Application of Neutrosophic Rough Set in Multi Criterion Decision Making

International Research Journal of Engineering and Technology (IRJET) Volume: 03 Issue: 10 | Oct -2016

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e-ISSN: 2395 -0056 p-ISSN: 2395-0072

Application of Neutrosophic Rough Set in Multi Criterion Decision Making on two universal sets C. Antony Crispin Sweety1 & I. Arockiarani2 Nirmala College for Women, Coimbatore- 641018 Tamilnadu, India.

----------------------------------------------------------------------------------------------------------------------------- -----------------------------1,2

Abstract -. The main objective of this study is to introduce a new hybrid intelligent structure called rough neutrosophic sets on the Cartesian product of two universe sets. Further as an application a multi criteria decision making problem is solved.

Key words: - Rough Set, Solitary Set, Relative Set, Accuracy, Neutrosophic Rough Set, Multi Criteria Decision Making.

I. INTRODUCTION The rough sets theory introduced by Pawlak [10] is an excellent mathematical tool for the analysis of uncertain, inconsistency and vague description of objects. Neutrosophic sets and rough sets are two different topics, none conflicts the other. While the neutrosophic set is a powerful tool to deal with indeterminate and inconsistent data, the theory of rough sets is a powerful mathematical tool to deal with incompleteness. By combining the Neutrosophic sets and rough sets the rough sets in neutrosophic approximation space [2] and Neutrosophic neutrosophic rough sets [4] were introduced Multi criterion decision making (MCDM) is a process in which decision makers evaluate each alternative according to multiple criteria. Many representative methods are introduced to solve MCDM problem in business and industry areas. However, a drawback of these approaches is that they mostly consider the decision making with certain information of the weights and decision values. This makes them much less useful when managing uncertain information. To this end, multi criteria fuzzy decision making has been studied in [4, 6, 7]. Several attempts have already been made to use the rough set theory to decision support . But, in many real life problems, an information system establishes relation between two universal sets. Multi criterion decision making on such information system is very challenging. This paper discusses how neutrosophic rough set on two universal sets can be employed on MCDM problems for taking decisions.

2. PRELIMINARIES Definition 2.1[13] A Neutrosophic set A on the universe of discourse X is defined as A = đ?‘Ľ, đ?‘‡đ??´ đ?‘Ľ , đ??źđ??´ đ?‘Ľ , đ??šđ??´ đ?‘Ľ , đ?‘Ľ ∈ đ?‘‹ , Where đ?‘‡, đ??ź, đ??š: đ?‘‹ →]-0,1+[ and



0  TA ( x)  I A ( x)  FA ( x)  3 .

Definition 2.2:[12] Let U be any non empty set. Suppose R is an equivalence relation over U. For any non null subset X of U, the sets A1(X) = {x: [x]R ďƒ? X}, A2(X) = {x: [x]R ďƒ‡ X≠∅} are called lower approximation and upper approximation respectively of X and the pair S= (U, R) is called approximation space. The equivalence relation R is called indiscernibility relation. The pair A(X) = (A1(X), A2(X)) is called the rough set of X in S. Here [x]R denotes the equivalence class of R containing x.

Definition 2.3[4]: Let U be a non empty universe of discourse. For an arbitrary fuzzy neutrosophic relation R over đ?&#x2018;&#x2C6; Ă&#x2014; đ?&#x2018;&#x2C6; the pair (U, R) is called fuzzy neutrosophic approximation space. For any đ??´ â&#x2C6;&#x2C6; đ??šđ?&#x2018; (đ?&#x2018;&#x2C6;), we define the upper and lower approximation with respect to (đ?&#x2018;&#x2C6;, đ?&#x2018;&#x2026;), denoted by đ?&#x2018;&#x2026; and đ?&#x2018;&#x2026; respectively. đ?&#x2018;&#x2026; đ??´ = {< đ?&#x2018;Ľ, đ?&#x2018;&#x2021;đ?&#x2018;&#x2026; đ??´ đ?&#x2018;Ľ , đ??źđ?&#x2018;&#x2026; đ??´ đ?&#x2018;Ľ , đ??šđ?&#x2018;&#x2026; đ??´ (đ?&#x2018;Ľ) >/đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2C6;} đ?&#x2018;&#x2026; đ??´ = {< đ?&#x2018;Ľ, đ?&#x2018;&#x2021;đ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ , đ??źđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ , đ??šđ?&#x2018;&#x2026;(đ??´) (đ?&#x2018;Ľ) > đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2C6;} đ?&#x2018;&#x2021;đ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

ď&#x192;&#x161; [đ?&#x2018;&#x2021;

đ?&#x2018;&#x2026;

ď&#x192;&#x2122; [đ??š

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ?&#x2018;&#x2021;đ??´ đ?&#x2018;Ś ], đ??źđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

yď&#x192;&#x17D;U

đ?&#x2018;&#x2021;đ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

đ?&#x2018;&#x2026;

đ?&#x2018;&#x2026;

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ??źđ??´ đ?&#x2018;Ś ], đ??šđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

ď&#x192;&#x2122; [1 â&#x2C6;&#x2019; đ??ź yď&#x192;&#x17D;U

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ?&#x2018;&#x2021;đ??´ đ?&#x2018;Ś ], đ??źđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

yď&#x192;&#x17D;U

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ď&#x192;&#x161; [đ??ź

đ?&#x2018;&#x2026;

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đ?&#x2018;&#x2026;

yď&#x192;&#x17D;U

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ??źđ??´ đ?&#x2018;Ś ], đ??šđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

yď&#x192;&#x17D;U

|

ď&#x192;&#x2122; [đ??š

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ?&#x2018;&#x2021;đ??´ đ?&#x2018;Ś ]

ď&#x192;&#x161; [đ?&#x2018;&#x2021;

đ?&#x2018;&#x2026;

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ??šđ??´ đ?&#x2018;Ś ]

yď&#x192;&#x17D;U

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The pair (đ?&#x2018;&#x2026; , đ?&#x2018;&#x2026;) is fuzzy neutrosophic rough set of A with respect to (U,R) and đ?&#x2018;&#x2026;, đ?&#x2018;&#x2026;:FN(U)â&#x2020;&#x2019;FN(U) are refered to as upper and lower Fuzzy neutrosophic rough approximation operators respectively.

3. NEUTROSOPHIC ROUGH SET ON TWO UNIVERSAL SETS

Now, we present the definitions, notations and results of neutrosophic rough set on two universal We define the basic concepts leading to neutrosophic rough set on two universal sets in which we denote for truth function TRN , indeterminacy

I RN and falsity function FRN for non membership functions that are associated with an neutrosophic rough set on two universal sets.

RN from U ď&#x201A;Ž V is an neutrosophic set of (U x V) characterized by the truth value function TRN , indeterminacy function and falsity function FRN where Definition 3.1:[11] Let U and V be two non empty universal sets. An neutrosophic relation

RN ď&#x20AC;˝ { ( x, y), TRN ( x, y ), I RN ( x, y ), FRN ( x, y), | x ď&#x192;&#x17D; U, y ď&#x192;&#x17D; V } with 0 ď&#x201A;Ł TRN ( x, y) ď&#x20AC;Ť I RN ( x, y) ď&#x20AC;Ť FRN ( x, y) ď&#x201A;Ł 3 for every (x,y) ď&#x192;&#x17D; U x V. Definition 3.2

[13] Let U and V be two non empty universal sets and

R N is a neutrosophic relation from U to V. If for xď&#x192;&#x17D;U, TRN ( x, y) ď&#x20AC;˝ 0 , I RN ( x, y) ď&#x20AC;˝ 0 and FRN ( x, y) ď&#x20AC;˝ 1 for all y ď&#x192;&#x17D; V , then x is said to be a solitary element with respect to R N .

The set of all solitary elements with respect to the relation

R N is called the solitary set S. That is, S ď&#x20AC;˝ {x | x ď&#x192;&#x17D; U , TRN ( x, y) ď&#x20AC;˝ 0, I RN ( x, y) ď&#x20AC;˝ 0, FRN ( x, y) ď&#x20AC;˝ 1, ď&#x20AC;˘ y ď&#x192;&#x17D; V

Definition 3.3:[13] Let U and V be two non empty universal sets and

R N is a neutrosophic relation from U to V. Therefore,

(U ,V , RN ) is called a neutrosophic approximation space. For Y ď&#x192;&#x17D; N (V ) an neutrosophic rough set is a pair ( R N Y , R N Y ) of neutrosophic set on U such that for every x ď&#x192;&#x17D; U.

R N (Y ) = { x, TRN

(Y )

( x), I RN

(Y )

( x), FRN

(Y )

R N (Y ) = { x, TR

(Y )

( x), I R

(Y )

( x), FR

(Y )

Where đ?&#x2018;&#x2021;đ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

ď&#x192;&#x161; [đ?&#x2018;&#x2021;

ď&#x192;&#x2122;

đ?&#x2018;&#x2026;

N

N

N

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ?&#x2018;&#x2021;đ??´ đ?&#x2018;Ś ] đ??źđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

yď&#x192;&#x17D;V

đ?&#x2018;&#x2021;đ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

(11)

( x) | x ď&#x192;&#x17D; U }

(12)

ď&#x192;&#x161; [đ??ź

ď&#x192;&#x2122;

đ?&#x2018;&#x2026;

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ??źđ??´ đ?&#x2018;Ś ] đ??šđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

yď&#x192;&#x17D;V

[đ??šđ?&#x2018;&#x2026; đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ?&#x2018;&#x2021;đ??´ đ?&#x2018;Ś ] đ??źđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

yď&#x192;&#x17D;V

The pair ( R N (Y ) ,

( x), | x ď&#x192;&#x17D; U } }

ď&#x192;&#x2122; [đ??š

đ?&#x2018;&#x2026;

yď&#x192;&#x17D;V

[1 â&#x2C6;&#x2019; đ??źđ?&#x2018;&#x2026; đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ??źđ??´ đ?&#x2018;Ś ] đ??šđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

yď&#x192;&#x17D;V

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ?&#x2018;&#x2021;đ??´ đ?&#x2018;Ś ]

ď&#x192;&#x161; [đ?&#x2018;&#x2021;

đ?&#x2018;&#x2026;

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ??šđ??´ đ?&#x2018;Ś ]

yď&#x192;&#x17D;V

R N (Y ) ) is called the neutrosophic rough set of Y with respect to (U ,V , RN ) where R N (Y ) , R N (Y ) :

N(U) ď&#x201A;Ž N(V) are referred as lower and upper neutrosophic rough approximation operators on two universal sets.

4.ALGEBRAIC PROPERTIES:

In this section, we discuss the algebraic properties of neutrosophic rough set on two universal sets through solitary set .

Proposition 4.1: Let U and V be two universal sets. Let solitary set with respect to (a)

R N be an neutrosophic relation from U to V and further let S be the

R N . Then for X, Y ď&#x192;&#x17D; N(V), the following properties holds:

R N (V) = U and R N ( ď Ś ) = ď Ś

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(b)

If X ď&#x192;? Y, then R N (X) ď&#x192;? R N (Y) and

(c)

R N (X) = R N (Xâ&#x20AC;˛))â&#x20AC;˛ and R N (X)= R N (Xâ&#x20AC;˛))â&#x20AC;˛

(d)

RN ď Ś ď&#x192;&#x160; S and RNV ď&#x192;? Sâ&#x20AC;˛ , where Sâ&#x20AC;˛ denotes the complement of S in U.

(e)

For any given index set J, Xi ď&#x192;&#x17D; N(V),

R N ( ď&#x192;&#x2C6; X i ) ď&#x192;&#x160; ď&#x192;&#x2C6; RN X i and iď&#x192;&#x17D;J

(f)

R N (X) ď&#x192;? R N (Y)

RN ( ď&#x192;&#x2021; X i ) ď&#x192;? iď&#x192;&#x17D;J

iď&#x192;&#x17D;J

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ď&#x192;&#x2021;R

iď&#x192;&#x17D;J

N

Xi

For any given index set J, Xi ď&#x192;&#x17D; N(V), R N

( ď&#x192;&#x2021; X i ) ď&#x20AC;˝ ď&#x192;&#x2021; RN X i and R N ( ď&#x192;&#x2C6; X i ) ď&#x20AC;˝ ď&#x192;&#x2C6; R N X i . iď&#x192;&#x17D;J

iď&#x192;&#x17D;J

iď&#x192;&#x17D;J

iď&#x192;&#x17D;J

Proof: First note that V is a neutrosophic set satisfying TY ( x) ď&#x20AC;˝ 1 , I Y ( x) ď&#x20AC;˝ 0 and FY ( x) ď&#x20AC;˝ 1 for all x ď&#x192;&#x17D;V . Thus, V can be represented as

V ď&#x20AC;˝ { x,1,1,0 I x ď&#x192;&#x17D; V }

Now, by definition we have đ?&#x2018;&#x2021;đ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

ď&#x192;&#x2122; yď&#x192;&#x17D;V

FRN ( x, y) ď&#x192;&#x161; TY ( y) ď&#x20AC;˝ 1 đ??źđ?&#x2018;&#x2026;

đ??´

đ?&#x2018;Ľ =

ď&#x192;&#x2122; yď&#x192;&#x17D;V

1 ď&#x20AC;­ I RN ( x, y) ď&#x192;&#x161; IY ( y) ď&#x20AC;˝ 1 đ??šđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ

=

ď&#x192;&#x161; yď&#x192;&#x17D;V

TRN ( x, y) ď&#x192;&#x2122; FY ( y) ď&#x20AC;˝ 0

Therefore we get,

RN (V ) ď&#x20AC;˝ { x, TRN

(V )

( x), I RN

(V )

( x), FRN

(V )

( x), | x ď&#x192;&#x17D; U } } = { x,1,1,0 I x ď&#x192;&#x17D; U }

ď Ś is a neutrosophic set satisfying TY ( x) ď&#x20AC;˝ 0 , I Y ( x) ď&#x20AC;˝ 1 and FY ( x) ď&#x20AC;˝ 0 for all x ď&#x192;&#x17D;V . Thus, ď Ś can be represented as ď Ś ď&#x20AC;˝ { x,0,0,1 I x ď&#x192;&#x17D; V } Similarly,

Now, by definition we have đ?&#x2018;&#x2021;đ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

ď&#x192;&#x161; [đ?&#x2018;&#x2021;

đ?&#x2018;&#x2026;

ď&#x192;&#x161; [đ??ź

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ?&#x2018;&#x2021;đ??´ đ?&#x2018;Ś ] = 0, đ??źđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

đ?&#x2018;&#x2026;

yď&#x192;&#x17D;V

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ??źđ??´ đ?&#x2018;Ś ] = 0, đ??šđ?&#x2018;&#x2026;(đ??´) đ?&#x2018;Ľ =

yď&#x192;&#x17D;V

Therefore we get, RN (ď Ś ) ď&#x20AC;˝ { x, TR

N

(ď Ś )

,IR

N

(ď Ś )

, FR

N

(ď Ś )

| x ď&#x192;&#x17D;U } }

=

ď&#x192;&#x2122; [đ??š

đ?&#x2018;&#x2026;

đ?&#x2018;Ľ, đ?&#x2018;Ś â&#x2C6;§ đ?&#x2018;&#x2021;đ??´ đ?&#x2018;Ś ]=1

yď&#x192;&#x17D;V

{ x,0,0,1 I x ď&#x192;&#x17D; U } = ď Ś .

(ii) First note that X ď&#x192;? Y if and only TX ( x) ď&#x201A;Ł TY ( x) , I X ( x) ď&#x201A;Ł I Y ( x) and FX ( x) ď&#x201A;Ł FY ( x) for all

x ď&#x192;&#x17D;V . Therefore, we have TRN ( X ) ( x) = ď&#x192;&#x2122; [ FRN ( x, y) ď&#x192;&#x161; TX ( y)] ď&#x201A;Ł

ď&#x192;&#x2122; [ F ( x, y) ď&#x192;&#x161; T ( y)] = T ( x) = ď&#x192;&#x2122; [1 ď&#x20AC;­ I ( x, y) ď&#x192;&#x161; I ( y)] ď&#x201A;Ł ď&#x192;&#x2122; [ I ( x, y) ď&#x192;&#x161; I ( y)] = I ( x) = ď&#x192;&#x161; [T ( x, y) ď&#x192;&#x2122; F ( y)] ď&#x201A;ł ď&#x192;&#x161; [T ( x, y) ď&#x192;&#x2122; F ( y)] = F yď&#x192;&#x17D;V

I RN ( X ) FRN ( X )

RN

yď&#x192;&#x17D;V

yď&#x192;&#x17D;V

RN

X

X

RN

yď&#x192;&#x17D;V

RN

yď&#x192;&#x17D;V

yď&#x192;&#x17D;V

RN (Y )

Y

Y

RN

RN (Y )

RN ( Y )

Y

( x) ( x)

( x)

Therefore, R N (X) ď&#x192;? R N (Y). Similarly, we have

TR

N

( x)

( x) = ď&#x192;&#x161; [TRN ( x, y) ď&#x192;&#x2122; TX ( y)] ď&#x201A;Ł yď&#x192;&#x17D;V

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ď&#x192;&#x161; [I

yď&#x192;&#x17D;V

RN

( x, y) ď&#x192;&#x2122; TY ( y)] = TR

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N

(Y )

( x)

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IR

N

FR

( x)

( x) =  [ I RN ( x, y)  I X ( y)] 

( x)

N

yV

( x) =  [ FRN ( x, y)  FX yV

 [I ( x, y)  I ( y)] = I ( y)]   [ F ( x, y)  F ( y)] = F

IR

N

FR

( x ')

RN

yV

Y

RN

yV

(iii) We know that RN ( X ' )  { x, TR N

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RN ( Y )

Y

( x)

RN (Y )

 [T  [I

( x) =

( X ')

( x), I R

N

( X ')

( x), FR

N

( X ')

( x) / x U } , where

( x, y)  TX ' ( y)] =  TRN ( x, y)  FY ( y)] = FRN ( X ) ( x)

RN

yV

N

yV

( x, y)  I X ' ( y)] =  I RN ( x, y)  1  I Y ( y)] = 1  I RN ( X ) ( x) = I RN ( X ) ( x)

( x ')

( x) =

( x)

( x) = [ FRN ( x, y)  FX ' ( y)] = [ FRN ( x, y)  TX ( y)] = TRN ( X ) ( x)

N

( x)

R N (X)  R N (Y).

Therefore,

TR

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RN

yV

yV

Therefore, we have

RN ( X ' )  { x, TR

N

( X ')

( x), I R

N

( X ')

( x), FR

N

( X ')

( x) / x  U }

= { x, FRN ( X ') ( x),1  I RN ( X ') ( x), TRN ( X ') ( x) / x  U } indicates that

R

R

(X ' )  { x, TRN



(X )

( x), I RN

(X )

( x), FRN

(X )

( x), | x  U and consequently

'

(X ' )  R N ( X )

N

R N ( X ' ) = { x, TRN TR



'

N

N

(X')

( X ')

( x), I RN

( X ')

( x), FRN

( X ')

( x), | x  U

( x) =  [ FRN ( x, y)  TX ' ( y)] =  [ FRN ( x, y)  FX ( y)] = FR ( X ) ( x) N yV

yV

I RN ( X ') ( x) =  [1  I RN ( x, y)  I X ' ( y)] =  [1  I RN ( x, y)  1  I X ( y)] = [ I RN ( x, y)  I Y ( y)] yV

=

IR

N

yV

(X )

( x)

FRN ( X ) ( x) =  [TRN ( x, y)  FX ' ( y)]  yV

R



'

N

(X ' ) = { x, TRN = { x, FRN

It indicates that

R



R

( X ')

(X )

( x), I RN

( x), I RN



( X ')

(X )

( x), FRN

( x), TRN

'

N

(X ' ) = { x, TR

 [T

N

(X )

RN

yV

(X )

( X ')

( x, y)  TY ( y)] = TR

N

(X )

( x) .

( x), | x  U

( x), | x  U

( x), I R

N

(X )

( x), FR

N

(X )

( x) / x U } and consequently

'

N

(X ' ) = R N ( X ) .

 is a neutrosophic set satisfying TY ( x)  0 , I Y ( x)  0 and FY ( x)  1 for all x V . Thus,  can be represented as (iv) First note that,

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  { x,0,0,1 I x  V } Also note that, S is a solitary set. This indicates that Therefore, we have for all x  S

 [F ( x, y)  T ( y)] =  [1  0] =1 ( x, y)  I  ( y)] =  [1  0] =1  ( x )   [1  I  ( x)   [T ( x, y )  F ( y )] =  [0  1] = 0

TRN ( ) ( x) 

I RN (

)

FRN (

RN

yV

yV

RN

yV

)

yV

RN

yV

yV

TRN ( ) ( x)  TRN ( x, y ) , I RN ( ) ( x)  I RN ( x, y ) and FRN ( ) ( x)  FRN ( x, y ) for x  S .

Hence, it is clear that

Therefore, by proposition (ii) we have

RN ( )  S .

Similarly, by proposition (iii) we have

R N ( X ) = RN (X ' ) .









'

' x V we get R N (V ) = RN (V ' ) .

On taking

But V '   .Again by proposition (iv), we have Therefore, we get

RN ( )  S . It implies that ( RN ( )) '  S ' .

R N (V )  S ' .

(v) From the properties of union, for any index set J={1,2,3….,n} X1  Xi , X2  Xi , X3  X i ,………… X n  Xi .





iJ



iJ



iJ

iJ

Therefore, by proposition (ii) we have

RN ( X 1 )  RN (  X i ) , RN ( X 2 )  RN (  X i ) ,……… RN ( X n )  RN (  X i ) iJ

iJ

iJ

It indicates that,

R

iJ

N

( X i )  RN (  X i ) , ie RN (  X i )   RN ( X i ) . iJ

iJ

iJ

Similarly, for any index set J={1,2,3….,n}  X i  X 1 ,  X i  X 2 ,  X i  X 3 ,…………  X i iJ

iJ

iJ

iJ

 Xn.

Therefore, by proposition (ii) we have

R N (  X i )  R N ( X 1 ) , R N (  X i )  R N ( X 2 ) ,……… R N (  X i )  R N ( X n ) iJ

iJ

iJ

It indicates that, R N (  X i )   R N ( X i ) . iJ

iJ

(vi) For any index set J ={1, 2, 3,...., n} ,

X i  N (V ) ,

RN (  X i ) = { x, TRN (  X i ) ( x), I RN (  X i ) ( x), FRN (  X i ) ( x) / x  U } . But, iJ

i J

i J

i J

TRN (  X i ) ( x) =  [ FRN ( x, y)  T(  X i ) ( y)] iJ

iJ

yV

 [F ( x, y)  (T ( y)  T ( y)  T ( y)  .........  T =  [( F ( x, y)  T ( y)  ( F ( x, y)  T ( y))  ......  ( F 

RN

yV

yV

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RN

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X1

X1

X2

RN

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X3

X n 1

X2

|

RN

( y)  TX n ( y))] ( x, y)  TX n ( y))]

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= TRN ( X1 ) ( x)  TRN ( X 2 ) ( x)  .....  TRN ( X n ) ( x) = Min{ TRN ( X 1 ) ( x ) } Similary, we can prove for indeterrminacy and false value functions.

Again, for any index set J ={1, 2, 3...., n}, X i  N (V ) ,

R N ( X 1 ) = { x, TRN

( X1 )

RN ( X 2 ) = { x, TRN

( x), I RN

( X1 )

( x), I RN

(X2 )

( x), FRN

( X2 )

( x), FRN

……………………………………………………. ……………. …………………..

R N ( X n ) = { x, TRN

(Xn )

( x), I RN

( X1 )

(Xn )

( x), | x  U

(X2 )

( x), | x  U

………

( x), FRN

(Xn )

( x), | x  U

Therefore, we have

 RN ( X i ) = { x, Min{TRN

iJ

Hence, it is clear that

TR

N

(  Xi ) i J

( x)}, Min{I RN

iJ

N

(  Xi ) iJ

( x ) = Max{ TR

( X1 )

RN ( X 2 ) = { x, TRN ……………. …………….

N

( X1 )

( x), I R

N

(  Xi )

( x) }, I R

iJ

N

( x), I RN

(X2 )

( X1 )

( x), I RN

(Xn )

( x), FR

(  Xi ) i J

( Xn )

( x)} | x  U

( X2 )

N

(  Xi )

( x) / x  U } .

iJ

( x ) =Max{ I R

N

( X1 )

( x) }, FR

N

(  Xi ) i J

( x) = Min{ FRN ( X1 ) ( x) }

X i  N (V ) ,

( x), FRN

( X1 )

( x), FRN

…………………..

R N ( X n ) = { x, TRN

( x)}, Max{FRN

iJ

Again, for any index set J={1, 2, 3,...., n},

R N ( X 1 ) = { x, TRN

(Xn )

RN (  X i ) =  RN ( X i ) . Similarly for any index set J ={1, 2, 3,...., n} , X i  N (V ) ,

R N (  X i ) = { x, TR iJ

( Xn )

( x), | x  U

(X2 )

( x), | x  U

………

( x), I RN

(Xn )

( x), FRN

(Xn )

( x), | x  U

Therefore, we have

 RN ( X i ) = { x, Min{TRN

iJ

Hence, it is clear that

( Xn )

( x)}, Min{I RN

(Xn )

( x)}, Max{FRN

( Xn )

( x)} | x  U

RN (  X i ) =  RN ( X i ) . iJ

iJ

4. AN APPLICATION TO MULTI CRITERION DECISION MAKING

In this section, we depict a real life application of neutrosophic rough set on two universal sets to multi criterion decision making. The model application is explained as neutrosophic rough set upper approximation. Let us consider the multi criteria decision making in the case of a online shop. However, it is observed that due to several factors such as, on time delivery, offers and discounts, quality of the product, Easy returns, Diccreet shoping. Therefore, from customer behaviour, clear review of the particular online shop can be obtained. Hence, neutrosophic relation better depicts the relation between the customers and supermarkets.

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Let us set the criteria U={U1, U2, U3, U4, U5, U6} in which U1 denotes the convenience in shopping, U2 denotes on time delivery of the products, U3 denotes offers and discounts given, U4 denote quality of the product, U5 denotes easy returns of the products, U6 denotes the discreetness in shoping. Let us consider the review rating, such as 5*,4*,3*,2*,1*,nil.

Several variety of customers and professionals are invited to the survey that only focuses on the criterion of best Easy returns in the online shop X. . DECISIONS ***** D5 **** D4 *** D3 ** D2 * D1 ----D0

YES(%) 15 28 46 40 26 34

NEUTRAL(%) 30 46 27 38 67 40

NO(%) 47 23 43 56 70 14

then the vector can be obtained as [(0.15,0.30,0.28),( 0.28,0.46,0.23),( 0.46,0.27,0.23), ( 0.40,0.38,0.34),(0.26,0.67,0.35) ,(0.34,0.40,0.14)]T where T represents the transpose.

Similarly, the decisions based on other criteria are obtained as follows: [(0.10,0.20,0.30), (0.30,0.30,0.21),(0.25,0.20,0.15),(0.10,0.40,0.8),(0.20,0.30,0.7),(0.25,0.20,0.25)]T [(0.55, 0.55, 0.41), (0.55,0.55,0.1), (0.20,0.30, 0.15), (0.50,0.40,0.6),( 0.30,0.10,0.3),(0.20,0.20, 0.4)]T [(0.1,0.10, 0.7),(0.10,0.10,0.4), (0.40,0.45,0.20) ,(0.20,0.25,0.30), (0.20,0.22,0.10),(0.35,0.10,0.10)]T [(0.24,0.10,0.20), (0.20,0.12,0.67) ,(0.15,0.13,0.59) ,(0.35,0.30,0.36),(0.5,0.40,0.36),(0.20,0.22,0.54)]T [(0.25,0.28,0.31),( 0.25,0.23,0.10),(0.25,0.23,0.20),(0.20,0.34,0.40),(0.10,0.35,0.40),(0.20,0.14,0.30)]T Based on the decision vectors, the neutrosophic relation from U to V is presented by the following matrix. We define the neutrosophic relation by the following matrix.

 (0.15,0.3,0.30)   (0.1, 0.2, 0.3)  (0.55,0.55, 0.41) RN    (0.10,0.10,0.70)  (0.24,0.10,0.20)   (0.25,0.28,0.31) 

(0.28,0.46,0.23)

(0.46,0.27,.23)

(0.4,0.38,0.34)

(0.30,0.30,0.21 (0.55,0.55, 0.1)

(0.25,0.20, 0.15) 0.2,0.3,0.15)

(0.1,0.40,0.80) (0.50,0,.40,0.60)

(0.10,0.10.40) ( 0.20,0.12,0.67)

(0.40,0.45,0.20) 0.15,0.13,0.59

(0.20,0.25,0.30) (0.35,0.30, 0.36)

(0.25,0.23,0.10)

( 0.25,0.23,0.20)

( 0.2,0.30,0.40)

(0.34,0.4,0.14)   ( 0.20,.30,0.70) (0.20,0.25,0.40)  (0.30,0.10,0.30) (0.20,0.20,0.40)   (0.20,0.22,0.10)] (0.35,0.10,0.10)  (0.50,0.4,0.36) (0.20,0.22,0.54)  (0.10,0.35,0.40) (0.20,0.14,0.30)  (0.26,0.67,0.35)

It is assumed that there are two categories of customers, where right weights for each criterion in V are

For,

Y1  d1.0.34,0.43.0.2,  d 2 ,0.23,0.45,0.67,  d 3 ,34,23,56, d 4 ,54,43,39,  d 5 ,45,27,39, d 6 ,17,28,34 

We can calculate,

  d 1 .0.40,0.45.0.28,  d 2 ,0.25,0.30,0.30,  d 3 ,34,34,36,   R N Y1     d 4 ,34,25,36,  d 5 ,35,30,36,  d 6 ,34,30,31  and according to the principle of maximum membership, the decision for the first category of customers is 5*.

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6. REFERENCES [1]

C. Antony Crispin Sweety and I.Arockiarani, “Fuzzy neutrosophic rough sets” GJRMA2(3), , 2014, 54-59

[2]

C.Antony Crispin Sweety and I.Arockiarani,"Rough sets in neutrosophic approximation space." , Annals of Fuzzy Mathematics and Informatics. [accepted].

[3]

I. Arockiarani and C. Antony Crispin Sweety, Rough Neutrosophic Relations on two universal sets, Bulletin of Mathematics and Statistical Research 4(1) (2016) 203-216.

[4] Chen S. M. and Tan J. M. Handing multi-criteria fuzzy decision-making problems based on vague set theory, Fuzzy Sets and Systems, 1994, 67, 163-172.

[5]

Dubois D, Prade H. Rough fuzzy sets and fuzzy rough sets. International Journal of General System, 1990, 17, 191-209.

[6]

Greco S., Matarazzo B. and Slowinski R. Rough sets theory for multicriteria decision analysis, European Journal of Operational Research,

[7]

Liu G. L. Rough set theory based on two universal sets and its applications, Knowledge Based Systems, 2010, 23,.110–115.

[8]

Pawlak Z. Rough sets. International Journal of Computer and Information Sciences, 1982, 11, 341-356.

[9]

Smarandache, F., A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic. Rehoboth: American Research Press, (1999).

[9] Zadeh L. A. Fuzzy sets, Information and Control, 1965, 8, p. 338–353.

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