Neutrosophic Sets and Systems Neutrosophic Sets and Systems

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Introduction
Zadeh [1] has been credited with having pioneered the development of the concept of fuzzy set in 1965.It is generally agreed that a major breakthrough in the evolution of the modern concept of uncertainty was achieved in defining fuzzy set, even though some ideas presented in the paper were envisioned in 1937 by Black [2].In order to define fuzzy set, Zadeh [1] introduced the concept of membership function with a range covering the interval [0, 1] operating on the domain of all possible values.It should be noted that the concept of membership in a fuzzy set is not a matter of affirmation or denial, rather a matter of a degree.Zadeh's original ideas blossomed into a comprehensive corpus of methods and tools for dealing with gradual membership and non-probabilistic uncertainty.In essence, the basic concept of fuzzy set is a generalization of classical set or crisp set [3,4].The field has experienced an enormous development, and Zadeh's seminal concept of fuzzy set [1] has naturally evolved in different directions.
Different sets have been derived in the literature such as Lfuzzy sets [5], flou sets [6], interval-valued fuzzy sets [7][8][9][10], intuitionistic fuzzy sets [11][12][13], two fold fuzzy sets [14], interval valued intuitionistic fuzzy set [15], intuitionistic L-fuzzy sets [16], etc. Interval-valued fuzzy sets are a special case of L-fuzzy sets in the sense of Goguen [5] and a special case of type 2 fuzzy set.Mathematical equivalence of intuitionistic fuzzy set (IFS) with interval-valued fuzzy sets was noticed by Atanassov [17], Atanassov and Gargov [15].Wang and He [18] proved that the concepts of IFS [11][12][13] and intuitionistic L-fuzzy sets [5] and the concept of L-fuzzy sets [5] are equivalent.Kerre [19] provided a summary of the links that exist between fuzzy sets [1] and other mathematical models such as flou sets [6], two-fold fuzzy sets [14] and L-fuzzy sets [5].Deschrijver and Kerre [20] established the relationships between IFS [11], L-fuzzy sets [5], interval-valued fuzzy sets [7], interval-valued IFS [15].Dubois et al. [21] criticized the term IFSs in the sense of [11][12][13], and termed it "to be unjustified, misleading, and possibly offensive to people in intui-Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making tionistic mathematics and logic" as it clashes with the correct usage of intuitionistic fuzzy set proposed by Takeuti and Titani [22].Dubois et al. [21] suggested changing the name of IFS as I-fuzzy set.Smarandache incorporated the degree of indeterminacy as independent component in IFS and defined neutrosophic set [23][24] as the generalization of IFSs.Georgiev [25] explored some properties of the neutrosophic logic and defined simplified neutrosophic set.A neutrosophic set is simplified [25] if its elements are comprised of singleton subsets of the real unit interval.Georgiev [25] concluded that the neutrosophic logic is not capable of maintaining modal operators, since there is no normalization rule for the components T, I and F. The author [25] claimed that the IFSs have the chance to become a consistent model of the modal logic, adopting all the necessary properties [26].However certain type of uncertain information such as indeterminate, incomplete and inconsistent information cannot be dealt with fuzzy sets as well as IFSs.Smarandache [27][28] re-established neutrosophic set as the generalization of IFS, which plays a key role to handle uncertain, inconsistent and indeterminacy information existing in real world.In this set [27][28] each element of the universe is characterized by the truth degree, indeterminacy degree and falsity degree lying in the nonstandard unit interval.The neutrosophic set [27][28] emerged as one of the research focus in many branches such as image processing [29][30][31], artificial intelligence [32], applied physics [33][34], topology [35] and social science [36].Furthermore, single valued neutrosophic set [37], interval neutrosophic set [38],neutrosophic soft set [39], neutrosophic soft expert set [40], rough neutrosophic set [41], interval neutrosophic rough set, interval valued neutrosophic soft rough set [42], complex neutrosophic set [43], bipolar neutrosophic sets [44] and neutrosophic cube set [45] have been studied in the literature which are connected with neutrosophic set.However, in this study, we have applied single valued neutrosophic set [37] (SVNS), a subclass of NS, in which each element of universe is characterized by truth membership, indeterminacy membership and falsity membership degrees lying in the real unit interval.Recently, SVNS has caught attention to the researcher on various topics such as similarity measure [46][47][48][49][50], medical diagnosis [51] and multi criteria/ attribute decision making [52][53][54][55][56][57][58], etc Aggregation of SVNS information becomes an important research topic for multi attribute decision making in which the rating values of alternatives are expressed in terms of SVNSs.Aggregation operators of SVNSs, usually taking the forms of mathematical functions, are common techniques to fuse all the input individual data that are typically interpreted as the truth, indeterminacy and the falsity membership degree in SVNS into a single one.Ye [59] proposed weighted arithmetic average operator and weighted geometric average operator for simplified neutrosophic sets.Peng et al. [60] developed some aggregation operators to aggregate single valued neutrosophic information, such as simplified neutrosophic number weighted averaging (SNNWA), simplified neutrosophic number weighted geometric (SNNWG), simplified neutrosophic number ordered weighted averaging (SNNOWA), simplified neutrosophic number ordered weighted geometric averaging (SNNOWG), simplified neutrosophic number hybrid ordered weighted averaging operator(SNNHOWA), simplified neutrosophic number hybrid ordered weighted geometric operator (SNNHOWG), generalised simplified neutrosophic number weighted averaging operator(GSNNWA) and generalised simplified neutrosophic number weighted geometric operator(GSNNGA) operators.Peng et al. [60] applied these aggregation operators in multi criteria group decision making problem to get an overall evaluation value for selecting the best alternative.Liu et al. [61] defined some generalized neutrosophic Hamacher aggregation operators and applied them to multi attribute group decision making problem.Liu and Wang [62] proposed a single valued neutrosophic normalized weighted Bonferroni mean operator for multi attribute decision making problem.
Application of SVNS has been extensively studied in multi-attribute decision making problem.However, in uncertain and complex situations, the truth membership, indeterminacy membership, and falsity membership degree of SVNS cannot be represented with exact real numbers or interval numbers.Moreover, triangular fuzzy number can handle effectively fuzzy data rather than interval number.Therefore, combination of triangular fuzzy number with SVNS can be used as an effective tool for handling incomplete, indeterminacy, and uncertain information existing in decision making problems.Recently, Ye [63] defined trapezoidal fuzzy neutrosophic set and developed trapezoidal fuzzy neutrosophic number weighted arithmetic averaging and trapezoidal fuzzy neutrosophic number weighted geometric averaging operators to solve multi attribute decision making problem.
Zhang and Liu [64] presented method for aggregating triangular fuzzy intuitionistic fuzzy information and its application to decision making.However, their approach cannot deal the decision making problems which involve indeterminacy.So new approach is essentially needed which can deal indeterminacy.Literature review reflects that this is the first time that aggregation operator of triangular fuzzy number neutrosophic values has been studied although this number can be used as an effective tool to deal with uncertain information.In this paper, we have first Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making presented triangular fuzzy number neutrosophic sets (TFNNS), score function and accuracy function of TFNNS.Then we have extended the aggregation method of triangular fuzzy intuitionistic fuzzy information [64] to triangular fuzzy number neutrosophic weighted arithmetic averaging (TFNNWA) operator and triangular fuzzy number neutrosophic weighted geometric averaging (TFNNWG) operator to aggregate TFNNSs.The proposed TFNNWA and TFNNWG operators are more flexible and powerful than their fuzzy and intuitionistic fuzzy counterpart as they are capable of dealing with uncertainty and indeterminacy.
The objectives of the study include to:  propose triangular fuzzy number neutrosophic sets (TFNNS), score function and accuracy function of TFNNS. propose two aggregation operators, namely, TFNNWA and TFNNWG. prove some properties of the proposed operators namely, TFNNWA and TFNNWG. establish a multi attribute decision making (MADM) approach based on TFNNWA and TFNNWG. provide an illustrative example of MADM problem.
The rest of the paper has been organized in the following way.In Section 2, a brief overview of IFS, SVNS have been presented.In Section 3, we have defined TFNNS, score function and accuracy function of TFNNS, and some operational rules of TFNNS.Section 4 has been devoted to propose two aggregation operators, namely, TFNNWA and TFNNWG operators to aggregate TFNNSs.In Section 5, applications of two proposed operators have been presented in multi attribute decision making problem.In Section 6, an illustrative example of MADM has been provided.Finally, conclusion and future direction of research have been presented in Section 7.

Preliminaries
In this section we recall some basic definitions of intuitionistic fuzzy sets, triangular fuzzy number intuitionistic fuzzy set (TFNIFS), score function and accuracy function of TFNIFS.

The numbers (x)
A

 and (x)
A  denote, respectively, the degree of membership degree and degree of non-membership of x in .1.

Single valued neutrosophic sets
In this section, some basic definitions of single valued neutrosophic sets are reviewed.
Definition 7. [37] Let X be a space of points (objects) with a generic element in X denoted by .
For convenience, SVNS A can be denoted by (x), (x), (x) Definition 8. [37] Assume that (x), (x), (x) B T I F  be two SVNSs in a universe of discourse X .Then the following operations are defined as follows:


3 Triangular fuzzy number neutrosophic set SVNS can represent imprecise, incomplete and inconsistent type information existing in the real world problem.However, decision maker often expresses uncertain information with truth, indeterminacy and falsity membership functions that are represented with uncertain numeric values instead of exact real number values.These uncertain numeric values of truth, indeterminacy and falsity membership functions of SVNSs can be represented in terms of triangular fuzzy numbers.
In this section, we combine triangular fuzzy numbers (TFNs) with SVNSs to develop triangular fuzzy number neutrosophic set (TFNNS) in which, the truth, indeterminacy and falsity membership functions are expressed with triangular fuzzy numbers.Definition 9. Assume that X be the finite universe of discourse and F [0, 1] be the set of all triangular fuzzy numbers on   0,1 .A triangular fuzzy number neutrosophic set (TFNNS) A in X is represented by Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making

  , ( ), ( ), ( ) | ,
( ) (x), (x), (x) ( ) (x), (x), (x) and () ( ), ( ), (x) , respectively, denote the truth membership degree, indeterminacy degree, and falsity membership degree of x in A and for every xX  : (x), (x), (x) ( , , ) and   A a b c e f g r s t A a b c e f g r s t  be two TFNNVs in the set of real numbers.Then the following operations are defined as follows: 1.

a a b b c c
A A e e e e f f f f g g g g r r r r s s s s t t t t

Score and accuracy function of TFNNV
In the following section, we define score function and accuracy function of TFNNV from Definition 5, Definition 6.
Definition 11.Assume that (a , b ,c ),(e ,f ,g ),(r ,s , t ) A  be a TFNNVs in the set of real numbers, the score function The value of score function of TFNNV (1,1,1), (0, 0, 0), (0, 0, 0) value of accuracy function of TFNNV (0, 0, 0), ( (a , b ,c ),(e ,f ,g ),(r ,s , t ) A  be a TFNNV in the set of real numbers, the accuracy function   1 HA of 1 A is defined as follows: The accuracy function determines the difference between truth and falsity.Larger the difference reflects the more affirmative of the TFNNV.The accuracy function (A ) (1,1,1), (0, 0, 0), (0, 0, 0) Based on Definition 11 and Definition 12, we present the order relations between two TFNNVs.Definition 13.Assume that (a , b ,c ),(e ,f ,g ),(r ,s , t ) A  be two TFNNVs in the set of real numbers.Suppose that (A ) i S and (A ) i H are the score and accuracy functions of TFNNS ( 1, 2)   i Ai , then the following order relations are defined as follows: Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making  21) and ( 22), we obtain the following results: 1. Score value of where [0,1] j w  is the weight vector of ( 1, 2,..., ) operator reduces to triangular fuzzy number neutrosophic averaging (TFNNA) operator: We can now establish the following theorem by using the basic operations of TFNNVs defined in Definition 10.
Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making Theorem 1.
j j j j j j j w w w j j j j j j w w w w w w j j j j j j j j j j j j bb e f g r s t Thus the theorem is true for n = 2 3.When n = k, we assume that Eq.( 27) is also true. Then, 4. When n = k + 1, we have Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making , .e , ., ., .
Therefore, by mathematical induction, we can say that Eq. ( 27) holds for all values of n.As the components of all three membership functions of It follows that the relation This completes the proof of the Theorem 1. Now, we highlight some necessary properties of TFNNWA operator.(a ,b ,c ),(e ,f ,g ),(r ,s , t )  min( ) max( ) (A , A ,..., A ) A wn TFNNWA  = (a, b, c), (e, f, g), (r,s, t) , then the score function of min( ) 2 min( ) min( ) 12 min( ) 2 min( ) min( )   Similarly, the score function of ( Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making
Theorem 2. Assume that (a ,b ,c ),(e ,f ,g ),(r ,s , t ) j j j j j j j j j j A  (j 1, 2,..., n)  be a collection TFNNVs in the set of real numbers.Then the aggregated value obtained from TFNNWG, is also a TFNNV, and then we have Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making ) .
Therefore, by mathematical induction, Eq. ( 49) holds for all values of n.
Since the components of all three membership functions of ( 1,2,..., ) (1 ) It follows that This completes the proof of Theorem 2. Now, we discuss some essential properties of TFNNWG operator for TFNNs., ), ( , , ), ( , , ) . a b c e f g r s t A  This completes the Property 4.
Step 4: Using Definition 11 to Definition 13, determine the ranking order of aggregated values obtained in Step 3.
Step 5: Select the best alternative in accordance with the ranking order obtained in Step 4.

Y Y Y Y
Step 5: The ranking order in Step 4 reflects that, 2 Y is the best medical representative.

Utilization of TFNNWG operator:
Step 1: Using Eq.( 49), we aggregate all the rating values of the alternative Y i (i= 1, 2, 3, 4) for the i-throw of the decision matrix  5.
Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making Thus the ranking order of the alternative is presented as follows:

. Y Y Y Y
Step 5: The ranking order in Step 4 reflects that 2 Y is the best medical representative.

Conclusions
MADM problems generally takes place in a complex environment and usually connected with imprecise data and uncertainty.The triangular neutrosophic fuzzy numbers are an effective tool for dealing with impreciseness and incompleteness of the decision maker's assessments over alternative with respect to attributes.We have first introduced TFNNs and defined some of its operational rules.Then we have proposed two aggregation operators called TFNNWAA and TFNNWGA operators and score function and applied them to solve multi attribute decision making problem under neutrosophic environment.Finally, the effectiveness and applicability of the proposed approach have been illustrated with medical representative selection problem.We hope that the proposed approach can be also applied in other decision making problems such as pattern recognition, personnel selection, medical diagnosis, etc.
the basic operations of IFNs are presented as follows: ) For notational convenience, we consider ( , , ),( , , ),( , , )A a b c e f g r s t as a trapezoidal fuzzy number neutrosophic values (TFNNV) where,

12 (
We prove the theorem by mathematical induction.1.When 1 n  , it is a trivial case When 2 n  , we have


Similar to arithmetic averaging operator, we can also prove the theorem by mathematical induction.1.When n = 1, the theorem is true.2.When n = 2, we have

Step 2 :
The aggregated rating values i u corresponding to the alternative i Y are shown in the Table TFNNVs in the set of real numbers.
Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making The proof of the Property 5 is similar to Property 2.

Table 1 .
-Triangular fuzzy number neutrosophic value based decision matrix For each alternative ( 1, 2,..., ), Au of the aggregated rating values obtained by TFNNWA or TFNNWG operators that are in Eqs. ( i Su and accuracy values () i

Table 2 .
Triangular fuzzy number neutrosophic value based rating valuesPranab Biswas, Surapati Pramanik, and Bibhas C. Giri; Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making

Table 4 .
Score and accuracy values of aggregated rating values

Table 6 .
Score and accuracy values of rating values