Journal of Mathematics

Journal of Mathematics / 2021 / Article
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Decision Making Based on Intuitionistic Fuzzy Sets and their Generalizations

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Research Article | Open Access

Volume 2021 |Article ID 6654657 |

Rana Muhammad Zulqarnain, Harish Garg, Imran Siddique, Rifaqat Ali, Abdelaziz Alsubie, Nawaf N. Hamadneh, Ilyas Khan, "Algorithms for a Generalized Multipolar Neutrosophic Soft Set with Information Measures to Solve Medical Diagnoses and Decision-Making Problems", Journal of Mathematics, vol. 2021, Article ID 6654657, 30 pages, 2021.

Algorithms for a Generalized Multipolar Neutrosophic Soft Set with Information Measures to Solve Medical Diagnoses and Decision-Making Problems

Academic Editor: Stanislaw Migorski
Received21 Oct 2020
Revised16 Mar 2021
Accepted20 Mar 2021
Published08 May 2021


The aim of this paper is to propose the generalized version of the multipolar neutrosophic soft set with operations and basic properties. Here, we define the AND, OR, Truth-Favorite, and False-Favorite operators along with their properties. Also, we define the necessity and possibility of operations for them. Later on, to extend it to solve the decision-making problems, we define some information measures such as distance, similarity, and correlation coefficient for the generalized multipolar neutrosophic soft set. Several desirable properties and their relationship between them are derived. Finally, based on these information measures, a decision-making algorithm is stated under the neutrosophic environment to tackle the uncertain and vague information. The applicability of the proposed algorithm is demonstrated through a case study of the medical-diagnosis and the decision-making problems. A comparative analysis with several existing studies reveals the effectiveness of the approach.

1. Introduction

Uncertainty plays a dynamic part in numerous fields of life such as modeling, medical, and engineering fields. However, a general question of how we can express and use the uncertainty concept in mathematical modeling is raised. A lot of researchers in the world proposed and recommended different approaches to use uncertainty theory. First of all, Zadeh developed the notion of fuzzy sets [1] to solve those problems which contain uncertainty and vagueness. It is observed that in some cases circumstances cannot be handled by fuzzy sets; to overcome such types of situations, Turksen [2] gave the idea of the interval-valued fuzzy set (IVFS). In some cases, we must deliberate membership unbiassed as the nonmembership values for the suitable representation of an object in uncertain and indeterminate conditions that could not be handled by fuzzy sets or by IVFS. To overcome these difficulties, Atanassov presented the notion of intuitionistic fuzzy sets (IFSs) in [3]. The theory that was presented by Atanassov only deals with the insufficient data considering both membership and nonmembership values; however, the IFSs theory cannot handle the overall incompatible as well as imprecise information. To address such incompatible as well as imprecise records, the idea of the neutrosophic set (NS) was developed by Smarandache [4]. A general mathematical tool was proposed by Molodtsov [5] to deal with indeterminate, fuzzy, and not clearly defined substances known as a soft set (SS). Maji et al. [6] extended the work on SS and defined some operations and their properties. Maji et al. [7] utilized the SS theory for decision-making. Ali et al. [8] revised the Maji approach to SS and developed some new operations with their properties. De Morgan’s Law on SS theory was proved in [9] by using different operators.

Maji [10] offered the idea of a neutrosophic soft set (NSS) with necessary operations and properties. The idea of the possibility NSS was developed by Karaaslan [11] and introduced a possibility of neutrosophic soft decision-making method to solve those problems which contain uncertainty based on And-product. Broumi [12] developed the generalized NSS with some operations and properties and used the proposed concept for decision-making. To solve MCDM problems with single-valued neutrosophic numbers (SVNNs) presented by Deli and Subas in [13], they constructed the concept of cut sets of SVNNs. On the basis of the correlation of IFS, the term correlation coefficient (CC) of SVNSs [14] was introduced. Ye [15] presented the simplified NSs introduced with some operational laws and aggregation operators such as weighted arithmetic and weighted geometric average operators. Therein, a multicriteria decision-making (MCDM) method was constructed based on proposed aggregation operators. Masooma et al. [16] progressed a new concept by combining the multipolar fuzzy set and neutrosophic set, which is known as the multipolar neutrosophic set. They also established various characterizations and operations with examples. Dey et al. [17] developed the grey relational projection method based on NSS to solve MADM complications. Pramanik et al. [18] extended the VIKOR technique to solve MAGDM problems under a bipolar neutrosophic set environment. Pramanik et al. [19] established the TOPSIS technique to solve MADM problems utilizing single-valued neutrosophic soft expert sets. Pramanik et al. [20] developed three different hybrid projection measures projection, bidirectional projection, and hybrid projection measures between bipolar neutrosophic sets.

Peng et al. [21] established the probability multivalued neutrosophic set by combining the multivalued neutrosophic set and probability distribution and used it for decision-making problems. Kamal et al. [22] proposed the idea of mPNSS with some important operations and properties; they also used the developed technique for decision-making. Garg [23] developed the MCDM method based on weighted cosine similarity measures under an intuitionistic fuzzy environment and used the proposed technique for pattern recognition and medical diagnoses. To measure the relative strength of IFS, Garg and Kumar [24] presented some new similarity measures. They also formulated a connection number for set pair analysis (SPA) and developed some new similarity measures and weighted similarity measures based on defined SPA. Garg and Rani [25] extended the IFS technique to complex intuitionistic fuzzy sets (CIFS) and developed the correlation and weighted correlation coefficient under the CIFS environment. To measure the relation between two Pythagorean fuzzy sets (PFS), Garg [26] proposed a novel CC and WCC and presented the numerical examples of pattern recognition and medical diagnoses to verify the validity of the proposed measures. Zulqarnain et al. [27] developed the aggregation operators for Pythagorean fuzzy soft sets and proposed a decision-making methodology using their developed aggregation operators. They also utilized their established decision-making technique for the selection of suppliers in green supply chain management. Zulqarnain et al. [28] extended the TOPSIS technique under Pythagorean fuzzy soft environment. Nguyen et al. [29] defined some similarity measures for PFS by using the exponential function for the membership and nonmembership degrees with its several properties and relations. Peng and Garg [30] presented some diverse types of similarity measures for PFS with multiple parameters. Wang and Li [31] introduced Pythagorean fuzzy interaction power Bonferroni mean (PBM) operators for solving MADM issues. Wang et al. [32] proposed the Pythagorean fuzzy interactive Hamacher power aggregation operators for assessment of express service quality with entropy weight. Saeed et al. [33] established the concept of mPNSS with its properties and operators; they also developed the distance-based similarity measures and used the proposed similarity measures for decision-making and medical diagnoses.

Gerstenkorn and Mafiko [34] proposed the functional measuring of the interrelation of IFSs, which is known nowadays as correlation, and developed its coefficient with properties. To measure the interrelation of fuzzy numbers, Yu [35] established the CC of fuzzy numbers. Evaluating the CC for fuzzy data had been developed by Chiang and Lin [36]. Hung and Wu [37] proposed the centroid method to calculate the CC of IFSs and extended the proposed method to interval-valued intuitionistic fuzzy sets (IVIFSs). Hong [38] and Mitchell [39] also established the CC for IFSs and IVIFSs, respectively. Ye [40] extended the work on IFSs and developed the CC of a single-valued neutrosophic set and developed a decision-making method for similarity measure. Xue et al. [41] developed the CC on a single-valued neutrosophic set and proposed a decision-making method for pattern recognition. Zulqarnain et al. [42] utilized the neutrosophic TOPSIS in the production industry for supplier selection. Garg and Arora [43] introduced the correlation measures on intuitionistic fuzzy soft sets and constructed the TOPSIS technique on developed correlation measures. In Iryna et al.’s work [44], an algorithm has been proposed to handle uncertainty in fault diagnoses by using single-valued neutrosophic sets. Faruk [45] established CC between possibility NSS and proved some properties. He also developed CC for a single-valued neutrosophic refined soft set, and it was used for clustering analysis [46]. A correlation measure of neutrosophic refined sets has been developed, which is the extension of the correlation measure of neutrosophic sets and intuitionistic fuzzy multisets [47].

In this era, professionals consider that the real life is moving in the direction of multipolarity. Thus, it projects as no surprise that multipolarity in information performs a significant part in flourishing numerous fields of science as well as technology. In neurobiology, multipolar neurons in the brain gather a good deal of information from other neurons. In information technology, multipolar technology could be used to control extensive structures. The motivation of the present research is extended and hybrid work is given step by step in the complete article. We demonstrate that different hybrid structures containing fuzzy sets are converted into the special privilege of mPNSS under whatsoever appropriate circumstances. The concept of a neutrosophic environment to a multipolar neutrosophic soft set is novel. We tend to discuss the effectiveness, flexibility, quality, and favorable position of our planned work and algorithms. The present research will be the most generalized form and is used to assemble data in considerable and appropriate medical, engineering, artificial intelligence, agriculture, and other everyday life complications. In the future, the present work might be gone competently for other approaches and different types of hybrid structures.

The remainder of the paper is organized as follows: in Section 2, we recollected some basic definitions which are used in the following sequel such as NS, SS, NSS, and multipolar neutrosophic set. In Section 3, we proposed the generalized version of mPNSS with its properties and operations, and we also developed the Truth-Favorite, False-Favorite, AND, and OR operators in this section. In Section 4, distance-based similarity measures have been developed by using Hamming distance and Euclidean distance between two generalized multipolar neutrosophic soft sets (GmPNSS). In Section 5, the idea of CC and WCC with their properties has been established. Finally, we use the developed distance-based similarity measures and CC for medical diagnoses and decision-making in Section 6. We also present the comparative study of our proposed similarity measures and CC with some already existing techniques in Section 7.

2. Preliminaries

In this section, we recollect some basic concepts such as neutrosophic set, soft set, neutrosophic soft set, and m-polar neutrosophic soft set, which are used in the following sequel.

Definition 1. (see [4]).
Let be a universe and let be an NS on defined as  = , where , , : , and  ≤  +  +  ≤ .

Definition 2. (see [5]).
Let be the universal set and let be the set of attributes concerning . Let be the power set of and . A pair () is called a soft set over and its mapping is given asIt is also defined as

Definition 3. (see [10]).
Let be the universal set and let be the set of attributes concerning . Let be the set of neutrosophic values of and . A pair is called a neutrosophic soft set over and its mapping is given as

Definition 4. (see [16]).
Let be the universal set and let be the set of attributes concerning ; then is said to be a multipolar neutrosophic set if , where :  [0, 1], and 0   3; . , , and represent the truth, indeterminacy, and falsity of the considered alternative.

3. Generalized Multipolar Neutrosophic Soft Set (GmPNSS) with Operators and Properties

In this section, we develop the concept of GmPNSS and introduce aggregate operators on GmPNSS with their properties.

Definition 5. Let and E be universal and set of attributes, respectively, and  ⊆ E, if there exists a mapping Φ such thatthen (Φ, ) is called GmPNSS over defined as follows:where  = , and 0   3 for all  1, 2, 3, …, ; and .

Definition 6. Let and be two GmPNSS over ; then is called a multipolar neutrosophic soft subset of , iffor all  1, 2, 3, …, m; eE and u .

Definition 7. Let and be two GmPNSS over , then  = , iffor all  1, 2, 3, …, ; and .

Definition 8. Let be a GmPNSS over , then empty GmPNSS can be represented as and defined as follows:

Definition 9. Let be a GmPNSS over , then universal GmPNSS can be represented as and defined as follows:

Definition 10. Let be a GmPNSS over , then the complement of GmPNSS is defined as follows:for all  1, 2, 3, …, ; E and .

Proposition 1. If is a GmPNSS, then(1) = (2) = (3) = 

Proof. LetThen, by using Definition 10, we getAgain, by using Definition 10,

Proof. Let be an empty GmPNSS over .Utilizing Definition 10,Similarly, we can prove 3.

Definition 11. Let and be two GmPNSS over . Then,

Proposition 2. Let , , and be GmPNSS over . Then,(1)  = (2)  = (3)  =  (4)( )  =   ( )

Proof. Letbe a GmPNSS. Then,By using Definition 11, we can easily prove the remaining properties.

Definition 12. Let and be GmPNSS over . Then,

Proposition 3. Let , , and be GmPNSS over . Then,(1)  = (2)  = (3)  = (4)  =  (5)( )  =   ( )

Proof. By using Definition 12, the proof is easy.

Proposition 4. Let and be GmPNSS over . Then,(1) = (2) = 

Proof. We know thatare two GmPNSS.
By using Definition 11,Now, by using Definition 10,Now,By using Definition 12,Hence,

Proof. We know thatare two GmPNSS.
Utilizing Definition 12,By Definition 10,Now,By using Definition 11,Hence,

Proposition 5. Let , , and be GmPNSS over . Then,(1)  ( ) = ( ) ( )(2)  ( ) = ( )  ( )(3)  ( ) = (4)  ( ) = 

Proof. We know thatHence,Similarly, we can prove other results.

Definition 13. Let and be GmPNSS, then their extended union is defined as

Example 1. Assume that  = {, } is a universe of discourse and let  = {, , , } be a set of attributes and  = {, } and  = {, } ⊆ . Consider and  G3-PNSS over can be represented as follows:Then,

Definition 14. Let and be GmPNSS; then their extended union is defined as

Definition 15. Let and be GmPNSS, then their difference is defined as follows:

Definition 16. Let and be GmPNSS, then their addition is defined as follows:

Definition 17. Let be a GmPNSS, then its scalar multiplication is represented as . , where  [0, 1] and it is defined as follows:

Definition 18. Let be a GmPNSS, then its scalar division is represented as /, where  [0, 1] and it is defined as follows:

Definition 19. Let be a GmPNSS over , then Truth-Favorite operator on can be represented by and it is defined as follows:

Proposition 6. Let and be GmPNSS over . Then,(1) = (2) ⊆ (3) ⊆ (4) = The proof of the above proposition is easily obtained by using Definitions 11, 12, 16, and 19.

Definition 20. Let be a GmPNSS over , then False-Favorite operator on can be represented by and it is defined as follows:

Proposition 7. Let and be GmPNSS over . Then,(1) = (2) ⊆ (3) ⊆