A Novel Approach for Assessing the Reliability of Data Contained in a Single Valued Neutrosophic Number and

Multiple criteria decision making is one of the many areas where neutrosophic sets have been successfully applied to solve various problems so far. Compared to a fuzzy set, and similar sets, neutrosophic sets use more membership functions which makes them decision making. On the other hand, the application of three membership functions makes evaluation somewhat more complex compared to evaluation using fuzzy sets. an impact on the selection of the appropriate solution/alternative. can be used to assess the reliability of information collected by surveying respond proposed approach is demonstrated in the numerical illustration of the supplier selection.


Introduction
Multiple criteria decision making (MCDM) started to emerge about 50 years ago, and until now it is used for solving a number of different decision choices in the presence of multiple confli associated with the need to use a larger number of criteria or use of more complex criteria that are later decomposed into sub-criteria [4][5]. However, an increase in the number of in cases where data should be collected by the survey Significant progress in using the MCDM methods for solving complex decision after Zadeh [7] proposed fuzzy sets, on then, many extensions of fuzzy sets theory have be intuitionistic fuzzy sets [12] and bipolar fuzzy sets neutrosophic sets, as a generalization of the fuzzy sets theory and their extensions. Multiple criteria decision making is one of the many areas where neutrosophic sets have been successfully applied to solve various problems so far. Compared to a fuzzy set, and similar sets, neutrosophic sets use more membership functions which makes them suitable for using complex evaluation criteria in multiple criteria decision making. On the other hand, the application of three membership functions makes evaluation somewhat more complex compared to evaluation using fuzzy sets. The reliability of the data used to solve a problem can have impact on the selection of the appropriate solution/alternative. Therefore, this paper discusses an approach that can be used to assess the reliability of information collected by surveying respond proposed approach is demonstrated in the numerical illustration of the supplier selection. neutrosophy, reliability, single-valued neutrosophic numbers, decision-making.
Multiple criteria decision making (MCDM) started to emerge about 50 years ago, and until now it is used for solving a number of different decision-making problems in different fields. MCDM can be defined as making choices in the presence of multiple conflicting criteria [1][2][3]. Solving complex decision-making problems is usually associated with the need to use a larger number of criteria or use of more complex criteria that are later decomposed . However, an increase in the number of criteria, as well as sub-criteria, can be less desirable in cases where data should be collected by the survey [6].
Significant progress in using the MCDM methods for solving complex decision-making problems was made proposed fuzzy sets, on which basis Bellman and Zadeh [8] proposed fuzzy MCDM then, many extensions of fuzzy sets theory have been developed, such as: interval bipolar fuzzy sets [13] . In 1999, Smarandache [14] generalization of the fuzzy sets theory and their extensions. Multiple criteria decision making is one of the many areas where neutrosophic sets have been successfully applied to solve various problems so far. Compared to a fuzzy set, and similar sets, neutrosophic sets use more suitable for using complex evaluation criteria in multiple criteria decision making. On the other hand, the application of three membership functions makes evaluation somewhat eliability of the data used to solve a problem can have Therefore, this paper discusses an approach that can be used to assess the reliability of information collected by surveying respondents. The usability of the proposed approach is demonstrated in the numerical illustration of the supplier selection.

making.
Multiple criteria decision making (MCDM) started to emerge about 50 years ago, and until now it is used for making problems in different fields. MCDM can be defined as making making problems is usually associated with the need to use a larger number of criteria or use of more complex criteria that are later decomposed criteria, can be less desirable making problems was made proposed fuzzy MCDM [9][10]. Since en developed, such as: interval-valued fuzzy sets [11], [14] introduced the concept of So far, neutrosophic sets are successfully used in the area of multi-criteria decision-making. Many extensions of the MCDM methods are proposed based on the use of neutrosophic numbers, such as: neutrosophic AHP [15]; neutrosophic TOPSIS [1]; neutrosophic MULTIMOORA [16]; neutrosophic WASPAS [17]; neutrosophic PROMETHEE [18]; neutrosophic VIKOR [19]; neutrosophic ARAS [20]; neutrosophic GRA [21]; neutrosophic EDAS [22], and so forth. Besides, it is worth mentioning newly-developed approaches, such as: the importance of neutrosophic soft matrices in decision-making [23], interval-valued neutrosophic soft sets in decision-making [24], as well as ivnpiv-neutrosophic soft sets for decision-making [25]. In general, neutrosophy so far is used in solving a number of decision-making problems [26][27][28][29][30][31][32]. introduces three membership functions that can be used to describe belonging to a set, that is; truth membership, indeterminacy membership, falsity membership. That is why neutrosophic sets could be more suitable for evaluating complex phenomena, events and problems.
However, the use of three membership functions can make evaluation somewhat more complex compared to evaluation using fuzzy sets. Therefore Smarandache et al. [33] proposed an approach that can be used to assess the reliability of information collected by surveying respondents. This approach is reviewed again in this article, and a new approach for determining the reliability of information contained in single valued neutrosophic numbers is also presented.
Therefore, the remainder of the article is organized as follows: in Section 2 basic elements of neutrosophic sets and single-valued neutrosophic numbers are considered. In Section 3 approaches for ranking single valued neutrosophic numbers are considered, and in Section 4 a numerical illustration is given in order to demonstrate the proposed approach. Finally, conclusions are given.

Basic Elements of Neutrosophic Sets and Single Valued Neutrosophic Numbers
Definition 1. Let X be a nonempty set, with a generic element in X denoted by x. Then, the Neutrosophic Set (NS) A in X is as follows [14]: with: where: T A (x), I A (x) and F A (x) are the truth-membership function, the indeterminacy-membership function and the falsity-membership function, respectively. Definition 2. Let X be a nonempty set. The Single Valued Neutrosophic Set (SVNS) A in X is as follows [14,34]: with:

Definition 3. For an SVNS
is as follows: where: w j is the element j of the weighting vector, ] 1 , 0 [ ∈ j w and 1

Determining the Reliability of the Information Contained in Single Valued Neutrosophic Numbers
Smarandache et al. [33] proposed an approach for accessing the reliability of the information r (x) contained in a SVNN, as follows: where: t, i, f denote the truth, the intermediacy and the falsity of information contained in SVNN Example: Assume that x =<0.9, 0.1, 0.3> is a SVNN. Then, the reliability of x is 55 .
In this approach, it is proposed to calculate reliability as follows: where: ].  Table 1.

A Numerical illustration
In order to briefly demonstrate the usability of the SVNNs for solving MCDM problems, an example of supplier selection is presented in this section. Assume that one company has to consider engaging with a new supplier. Therefore, a team of three experts if formed with the aim to select the most appropriate supplier from three alternatives, denoted as A 1 -A 3 , on the basis on the following criteria: • C 1 -Delivery, The ratings obtained from three experts are shown in Tables 1 to 3. Table 2. The ratings obtained from the first of three experts  Table 3. The ratings obtained from the second of three experts  Table 4. The ratings obtained from the third of three experts  Tables 5 and 6. Table 5. The reliability of ratings obtained from the first expert using Eq. (5)  The average reliability of responses obtained from all three decision makers, calculated using Eq. (6), are accounted for in Table 7. As can be seen from Table 7, all three experts provide relatively consistent responses, and therefore their ratings can be used for further evaluation of alternatives. In contrast, if the average reliability of ratings obtained from a respondent has low value, his or her responses must be rejected from further evaluation of the alternatives or his or her responses must be re-considered again until adequate reliability is achieved.
A possible scenario of the evaluation of alternatives is discussed below. A group decision matrix, shown in , as it is shown in Table 9. The ideal point is also shown in Table 9.    Table 10.

Conclusion
Neutrosophic sets theory introduces three membership functions that is why single-valued neutrosophic numbers could be suitable for evaluating alternatives in relation to the complex evaluation criteria in multiple criteria decision making. However, the use of three membership functions can make evaluation somewhat complex especially when the evaluation is based on data collected by the survey.
The reliability of the data used to solve a problem can have an impact on the final selection of the appropriate alternative. In this manuscript, an improved procedure for estimating the reliability of the collected data is proposed. Therefore, Smarandache et al. [33] has proposed an approach that can be used to assess the reliability of information collected by surveying respondents.
Compared to the previous approach, in the new approach reliability and information belong to the interval [0, 1], unlike the previously proposed approach where reliability belongs to the interval [-1, 1], which makes new application easier for using.
By using the proposed procedure, the reliability of data could be easily determined. In this paper, the usability and efficiency of the proposed approach is successfully demonstrated on an illustrative example of the supplier selection.