={|TψD(u)+IψD(u)-FψD(u)|;u∈ΓψD(v)0;u∉ΓψD(v)
Definition 5.12. If ΨD∈ΩNPNHS(X) with neutrosophic decision set ΨDD then reduced fuzzy set of ΨDD is a fuzzy set represented as
ℝ(ΨDD)={ζΨDD(u)/u:u∈X}
where ζΨDD:X→I with ζΨDD(u)=TDD(u)+IDD(u)-FDD(u).
Proposed Algorithm
Once ΨDD has been established, it may be indispensable to select the best single substitute from the options. Therefore, decision can be set up with the help of following algorithm:
Hand sanitizer is a liquid or gel mostly used to diminish infectious agents on the hands. According to the World Health Organization (WHO), in current epidemic circumstances of COVID-19, high-quality sanitation and physical distancing are the best ways to protect ourselves and everyone around us from this virus. This virus spreads by touching an ailing person. We cannot detach ourselves totally being cautious from this virus. So, high-quality sanitation can be the ultimate blockade between us and the virus. Alcohol-based hand sanitizers are recommended by WHO to remove the novel corona virus. Alcohol-based hand sanitizers avert the proteins of germs including bacteria and some viruses from functioning normally. Demand of a hand sanitizer has been increased terrifically in such serious condition of COVID-19. Therefore, it is tricky to have good and effectual hand sanitizers in local markets. Low quality hand sanitizers have also been introduced due to its increasing demand. The core motivation of this application is to select an effectual sanitizer to alleviate the spread of corona virus by applying the NPNHS-set theory.
Example 5.2. Suppose that Mr. William wants to purchase an effective hand sanitizer from the local market. There are eight kinds of Hand Sanitizer (options) which form the set of discourse
X={H1,H2,H3,H4,H5,H6,H7,H8}.
The best selection may be evaluated by observing the attributes i.e., k1 = Manufacturer, k2 = Quantity of Ethanol (percentage), k3 = Quantity of Distilled Water (percentage), k4 = Quantity of Glycerol (percentage), and k5 = Quantity of Hydrogen peroxide (percentage). The attribute-valued sets corresponding to these attributes are:
K1={k11=ProcterandGamble,k12=Unilever}K2={k21=75.15,k22=80}K3={k31=23.425,k32=18.425}K4={k41=1.30,k42=1.45}K5={k51=0.125}
then ℙ=K1×K2×K3×K4×K5ℙ={p1,p2,p3,p4,…,p16} where each pi,i=1,2,…,16,is a 5-tuples element.
Step 1:
From Tabs. 17–19, we can construct D as
D={<0.1,0.2,0.3>/p1,<0.2,0.3,0.4>/p2,<0.3,0.4,0.5>/p3,<0.4,0.5,0.6>/p4,<0.5,0.6,0.7>/p5,<0.6,0.7,0.8>/p6,<0.7,0.8,0.9>/p7,<0.8,0.9,0.1>/p8,<0.9,0.1,0.2>/p9,<0.16,0.27,0.37>/p10,<0.25,0.35,0.45>/p11,<0.45,0.55,0.65>/p12,<0.35,0.45,0.55>/p13,<0.75,0.85,0.95>/p14,<0.65,0.75,0.85>/p15,<0.85,0.95,0.96>/p16}.
Degrees of membership TD(pi)
TD(pi)
Degree
TD(pi)
Degree
TD(p1)
0.1
TD(p9)
0.9
TD(p2)
0.2
TD(p10)
0.16
TD(p3)
0.3
TD(p11)
0.25
TD(p4)
0.4
TD(p12)
0.45
TD(p5)
0.5
TD(p13)
0.35
TD(p6)
0.6
TD(p14)
0.75
TD(p7)
0.7
TD(p15)
0.65
TD(p8)
0.8
TD(p16)
0.85
Degrees of indeterminacy ID(pi)
ID(pi)
Degree
ID(pi)
Degree
ID(p1)
0.2
ID(p9)
0.1
ID(p2)
0.3
ID(p10)
0.27
ID(p3)
0.4
ID(p11)
0.35
ID(p4)
0.5
ID(p12)
0.55
ID(p5)
0.6
ID(p13)
0.45
ID(p6)
0.7
ID(p14)
0.85
ID(p7)
0.8
ID(p15)
0.75
ID(p8)
0.9
ID(p16)
0.95
Degrees of non-membership FD(pi)
FD(pi)
Degree
FD(pi)
Degree
FD(p1)
0.3
FD(p9)
0.2
FD(p2)
0.4
FD(p10)
0.37
FD(p3)
0.5
FD(p11)
0.45
FD(p4)
0.6
FD(p12)
0.65
FD(p5)
0.7
FD(p13)
0.55
FD(p6)
0.8
FD(p14)
0.95
FD(p7)
0.9
FD(p15)
0.85
FD(p8)
0.1
FD(p16)
0.96
Step 2:
Tab. 20 presents ψD(pi) corresponding to each element of G.
ΨD can be constructed with the help of Step 1 and Step 2 same as done in Step 3 of Section 3.
Step 4:
From Tabs. 21–24, we can construct ℝ(ΨDD) as
ℝ(ΨDD)={0.0344/H1,0.1600/H2,0.1500/H3,0.1289/H4,0.1367/H5,0.0749/H6,0.1538/H7,0.1006/H8}.
Membership values TDD(Hi)
Hi
TDD(Hi)
Hi
TDD(Hi)
H1
0.0431
H5
0.1656
H2
0.1825
H6
0.0964
H3
0.1588
H7
0.1588
H4
0.1606
H8
0.1231
Indeterminacy values IDD(Hi)
Hi
IDD(Hi)
Hi
IDD(Hi)
H1
0.0519
H5
0.1956
H2
0.1713
H6
0.1203
H3
0.1900
H7
0.1081
H4
0.1969
H8
0.0788
Non-membership values IDD(Hi)
Hi
FDD(Hi)
Hi
FDD(Hi)
H1
0.0606
H5
0.2245
H2
0.1938
H6
0.1418
H3
0.1988
H7
0.1131
H4
0.2286
H8
0.1013
Reduced fuzzy membership ζΨDD(Hi)
Hi
ζΨDD(Hi)
Hi
ζΨDD(Hi)
H1
0.0344
H5
0.1367
H2
0.1600
H6
0.0749
H3
0.1500
H7
0.1538
H4
0.1289
H8
0.1006
The graphical representation of this decision system is presented in Fig. 3.
Neutrosophic decision system on npnhs-set
Step 5:
Since maximum of ζΨDD(Hi) is 0.1600 so the Hand Sanitizer H2 is selected.
Discussion
The development and stability of any society depends on its justice system and the judges, lawyers and plaintiffs play a key role in its basic components. The lawyer prepares the writ petition at the request of the plaintiff but when filing the case in the Court of Justice, he/she is in a state of uncertainty for its success. This uncertain condition can be of fuzzy, intuitionistic fuzzy or even neutrosophic. And after the case is submitted, the judge concerned writes his/her decision in the light of the facts, but usually all facts have some kind of uncertainty. Such factual vagueness again may be of fuzzy, intuitionistic fuzzy or neutrosophic nature. So when initial stage (submission stage) and final stage (decisive stage) are neutrosophic valued and the process is executed with the help of parameterized data (collections of parametric values) then we say that we are tackling such problem with the help of neutrosophic parameterized neutrosophic hypersoft set (npnhs-set). Since decision makers always face some sort of uncertainties and any decision taken by ignoring uncertainty may have some extent of inclination. Indeterminacy and uncertainty are both interconnected. In this study, it has been shown (i.e., see Fig. 4) that how results are affected when indeterminacy is ignored or considered. Our proposed structure npnhs-set is very useful in dealing with many decisive systems and it is the generalization of:
Neutrosophic Parameterized Intuitionistic Fuzzy Hypersoft Set (npifhs-set) if indeterminacy is ignored and remaining two are made interdependent within closed unit interval in approximate function of npnhs-set,
Neutrosophic Parameterized Fuzzy Hypersoft Set (npfhs-set) if indeterminacy and falsity are ignored and remaining be restricted within closed unit interval in approximate function of npnhs-set,
Neutrosophic Parameterized Hypersoft Set (nphs-set) if all uncertain components are ignored and approximate function of npnhs-set is a subset of universe of discourse,
Neutrosophic Parameterized Neutrosophic Soft Set (npns-set) if attribute-valued sets are replaced with only attributes in npnhs-set,
Neutrosophic Parameterized Intuitionistic Fuzzy Soft Set (npifs-set) if attribute-valued sets are replaced with only attributes and indeterminacy is ignored and remaining two are made interdependent within closed unit interval in approximate function of npnhs-set,
Neutrosophic Parameterized Fuzzy Soft Set (npfs-set) if attribute-valued sets are replaced with only attributes and indeterminacy, falsity are ignored and remaining be restricted within closed unit interval in approximate function of npnhs-set,
Neutrosophic Parameterized Soft Set (nps-set) if attribute-valued sets are replaced with only attributes and all uncertain components are ignored with approximate function of npnhs-set as a subset of universe of discourse.
Fig. 5 presents the pictorial view of the generalization of the proposed structure.
Comparison of neutrosophic decision system on npfhs-set, npifhs-set and npnhs-set
Generalization of npnhs-set
Conclusion
In this study, neutrosophic parameterized hypersoft set is conceptualized for the environments of fuzzy set, intuitionistic fuzzy set and neutrosophic set along with some of their elementary properties and theoretic operations. Novel algorithms are proposed for decision making and are validated with the help of illustrative examples for appropriate purchasing of suitable products i.e., Mobile Tablet, Washing Machines and Hand Sanitizers, from the local market. Future work may include the extension of this work for:
The development of algebraic structures i.e., topological spaces, vector spaces, etc.,
The development of hybrid structures with fuzzy-like environments,
Dealing with decision making problems with multi-criteria decision making techniques,
Applying in medical diagnosis and optimization for agricultural yield,
Investigating and determining similarity, distance, dissimilarity measures and entropies between the proposed structures.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
ReferencesZadeh, L. (1965). Fuzzy sets. ,8(3),338–353. DOI 10.1016/S0019-9958(65)90241-X.Atanassov, K. (1986). Intuitionistic fuzzy sets. ,20(1),87–96. DOI 10.1016/S0165-0114(86)80034-3.Smarandache, F. (1998). . Rehoboth: American Research Press.Smarandache, F. (1999). A Unifying field in logics. . Rehoboth: American Research Press.Smarandache, F. (2005). Neutrosophic set, a generalization of intuitionistic fuzzy sets. ,24(3),287–297.Smarandache, F. (2006). Neutrosophic set—A generalization of intuitionistic fuzzy set. International Conference on Granular Computing, pp. 38–42. Atlanta, Georgia, USA, IEEE.Wang, H., Rogatko, A., Smarandache, F., Sunderraman, R. (2006). A Neutrosophic description logic. International Conference on Granular Computing, pp. 305–308. Atlanta, Georgia, USA, IEEE. DOI 10.1109/GRC.2006.1635801.Smarandache, F. (2013). n-Valued refined neutrosophic logic and its applications in physics. ,4,143–146. DOI 10.5281/zenodo.49149.Smarandache, F. (2016a). Operators on single valued neutrosophic sets, neutrosophic undersets, and neutrosophic offsets. ,5,63–67. DOI 10.5281/zenodo.57412.Smarandache, F. (2016b). . Brussels: Pons Editions. DOI 10.5281/zenodo.57410.Broumi, S., Bakali, A., Talea, M., Smarandache, F., Karaaslan, F. (2016). Interval valued neutrosophic soft graphs. ,2,214–251. DOI 10.5281/zenodo.1410271.Alhabib, R., Ranna, M. M., Farah, H., Salama, A. A. (2018). Some neutrosophic probability distributions. ,22,30–38. DOI 10.5281/zenodo.2160479.Pramanik, S., Dey, P. P., Smarandache, F. (2018). Correlation coefficient measures of interval bipolar neutrosophic sets for solving multi-attribute decision making problems. ,19,70–79. DOI 10.5281/zenodo.1235151.Broumi, S., Smarandache, F. (2013). Intuitionistic neutrosophic soft set. ,8(2),130–140. DOI 10.5281/zenodo.2861554.Broumi, S., Smarandache, F. (2013). More on intuitionistic neutrosophic soft set. ,1(4),257–268. DOI 10.13189/csit.2013.010404.Broumi, S. (2013). Generalized neutrosophic soft set. ,3(2),17–30. DOI 10.5121/ijcseit.2013.3202.Broumi, S., Deli, I., Smarandache, F. (2014). Relations on interval valued neutrosophic soft sets. ,5,1–20. DOI 10.5281/zenodo.30306.Deli, I. (2017). Interval-valued neutrosophic soft sets and its decision making. ,8(2),665–676. DOI 10.1007/s13042-015-0461-3.Kharal, A. (2013). A neutrosophic multicriteria decision making method. ,10(2),143–162. DOI 10.1142/S1793005714500070.Edalatpanah, S. A. (2020). Systems of neutrosophic linear equations. ,33,92–104. DOI 10.5281/zenodo.3782826.Kumar, R., Edalatpanah, S. A., Jha, S., Singh, R. (2019). A novel approach to solve gaussian valued neutrosophic shortest path problems. ,8(3),347–353. DOI 10.35940/ijeat.E1049.0585C19.Molodtsov, D. (1999). Soft set theory—First results. ,37,19–31. DOI 10.1016/S0898-1221(99)00056-5.Maji, P. K., Biswas, R., Roy, A. R. (2003). Soft set theory. ,45,555–562. DOI 10.1016/S0898-1221(03)00016-6.Maji, P. K., Biswas, R., Roy, A. R. (2001). Fuzzy soft sets. ,9(3),589–602.Pei, D., Miao, D. (2005). From soft set to information system. International Conference of Granular Computing, vol. 2, pp. 617–621. IEEE, DOI 10.1109/GRC.2005.1547365.Ali, M. I., Feng, F., Liu, X., Min, W. K., Sabir, M. (2009). On some new operations in soft set theory. ,57,1547–1553. DOI 10.1016/j.camwa.2008.11.009.Babitha, K. V., Sunil, J. J. (2010). Soft set relations and functions. ,60,1840–1849. DOI 10.1016/j.camwa.2010.07.014.Babitha, K. V., Sunil, J. J. (2011). Transitive closure and ordering in soft set. ,61,2235–2239. DOI 10.1016/j.camwa.2011.07.010.Sezgin, A., Atagün, A. O. (2011). On operations of soft sets. ,61(5),1457–1467. DOI 10.1016/j.camwa.2011.01.018.Ge, X., Yang, S. (2011). Investigations on some operations of soft sets. ,5(3),370–373.Li, F. (2011). Notes on soft set operations. ,1(6),205–208.Maji, P. K., Biswas, R., Roy, A. R. (2001). Intuitionistic fuzzy soft sets. ,9(3),677–692.Çağman, N., Enginoğlu, S., Çitak, F. (2011). Fuzzy soft set theory and its applications. ,8(3),137–147. DOI 10.22111/IJFS.2011.292.Çağman, N., Karataş, S. (2013). Intuitionistic fuzzy soft set theory and its decision making. ,24(4),829–836. DOI 10.3233/IFS-2012-0601.Maji, P. K. (2013). Neutrosophic soft set. ,5(1),157–168.Mandal, D. (2015). Comparative study of intuitionistic and generalized neutrosophic soft sets. ,9(2),111–114. DOI 10.5281/zenodo.1100511.Smarandache, F. (2018). Extension of soft set of hypersoft set, and then to plithogenic hypersoft set. ,22,168–170. DOI 10.5281/zenodo.2159755.Saeed, M., Ahsan, M., Siddique, M. K., Ahmad, M. R. (2020). A study of the fundamentals of hypersoft set theory. ,11(1),320–329.Saeed, M., Rahman, A. U., Ahsan, M., Smarandache, F. (2021). An inclusive study on fundamentals of hypersoft set. , pp. 1–23. Brussels: Pons Publishing House.Abbas, F., Murtaza, G., Smarandache, F. (2020). Basic operations on hypersoft sets and hypersoft points. ,35,407–421. DOI 10.5281/zenodo.3951694.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. ,32,317–329. DOI 10.5281/zenodo.3723165.Saqlain, M., Moin, S., Jafar, N., Saeed, M., Smarandache, F. (2020). Aggregate operators of neutrosophic hypersoft sets. ,32,294–306. DOI 10.5281/zenodo.3723155.Saqlain, M., Saeed, M., Ahmad, M. R., Smarandache, F. (2020). Generalization of TOPSIS for neutrosophic hypersoft sets using accuracy function and its application. ,27,131–137. DOI 10.5281/zenodo.3275533.Martin, N., Smarandache, F. (2020). Concentric plithogenic hypergraph based on plithogenic hypersoft sets a novel outlook. ,33,78–91. DOI 10.5281/zenodo.3782824.Rahman, A. U., Saeed, M., Smarandache, F., Ahmad, M. R. (2020). Development of hybrids of hypersoft set with complex fuzzy set, complex intuitionistic fuzzy set and complex neutrosophic set. ,38,335–354. DOI 10.5281/zenodo.4300520.Rahman, A. U., Saeed, M., Smarandache, F. (2020). Convex and concave hypersoft sets with some properties. ,38,497–508. DOI 10.5281/zenodo.4300580.Deli, I. (2020). Hybrid set structures under uncertainly parameterized hypersoft sets: Theory and applications. , pp. 24–49. Brussels: Pons Publishing House.Gayen, S., Smarandache, F., Jha, S., Singh, M. K., Broumi, S.et al. (2020). Introduction to plithogenic hypersoft subgroup. ,33(1),14. DOI 10.5281/zenodo.3782897.Çağman, N., Çitak, F., Enginoğlu, S. (2010). Fuzzy parameterized fuzzy soft set theory and its applications. ,1(1),21–35.Deli, I., Çağman, N. (2015). Intuitionistic fuzzy parameterized soft set theory and its decision making. ,28,109–113. DOI 10.1016/j.asoc.2014.11.053.Karaaslan, F., Karataş, S. (2013). OR and AND-products of ifp-intuitionistic fuzzy soft sets and their applications in decision making. ,31(3),1427–1434. DOI 10.3233/IFS-162209.Çağman, N., Deli, I. (2012). Products of FP-soft sets and their applications. ,41(3),365–374.Çağman, N., Deli, I. (2012). Means of FP-soft sets and their applications. ,41(5),615–625.Broumi, S., Deli, I., Smarandache, F. (2014). Neutrosophic parametrized soft set theory and its decision making. ,1,1–10. DOI 10.18052/www.scipress.com/IFSL.1.1.