Cantor set is introduced to present the well defined countable/uncountable objects. The characteristic function is introduced that allow the users to take only two values 0(for non-belongingness) and 1(for belongingness) of an object. To measure any degree of imprecision, the characteristic function is replaced by membership function which is a basic constituent of fuzzy set (FS) introduced by Zadeh [1]. Later on, to measure the incomplete information of an object, the notion of intuitionistic fuzzy set (IFS) [2] is forwarded. In IFS, the sum of the membership and the non-membership degree is limited to 1. But, what we do when their sum exceed 1. Realizing the importance to defend such situation that is inspired by nature due to computation intelligence, Yager [3] introduced the Pythagorean fuzzy set (PyFS). In PyFS, there is more liberty in the selection of membership and non-membership values with a condition that the sum of their squares does not exceed 1. So, by PyFS, we measure both the membership and non-membership degrees of an object with better accuracy and precision. For practical decision-making, PyFS provides more information than IFS. Cuong et al. [4] tried to handle uncertainty in a different way with an aid of picture fuzzy set (PFS). PFS can be regarded as a direct extension of FS and IFS that is promised to address more complex phenomena found in human opinions.  The decision-makers decision somehow restricted under the PFS because of its structural system. To enlarge the space of the membership degrees of PFS, in 2019, Ashraf et al.[5] introduced the spherical fuzzy set(SFS) as an extension of PyFS.  Furthermore, the PyFS has been generalized by introducing the q-rung orthopair fuzzy set (q-ROFS) [6]. Also, Senapati et al.[7] introduced the Farmatean fuzzy set(FFS) as a special case of q-ROFS.

Indeterminacy is an important component of uncertainty that we encounter in real-world. The concept of FS, IFS, and PyFS are not sufficient to tackle the indeterminate information. In 2005, Smarandache[8] introduced the neutrosophic set(NS) as a generalization of IFS. In NS, every object has three neutrosophic components denoted by , where for truth-component, for indeterminate-component, and is false-component. Here, such that . Though, it is sensible to use non-standard unit interval in NS for describing the linguistic and philosophical concept but it is not useful for scientific modeling as it demands specified membership values. For this purpose, the single-valued neutrosophic set (SVNS)[9] is introduced where the neutrosophic components belong to the standard unit interval . Normally in NS, the components are independent components, but to reach different target problem, researchers put various restrictions on the components and utilize them to solve certain types of real problems. In 2019, Jansi et al.[10] presented the Pythagorean neutrosophic set(PNS) as a special type of NS with dependent components and apply in medical diagnosis by using its correlation measure. Ajay et al.[11] gives an introduction of Pythagorean neutrosophic fuzzy graph.  Veerappan et al. [12] proposed the Pythagorean neutrosophic ideal in semi groups. Rajan et al.[13] have introduced the similarity measures of PNS. In [14], Jansi et al. introduced the pairwise Pythagorean neutrosophic P-spaces. So far several works associated to PNS have been successfully carried out and thus the PNS has a rich potential in many practical areas and so there is a natural question arises how to generalize it further. In view of the present discussion, it is worthy to introduce RNS as a special type of NS that is undertaken to generalize the PNS to cover up the information gap realized by the decision-makers.

The hierarchy structure of RNS is shown in Fig 1.

                                                                       Fig. 1. Hierchy formation of RNS

To show the feasibility of the proposed study, we make a comparison of SVNS, PNS, and RNS in the following Table 1.

          SVNS                                                                         PNS                                                                        RNS                         

    such that                   and          such that

.                                =                    and =1-

  =                                                                                       

                                         Table 1. Comparison of SVNS, PNS, and RNS

This paper is structured in the following manner: Section 2 contains some basic definitions that are relevant to the proposed topic. Section 3 includes the introduction of RNS and its various properties. Correlation measures of two RNSs are introduced in section 4. Section 5 includes the application of RNS in medical decision-making. In the last section i.e. in section 6, the conclusion and the future study of the current topic has been briefly discussed

2. Preliminaries

In the context of the proposed study, in this section, we recollect some basic notions. Throughout the section, the set of universe is denoted by .

Definition 2.1 [1] A fuzzy set over is denoted by an ordered pair of the form

, where represents the membership function that is defined by the mapping .

Definition 2.2 [2] An intuitionistic fuzzy set over is defined in the following form:

, where and represent the membership and the non-membership degree respectively, and , such that .

Definition 2.3 [4] A picture fuzzy set over is an object of the form

, where denote the degree of acceptance membership-degree, neutral membership-degree, and rejection membership-degrees respectively such that .

Definition 2.4 [5] A spherical fuzzy set over can be written as;

, where indicate the positive, neutral, and negative membership degrees such that .

Definition 2.5 [3] A Pythagorean fuzzy set over is an object of the form given as;

, where the functions , defined the degree of membership and the degree of non-membership respectively such that .

Definition 2.6 [9] A single-valued neutrosophic set over is an object that can be written as;

, where the functions , , defined the degree of truth membership , indeterminate membership , and false membership such that

Definition 2.7 [10] A Pythagorean neutrosophic set over  is defined as;

, where denote the degrees of truth membership, indeterminate membership, and false membership respectively with the conditions and

From the above discussion, an information gap has been identified that cannot be filled up by the existing theories. Such research gap has been produced due to the human intelligence. To make it visible to the readers, we discuss the following:

Suppose there exist a certain domain where an information provided by a decision-maker is denoted by a triplet where , , and . Such information cannot be answered by the SFS and PNS. Though it can be easily defined by the SVNS, but we are looking for another robust system where the same information can be easily accommodated within a small range. This lead to the introduction of RNS.

3. Restricted Neutrosophic Set

Definition 3.1 Let  be a set of universe. A restricted neutrosophic set (RNS) over is characterized by the degrees of truth membership , indeterminate membership , and false membership and it is defined as;

, where , , with the condition .

The hesitation degree is denoted by and it is defined as . For , .

The set of all RNSs over  is denoted by . Also, for any , the restricted neutrosophic number (RNN) is denoted by .

Example 3.1.1 Let be a RNS with , , and . Then, .

Definition 3.2 For any two RNNs and , we have the following properties:

1. , , and .

2. and

3. If , then its complement is defined as

4.

5.

6. ;

Definition 3.3 For any two RNNs and , we define the following operators:

1. (t-norm)

2. (t-co norm)

3. (Scalar t-norm)

For any scalar ,

4. (Scalar t-co norm)

For any scalar ,

Definition 3.4 (Score function) Let be any RNN. Then the score function defined on is a mapping such that =1- . - . .

Definition 3.5 (Accuracy function) Let be any HNN. Then the accuracy function defined on  is a mapping such that = .

Proposition 3.6 Let . Then, , we have the following properties:

1. and ; OR and

2.  or =0 or =

3. =

Proposition 3.7 Let . Then, , we have the following properties:

1.  

2.

Definition 3.8 Suppose  and be two RNNs over . Their score and accuracy functions are denoted by , and , respectively. Then, we have the following properties:

1. If , then .

2. If , then .

3. If , then we compare their accuracy functions as:

i. If , then .

ii. If , then .

iii. If , then .

Theorem 3.9 Let us consider the three RNNs over  as  , , and  . Then, the following properties hold true:

1.

2.

3. for any scalar ,  

Note: It has been observed that

Definition 3.10 Let  defined over represent a family of RNNs, where . Then, the restricted neutrosophic weighted average (RNWA) t-norm operator with weight vector , where and is given by .

Definition 3.11 Let  defined over represent a family of RNNs, where . Then, the restricted neutrosophic weighted geometric (RNWG) t-co norm operator with weight vector , where and is given by .

Theorem 3.12 If , then the score function defined by  is always monotonically decreasing. 

Proof. We have,

Then,

Also,

This proves the statement.

Theorem 3.13 If , then the accuracy function defined by = .

 is always monotonically increasing. 

Proof. It is left for the readers.

Proposition 3.14 Let .If then .

Example 3.14.1 Let  and  such that .

Then, and . Therefore, .

Proposition 3.15 Let .If then .

4. Correlation Measures of Two RNSs

Correlation measures between two variables describe how the two variables close to each other or relate to each other. That is, it measures the degree of closeness between two variables. In practical scenario, there are many instances where we are eager to know about the relationship between two variables. We classify the relationship between two variables as positive correlation (when both increase/decrease), negative correlation (when one increase/decrease due to the decrease/increase to other), zero correlation (when there is no relation between the variables). For example, during pandemic situation, price of an item will increase with the crease of its demand, so price and demand are positively correlated. Increase in lockdown period will reduce the viral infection is an example of negative correlation between lockdown period and the number of infection. But there is no correlation between the ages of a player with the run scored by him/her. In this section, our attempt is to define the correlation measures between two RNNs and established some properties related to these measures. This topic will surely gives an idea about the closeness between two RNNs. Motivating from the article [10], we discuss the following definitions and results that are appropriate for the proposed study:

Definition 4.1 Let  be an initial universe and . We defined the two RNSs given by

 and . Then, we defined the correlation coefficient between and as;

, where

By putting in , we can easily get and .

Proposition 4.2 Let denotes the correlation coefficient between two RNSs and . Then, we have the following properties:

(1).

(2).

(3).

(4).

Proof. , , and are obvious. We only discuss .

(4)

Using the Cauchy-Schwarz inequality, by squaring the above expression, we have

=

Thus, =

Putting it in (i), we obtain the result .

Definition 4.3 Let denotes the weight vector of elements with  and , then we defined the weighted correlation coefficient as follows:

Where

Putting , we can easily obtain and

Note. If , then equation (ii) reduces to equation (i).

Proposition 4.4 Let be the weighted correlation coefficient. Then, we consider the following properties:

(1)

(2)

(3)

Proof. (2) and (3) are straightforward. We only concentrate on .

(1) We have,

By using Cauchy-Schwarz inequality, we obtain

Therefore, =

This proves the result .

5. Application of RNS in Medical Decision Making

In this section, we construct an algorithm based on score function under the RNS environment and apply them in medical decision-making.

Algorithm 1

Step 1- Input the patient-symptom and the symptom-disease data set provided by the expert in the form of hyper neutrosophic triplet. And formulate their corresponding decision matrices and .

Step 2- Next we normalize and in the following process:

Let and , where .

Then their corresponding normalized matrices are represented by

And

Step 3- Perform the maxmin-maxmin-minmax composition between the two normalized matrices and obtain the Patient-disease matrix .

Step 4- Using the definition 3.4, obtained the score value of each entry in .

Step 5- Finally, the patient with the largest score value in each row is likely to suffer from that disease. In case of a tie, there is a chance that a patient will suffer from more than one disease.

5.1 Example

Let us consider a set of five patients denoted by with symptoms Red Skin, Headache, Cough, Joint Pain, and Breathing Difficulty. Let the possible diseases relating to the above symptoms be Dengue, Pneumonia, Asthma, cholera, Viral Fever. A medical investigator is being appointed for different types of medical tests according to the symptoms are concerned. Though, it is a complicated procedure due to the information provided by the patients is ill defined. So, the entire procedure contains a lot of uncertainty. To minimize the level of uncertainty in the detection of the diseases, the decision-maker or expert follows up the RNS environment. For this we use the above algorithm in the following manner:

Step 1-

Input the patient-symptom relation matrix in the form of Table 2

                                          Table 2. The patient-symptom relation matrix

Input the symptom-disease relation matrix in the form of Table 3

                                          Table 3. The symptom-disease relation matrix

Step2-

The normalized patient-symptom matrix is given by the Table 4

                                          Table 4. The normalized patient-symptom matrix

The normalized symptom-disease matrix is given by the Table 5

                                          Table 5. The normalized symptom-disease matrix

Step3-

The resultant patient-disease matrix formed by combining the two normalized matrices obtained in step 2 is given in Table6

                                          Table 6. The resultant patient-disease matrix

Step4-The score values of all the entries of the resultant patient-disease matrix obtained in the following Table 7

                                                   Table 7. The score values of the resultant patient-disease matrix

Step5-

In the above table, the maximum score values along each row is highlighted with yellow mark and from which we make a decision that is suffering from Asthma, is suffering from both Dengue and Pneumonia,  is suffering from Asthma, is suffering from Dengue, and is suffering from Viral Fever. There is no patient suffering from the disease Cholera.

Note. For the sake of comparison, we may obtain the accuracy values of the patient-disease matrix. Also, we use the correlation measures to detect the possible disease/s of a patient having certain symptoms. This part left for the readers.

6. Conclusion and Scope

In the present paper, we have introduced the RNS, as a new type of neutrosophic oriented set in a sense that we have not seen such topic in any research paper until now. Like SVNS and PNS, the RNS also a subclass of NS. The main feature of the RNS is the modification of the restricted condition which makes it a robust model. So, RNS is an extension of PNS. There is no such model ever introduced that is capable to regulate uncertainty perfectly. That’s why the researchers are toiling hard with a believe that they will be capable to produce more advanced new model in near future and thus reduce the level of uncertainty with more precision. This actually the main reason to introduce RNS. Then we discuss some basic set-theoretic properties on RNSs. We also defined their t-norm, t-co norm, RNWA, and RNWG- operators, score and accuracy function. Some important properties of score and accuracy functions are addressed. Furthermore, we discuss the correlation measures of RNSs and established some useful results. An algorithm is introduced and applies it in solving a medical decision-making problem.

In the future, the present topic has a rich potential to utilize it in various types of MCDM, MADM, MCGDM problems in different fields such as risk management, weather forecasting, linear programming, game theory, green supplier selection, robotics by using TOPSIS, MULTIMOORA, ELECTRE, PROMETHEE, AHP, VIKOR, DEA, ANP methods.   

Funding: “This research received no external funding”

Conflicts of Interest: “The authors declare no conflict of interest.”

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