Decision Making Algorithmic Approaches Based on Parameterization of Neutrosophic Set under HypersoftSet Environment with Fuzzy,Intuitionistic Fuzzy and Neutrosophic Settings

Atiqe Ur Rahman,Muhammad Saeed,Sultan S.Alodhaibi and Hamiden Abd El-Wahed Khalifa

1Department of Mathematics,University of Management and Technology,Lahore,54000,Pakistan

2Department of Mathematics,College of Science and Arts,Qassim University,Al-Rass,51921,Saudi Arabia

3Department of Operations Research,Faculty of Graduate Studies for Statistical Research,Cairo University,Giza,12613,Egypt

4Department of Mathematics,College of Science and Arts,Qassim University,Al-Badaya,51951,Saudi Arabia

ABSTRACT Hypersoftset is an extension of softset 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 softset 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 softset in line with the need of multi-attribute approximate function.Firstly,we conceptualize the neutrosophic parameterized hypersoftsets 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.

KEYWORDS Neutrosophic set; hypersoftset; neutrosophic hypersoftset; parameterized softset; parameterized hypersoftset

1 Introduction

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 neutrosophiclike 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.

1.1 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.

1.2 Organization of Paper

The rest of the paper is systemized as:

Section 2Some essential definitions and terminologies are recalled.Section 3Theory of neutrosophic parameterized fuzzy hypersoft set is developed with suitable examples.Section 4Theory of neutrosophic parameterized intuitionistic fuzzy hypersoft set is characterized with suitable examples.Section 5Theory of neutrosophic parameterized neutrosophic hypersoft set is investigated with suitable examples.Section 6Analysis of proposed structure is discussed.Section 7Paper is summarized with future directions.

2 Preliminaries

3 Neutrosophic Parameterized Fuzzy Hypersoft Set(npfhs-Set)with Application

3.1 Neutrosophic Decision Set of npfhs-Set

An algorithm is presented with the help of characterization of neutrosophic decision set onnpfhs-set which based on decision making technique and is explained with example.

Definition 3.11.LetΨA∈ΩNPFHS(X)then a neutrosophic decision set ofΨA(i.e.,)is represented as

Step 2:

Tab.4 presentsψA(ri)corresponding to each element of G.

Step 3:

With the help of Step 1 and Step 2,we can constructΨAas

The graphical representation of this decision system is presented in Fig.1.

Step 5:

Figure 1:Neutrosophic decision system on npfhs-set

4 Neutrosophic Parameterized Intuitionistic Fuzzy Hypersoft Set(npifhs-set)with Application

4.1 Neutrosophic Decision Set of npifhs-Set

Here an algorithm is presented with the help of characterization of neutrosophic decision set onnpifhs-set which based on decision making technique and is explained with example.

Definition 4.11.LetΨB∈ΩNPIFHS(X)then a neutrosophic decision set ofΨB(i.e.,)is represented as

4.2 Proposed Algorithm

OnceΨDBhas 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:

Step 2 FindψB(g)

Step 3 ConstructΨBover X,

Step 4 Compute

Step 5 Choose the maximum of

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 discourseThe best selection may be evaluated by observing the attributes i.e.,b1= Company,b2= Power in Watts,b3= Voltage,b4= Capacity in kg,andb5= 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}

Step 2:

Tab.12 presentsψB(qi)corresponding to each element of G.

Step 3:With the help of Step 1 and Step 2,we can constructΨBas performed in step of Section 3.

Step 4:

From Tabs.13–16,we can constructas

The graphical representation of this decision system is presented in Fig.2.

Step 5:

5 Neutrosophic Parameterized Neutrosophic Hypersoft Set(npnhs-Set)with Application

5.1 Neutrosophic Decision Set of npnhs-Set

Here an algorithm is presented with the help of characterization of neutrosophic decision set onnpnhs-set which based on decision making technique and is explained with example.represented as

Defniition5.11.LetΨD∈ΩNPNHS(X)then a neutrosophic decision set ofΨD(i.e.,is

5.2 Proposed Algorithm

Step 2 FindψD(g)

Step 3 ConstructΨDover X,

Step 5 Choose the maximum of

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),andk5= Quantity of Hydrogen peroxide (percentage).The attributevalued sets corresponding to these attributes are:

K1={k11=Procter and Gamble,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×K5

P={p1,p2,p3,p4,...,p16}where eachpi,i=1,2,...,16,is a 5-tuples element.

Step 1:

From Tabs.17–19,we can constructDas

Step 2:

Tab.20 presentsψD(pi)corresponding to each element of G.

Step 3:

ΨDcan 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

The graphical representation of this decision system is presented in Fig.3.

Step 5:

Since maximum ofis 0.1600 so the Hand SanitizerH2is selected.

Figure 3:Neutrosophic decision system on npnhs-set

6 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 (npnhsset).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:

(i)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,

(ii)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,

(iii)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,

(iv)Neutrosophic Parameterized Neutrosophic Soft Set (npns-set)if attribute-valued sets are replaced with only attributes in npnhs-set,

(v)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,

(vi)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,

(vii)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.

Figure 4:Comparison of neutrosophic decision system on npfhs-set,npifhs-set and npnhs-set

Figure 5:Generalization of npnhs-set

7 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:

Table 1:Degrees of membership TA(ri)

Table 2:Degrees of indeterminacy

Table 2:Degrees of indeterminacy

IA(ri)DegreeIA(ri)Degree IA(r1)0.2IA(r9)0.1 IA(r2)0.3IA(r10)0.27 IA(r3)0.4IA(r11)0.35 IA(r4)0.5IA(r12)0.55 IA(r5)0.6IA(r13)0.45 IA(r6)0.7IA(r14)0.85 IA(r7)0.8IA(r15)0.75 IA(r8)0.9IA(r16)0.95

Table 3:Degrees of non-membership

Table 3:Degrees of non-membership

FA(ri)DegreeFA(ri)Degree FA(r1)0.3FA(r9)0.2 FA(r2)0.4FA(r10)0.37 FA(r3)0.5FA(r11)0.45 FA(r4)0.6FA(r12)0.65 FA(r5)0.7FA(r13)0.55 FA(r6)0.8FA(r14)0.95 FA(r7)0.9FA(r15)0.85 FA(r8)0.1FA(r16)0.96

Table 4:Approximate functions ψA(ri)

Table 5:Membership values

Table 5:Membership values

ˆTiT DA(ˆTi)ˆTiT DA(ˆTi)ˆT10.0413 0.1700 ˆT50.2456 ˆT2ˆT60.2034 ˆT30.2006ˆT70.1945 ˆT40.1331ˆT80.1055

Table 6:Indeterminacy values

Table 6:Indeterminacy values

ˆTiIDA(ˆTi)ˆTiIDA(ˆTi)ˆT10.0500ˆT 5 650ˆT60.381ˆT70.644ˆT80.0.2856 ˆT20.12474 ˆT30.21628 ˆT40.10685

Table 7:Non-membership values

Table 7:Non-membership values

ˆTiFDA(ˆTi)ˆTiFDA(ˆTi)ˆT10.0588ˆT50.3200 ˆT20.193 8 ˆT6 9 ˆT7ˆ 0.2833 ˆT30.2410.1415 ˆT40.1923T80.0873

Table 8:Reduced fuzzy membership

Table 8:Reduced fuzzy membership

ˆTiζΨDA(ˆTi)ˆTiζΨDA(ˆTi)ˆT10.0325 0.1412 ˆT50.2112 ˆT2ˆT60.1675 ˆT30.1968ˆT70.2158 ˆT40.1052ˆT80.0867

Table 9:Degrees of membership TB(qi)

Table 10:Degrees of indeterminacy IB(qi)

Table 11:Degrees of non-membership FB(qi)

Table 12:Approximate functions ψB(qi)

Table 13:Membership values

ˆWiT DB ( ˆWi)ˆWiT DB ( ˆWi)ˆW10.0406ˆW50.1006 ˆW20.0950ˆW60.1676 ˆW30.1006ˆW70.0728 ˆW40.0800ˆW80.0358

Table 14:Indeterminacy values

Table 14:Indeterminacy values

ˆWiIDB( ˆWi)ˆWiIDB( ˆWi)ˆW10.0481ˆW50.1169 ˆW2 0.1025ˆW60.2028 ˆW30.1219ˆW70.0655 ˆW40.0975ˆW80.0309

Table 15:Non-membership values

Table 15:Non-membership values

ˆWiFDB( ˆWi)ˆWiFDB( ˆWi)ˆW10.0556ˆW50.1320 ˆW2 0.0875 0.1206 0.1116ˆW60.2310 ˆW3ˆW70.0693 ˆW4ˆW80.0371

Table 16:Reduced fuzzy membership

Table 16:Reduced fuzzy membership

ˆWiζΨD B 0.0331 0.1100( ˆWi)ˆWiζΨDB( ˆWi)ˆW1ˆW50.0855 ˆW2ˆW60.1394 ˆW30.1019ˆW70.0690 ˆW40.0659ˆW80.0296

Table 17:Degrees of membership TD(pi)

Table 18:Degrees of indeterminacy ID(pi)

Table 19:Degrees of non-membership FD(pi)

Table 20:Approximate functions ψD(pi)

Table 21:Membership values

Table 21:Membership values

HiT DD(Hi)HiT DD(Hi)H10.04 31H5 25H6 88H70.1656 H20.180.0964 H30.150.1588 H40.1606H80.1231

Table 22:Indeterminacy values

Table 22:Indeterminacy values

HiIDD(Hi)Hi H5 H6IDD(Hi)H10.05190.1956 H20.17130.1203 H30.1900H70.1081 H40.1969H80.0788

Table 23:Non-membership values

Table 23:Non-membership values

HiFDD(Hi)HiFDD(Hi)H10.06 06H5 38H60.2245 H20.190.1418 H30.1988H70.1131 H40.2286H80.1013

Table 24:Reduced fuzzy membership ζΨDD(Hi)

• 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.