A Single-Valued Neutrosophic Extension of the EDAS Method

: This manuscript aims to propose a new extension of the EDAS method, adapted for usage with single-valued neutrosophic numbers. By using single-valued neutrosophic numbers, the EDAS method can be more efﬁcient for solving complex problems whose solution requires assessment and prediction, because truth-and falsity-membership functions can be used for expressing the level of satisfaction and dissatisfaction about an attitude. In addition, the indeterminacy-membership function can be used to point out the reliability of the information given with truth-and falsity-membership functions. Thus, the proposed extension of the EDAS method allows the use of a smaller number of complex evaluation criteria. The suitability and applicability of the proposed approach are presented through three illustrative examples.


Introduction
Multicriteria decision making facilitates the evaluation of alternatives based on a set of criteria.So far, this technique has been used to solve a number of problems in various fields [1][2][3][4][5][6].
Notable advancement in solving complex decision-making problems has been made after Bellman and Zadeh [7] introduced fuzzy multiple-criteria decision making, based on fuzzy set theory [8].
In fuzzy set theory, belonging to a set is shown using the membership function µ(x) ∈ [0, 1].Nonetheless, in some cases, it is not easy to determine the membership to the set using a single crisp number, particularly when solving complex decision-making problems.Therefore, Atanassov [9] extended fuzzy set theory by introducing nonmembership to a set ν(x) ∈ [0, 1].In Atanassov's theory, intuitionistic sets' indeterminacy is, by default, 1 − µ(x) − ν(x).
Smarandache [10,11] further extended fuzzy sets by proposing a neutrosophic set.The neutrosophic set includes three independent membership functions, named the truthmembership T A (x), the falsity-membership F A (x) and the indeterminacy-membership I A (x) functions.Smarandache [11] and Wang et al. [12] further proposed a single-valued neutrosophic set, by modifying the conditions T A (x), I A (x) and F A (x) ∈ [0, 1] and 0 ≤ T A (x) + I A (x) + F A (x) ≤ 3, which are more suitable for solving scientific and engineering problems [13].
When solving some kinds of decision-making problems, such as problems related to estimates and predictions, it is not easy to express the ratings of alternatives using crisp values, especially in cases when ratings are collected through surveys.The use of fuzzy sets, intuitionistic fuzzy sets, as well as neutrosophic fuzzy sets can significantly simplify the solving of such types of complex decision-making problems.However, the use of fuzzy sets and intuitionistic fuzzy sets has certain limitations related to the neutrosophic set theory.By using three mutually independent membership functions applied in neutrosophic set theory, the respondent involved in surveys has the possibility of easily expressing their views and preferences.The researchers recognized the potential of the neutrosophic set and involved it in the multiple-criteria decision-making process [14,15].
Several extensions are also proposed for the EDAS method, such as: a fuzzy EDAS [19], an interval type-2 fuzzy extension of the EDAS method [18], a rough EDAS [20], Grey EDAS [28], intuitionistic fuzzy EDAS [29], interval-valued fuzzy EDAS [30], an extension of EDAS method in Minkowski space [23], an extension of the EDAS method under q-rung orthopair fuzzy environment [31], an extension of the EDAS method based on intervalvalued complex fuzzy soft weighted arithmetic averaging (IV-CFSWAA) operator and the interval-valued complex fuzzy soft weighted geometric averaging (IV-CFSWGA) operator with interval-valued complex fuzzy soft information [32], and an extension of the EDAS equipped with trapezoidal bipolar fuzzy information [33].
In order to enable the usage of the EDAS method for solving complex decision-making problems, a novel extension that enables usage of single-valued neutrosophic numbers is proposed in this article.Therefore, the rest of this paper is organized as follows: In Section 2, some basic definitions related to the single-valued neutrosophic set are given.In Section 3, the computational procedure of the ordinary EDAS method is presented, whereas in Section 3.1, the single-valued neutrosophic extension of the EDAS method is proposed.In Section 4, three illustrative examples are considered with the aim of explaining in detail the proposed methodology.The conclusions are presented in the final section.

Preliminaries
Definition 1.Let X be the universe of discourse, with a generic element in X denoted by x.A Neutrosophic Set (NS) A in X is an object having the following form [11]: 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, T A (x), , and ] − 0, 1 + [ denotes bounds of NS.
Definition 2. Let X be a space of points, with a generic element in X denoted by x.A Single-Valued Neutrosophic Set (SVNS) A over X is as follows [12]: 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, T A (x), Definition 3. A Single-Valued Neutrosophic Numbera = t a , i a , f a is a special case of an SVNS on the set of real numbers , where t a , i a , f a ∈ [0, 1] and 0 ≤ t a + i a + f a ≤ 3 [12].
f 2 be two SVNNs and λ > 0. The basic operations over two SVNNs are as follows: Definition 5. Let x =< t i , i i , f i > be an SVNN.The score function s x of x is as follows [44]: where Definition 6.Let a j ≤ t j , i j , f j > (j = 1, . . ., n) be a collection of SVNSs and W = (w 1 , w 2 , . . . ,w n ) T e an associated weighting vector.The Single-Valued Neutrosophic Weighted Average (SVNWA) operator of a j is as follows [40]: where: w j is the element j of the weighting vector, w j ∈ [0, 1] and ∑ n j=1 w j = 1.
Definition 7. Let x =< t i , i i , f i > be an SVNN.The reliability r i of x is as follows [45]: Definition 8. Let D be a decision matrix, dimension m x n, whose elements are SVNNs.The overall reliability of the information contained in the decision matrix is as follows:

The EDAS Method
The procedure of solving a decision-making problem with m alternatives and n criteria using the EDAS method can be presented using the following steps: Step 1. Determine the average solution according to all criteria, as follows: with: where: x ij denotes the rating of the alternative i in relation to the criterion j.
Step 2. Calculate the positive distance from average (PDA) d + ij and the negative distance from average (NDA) d − ij , as follows: where: Ω max and Ω min denote the set of the beneficial criteria and the nonbeneficial criteria, respectively.
Step 3. Determine the weighted sum of PDA, Q + i , and the weighted sum of NDS, Q − i , for all alternatives, as follows: where w j denotes the weight of the criterion j.
Step 4. Normalize the values of the weighted sum of the PDA and NDA, respectively, for all alternatives, as follows: where: S + i and S − i denote the normalized weighted sum of the PDA and the NDA, respectively.
Step 5. Calculate the appraisal score S i for all alternatives, as follows: Step 6. Rank the alternatives according to the decreasing values of appraisal score.The alternative with the highest S i is the best choice among the candidate alternatives.

The Extension of the EDAS Method Adopted for the Use of Single-Valued Neutrosophic Numbers in a Group Environment
Let us suppose a decision-making problem that include m alternatives, n criteria and k decision makers, where ratings are given using SVNNs.Then, the computational procedure of the proposed extension of the EDAS method can be expressed concisely through the following steps: Step 1. Construct the single-valued neutrosophic decision-making matrix for each decision maker, as follows: whose elements Step2.Construct the single-valued neutrosophic decision making using Equation ( 8): Step 3. Determine the single-valued average solution (SVAS) x * j according to all criteria, as follows: where: Step 4. Calculate a single-valued neutrosophic PDA (SVNPDA), where: For a decision-making problem that includes only beneficial criteria, the SVNPDA and SVNNDA can be determined as follows: Step 5. Determine the weighted sum of the SVNPDA, , for all alternatives.Based on Equations ( 5) and ( 8) the weighted sum of the SVNPDA, Q + i , and the weighted sum of the SVNNDA, Q − i , can be calculated as follows: Step 6.In order to normalize the values of the weighted sum of the single-valued neutrosophic PDA and the weighted sum of the single-valued neutrosophic NDA, these values should be transformed into crisp values.This transformation can be performed using the score function or similar approaches.After that, the following three steps remain the same as in the ordinary EDAS method.
Step 7. Normalize the values of the weighted sum of the SVNPDA and the singlevalued neutrosophic SVNNDA for all alternatives, as follows: Step 8. Calculate the appraisal score S i for all alternatives, as follows: Step 9. Rank the alternatives according to the decreasing values of the appraisal score.The alternative with the highest S i is the best choice among the candidate alternatives.

A Numerical Illustrations
In this section, three numerical illustrations are presented in order to indicate the applicability of the proposed approach.The first numerical illustration shows in detail the procedure for applying the neutrosophic extension of the EDAS method.The second numerical illustration shows the application of the proposed extension in the case of solving MCDM problems that contain nonbeneficial criteria, while the third numerical illustration shows the application of the proposed approach in combination with the reliability of the information contained in SVNNs.

The First Numerical Illustration
In this numerical illustration, an example adopted from Biswas et al. [46] is used to demonstrate the proposed approach in detail.Suppose that a team of three IT specialists was formed to select the best tablet from four initially preselected tablets for university students.The purpose of these tablets is to make university e-learning platforms easier to use.
The preselected tablets are evaluated based on the following criteria: Features-C 1 , Hardware-C 2 , Display-C 3 , Communication-C 4 , Affordable Price-C 5 , and Customer care-C 6 .The ratings obtained from three IT specialists are shown in Tables 1-3.
Table 1.The ratings of three tablets obtained from the first of three IT specialist.

The Second Numerical Illustration
The second numerical illustration shows the application of the NS extension of the EDAS method in the case of solving MCDM problems that include nonbeneficial criteria.
An example taken from Stanujkic et al. [47] was used for this illustration.In the given example, the evaluation of three comminution circuit designs (CCDs) was performed based on five criteria: Grinding efficiency-C 1 , Economic efficiency-C 2 , Technological reliability-C 3 , Capital investment costs-C 4 , and Environmental impact-C 5 .The group decision-making matrix, as well as the types of criteria, are shown in Table 8.
A similar order of alternatives was obtained in Stanujkic et al. [45] using the Neutrosophic extension of the MULTIMOORA method, where the following order of alternatives was achieved A 3 > A 2 > A 1 .

The Third Numerical Illustration
The third numerical illustration shows the use of a newly proposed approach with an approach that allows for determining the reliability of data contained in SVNNs, proposed by Stanujkic et al. [43].Using this approach, inconsistently completed questionnaires can be identified and, if necessary, eliminated from further evaluation of alternatives.
In order to demonstrate this approach, an example was taken from Stanujkic et al. [48].In this example, the websites of five wineries were evaluated based on the following five criteria: Content-C 1 , Structure and Navigation-C 2 , Visual Design-C 3 , Interactivity-C 4 , and Functionality-C 5 .
The ratings obtained from the three respondents are also shown in Tables 12-14.
Table 12.The ratings obtained from the first of three respondents.
Table 18.The group decision-making matrix.

Conclusions
A novel extension of the EDAS method based on the use of single-valued neutrosophic numbers is proposed in this article.Single-valued neutrosophic numbers enable simultaneous use of truth-and falsity-membership functions, and thus enable expressing the level of satisfaction and the level of dissatisfaction about an attitude.At the same time, using the indeterminacy-membership function, decision makers can express their confidence about already-given satisfaction and dissatisfaction levels.
The evaluation process using the ordinary EDAS method can be considered as simple and easy to understand.Therefore, the primary objective of the development of this extension was the formation of an easy-to-use and easily understandable extension of the EDAS method.By integrating the benefits that can be obtained by using single-valued neutrosophic numbers and simple-to-use and understandable computational procedures of the EDAS method, the proposed extension can be successfully used for solving complex decision-making problems, while the evaluation procedure remains easily understood for decision makers who are not familiar with neutrosophy and multiple-criteria decision making.
Finally, the usability and efficiency of the proposed extension is demonstrated on an example of tablet evaluation.

Table 2 .
The ratings of three tablets obtained from the second of three IT specialist.

Table 4 .
The group evaluation matrix.

Table 7 .
Computational details and ranking order of considered tablets.

Table 11 .
Computational details and ranking order of considered GCDs.

Table 15 .
The reliability of information obtained from the first of three respondents.

Table 16 .
The reliability of information obtained from the second of three respondents.

Table 17 .
The reliability of information obtained from the third of three respondents.

Table 19 .
Computational details and ranking order of considered websites.