AN IMPROVED SCORE FUNCTION FOR RANKING NEUTROSOPHIC SETS AND ITS APPLICATION TO DECISION-MAKING PROCESS

The neutrosophic set (NS) is a more general platform which generalizes the concept of crisp, fuzzy, and intuitionistic fuzzy sets to describe the membership functions in terms of truth, indeterminacy, and false degree. Under this environment, the present paper proposes an improved score function for ranking the single as well as interval-valued NSs by incorporating the idea of hesitation degree between the truth and false degrees. Shortcomings of the existing function have been highlighted in it. Further, the decision-making method has been presented based on proposed function and illustrates it with a numerical example to demonstrate its practicality and effectiveness.


INTRODUCTION
Decision making is one of the most widely used phenomena in our day-to-day life. Almost all decisions take several steps to reach the final destination and some of them may be vague in nature. On the other hand, with the growing complexities of the systems day-by-day, it is difficult for the decision maker to make a decision within a reasonable time by using uncertain, imprecise, and vague information. For handling this, researchers pay more attention to the fuzzy set (FS) theory [1] and corresponding extensions such as intuitionistic fuzzy set (IFS) theory [2], interval-valued IFS (IVIFS) [3] , Neutrosophic set (NS) [4], etc. To date, IFSs and IVIFSs have been widely applied by the various researchers in different decision-making problems. For instance, some authors [5][6][7][8][9] proposed an aggregation operator for handling the different preferences of the decision makers towards the alternatives under IFS environment. Ye [10] proposed a novel accuracy function for ranking the different alternatives of IFS or IVIFS. A ranking method on IVIFSs has been presented in Mitchell [11] and Nayagam et al. [12]. Garg [13] presented a generalized score function for ranking the IVIFSs. Xu [14] and Liu and Xie [15] developed a weighted score and accuracy function to rank the IVIFSs. Garg [16] presented some series of geometric aggregation operator under an intuitionistic multiplicative set environment. Garg [17] presented an accuracy function for interval-valued Pythagorean fuzzy sets. A novel correlation coefficient between the Pythagorean fuzzy sets has been studied by Garg [18]. Recently, Xu and Zhao [19] presented a comprehensive analysis of these methods under IFSs and/or IVIFSs and their corresponding applications in multicriteria decision-making (MCDM) problems.
Although the FSs or IFSs have been widely used by the researchers they cannot deal with indeterminate and inconsistent information. For example, if an expert takes an opinion from a certain person about a certain object, then the person may say that 0.5 is the possibility that statement is true, 0.7 say that the statement is false, and 0.2 say that he or she is not sure about it. This issue is not handled by the FSs or IFSs. To resolve this, Smarandache [4] proposed neutrosophic sets (NSs) which zre characterized by three independent components lying in ]0 + , 1 + [ namely the "truth degree" (T ), "indeterminacy degree" (I), and "falsity degree" (F ), rather than only two T and F in IFS and only T in FS theory. In NSs, the indeterminate information is dependent on T and F whereas the inconsistent information is dependent on the degree of the belongings and nonbelongings of IFSs. However, without specification, NSs are difficult to apply in real-life problems. Thus, an extension of the NS, called a single-valued NS (SNS) and interval-valued NSs (INSs) were proposed by Wang et al. [20,21], respectively. Majumdar and Samant [22] and Ye [23] proposed entropy and similarity measures of SVNSs and IVNSs, respectively. Ye [24] and Broumi and Smarandache [25] proposed a correlation coefficient of SVNSs and IVNSs. Ye [26] and Zhang et al. [27] proposed an aggregation operator for SVNSs and IVNSs. Later on, Peng et al. [28] showed that some operations in Ye [26] may be unrealistic and hence define the novel operations and aggregation operators for MCDM problems. Sahin [29] proposed score and accuracy function for SVNSs and IVNSs. However, the most important task for the decision maker is to rank the objects so as to get the desired decision(s). Thus, the objective of the present work is to present a novel score function for ranking the NSs and INSs by overcoming certain shortcomings of the existing functions. Furthermore, a ranking approach to MCDM problems is proposed in which preferences related to different alternatives with respect to each criterion are represented in the form of NSs.
The rest of the paper is organized as follows. Section 2 briefly introduces the concepts, operations, and operators of NSs, SVNSs, and INSs. Section 3 proposes an improved score function for SVNSs and INSs by overcoming the shortcoming of the existing functions. In Section 4, a novel approach to MCDM problems with SVNSs and INSs is developed. Then in Section 5, two examples are presented to illustrate the proposed methods and a comparative analysis is provided. Finally, Section 6 concludes the paper.

PRELIMINARIES
An overview of NS, SNS, and INS has been addressed here on the universal set X.

Definition 2.1. [4] A NS
A in X is defined by its "truth membership function" (T A ), an "indeterminacy-membership function" (I A (x)), and a "falsity membership function" Ye [26] reduced the NSs to SNSs, defined as below.

Definition 2.3. [29] Let
(ii) Neutrosophic weighted geometric (NWG) operator is defined as where ω i is the weight vector of it such that ω i ∈ [0, 1] and

Definition 2.5. [29] Let
(ii) An interval-valued neutrosophic weighted geometric (INWG) operator is defined a where ω i is the weight vector of it such that ω i ∈ [0, 1] and n i=1 ω i = 1. Definition 2.6. Consider SNS A = a, b, c ; then in order to rank the NS, score functions [29] have been defined as Therefore, there is a need to modify the existing functions for ranking the different alternative(s). Hence, a modified score function has been proposed for NSs as follows.

PROPOSED SCORE FUNCTION
In this section, a new score function for ranking NS and INS by overcoming the shortcoming of the above functions has been proposed. Definition 3.1. Let A = a, b, c be a SNS, a score function N (·), based on the "truth-membership degree" (a), "indeterminacy-membership degree" (b), and "falsity membership degree" (c) which is defined as Clearly, if in some cases SNS has a + c = 1 then N (A) reduces to K(A). Based on it, a prioritized comparison method for any two SNSs A 1 and A 2 is defined as The proposed function can easily solve the above shortcomings of the above functions which can be demonstrated as below.  Thus, the proposed function is reasonable and provides an effective algorithm for the decision analysis process.
Proof. By the definitions of N (A) and K(A), we have hence the result. Proof. Proof follows from Eq. (3).
Furthermore, we can have the following conclusions: • When A = 1, 0, 0 then the value of N (A) reaches the maximum value 1.
• When A = 0, 0, 1 then the value of N (A) reaches the minimum value 0.
Furthermore, for decision makers it is not always possible to present their preferences in terms of exact numbers and hence corresponding to it they will prefer to give them in the form of interval-valued numbers. Then Here, we introduce the score function for the INSs: be an INS. A score function M (·) of an INS based on the "truth-membership degree" interval, "indeterminacy-membership degree" interval, and "falsity-membership degree" interval is defined as In particular, when a L = a U , b L = b U , and c L = c U , an INS is degenerated to NS and their corresponding score function reduces to Eq.

PROPOSED APPROACH
In The pairs of these neutrosophic numbers for C j are denoted by α ij = a ij , b ij , c ij . Therefore, the overall collective neutrosophic matrix is D = (α ij ) m×n . Since the different criteria may be of different types, namely, benefit or cost, then there is a need to normalize it. For this, the value of the benefit type is converted into the cost type by using the following equation: and hence the matrix D is converted into matrix R = (r ij ) n×m where α c ij = c ij , 1 − b ij , a ij is the complement of SNS α ij . Then, we have the following methods for MCDM based on proposed functions.

Approach I:
Step 1: Obtain the decision matrices R from D, if required, by using Eq. (5).
Step 2: Aggregate the SNSs r ij for each A i into an overall neutrosophic value r i by using the N W A or N W G operator.
Step 3: Utilize Eq. (3) to compute the score value of r i .
Step 4: Rank the r i according to score values and hence choose the best alternative based on better value.

Volume 6, Issue 5, 2016
Approach II: If the decision-makers give their preferences towards each A i (i = 1, 2, . . . , n) in the form of interval numbers rather than the crisp number a ij , b ij , c ij then we have the following method.
Step 1: Obtain the decision matrices R from D, if required, by using Eq. (5).
Step 2: Aggregate the INSs r ij for each A i into an overall interval-valued neutrosophic number value r i by using the IN W A or IN W G operator.
Step 4: Find the best one(s) based on its ranking value.

PRACTICAL EXAMPLE
In this section, two illustrative examples have been given for demonstrating as well as validating the proposed method.
Example 5.1. Consider a multicriteria decision making problem of an investment company where an investor wants to invest some money. To do that, they set a panel for four possible alternatives A = {A 1 , A 2 , A 3 , A 4 }, namely, food, transport, electronic, and tire company, respectively, to invest the money under the three different criteria denoted by C 1 , C 2 , and C 3 with weight ω = (0.35, 0.25, 0.40) T . Then by utilizing the proposed Approach I we obtain the most desirable alternative(s) as follows.
Step 1: The decision makers evaluate these alternatives w.r.t. each criterion and give their rating in terms of the SNSs r ij = a ij , b ij , c ij , (i = 1, 2, 3, 4; j = 1, 2, 3). The normalized information for these is listed in Table 1.
Step 2: Compute the neutrosophic number r i for each alternative A i by using the NWA operator as follows. Step 4: Based on the score values, we observe that A 4 , i.e., the tire company, is the most desirable alternative.
If we utilize existing score functions to rank these alternatives then we get I(r   Example 5.2. Interval-valued neutrosophic numbers: Consider the MCDM problem as described in the above example under the INSs environment. Then the following steps of Approach II have been performed to get the best alternative(s).
Step 1: Assume that an expert gives their preferences towards each alternative under each criterion in terms of INSs and their values are summarized in Table 2.
Step Step 3: By using Eq. Step 4: Rank these alternatives and hence we find A 4 is the best company for investment.
If we utilize the INWG operator then we have Step Step 3: We obtain M (r i )(i = 1, 2, 3, 4) as Step 4: Rank these alternatives and hence we find A 4 is the most desirable company.

CONCLUSION
In the present manuscript, an attempt has been made to present an improved score function for ranking the NSs or INSs. From the existing studies, it has been observed that the existing score functions are unable to give the best alternative(s) under some situations. For overcoming this, the present manuscript considers the hesitation degree between the membership functions and hence proposes a novel score function for ranking NSs. Further, a ranking method for solving MCDM problems based on it has been presented and illustrated with a numerical example in the SNSs and INSs environment. From the results, it has been observed that it can equivalently solve the decision-making problem efficiently. In the future, we will extend this approach to the other domains.