# Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making.

1 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 L-fuzzy sets [5], flou sets [6], interval-valued fuzzy sets [7-10], intuitionistic fuzzy sets [11-13], 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 [11-13] 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 [I] 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 [II], 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 [11-13], and termed it "to be unjustified, misleading, and possibly offensive to people in intuitionistic 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 [29-31], 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 [46-50], medical diagnosis [51] and multi criteria/ attribute decision making [52-58], 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 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.

2 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.

2.1 Intuitionistic fuzzy sets

Definition 1. (Intuitionistic fuzzy set [13]) An intuitionistic fuzzy set A in finite universe of discourse

X = {[x.sub.1], [x.sub.2], ..., [x.sub.n]} is given by

A = {(x, [[mu].sub.A] (x), [v.sub.A] (x))| x [member of] X}, (1)

where [[mu].sub.A]: X [left arrow] [0,1] and [v.sub.A] : X [left arrow] -[0,1] with the condition 0 [less than or equal to] [[mu].sub.A] (x) + [v.sub.A] (x) [less than or equal to] 1. The numbers [[mu].sub.A] (x) and [v.sub.A] (x) denote, respectively, the degree of membership degree and degree of non-membership of x in A. In addition [[pi].sub.A](x) = 1 - [[mu].sub.A](x) - [v.sup.A](x) is called a hesitancy degree of x [member of] X in A. For convenience, A = ([[mu].sub.A] (x), [v.sub.A] (x)) is considered as an intuitionistic fuzzy number (IFN).

Definition 2. (Operations rules of IFNs [65-67])

Let A = ([[mu].sub.A] (x), [v.sub.A] (x)) and B = ([[mu].sub.B] (x), [v.sub.B] (x)) be two

IFNs, then the basic operations of IFNs are presented as follows:

1. A [direct sum] B = ([[mu].sub.A] (x) + [V.sub.B] (x)-[[mu].sub.A] (x)[V.sub.B] (x), [V.sub.A] (x) [V.sub.B] (x)), (2)

2. A [cross product] B = ([[mu].sub.A] (x) [V.sub.B] (x), [V.sub.A] (x)[V.sub.B] (x) - [V.sub.A] (x) [V.sub.B] (x)), (3)

3. [lambda]A = [(1 -(1 - [[mu].sub.A] (x)).sup.[lambda]],[([v.sub.A] (x)).sup.[lambda]]) for [lambda] > 0, (4)

4. [A.sup.[lambda]] =([([[mu].sub.A] (x)).sup.[lambda]], 1 -[(1-[V.sub.A] (x)).sup.[lambda]]) for [lambda] > 0. (5)

Definition 3. [68] Let X be a finite universe of discourse and.F[0,1] be the set of all triangular fuzzy numbers on [0,1]. A triangular fuzzy number intuitionistic fuzzy set (TFNIFS) A inX is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The triangular fuzzy numbers

[[??].sub.A] (x) = ([[mu].sup.1.sub.A](x), [[mu].sup.2.sub.A](x), [[mu].sup.3.sub.A] (x)) and

[[??].sub.A] (x) = ([v.sup.1.sub.A](x), [v.sup.2.sub.A](x), [v.sup.3.sub.A] (x)), respectively, denote the membership degree and non-membership degree of x in A and for every x [member of] X:

0 [greater than or equal to] [[mu].sup.3.sub.A] (x) + [v.sup.3.sub.Z](x) [less than or equal to] 1.

For convenience, we consider A = <(a,b, c), (e,f, g)> as the trapezoidal fuzzy number intuitionistic fuzzy values (TFNIFV) where,

([[mu].sup.1.sub.A](x), [[mu].sup.2.sub.A](x), [[mu].sup.3.sub.A] (x)) = (a, b, c) and ([v.sup.1.sub.A](x), [v.sup.2.sub.A](x), [v.sup.3.sub.A] (x)) = (e, f, g).

Definition 4. [69-70] Let [A.sub.1] = <([a.sub.1],[b.sub.1],[c.sub.1]),([e.sub.1],[f.sub.1],[g.sub.1])> and [A.sub.2] = <([a.sub.2],[b.sub.2],[c.sub.2]),([e.sub.2],[f.sub.2],[g.sub.2]) be two TFNIFVs, then the following operations are valid:

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (6)

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (7)

3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [lambda] > 0, and (8)

4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; for [lambda] > 0. (9)

Definition 5. [69-70]Let [A.sub.1] = <([a.sub.1],[b.sub.1],[c.sub.1]),([e.sub.1],[f.sub.1],[g.sub.1])> be a TFNIFV, the score function S([A.sub.1]) of [A.sub.1] is defined as follows:

S([A.sub.1]) = 1/4 [([a.sub.1] + 2[b.sub.1] + [c.sub.1])-([e.sub.1] + 2 [f.sub.1] + [g.sub.1])], S([A.sub.1]) [member of] [-1,1] (10)

The score function S ([A.sup.+]) = 1 for the TFNIFV

[A.sup.+] = <(1,U),(0,0,0)> and S([A.sup.-]) = -1 for the TFNIFV [A.sup.-] = ((0,0,0), (1,1,1)).

Definition 6. [69-70] Let [A.sub.1] = <([a.sub.1],[b.sub.1],[c.sub.1]),([e.sub.1],[f.sub.1],[g.sub.1])> be a TFNIFV, the accuracy function H ([A.sub.1] ) is of [A.sub.1] is defined as follows:

H([A.sub.1]) = 1/4 [([a.sub.1] + 2[b.sub.1] + [c.sub.1])+([e.sub.1] + 2[f.sub.1] + [g.sub.1])], H([A.sub.1]) [member of] [-0,1] (11)

2.2 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 x. A single valued neutrosophic set A in X is characterized by a truth membership function [T.sub.A] (x), an indeterminacy membership function [I.sub.A] (x), and a falsity membership function [F.sub.A] (x) and is denoted by

[??] = {x, <[T.sub.[??]] (x), [I.sub.[??]] (x), [F.sub.[??]] (x)> | x [member of] X}.

Here [T.sub.[??]] (x), [I.sub.[??]] (x) and [F.sub.[??]](x) are real subsets of [0, 1] that is [T.sub.[??]] (x): X [left arrow] [0,l], [I.sub.[??]](x): X [left arrow] [0,1] and [F.sub.[??]] (x): X [left arrow] [0,1]. The sum of [T.sub.[??]]{x), [I.sub.[??]](x) and [F.sub.[??]](x) lies in [0, 3] that is 0 [less than or equal to] sup [T.sub.[??]] (x) + sup [I.sub.[??]] (x) + sup [F.sub.[??]] (x) < 3.

For convenience, SVNS [??] can be denoted by [??] = {[T.sub.[??]](x) [I.sub.[??]] (x),[F.sub.[??]](x)} for all x in X.

Definition 8. [37] Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be

two SVNSs in a universe of discourse X. Then the following operations are defined as follows:

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (12)

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (13)

3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (14)

4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (15)

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) [??] in X is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The triangular fuzzy numbers

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively, denote the truth membership degree, indeterminacy degree, and falsity membership degree of x in [??] and for every x [member] X:

0 [less than or equal to] [T.sup.3.sub.A](x) + [I.sup.3.sub.A](x) + [F.sup.3.sub.A] [less than or equal to] 3. (16)

For notational convenience, we consider

[??] = <(a,b,c), (e,f,g), (r, s, t)> as a trapezoidal fuzzy number neutrosophic values (TFNNV) where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 10. Let [[??].sub.1] = <([a.sub.1],[b.sub.1],[c.sub.1]),([e.sub.1],[f.sub.1],[g.sub.1]), ([r.sub.1],[s.sub.1],[t.sub.1])> and [[??].sub.2] = <([a.sub.2],[b.sub.2],[c.sub.2]),([e.sub.2],[f.sub.2],[g.sub.2]), ([r.sub.2],[s.sub.2],[t.sub.2])> be two TFNNVs in the set of real numbers. Then the following operations are defined as follows:

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (17)

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (18)

3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [lambda] > 0 and (19)

4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [lambda] > 0. (20)

The operations defined in Definition 10 satisfy the following properties:

1. [[??].sub.1] [direct sum] [[??].sub.2] = [[??].sub.2] [direct sum] [[??].sub.1], [[??].sub.1] [cross product] [[??].sub.2] = [[??].sub.2] [cross product] [[??].sub.1];

2. [lambda]([[??].sub.1] [direct sum] [[??].sub.2]) = [lambda][[??].sub.1] [direct sum] [[??].sub.2], [([[??].sub.1] [cross product] [[??].sub.2]).sup.[lambda]] = [[??].sub.1.sup.[lambda]] [cross product] [[??].sub.2.sup.[lambda]] for [lamda] > 0, and

3.[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3.1 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

[??].sub.1] = <([a.sub.1],[b.sub.1],[c.sub.1]),([e.sub.1],[f.sub.1],[g.sub.1]), ([r.sub.1],[s.sub.1],[t.sub.1])> be a TFNNVs in the set of real numbers, the score function S([[??].sub.1]) of [[??].sub.1] is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. 21

The value of score function of

TFNNV [A.sup.+] = <(1,1,1),(0,0,0),(0,0,0)> is S([A.sup.+]) = 1 and value of accuracy function of

TFNNV [A.sup.-] = <(0,0,0),(1,1,1),(1,1,1)> is S([A.sup.-]) = -1.

Definition 12. Assume

that [[??].sub.1] = <([a.sub.1],[b.sub.1],[c.sub.1]),([e.sub.1],[f.sub.1],[g.sub.1]),([r.sub.1],[s.sub.1],[t.sub.1])> be a TFNNV in the set of real numbers, the accuracy function H ([[??].sub.1]) of [[??].sub.1] is defined as follows:

H([[??].sub.1]) = 1/4[([a.sub.1] + 2[b.sub.1] + [c.sub.1]) - ([r.sub.1] + 2[s.sub.1] + [t.sub.1])]. (22)

The accuracy function H([[??].sub.1]) [member of] [-1.1] determines the difference between truth and falsity. Larger the difference reflects the more affirmative of the TFNNV. The accuracy function H([[??].sup.+])= 1 for [A.sup.+] = ((1,1,1), (0,0,0), (0,0,0)) and H([[??].sup.-]) = -1 for the TFNNV [A.sup.-] = <(0,0,0), (1,1,1), (1,1,1)>. Based on Definition 11 and Definition 12, we present the order relations between two TFNNVs.

Definition 13. Assume that

[[??].sub.1] = <([a.sub.1],[b.sub.1],[c.sub.1]),([e.sub.1],[f.sub.1],[g.sub.1]), ([r.sub.1],[s.sub.1],[t.sub.1])> and [[??].sub.2] = <([a.sub.2],[b.sub.2],[c.sub.2]),([e.sub.2],[f.sub.2],[g.sub.2]), ([r.sub.2],[s.sub.2],[t.sub.2])> be two TFNNVs in the set of real numbers. Suppose that S([[??].sub.i]) and H([[??].sub.i]) are the score and accuracy functions of TFNNS [[??].sub.i](I = 1,2), then the following order relations are defined as follows:

1. If S([[??].sub.1]) > S([[??].sub.2]), then [[??].sub.1], is greater than [[??].sub.2] that is [[??].sub.1] [??] [A.sub.2];

2. If S([[??].sub.1]) = S([[??].sub.2]) and H[[??].sub.2], is greater than [[??].sub.1] is greater than [[??].sub.2] that is, [[??].sub.1] [??][[??].sub.2]:

3. If S([[??].sub.1]) = S([[??].sub.2]) and H[[??].sub.1] = H[[??].sub.2] then [[??].sub.1] is indifferent to [[??].sub.2], i e. [[??].sub.1] [approximately equal to] [[??].sub.2].

Example 1. Consider two TFNNVs in the set of real numbers:

[[??].sub.1] = <(0.70,0.75,0.80), (0.15,0.20,0.25), (0.10,0.15,0.20)>,

[[??].sub.2] = <(0.40,0.45,0.50), (0.40,0.45,0.50), (0.35,0.40,0.45)>.

Then from Eqs. (21) and (22), we obtain the following results:

1. Score value of S([[??].sub.1]) = (8 + 3-0.8-0.6) /12 = 0.80, and S([[??].sub.2]) = (8+ 1.8-1.8-1.6)/12 [approximately equal to] 0.53;

2. Accuracy value of H([[??].sub.1]) = (3 - 0.6) / 4 = 0.60, and H([[??].sub.2]) = (1.8 - 1.6)/4 = 0.05.

Therefore from Definition 13, we obtain [A.sub.1] [??] [A.sub.2].

Example 2. Consider two TFNNVs in the set of real numbers:

[[??].sub.1] = <(0.50,0.55,0.60), (0.25,0.30,0.35), (0.20,0.25,0.30)>

[[??].sub.2] = <(0.40,0.45,0.50), (0.40,0.45,0.50), (0.35,0.40,0.45)>.

Using Eqs. (21) and (22), we obtain the following results:

1. Score value of S([[??].sub.1]) = (8 + 2.2-1.2-1.0) /12 [approximately equal to] 0.67, and S([[??].sub.2]) = (8 + 1.8-1.8-1.6)/12 [approximately equal to] 0.53;

2. Accuracy value of H([[??].sub.1]) = (2.2 - 1.2) / 4 = 0.25, and H([[??].sub.2]) = (1.8 - 1.6)/4 = 0.05.

Therefore from Definition 13, we have [[??].sub.1] > [A.sub.2].

4 Aggregation of triangular fuzzy number neutrosophic sets

In this section, we first recall some basic definitions of aggregation operators for real numbers.

Definition 14. [72] Assume that W: [(Re).sup.n] [left arrow] Re, and [a.sub.j](j = 1, 2, ..., n) be a collection of real numbers. The weighted averaging operator [WA.sub.w] is defined as

[WA.sub.w] ([a.sub.1], [a.sub.2], ..., [a.sub.n]) = [n.summation over (j=1)][w.sub.j][a.sub.j] (23)

where Re is the set of real numbers, w = [([w.sub.1],[w.sub.2], ..., [w.sub.n]).sup.T] is the weight vector of [a.sub.j] (j = 1,2, ..., n) such that [w.sub.j] [member of] [0,1] (j = 1,2, ..., n) and [[summation].sup.n.sub.j=1] = 1.

Definition 15. [73] Assume that W: [(Re).sup.n] [left arrow] Re, and [a.sub.j](j = 1,2, ..., n) be a collection of real numbers. The weighted geometric operator WGw is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

where Re is the set of real numbers, w = [([w.sub.1],[w.sub.2], ..., [w.sub.n]).sup.T] is the weight vector of [a.sub.j] (j = 1,2, ..., n) with [w.sup.j] [member of][0,1] (j = 1,2, ..., n) and [[summation].sup.n.sub.j=1] [w.sub.j] = 1

Based on Definition 14 and Definition 15, we propose the following two aggregation operators of TFNNSs to be used in decision making.

4.1 Triangular fuzzy number neutrosophic arithmetic averaging operator

Definition 16. Assume

that [[??].sub.1] = <([a.sub.1],[b.sub.1],[c.sub.1]),([e.sub.1],[f.sub.1],[g.sub.1]), ([r.sub.1],[s.sub.1],[t.sub.1])> (j = 1,2, ..., n)

be a collection TFNNVs in the set of real numbers and let TFNNWA: [[THETA].sup.n] [left arrow] [THETA]. The triangular fuzzy number neutrosophic weighted averaging (TFNNWA) operator denoted by TFNNWA([[??].sub.1], [[??].sub.2], ..., [[??].sub.n]) is defined as TFNNWA ([[??].sub.1], [[??].sub.2], ..., [[??].sub.n])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

where [w.sub.j] [member of] [0,1] is the weight vector of [A.sub.j] (j = 1,2, ..., n) such that [n.summation over (j=1)][w.sub.j]= 1

In particular, if w = [(1/n, 1/n, ..., 1/n).sup.T] then the TFNNWA([[??].sub.1], [[??].sub.2], ..., [[??].sub.n]) operator reduces to triangular fuzzy number neutrosophic averaging (TFNNA) operator: TFFW4([[??].sub.1], [[??].sub.2], ..., [[??].sub.n]) = 1/n ([[??].sub.1] [direct sum] [[??].sub.2] [direct sum] ... [direct sum] [[??].sub.n]) (26)

We can now establish the following theorem by using the basic operations of TFNNVs defined in Definition 10.

Theorem 1.

Let [A.sub.j] = <([a.sub.1],[b.sub.1],[c.sub.1]),([e.sub.1],[f.sub.1],[g.sub.1]), ([r.sub.1],[s.sub.1],[t.sub.1])> (j = 1,2, ..., n) be a collection TFNNVs in the set of real numbers. Then the aggregated value obtained by TFNNWA, is also a TFNNV, and

[TFNNWA.sub.w]([[??].sub.1], [[??].sub.2], ..., [[??].sub.n])

where [w.sub.j] [member of] [0,1] is the weight vector of TFNNV [A.sub.j] (j = 1,2, ...,) such that [n.summation over (j=1)][w.sub.j] = 1

Proof: We prove the theorem by mathematical induction.

1. When n = 1, it is a trivial case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

When n = 2, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

Then the theorem is true for n = 2

3. When n = k, we assume that Eq. (27) is also true.

Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

4. When n = k + 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. 30

We observe that the theorem is true for n = k + 1. 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 [[??].sub.j] belong to [0, 1], the following relations are valid

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

It follows that the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also valid.

This completes the proof of the Theorem 1.

Now, we highlight some necessary properties of TFNNWA operator.

Property l.(Idempotency): If all [[??].sub.j] (j = 1, 2.....n) are equal i.e. [[??].sub.j] = [??] = <(a,b,c),(e,f,g),(r,s,t)>, for all j, then TFNNWA ([[??].sub.j],[[??].sub.2], ..., [[??].sub.k]) = [??].

Proof: From Eq.(27), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

= <(a,b,c), (e,f,g), (r,s,t)> = [??].

This completes the proof the Property 1.

Property 2. (Boundedness)

Let [[??].sub.j] = <([a.sub.j],[b.sub.j],[c.sub.j]),([e.sub.j],[f.sub.j],[g.sub.j]), ([r.sub.j],[s.sub.j],[t.sub.j])> (j = 1,2, ..., n) be a collection TFNNVs in the set of real numbers.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all

j = 1, 2, ... n.

Then [[??].sup.-] [less than or equal to] TFNNWA ([[??].sub.1],[[??].sub.2],... [[??].sub.n]) [less than or equal to] ([[??].sup.+]. (32)

Proof: We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Similarly, the score function of [??]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, we consider the following cases:

1. If S([??]) < S([([??].sup.+]) and S(([??]) > S([([??].sup.-]) then we have [([??].sup.-] <TFNNWA([[??].sub.1],[[??].sub.2], ..., [[??].sub.n]) < [k.sup.+]. (35)

2. If S([??]) = S([[??].sup.+]), then we can take

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore the accuracy function of [??]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], 36

From (36), we have TFNNWA ([[??].sub.1],[[??].sub.2],... [[??].sub.n]) = ([[??].sup.+] (*37

Similarly, for s([??]) = S([??]), the accuracy function of [??]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

From (38), we have TNFNWA ([[??].sub.1],[[??].sub.2],... [[??].sub.n]) = ([[??].sup.-]

Combining Eqs. (35), (37) and (39), we obtain the following result

[[??].sup.-] [less than or equal to] TFNNWA([[??].sub.1],[[??].sub.2],... [[??].sub.n]) [less than or equal to] ([[??].sup.+]

This proves the Property 2.

Property 3. (Monotonicity) Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a collection of two TFNNVs in the set of real numbers.

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ..., n then

TFNNWA [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (41)

Proof: We first considered, [c.sup.1.sub.j], [g.sup.1.sub.j], [t.sup.1.sub.j] of [[??].sup.1.sub.j] and [c.sup.2.sub.j], [g.sup.2.sub.j], [t.sup.2.sub.j] of [[??].sup.2.sub.j] to prove the property 3.

We can consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (42

Now we consider the score function of [[??].sub.1]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we consider the following two cases:

Case 1. If S([[??].sup.1]) < S([[??].sup.2]), from Definition-13, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (43)

Case 2. If S([[??].sup.1]) = S([[??].sup.2]), then by Eq.(21) we can consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, the accuracy function of [[??].sup.1] yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (44)

Thus from Definition-13, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (45)

Finally, from Eqs. (43) and (45), we have the following result

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This completes the proof of Property 3.

Example3.We consider the following four TFNNVs:

[[??].sub.1] = ((0.80,0.85,0.90), (0.10,0.15,0.20), (0.05,0.10,0.15)); [[??].sub.2] = ((0.70,0.75,0.80), (0.15,0.20,0.25), (0.10,0.15,0.20)); [[??].sub.3] = ((0.40,0.45,0.50), (0.40,0.45,0.50), (0.35,0.40,0.45)) and [[??].sub.4] = ((0.70,0.75,0.80), (0.15,0.20,0.25), (0.10,0.15,0.20)).

Using TFNNWA operator defined in Eq.(27), we can aggregate [[??].sub.1], [[??].sub.2], [[??].sub.3] and [[??].sub.1] with weight vector w = (0.30,0.25,0.25,0.20) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (46)

4.2 Triangular fuzzy number neutrosophic geometric averaging operator

Definition 17. Suppose that

that [[??].sub.j] = <([a.sub.j],[b.sub.j],[c.sub.j]),([e.sub.j],[f.sub.j],[g.sub.j]), ([r.sub.j],[s.sub.j],[t.sub.j])> (j = 1,2, ..., n) be a collection TFNNVs in the set of real numbers and TFNNWG: [[THETA].sup.n] [left arrow] [THETA]. The triangular fuzzy number neutrosophic weighted geometric (TFNNWG) operator denoted by [TFNNWG.sub.w] ([[??].sub.1], [[??].sub.2], ..., [[??].sub.n]) is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

where [w.sub.j] [member of] [0,1] is the exponential weight vector of [[??].sub.j] (j = 1, 2, ..., n) such that [n.summation over (j=1)] [w.sub.j] = 1. In particular, if w = [(l/n, l/n, ..., l/n).sup.T] then the

TFNNWG ([[??].sub.1], [[??].sub.2], ... [[??].sub.n]) operator reduces to triangular fuzzy neutrosophic geometric(TNFG) operator:

[TFNNMWG.sub.w] ([[??].sub.1], [[??].sub.2], ..., [[??].sub.n]) = ([[[??].sub.1] [cross product] [[??].sub.2] [cross product] ... [cross product] [[??].sub.n]).sup.1/n] . (48)

We now establish the following theorem with the basic operations of TFNNV defined in Definition 10.

Theorem 2. Assume that

[??].sub.j] = <([a.sub.j],[b.sub.j],[c.sub.j]),([e.sub.j],[f.sub.j],[g.sub.j]), ([r.sub.j],[s.sub.j],[t.sub.j])> (j = 1,2, ..., n) be a collection TFNNVs in the set of real numbers. Then the

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

aggregated value obtained from TFNNWG, is also a TFNNV, and then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 49

where [w.sub.j] [member of] [0,1] is the weight vector of TFNNV [[??].sub.j](j = 1, 2,..., n) such that [n.summation over (j=1) [w.sub.j] = 1.

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

3. When n = k. we assume that Eq.(49) is true then.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

4. When n = k+1. we can consider the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (52)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (53)

We observe that the theorem is also true for n = k+1.

Therefore, by mathematical induction, Eq. (49) holds for all values of n.

Since the components of all three membership functions of [[??].sub.j](j = 1, 2 belong to [0, 1] the following relations are valid

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (54)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This completes the proof of Theorem 2.

Now, we discuss some essential properties of TFNNWG operator for TFNNs.

Property 4.(Idempotency): If all [[??].sub.j] (j = 1, 2, ..., n) are equal that is [[??].sub.j] =((a,b,c),(e,f,g),(r,s,t)), for all j, then [TFNNWG.sub.w]([[??].sub.1],[[??].sub.2], ..., [[??].sub.k]) = [??].

Proof: From Eq.(49), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This completes the Property 4.

Property 5. (Boundedness).

Let [[??].sub.j] = <([a.sub.j],[b.sub.j],[c.sub.j]),([e.sub.j],[f.sub.j],[g.sub.j]), ([r.sub.j],[s.sub.j],[t.sub.j])> (j = 1,2, ..., n) be a collection TFNNs in the set of real numbers. Assume

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all j = 1, 2, ..., n. Then

[[??].sup.-] [less than or equal to] [TFNNWG.sub.w]([[??].sub.1],[[??].sub.2], ..., [[??].sub.n]) [less than or equal to] [[??].sup.+]. (55)

Proof: The proof of the Property 5 is similar to Property 2.

Property 6. (Monotonicity).

Let [[??].sup.1.sub.j] = <([a.sup.1.sub.j], [b.sup.1.sub.j], [c.sup.1.sub.j]), ([e.sup.1.sub.j], [f.sup.1.sub.j], [g.sup.1.sub.j]), ([r.sup.1.sub.j], [s.sup.1.sub.j], [t.sup.1.sub.j])> and [[??].sup.2.sub.j] = <([a.sup.2.sub.j], [b.sup.2.sub.j], [c.sup.2.sub.j]), ([e.sup.2.sub.j], [f.sup.2.sub.j], [g.sup.2.sub.j]), ([r.sup.2.sub.j], [s.sup.2.sub.j], [t.sup.2.sub.j])> (j = 1,2, ..., n) be a collection of two TFNNVs in the set of real numbers. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (56)

Proof: Property 6 can be proved by a similar argument of Property 3. Therefore, we do not discuss again to avoid repetition.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 4. Assume that

[[??].sub.1] =((0.80,0.85,0.90), (0.10,0.15,0.20), (0.05,0.10,0.15)); [[??].sub.2]] = ((0.70,0.75,0.80), (0.15,0.20,0.25), (0.10,0.15,0.20)); [[??].sub.3] =((0.40,0.45,0.50), (0.40,0.45,0.50), (0.35,0.40,0.45)) and [[??].sub.4] = ((0.70,0.75,0.80), (0.15,0.20,0.25), (0.10,0.15,0.20)) are four TFNNVs.

Then using TFNNWG operator defined in Eq.(49), we can aggregate [[??].sub.1], [[??].sub.2], [[??].sub.3] and [[??].sub.4] with the considered weight vector w = (0.30,0.25,0.25,0.20) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5 Application of TFNNWA and TFNNWG operators to multi attribute decision making

Consider a multi attribute decision making problem in which Y = {[Y.sub.1], [Y.sub.2], ..., [Y.sub.m]} be the set of n feasible alternatives and C = {[C.sub.1], [C.sub.2], ..., [C.sub.n]} be the set of attributes. Assume that w = [([w.sub.1], [w.sub.2], ..., [w.sub.n]).sup.T] be the weight vector of the attributes, where [w.sub.j] denotes the importance degree of the attribute [C.sub.j] such that [w.sub.j] [greater than or equal to] 0 and [[summation].sup.n.sub.j=1] = 1 for j = 1, 2, ..., n.

The ratings of all alternatives [Y.sub.i] (i = 1, 2, ..., m) with respect to the attributes [C.sub.j] (j = 1, 2,..., n) have been presented in a TFNNV based decision matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (see the Table 1). Furthermore, in the decision matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the rating [u.sub.ij] = <([a.sub.ij], [b.sub.ij], [c.sub.ij]), ([e.sub.ij], [f.sub.ij], [g.sub.ij]), ([r.sub.ij], [s.sub.ij], [t.sub.ij])> represents a TFNNV, where the fuzzy number ([a.sub.ij], [b.sub.ij], [c.sub.ij]) represents the degree that the alternative [Y.sub.t](i = 1, 2, ..., m) satisfies the attribute [C.sub.j] (j = 1, 2, ..., n), the fuzzy number ([e.sub.ij], [f.sub.ij], [g.sub.ij]) represents the degree that the alternative [Y.sub.1] is uncertain about the attribute [C.sub.j]. and fuzzy number ([r.sub.ij], [s.sub.ij], [t.sub.ij]) indicates the degree that the alternative [Y.sub.1] does not satisfy the attribute C such that

0 [less than or equal to] [c.sub.ij] + [g.sub.ij] + [t.sub.ij] [less than or equal to] 3, for i = 1, 2,..., m and j = 1, 2, ..., n.

Based on the TFNNWA and TFNNWG operators, we develop a practical approach for solving MADM problems, in which the ratings of the alternatives over the attributes are expressed with TFNNVs. The schematic diagram of the proposed approach for MADM is depicted in the Figure-1.

Therefore, we design the proposed approach in the following steps:

Step 1: First aggregate all rating values [p.sub.ij] (j = 1, 2,...,) of the i-th row of the decision matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined in Table 1.

Step 2: Determine the aggregation value ui corresponding to the alternative [Y.sub.1] obtained from TFNNWA operator:

[u.sub.i] = <([a.sub.i],[b.sub.i],[c.sub.i]),([e.sub.i],[f.sub.i],[g.sub.i]), ([r.sub.i],[s.sub.i],[t.sub.i])> = [TFNNWA.sub.w] ([p.sup.i1], [p.sup.i2], ..., [p.sup.in]) (57)

or by the TFNNWG operator as

[u.sub.i] = <([a.sub.i],[b.sub.i],[c.sub.i]),([e.sub.i],[f.sub.i],[g.sub.i]), ([r.sub.i],[s.sub.i],[t.sub.i])> = [TFNNWG.sub.w] ([p.sup.i1], [p.sup.i2], ..., [p.sup.in]) (58)

Step 3: For each alternative [A.sub.i] (i = 1,2,..., m), calculate the score values S([u.sub.i]) and accuracy values A([u.sub.i]) of the aggregated rating values obtained by TFNNWA or TFNNWG operators that are in Eqs. (21) and (22).

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.

6 An illustrative example of multi attribute decision making

In this section, we consider an illustrative example of medical representative selection problem to demonstrate and applicability of the proposed approach to multi attribute decision making problem.

Assume that a pharmacy company wants to recruit a medical representative. After initial scrutiny four candidates [Y.sub.i] (i = 1,2,3,4) have been considered for further evaluation with respect to the five attributes [C.sub.j] (j = 1,2,3,4,5) namely,

1. oral communication skill ([c.sub.1]);

2. past experience ([c.sub.2]);

3. general aptitude ([c.sub.3]);

4. willingness ([c.sub.4]) and

5. self confidence ([c.sub.5]).

The ratings of the alternatives [Y.sup.i] (i = 1,2,3,4) with respect to the attributes [c.sub.j] (j = 1,2,3,4,5) are expressed with TFNNVs shown in the decision matrix P = [([p.sub.ij]).sub.4x5] (see Table 2.). Assume w = [(0.10, 0.25, 0.25, 0.15, 0.25).sup.T] be the relative weight vector of all attributes [C.sub.j] (j = 1,2,3,4,5).

Here, we apply two proposed aggregation operators TFNNWA and TFNNWG to solve the medical representative selection problem by using the following steps.

6.1 Utilization of TFNNWA operator:

Step 1: Aggregate the rating values of the alternative [Y.sub.i] (I = 1, 2, 3, 4) defined in the i-th row of decision matrix P = ([p.sub.ij])4x5 (see Table 2.) with TFNNWA operator.

Step 2: The aggregated rating values [u.sub.i] corresponding to the alternative [Y.sub.1] are determined by Eq.(27) and the values are shown in Table 3.

Step 3: The score and accuracy values of alternatives [Y.sub.1] (i= 1, 2, 3, 4) are determined by Eq.(21) and Eq.(22) in Table 4.

Step 4: The order of the alternatives [Y.sub.1] (i= 1, 2, 3, 4) is determined according to the descending order of the score and accuracy values shown in Table 4. Thus the ranking order of the alternatives is presented as follows:

[Y.sub.2] [??] [Y.sub.1] [??] [Y.sub.4] [??] [Y.sub.3].

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

6.2 Utilization of TFNNWG operator:

Step 1: Using Eq.(49), we aggregate all the rating values of the alternative [Y.sub.1] (i= 1, 2, 3, 4) for the i throw of the decision matrix P = [([p.sub.ij]).sub.4x5] (see Table 2.).

Step 2: The aggregated rating values [u.sub.i] corresponding to the alternative [Y.sub.1] are shown in the Table 5.

Step 3: The score and accuracy values of alternatives [Y.sub.i] (I = 1, 2, 3, 4) are determined by Eqs.(21) and (22) and the results are shown in the Table 6.

Step 4: The order of alternatives [Y.sub.i] (i = 1, 2, 3, 4) has been determined according to the descending order of score and accuracy values shown in Table 4.

Thus the ranking order of the alternative is presented as follows:

[Y.sub.2] [??] [Y.sub.1] [??] [Y.sub.4] [??] [Y.sub.3].

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

7 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.

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Received: March 15, 2016. Accepted: June 12, 2016.

Pranab Biswas (1), Surapati Pramanik (2) *, and Bibhas C. Giri (3)

(1) Department of Mathematics, Jadavpur University, Kolkata, 700032, India. E-mail: paldam2010@gmail.com

(2) Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District-North 24 Parganas, West Bengal, PIN-743126, India. Email: sura_pati@yahoo.co.in

(3) Department of Mathematics, Jadavpur University, Kolkata, 700032, India. Email:bcgiri.jumath@gmail.com

* Corresponding author's email: sura_pati@yahoo.co.in

Caption: Figure-1. Framework for the proposed MADM method
```Table 1. Triangular fuzzy number neutrosophic value based decision
matrix

[C.sub.1]

[Y.sub.1]   <([a.sub.11], [b.sub.11] [c.sub.11]),
([e.sub.11], [f.sub.11] [g.sub.11]),
([r.sub.11], [s.sub.11] [t.sub.11])>

[Y.sub.2]   <([a.sub.21], [b.sub.21] [c.sub.21]),
([e.sub.21], [f.sub.21] [g.sub.21]),
([r.sub.21], [s.sub.21] [t.sub.21])>

...                          ...

[Y.sub.m]   <([a.sub.m1], [b.sub.m1] [c.sub.m1]),
([e.sub.m1], [f.sub.m1] [g.sub.m1]),
([r.sub.m1], [s.sub.m1] [t.sub.m1])>

[C.sub.2]                  ...

[Y.sub.1]   <([a.sub.12], [b.sub.12] [c.sub.12]),   ...
([e.sub.12], [f.sub.12] [g.sub.12]),
([r.sub.12], [s.sub.12] [t.sub.12])>

[Y.sub.2]   <([a.sub.22], [b.sub.22] [c.sub.22]),   ...
([e.sub.22], [f.sub.22] [g.sub.22]),
([r.sub.22], [s.sub.22] [t.sub.22])>

...                          ...

[Y.sub.m]   <([a.sub.m2], [b.sub.m2] [c.sub.m2]),
([e.sub.m2], [f.sub.m2] [g.sub.m2]),
([r.sub.m2], [s.sub.m2] [t.sub.m2])>

...                [C.sub.n]

[Y.sub.1]            <([a.sub.1n], [b.sub.1n] [c.sub.1n]),
...     ([e.sub.1n], [f.sub.1n] [g.sub.1n]),
([r.sub.1n], [s.sub.1n] [t.sub.1n])>

[Y.sub.2]            <([a.sub.2n], [b.sub.2n] [c.sub.2n]),
...     ([e.sub.2n], [f.sub.2n] [g.sub.2n]),
([r.sub.2n], [s.sub.2n] [t.sub.2n])>

...                          ...

[Y.sub.m]            <([a.sub.mn], [b.sub.mn] [c.sub.mn]),
...     ([e.sub.mn], [f.sub.mn] [g.sub.mn]),
([r.sub.mn], [s.sub.mn] [t.sub.mn])>

Table 2. Triangular fuzzy number neutrosophic value based rating values

[C.sub.1]              [C.sub.2]

[Y.sub.1]     <(0.80,0.85,0.90)      <(0.50,0.55,0.60)
(0.10,0.15,0.20)       (0.25,0.30,0.35)
(0.05,0.10,0.15)>      (0.20,0.25,0.30)>

[Y.sub.2]     <(0.50,0.55,0.60)      <(0.70,0.75,0.80)
(0.25,0.30,0.35)       (0.15,0.20,0.25)
(0.20,0.25,0.30)>      (0.10,0.15,0.20)>

[Y.sub.3]     <(0.40,0.45,0.50)      <(0.50,0.55,0.60)
(0.40,0.45,0.50)       (0.25,0.30,0.35)
(0.35,0.40,0.45)>      (0.20,0.25,0.30)>

[Y.sub.4]     <(0.40,0.45,0.50)      <(0.50,0.55,0.60)
(0.40,0.45,0.50)       (0.25,0.30,0.35)
(0.35,0.40,0.45)>      (0.20,0.25,0.30)>

[C.sub.3]              [C.sub.4]

[Y.sub.1]     <(0.70,0.75,0.80)      <(0.80,0.85,0.90)
(0.15,0.20,0.25)       (0.10,0.15,0.20)
(0.10,0.15,0.20)>      (0.05,0.10,0.15)>

[Y.sub.2]     <(0.80,0.85,0.90)      <(0.70,0.75,0.80)
(0.10,0.15,0.20)       (0.15,0.20,0.25)
(0.05,0.10,0.15)>      (0.10,0.15,0.20)>

[Y.sub.3]     <(0.40,0.45,0.50)      <(0.40,0.45,0.50)
(0.40,0.45,0.50)       (0.40,0.45,0.50)
(0.35,0.40,0.45)>      (0.35,0.40,0.45)>

[Y.sub.4]     <(0.40,0.45,0.50)      <(0.70,0.75,0.80)
(0.40,0.45,0.50)       (0.15,0.20,0.25)
(0.35,0.40,0.45)>      (0.10,0.15,0.20)>

[C.sub.5]

[Y.sub.1]     <(0.70,0.75,0.80)
(0.15,0.20,0.25)
(0.10,0.15,0.20)>

[Y.sub.2]     <(0.70,0.75,0.80)
(0.15,0.20,0.25)
(0.10,0.15,0.20)>

[Y.sub.3]     <(0.50,0.55,0.60)
(0.25,0.30,0.35)
(0.20,0.25,0.30)>

[Y.sub.4]     <(0.70,0.75,0.80)
(0.15,0.20,0.25)
(0.10,0.15,0.20)>

Table 3. Aggregated TFNNV based ratings
Aggregated ratings

[u.sub.1] <(0.6920, 0.7451, 0.8000), (0.1540, 0.2026, 0.2572),
(0.1000, 0.1540, 0.2060)>

[u.sub.2] <(0.7147, 0.7667, 0.8197), (0.1426, 0.1938, 0.2445),
(0.0901, 0.1426, 0.1938)>

[u.sub.3] <(0.4523, 0.5025, 0.5528), (0.3162, 0.3674, 0.4183),
(0.2646, 0.3162, 0.3674)>

[u.sub.4] <(0.5655, 0.6184, 0.6722), (0.2402, 0.2940, 0.3466),
(0.1844, 0.2402, 0.2940)>

Table 4. Score and accuracy values of aggregated rating values

Alternative            Score                 Accuracy

values S ([u.sub.i])     values A([u.sub.i])

[Y.sub.1]             0.7960                  0.5921
[Y.sub.2]             0.8103                  0.6247
[Y.sub.3]             0.6464                  0.1864
[Y.sub.4]             0.6951                  0.3789

Table 5. Aggregated TFNN based rating values

Aggregated rating values

[u.sub.1] <(0.6654, 0.7161, 0.7667), (0.1643, 0.2144, 0.2646),
(0.1142, 0.1643, 0.2144)>

[u.sub.2] <(0.6998, 0.7502, 0.8002), (0.1485, 0.1986, 0.2486),
(0.0984, 0.1485, 0.1986)>

[u.sub.3] <(0.4472, 0.4975, 0.5477), (0.3292, 0.3795, 0.4299),
(0.2789, 0.3292, 0.3795)>

[u.sub.4] <(0.5291, 0.5804, 0.6316), (0.2707, 0.3214, 0.3721),
(0.2202, 0.2707, 0.3214)>

Table 6. Score and accuracy values of rating values

Alternative            Score                 Accuracy

values S ([u.sub.i])     values A([u.sub.i])

[Y.sub.1]             0.7791                  0.5518
[Y.sub.2]             0.8010                  0.6016
[Y.sub.3]             0.5962                  0.1683
[Y.sub.4]             0.6627                  0.3096
```
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