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

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.


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]. e 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. e 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. erein, a multicriteria decisionmaking (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. ey 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. ey 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. ey 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. us, 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. e 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. e 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. e 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. e 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.

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 U be a universe and let A be an NS on U defined as Definition 2 (see [5]).
Let U be the universal set and let E be the set of attributes concerning U. Let P(U) be the power set of U and A⊆E. A pair (F, A) is called a soft set over U and its mapping is given as It is also defined as Definition 3 (see [10]). Let U be the universal set and let E be the set of attributes concerning U. Let P(U) be the set of neutrosophic values of U and A⊆E. A pair (F, A) is called a neutrosophic soft set over U and its mapping is given as Definition 4 (see [16]). Let U be the universal set and let E be the set of attributes concerning U; then F E is said to be a multipolar . . , m. u e , v e , and w e represent the truth, indeterminacy, and falsity of the considered alternative.

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 U and E be universal and set of attributes, respectively, and A ⊆ E, if there exists a mapping Φ such that then (Φ, A) is called GmPNSS over U defined as follows: Definition 10. Let F A be a GmPNSS over U, then the complement of GmPNSS is defined as follows: for all i ∈ 1, 2, 3, . . ., m; e ∈ E and u ∈ U.
en, by using Definition 10, we get Again, by using Definition 10, □ Proof. Let F � 0 be an empty GmPNSS over U. 4

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Utilizing Definition 10, Similarly, we can prove 3.
□ Definition 11. Let F A(e) and G B(e) be two GmPNSS over U. en, be a GmPNSS. en, By using Definition 11, we can easily prove the remaining properties. en, and H � C be GmPNSS over U. en, Proof. By using Definition 12, the proof is easy. □ Proposition 4. Let F � A and G � B be GmPNSS over U. en, Proof. We know that are two GmPNSS. By using Definition 11, Now, by using Definition 10, Now, By using Definition 12, Hence, □ 6

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Proof. We know that are two GmPNSS. Utilizing Definition 12, By Definition 10, Now, By using Definition 11, Hence, □ Journal of Mathematics , and H � C(e) be GmPNSS over U. en, Proof. We know that Similarly, we can prove other results.
□ Definition 13. Let F � A and G � B be GmPNSS, then their extended union is defined as en, Definition 14. Let F � A and G � B be GmPNSS; then their extended union is defined as Definition 15. Let F� A and G� B be GmPNSS, then their difference is defined as follows: (56) be GmPNSS, then their addition is defined as follows: (57) where � a ∈ [0, 1] and it is defined as follows: and it is defined as follows: Definition 19. Let F � A be a GmPNSS over U, then Truth-Favorite operator on F� A can be represented by ΔF� A and it is defined as follows: 10 Journal of Mathematics e proof of the above proposition is easily obtained by using Definitions 11,12,16,and 19. Definition 20. Let F � A be a GmPNSS over U, then False-Favorite operator on F � A can be represented by ∇ F � A and it is defined as follows: Proposition 7. Let F� A and G� B be GmPNSS over U. en, e proof of the above proposition is easily obtained by using Definitions 11,12,16,and 20. Definition 21. Let F� A and G� B be GmPNSS; then their AND operator is represented by F� A ∧ G� B and it is defined as Example 2. Reconsider Example 1.
Proof. We can prove this easily by using Definitions 10, 21, and 22.
□ Definition 23. Let F � A be a GmPNSS; then necessity operation on GmPNSS is represented by ⊕ F� A and defined as follows:

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Definition 24. Let F � A be a GmPNSS; then possibility operation on GmPNSS is represented by ⊗F � A and defined as follows: (66) are two GmPNSS.
By using Definition 23, where ⊕F � A and ⊕G � B are given as follows by using the definition of necessity operation: en, by using Definition 13, Consequently, ⊕(H � C ) and ℵ are the same, so Similarly, we can prove 2.
□ Proposition 10. Let F � A and G � B be two GmPNSS. en, e proof is similar to that of Proposition 9.
□ Proposition 11. Let F� A and G� B be GmPNSS, then we have the following: Proof. We know that F� A and G� B are GmPNSS: H � C � e i , e j , u, min By using Definition 23, . Similar to Assertion 1, we can prove 2, 3, and 4.

Distance and Similarity Measure of Generalized Multipolar Neutrosophic Soft Set
In this section, we introduce the Hamming distance and Euclidean distance between two GmPNSS and develop the similarity measure by using these distances.
Definition 25. U and E are a universal set and a set of attributes, respectively; assume that GmPNSS(U) represents the collection of all GmPNSS. Consider (Φ F , E) as well as (φ G , E) ∈ GmPNSS and there exists a mapping Φ F , φ G : E ⟶ GmPNSS(U); after that, we tend to establish the distances between (Φ F , E) and (φ G , E) as follows.
Definition 26. U and E are a universal set and a set of attributes, respectively, and (Φ F , E) and (φ G , E) are two GmPNSS(U). en similarity measure based on Definition 25 between (Φ F , E) and (φ G , E) is defined as follows: Another similarity measure between (Φ F , E) and (φ G , E) is defined as where β is a steepness measure and a positive real number.

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Definition 28. Similarity measure between two GmPNSS (Φ F , E) and (φ G , E) based on the weighted distance of (Φ F , E) and (φ G , E) is defined as follows: Definition 29. Let Φ F and φ G be GmPNSS over the universal set; then Φ F and φ G are said to be α-similar if and only if S GmPNSS (Φ F , φ G ) ≥ α for α ∈ (0, 1). If S GmPNSS (Φ F , φ G ) > 1/2, then we can say that Φ F and φ G are significantly similar.

Correlation Coefficient of Generalized Multipolar Neutrosophic Soft Set
In this section, we propose the concept of correlation coefficient and weighted correlation coefficient of GmPNSS with some properties.

Definition 30.
Let be two GmPNSS over a set of parameters en, informational neutrosophic energies of two GmPNSS can be expressed as follows: Definition 31. e correlation of two GmPNSS can be presented as follows: Definition 32. Let F A ⌣ and G B ⌣ be two GmPNSS; then the CC between them can be defined as follows: (106)

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By using Cauchy-Schwarz inequality, we get . Hence, by using Definition 32, we get R GmPNSS (F As we know that , for all j, k, by using Definition 32, we have 18 Journal of Mathematics (109) □ Definition 33. Let F A ⌣ and G B ⌣ be two GmPNSS; then the CC between them also can be defined as follows: between them satisfies the following properties: Proof. e proof is easy according to Definition 33. Nowadays, considering that the weight of GmPNSS is very necessary for practical applications, the result of decisions may vary, whenever decision-makers adjust the different weight to every element in the universe of discourse. Consequently, it is particularly significant to plan the weight before decision-making. Let ώ � ώ 1 , ώ 2 , ώ 3 , . . . , ώ t } be a weight vector for experts such as ώ k > 0 and t k�1 ώ k � 1, and let δ � δ 1 , δ 2 , δ 3 , . . . , δ n be a weight vector for parameters such as δ j > 0 and z j�1 δ j � 1. In the following, we develop the WCC between GmPNSS by extending Definitions 32 and 33. □ Definition 34. For two GmPNSS F A ⌣ and G B ⌣ , the WCC between them can be defined as follows: Definition 35. Let F A ⌣ and G B ⌣ be two GmPNSS, then the WCC between them can be defined as follows: If we consider ώ � 1/t, t1/tn, q . . . h, 1/t} and δ � 1 Step 5. Calculate the CC between any two GmPNSS by using the following formula: Step 6. Analyze the results.
e flow chart of the presented algorithm can be seen in Figure 2.

Application of Correlation Coefficient.
We use the proposed algorithm for medical diagnoses. For this, we consider that F� A and G� B are G3-PNSS which are described in Section 6.2.1 in Tables 1 and 2, respectively. By using equation (12), we can find CC against the values of the universal set given as follows: R G3−PNSS(u 1 ) (F� A , G� B ) � 0.9967 and R G3−PNSS(u 2 ) (F� A , G� B ) � 0.9925. By the above calculation, we analyze the results and get R G3−PNSS( , which shows that patient suffers from typhoid. Step 1 (i) Input GmPNSS according to requirement Step 2 (ii) Construct the evaluation report for each alternative in the form of GmPNSS Step 3 (iii) Compute the distance between GmPNSS Step 4 (iv) Compute the similarity measures Step 5 (v) Choose the alternative with highest similarity Step 6 (vi) Analyze the ranking

Application of GmPNSS for Decision-Making.
Assume that S � {S 1 , S 2 , S 3 , S 4 } is a set of scholars who are shortlisted for interview and E � {x 1 � research paper, x 2 � research quality, x 3 � interview} is a set of parameters for the selection of scholarship. Let F and G ⊆ E; then we construct the G3-PNSS Φ F (x) according to the requirement of the scientific discipline department. Now we will construct the G3-PNSS φ t G according to four experts, where t � 1, 2, 3, 4.
By using equation (3), we calculate the Euclidean distance between Φ F and φ t G as follows: 1132 Similarity measures of Φ F and φ t G can be calculated as follows: .4732 According to the proposed similarity measure, ranking of the alternatives is S 3 > S 4 > S 2 > S 1 , and it is clear that S G3−PNSS (Φ F , φ 3 G ) � 0.6850 > 0.5, which shows that Φ F and φ 3 G are significantly similar to G3-PNSS. So S 3 is the best scholar for the postdoctoral position. Graphical representation of alternatives ranking can be seen in Figure 3. Tables 3-7, we can find the correlation coefficient for each alternative by using equation (12) given as R GmPNSS

Solution by Using Algorithm 2. Now, by using
. Hence, S 3 is the best scholar for a postdoctoral position. In Figure 3, we can see the graphical representation of alternatives ranking.

Result Discussion and Comparative Analysis
In the following section, we will discuss the effectiveness, naivety, flexibility, and advantages of the proposed methods and algorithms. We also conducted a brief comparative analysis of the following: suggested methods and existing methods.

Advantages and Flexibility of the Proposed Approach.
e recommended technique is effective and applicable to all forms of input data. Here, we introduce two novel algorithms based on GmPNSS: one is CC, and the other is similarity measures. Both algorithms are effective and can provide the best results in MCDM problems. e recommended algorithm is simple and easy to understand, can deepen understanding, and is suitable for many types of choices and indicators. Developed algorithms are flexible and easy to change to suit different situations, inputs, and outputs. ere are subtle differences between the rankings of the suggested methods because different techniques have different ranking methods, so they can be afforded according to their considerations.

Superiority of the Proposed Method.
rough this research and comparative analysis, we have concluded that the results obtained by the proposed method are more general than the prevailing methods. However, in the decision-making process, compared with the existing decisionmaking methods, it contains more information to deal with the uncertainty in the data. Moreover, the mixed structure of many FS has become a special case of GmPNSS, by adding some suitable conditions. Among them, the information related to the object can be expressed more accurately and empirically, so it is a convenient tool for combining inaccurate and uncertain information in the decision-making process. erefore, our proposed method is effective, flexible, simple, and superior to other hybrid structures of fuzzy sets.
Step 1 (i) Input GmPNSS according to requirement Step 2 (ii) Construct the evaluation report for each alternative in the form of GmPNSS Step 3 (iii) Compute the neutrosophic informational energies Step 4 (iv) Compute the correlation between GmPNSS Step 5 (v) Calculate the correlation coefficient Step 6 (vi) Choose the alternative with maximum score value Step 7 (vii) Ranking alternatives        It turns out that this is a contemporary issue. Why do we have to express novel algorithms based on the current novel structure? Many indications are compared with other existing methods, and the recommended methods are surely competent. We remember the following fact: the mixed structures have some limitations in IFS, picture fuzzy sets, FS, hesitation fuzzy sets, NS, and other fuzzy sets, so complete information about the situation cannot be provided. But our m-polar model GmPNSS can be the most suitable for MCDM because it can deal with truth, indeterminacy, and falsity. Due to the exaggerated multipolar neutrosophy, these three degrees are independent of each other and provide a lot of information about alternative norms. Other similarity measures of available hybrid structures are converted into special cases of GmPNSS. A comparative analysis of some existing techniques is listed in Table 8.
erefore, compared with intuitionistic, neutrosophy, hesitant, image, and ambiguity substitution, this model is more versatile and can easily resolve complications. e similarity measures established for GmPNSS become better than the existing similarity measures for MCDM. 7.1.2. Discussion. By using the technique of Chen et al. [48], we deal with the multipolar information of fuzzy sets, but, with this method, we cannot deal with the indeterminacy and falsity objects of alternatives. By using the methodologies of Xu et al. [49] and Zhang et al. [50], we cannot deal with the multipolar information and uncertainty part of the alternative. But, on the other hand, the methodology we established involves the truthiness, indeterminacy, and falsity of alternatives with multiple data. erefore, the technique we developed is more efficient and can provide better results for decision-makers through various information. Ali et al.'s method [51] dealt with the truthiness, indeterminacy, and falsity levels of alternatives, but these techniques cannot manage multiple data. Instead, the method we developed is an advanced technique that can handle alternatives with multiple types of information. It can be seen in Table 8.

Comparative Analysis.
In this article, we propose two types of algorithms. First, an algorithm is proposed based on the correlation coefficient, and the other is based on similarity measures for GmPNSS. Next, both algorithms are utilized to solve practical problems in real life, that is, for the selection of a postdoctoral position. e graphical representation of results obtained by both algorithms is shown in Figure 3. e results show that the proposed technique is effective and practical. Finally, the ranking of all alternatives using the existing methodologies gives the same final decision; that is, the "postdoctoral" position is selected as S 3 . All rankings are also calculated by applying existing methods with the same case study. e proposed method is also compared with other existing methods: Saeed et al. [33], Masooma et al. [16], Riaz et al. [52], and Kamal et al. [22]. e comparison results are listed in Table 9, which shows the final ranking of the top 4 alternatives. It can be observed that the best selections made by the proposed methods are compared with the already established methods which are expressive in themselves and approve the reliability and validity of the proposed method.

Conclusion
In this paper, we study the mPNSS and propose a generalized version of mPNSS with some basic operations and properties. We also develop the AND operator, OR operator, Truth-Favorite operator, and False-Favorite operator with properties and examples. e concepts of necessity and possibility operations with their properties are developed in this research. e distance-based similarity measures on GmPNSS are established by using the Hamming and Euclidean distances with their properties, and a decisionmaking approach is presented to solve multicriteria decision-making problems. We also established the correlation coefficient and the weighted correlation coefficient of GmPNSS with the decision-making technique. Furthermore, a numerical illustration has been described to solve the MCDM problem by using the proposed decision-making approaches for medical diagnoses and decision-making. A comparative analysis is presented to verify the validity and demonstration of the proposed method. Finally, the suggested techniques showed higher stability and practicality for decision-makers in the decision-making process. Based on the results obtained, it is concluded that the proposed method is most suitable for solving the MCDM problem in today's life. e presented technique is unable to handle the scenario when the information of truth, falsity, and indeterminacy is given in intervals. In the future, the concept of mPNSS will be extended to interval-valued mPNSS and the