Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2021.016736

ARTICLE

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

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

*Corresponding Author: Atiqe Ur Rahman. Email: aurkhb@gmail.com

Received: 23 March 2021; Accepted: 07 May 2021

Abstract: 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.

Keywords: Neutrosophic set; hypersoft set; neutrosophic hypersoft set; parameterized soft set; parameterized hypersoft set

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

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.

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.

The rest of the paper is systemized as:

Here some basic terms are recalled from existing literature to support the proposed work. Throughout the paper,

Definition 2.1. [1]

A fuzzy set

Definition 2.2. [2]

An intuitionistic fuzzy set

Definition 2.3. [3]

A neutrosophic set

Definition 2.4. [22]

A pair

For more detail on soft set, see [23–32].

Definition 2.5. [37]

The pair

For more definitions and operations of hypersoft set, see [38–40].

3 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

i)

ii)

iii)

iv)

Note that collection of all npfhs-sets is represented by

Definition 3.2. Let

Definition 3.3. Let

Example 3.1. Consider

Case 1.

If

Case 2.

If

Case 3.

If

Case 4.

If

Case 5.

If

Definition 3.4. Let

Definition 3.5. Let

Definition 3.6. Let

Proposition 3.1. Let

1.

2.

Definition 3.7. Let

i)

ii)

iii)

iv)

Definition 3.8. Let

i)

ii)

iii)

iv)

Remark 3.1. Let

Proposition 3.2. Let

1.

2.

Proof. For all

also

Proposition 3.3. Let

1.

2.

Proof. For all

In the same way, (2) can be proved.

Definition 3.9. Let

i)

ii)

iii)

iv)

Definition 3.10. Let

i)

ii)

iii)

iv)

Proposition 3.4. Let

1.

2.

3.

3.1 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

Definition 3.12. If

Algorithm 3.1. Once

Step 1 Determine

Step 2 Find

Step 3 Construct

Step 4 Compute

Step 5 Choose the maximum of

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

Step 1:

From Tabs. 1–3, we can construct

Step 2:

Tab. 4 presents

Step 3:

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

Step 4:

From Tabs. 5–8, we can construct

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

Step 5:

Since maximum of

4 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

i)

ii)

iii)

iv)

Note that collection of all npifhs-sets is represented by

Definition 4.2. Let

Definition 4.3. Let

Example 4.1. Consider

Case 1.

If

Case 2.

If

Case 3.

If

Case 4.

If

Case 5.

If

Definition 4.4. Let

Definition 4.5. Let

Definition 4.6. Let

Proposition 4.1. Let

1.

2.

Definition 4.7. Let

i)

ii)

iii)

iv)

Definition 4.8. Let

i)

ii)

iii)

iv)

Remark 4.1. Let

Proposition 4.2. Let

1.

2.

Proof. For all

Proposition 4.3. Let

1.

2.

Proof. For all

In the same way, (2) can be proved.

Definition 4.9. Let

i)

ii)

iii)

iv)

Definition 4.10. Let

i)

ii)

iii)

iv)

Proposition 4.4. Let

1.

2.

3.

4.1 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

Definition 4.12. If

Once

Step 1 Determine

Step 2 Find

Step 3 Construct

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 discourse

Step 1:

From Tabs. 9–11, we can construct

Step 2:

Tab. 12 presents

Step 3: With the help of Step 1 and Step 2, we can construct

Step 4:

From Tabs. 13–16, we can construct

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

Step 5:

Since maximum of

5 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

i)

ii)

iii)

iv)

Note that collection of all npnhs-sets is represented by

Definition 5.2. Let

Definition 5.3. Let

Example 5.1. Consider

Case 1.

If

Case 2.

If

Case 3.

If

Case 4.

If

Case 5.

If

Definition 5.4. Let

Proposition 5.1. Let

1.

2.

3.

4. if

Definition 5.5. Let

Proposition 5.2. Let

1. if

2. if

Definition 5.6. Let

Proposition 5.3. Let

1.

2.

Definition 5.7. Let

i)

ii)

iii)

iv)

Proposition 5.4. Let

1.

2.

3.

4.

5.

Definition 5.8. Let

i)

ii)

iii)

iv)

Proposition 5.5. Let

1.

2.

3.

4.

5.

Note: It is pertinent to mention here that Propositions 5.1, 5.2, 5.4 and 5.5 are also valid for elements of

Remark 5.1. Let

Proposition 5.6. Let

1.

2.

Proof. For all

Proposition 5.7. Let

1.

2.

Proof. For all

In the same way, (2) can be proved.

Definition 5.9. Let

i)

ii)

iii)

iv)

Definition 5.10. Let

i)

ii)

iii)

iv)

Proposition 5.8. Let

1.

2.

3.

5.1 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

Definition 5.12. If

Once

Step 1 Determine

Step 2 Find

Step 3 Construct

Step 4 Compute

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

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:

Step 1:

From Tabs. 17–19, we can construct

Step 2:

Tab. 20 presents

Step 3:

Step 4:

From Tabs. 21–24, we can construct

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

Step 5:

Since maximum of

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:

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.

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.

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