Decision-Making Approach Based on Neutrosophic Rough Information

Article

Decision-Making Approach Based on Neutrosophic Rough Information Muhammad Akram 1 1 2

*

ID

, Nabeela Ishfaq 1 , Sidra Sayed 1 and Florentin Smarandache 2, *

ID

Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Pakistan; m.akram@pucit.edu.pk (M.A.); nabeelaishfaq123@gmail.com (N.I.); sidratulmuntha228@yahoo.com (S.S.) Mathematics & Science Department, University of New Mexico, Gallup, NM 87301, USA Correspondence: fsmarandache@gmail.com

Received: 13 February 2018; Accepted: 1 May 2018; Published: 3 May 2018

Abstract: Rough set theory and neutrosophic set theory are mathematical models to deal with incomplete and vague information. These two theories can be combined into a framework for modeling and processing incomplete information in information systems. Thus, the neutrosophic rough set hybrid model gives more precision, flexibility and compatibility to the system as compared to the classic and fuzzy models. In this research study, we develop neutrosophic rough digraphs based on the neutrosophic rough hybrid model. Moreover, we discuss regular neutrosophic rough digraphs, and we solve decision-making problems by using our proposed hybrid model. Finally, we give a comparison analysis of two hybrid models, namely, neutrosophic rough digraphs and rough neutrosophic digraphs.

Keywords: neutrosophic rough hybrid model; decision-making method

regular neutrosophic rough digraphs;

1. Introduction The concept of a neutrosophic set, which generalizes fuzzy sets [1] and intuitionistic fuzzy sets [2], was proposed by Smarandache [3] in 1998, and it is defined as a set about the degree of truth, indeterminacy, and falsity. A neutrosophic set A in a universal set X is characterized independently by a truth-membership function (TA ( x )), an indeterminacy-membership function (I A ( x )), and a falsity-membership function (FA ( x )). To apply neutrosophic sets in real-life problems more conveniently, Smarandache [3] and Wang et al., [4] defined single-valued neutrosophic sets which take the value from the subset of [0, 1].

Rough set theory was proposed by Pawlak [5] in 1982. Rough set theory is useful to study the intelligence systems containing incomplete, uncertain or inexact information. The lower and upper approximation operators of rough sets are used for managing hidden information in a system.

Therefore, many hybrid models have been built such as soft rough sets, rough fuzzy sets, fuzzy rough sets, soft fuzzy rough sets, neutrosophic rough sets and rough neutrosophic sets for handling uncertainty and incomplete information effectively. Dubois and Prade [6] introduced the notions of rough fuzzy sets and fuzzy rough sets. Liu and Chen [7] have studied different decision-making methods. Mordeson and Peng [8] presented operations on fuzzy graphs. Akram et al., [9–12] considered several new concepts of neutrosophic graphs with applications. Rough fuzzy digraphs with applications are presented in [13,14]. To get the extended notion of neutrosophic sets and rough sets, many attempts have been made. Broumi et al., [15] introduced the concept of rough neutrosophic sets. Yang et al., [16] proposed single valued neutrosophic rough sets by combining single valued neutrosophic sets and rough sets, and established an algorithm for the decision-making problem based on single valued neutrosophic rough sets on two universes. Nabeela et al., [17] Algorithms 2018, 11, 59; doi:10.3390/a11050059

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and Sayed et al., [18] introduced rough neutrosophic digraphs, in which they have approximated neutrosophic set under the influence of a crisp equivalence relation. In this research article, we apply another hybrid set model, neutrosophic rough, to graph theory. We deal with regular neutrosophic rough digraphs and then solve the decision-making problem by using our proposed hybrid model.

Our paper is organized as follows: Firstly, we develop the notion of neutrosophic rough digraphs and present some numerical examples. Secondly, we explore basic properties of neutrosophic rough digraphs. In particular, we investigate the regularity of neutrosophic rough digraphs. We describe novel applications of our proposed hybrid decision-making method. To compare the two notions, rough neutrosophic digraphs and neutrosophic rough digraphs, we give a comparison analysis.

Finally, we end the paper by concluding remarks.

2. Neutrosophic Rough Information Definition 1. [4] Let Z be a nonempty universe. A neutrosophic set N on Z is defined as follows: N = {< x : TN ( x ), IN ( x ), FN ( x ) >: x ∈ Z }, where the functions T, I, F:Z → [0, 1] represent the degree of membership, the degree of indeterminacy and the degree of falsity.

Definition 2. [5] Let Z be a nonempty universe and R an equivalence relation on Z. A pair ( Z, R) is called an approximation space. Let N ∗ be a subset of Z and the lower and upper approximations of N ∗ in the approximation space ( Z, R) denoted by RN ∗ and RN ∗ are defined as follows: RN ∗ = { x ∈ Z |[ x ] R ⊆ N ∗ }, RN ∗ = { x ∈ Z |[ x ] R ∩ N ∗ 6= φ}, where [ x ] R denotes the equivalence class of R containing x. A pair ( RN ∗ , RN ∗ ) is called a rough set.

For other notations and applications, readers are referred to [19–32].

Definition 3. [16] Let X ∗ be a nonempty universe and R a neutrosophic relation on X ∗ . Let X be a neutrosophic set on X ∗ , defined as X = {< x, TX ( x ), IX ( x ), FX ( x ) >: x ∈ X ∗ }.

Then the lower and upper approximations of X represented by RX and RX, respectively, are characterized as neutrosophic sets in X ∗ such that, ∀ x ∈ X ∗ RX = {< x, TR(X ) ( x ), IR(X ) ( x ), FR(X ) ( x ) >: y ∈ X ∗ }, RX = {< x, TR(X ) ( x ), IR(X ) ( x ), FR(X ) ( x ) >: y ∈ X ∗ }, where TRX ( x ) =

^

y∈ X

IRX ( x ) =

_

y∈ X

FRX ( x ) =

_

y∈ X

FR ( x, y) ∨ TX (y) ,

1 − IR ( x, y) ∧ IX (y) ,

TR ( x, y) ∧ FX (y) ,

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and TRX ( x ) =

_

y∈ X

IRX ( x ) =

^

y∈ X

FRX ( x ) =

^

y∈ X

TR ( x, y) ∧ TX (y) ,

IR ( x, y) ∨ IX (y) ,

FR ( x, y) ∨ FX (y) .

A pair ( RX, RX ) ia called a neutrosophic rough set.

Definition 4. Let X ∗ be a nonempty set and R a neutrosophic tolerance relation on X ∗ . Let X be a neutrosophic set on X ∗ defined as: X = {< x, TX ( x ), IX ( x ), FX ( x ) >: x ∈ X ∗ }.

Then the lower and upper approximations of X represented by RX and RX, respectively, are characterized as neutrosophic sets in X ∗ such that, ∀ x ∈ X ∗ RX = {< x, TRX ( x ), IRX ( x ), FRX ( x ) >: y ∈ X ∗ }, RX = {< x, TRX ( x ), IRX ( x ), FRX ( x ) >: y ∈ X ∗ },

where TRX ( x ) =

^

y∈ X ∗

IRX ( x ) =

^

y∈ X ∗

FRX ( x ) =

_

y∈ X ∗

FR ( x, y) ∨ TX (y) ,

1 − IR ( x, y) ∨ IX (y) ,

TR ( x, y) ∧ FX (y) ,

and TRX ( x ) = IRX ( x ) = FRX ( x ) =

_

y∈ X ∗

_

y∈ X ∗

^

y∈ X ∗

TR ( x, y) ∧ TX (y) ,

IR ( x, y) ∧ IX (y) ,

FR ( x, y) ∨ FX (y) .

Let Y ∗ ⊆ X ∗ × X ∗ and S a neutrosophic tolerance relation on Y ∗ such that TS (( x1 , x2 )(y1 , y2 )) = min{ TR ( x1 , y1 ), TR ( x2 , y2 )}, IS (( x1 , x2 )(y1 , y2 ) = min{ IR ( x1 , y1 ), IR ( x2 , y2 )},

FS (( x1 , x2 )(y1 , y2 )) = max{ FR ( x1 , y1 ), FR ( x2 , y2 )}.

Let Y be a neutrosophic set on Y ∗ defined as: Y = {< xy, TY ( xy), IY ( xy), FY ( xy) >: xy ∈ Y ∗ }, such that TY ( xy) ≤ min{ TRX ( x ), TRX (y)}, IY ( xy) ≤ min{ IRX ( x ), IRX (y)},

FY ( xy) ≤ max{ FRX ( x ), FRX (y)} ∀ x, y ∈ X ∗ .

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Then the lower and the upper approximations of Y represented by SY and SY, are defined as follows: SY = {< xy, TSY ( xy), ISY ( xy), FSY ( xy) >: xy ∈ Y ∗ },

SY = {< xy, TSY ( xy), ISY ( xy), FSY ( xy) >: xy ∈ Y ∗ }, where TSY ( xy) = ISY ( xy) = FSY ( xy) =

^

wz∈Y ∗

^

wz∈Y ∗

_

wz∈Y ∗

FS (( xy), (wz)) ∨ TY (wz) ,

(1 − IS (( xy), (wz))) ∨ IY (wz) ,

TS (( xy), (wz)) ∧ FY (wz) ,

and TSY ( xy) = ISY ( xy) = FSY ( xy) =

_

wz∈Y ∗

_

wz∈Y ∗

^

wz∈Y ∗

TS (( xy), (wz)) ∧ TY (wz) ,

IS (( xy), (wz)) ∧ IY (wz) ,

FS (( xy), (wz)) ∨ FY (wz) .

A pair SY = (SY, SY ) is called neutrosophic rough relation.

Definition 5. A neutrosophic rough digraph on a nonempty set X ∗ is a 4-ordered tuple G = ( R, RX, S, SY ) such that (a) (b) (c) (d) (e)

R is a neutrosophic tolerance relation on X ∗ , S is a neutrosophic tolerance relation on Y ∗ ⊆ X ∗ × X ∗ , RX = ( RX, RX ) is a neutrosophic rough set on X ∗ , SY = (SY, SY ) is a neutrosophic rough relation on X ∗ , ( RX, SY ) is a neutrosophic rough digraph where G = ( RX, SY ) and G = ( RX, SY ) are lower and upper approximate neutrosophic digraphs of G such that

TSY ( xy) ≤ min{ TRX ( x ), TRX (y)}, ISY ( xy) ≤ min{ IRX ( x ), IRX (y)},

FSY ( xy) ≤ max{ FRX ( x ), FRX (y)}, TSY ( xy) ≤ min{ TRX ( x ), TRX (y)}, ISY ( xy) ≤ min{ IRX ( x ), IRX (y)},

FSY ( xy) ≤ max{ FRX ( x ), FRX (y)} ∀ x, y ∈ X ∗ .

Example 1. Let X ∗ = { p, q, r, s, t} be a nonempty set and R a neutrosophic tolerance relation on X ∗ is given as:

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R p p (1,1,0) (0.5,0.2,0.3) q R p r (0.1,0.9,0.4) p (1,1,0) s (0.6,0.5,0.2) q (0.5,0.2,0.3) t (0.2,0.1,0.8)

q (0.5,0.2,0.3) (1,1,0) q (0.3,0.7,0.5) (0.5,0.2,0.3) (0.1,0.9,0.6) (1,1,0) (0.6,0.5,0.1)

r s t (0.1,0.9,0.4) (0.6,0.5,0.2) (0.2,0.1,0.8) (0.3,0.7,0.5) (0.1,0.9,0.6) (0.6,0.5,0.1) r s t (1,1,0) (0.2,0.8,0.7) (0.1,0.9,0.6) (0.1,0.9,0.4) (0.6,0.5,0.2) (0.2,0.1,0.8) (0.2,0.8,0.7) (1,1,0) (0.2,0.3,0.1) (0.3,0.7,0.5) (0.1,0.9,0.6) (0.6,0.5,0.1) (0.1,0.9,0.6) (0.2,0.3,0.1) (1,1,0)

r (0.1,0.9,0.4) (0.3,0.7,0.5) (1,1,0) (0.2,0.8,0.7) (0.1,0.9,0.6) Let X1 = s {((0.6,0.5,0.2) p, 0.2, 0.1, 0.7),(0.1,0.9,0.6) (q, 0.4, 0.5, 0.6(0.2,0.8,0.7) ), (r, 0.7, 0.8, 0.9)(1,1,0) , (s, 0.2, 0.9, (0.2,0.3,0.1) 0.1), (t, 0.6, 0.8, 0.4)} be t (0.2,0.1,0.8) (0.6,0.5,0.1) (0.1,0.9,0.6) (0.2,0.3,0.1) (1,1,0) ∗ a neutrosophic set on X . The lower and upper approximations of X1 are given as: Let X1 = {(p, 0.2, 0.1, 0.7), (q, 0.4, 0.5, 0.6), (r, 0.7, 0.8, 0.9), (s, 0.2, 0.9, 0.1), (t, 0.6, 0.8, 0.4)} be a neutrosophic set RXlower p, 0.2, 0.1, 0.7 ), (q, 0.3, 0.5, 0.6 0.4,given 0.1, 0.9 1 = {(and on X ∗ . The upper approximations of )X, (1r,are as:), (s, 0.2, 0.5, 0.6), (t, 0.2, 0.5, 0.6)}, RX1 = {( p, 0.4, 0.2, 0.8), (q, 0.6, 0.9, 0.4), (r, 0.7, 0.8, 0.6), (s, 0.2, 0.9, 0.1), (t, 0.6, 0.8, 0.1)}.

RX1 = {(p, 0.2, 0.1, 0.7), (q, 0.3, 0.5, 0.6), (r, 0.4, 0.1, 0.9), (s, 0.2, 0.5, 0.6), (t, 0.2, 0.5, 0.6)}, ∗ = {= ∗ and Let YRX qs,0.4, rt, sp, } ⊆ (q, X ∗0.6, × X0.9, S a(r,neutrosophic relation given as: {(p, 0.2,tq0.8), 0.4), 0.7, 0.8, 0.6),tolerance (s, 0.2, 0.9, 0.1), which (t, 0.6,is0.8, 0.1)}.

1 pr, Let Y ∗ = {pr, qs, rt,Ssp, tq} ⊆prX ∗ × X ∗ and tolerance is given as: qs S a neutrosophic rt sp relation which tq pr

(1,1,0)

(0.2,0.2,0.7)

(0.1,0.9,0.6)

(0.1,0.5,0.4)

(0.2,0.1,0.8)

S pr qs rt sp tq qs (0.2,0.2,0.7) (1,1,0) (0.2,0.3,0.5) (0.1,0.5,0.6) (0.1,0.5,0.6) pr (1,1,0) (0.2,0.2,0.7) (0.1,0.9,0.6) (0.1,0.5,0.4) (0.2,0.1,0.8) (0.1,0.9,0.6) (0.2,0.3,0.5) (1,1,0) (0.2,0.1,0.8) (0.1,0.5,0.6) qsrt (0.2,0.2,0.7) (1,1,0) (0.2,0.3,0.5) (0.1,0.5,0.6) (0.1,0.5,0.6) sp (0.1,0.5,0.4) (0.1,0.5,0.6) (0.2,0.1,0.8) (1,1,0) (0.2,0.2,0.3) rt (0.1,0.9,0.6) (0.2,0.3,0.5) (1,1,0) (0.2,0.1,0.8) (0.1,0.5,0.6) (0.2,0.1,0.8) (0.1,0.5,0.6) (0.1,0.5,0.6) (0.2,0.2,0.3) (1,1,0) tq (1,1,0) (0.2,0.2,0.3) sp (0.1,0.5,0.4) (0.1,0.5,0.6) (0.2,0.1,0.8) tq (0.2,0.1,0.8) (0.1,0.5,0.6) (0.1,0.5,0.6) (0.2,0.2,0.3) (1,1,0) Let Y1 = {( pr, 0.2, 0.1, 0.5), (qs, 0.1, 0.3, 0.3), (rt, 0.2, 0.1, 0.4), (sp, 0.1, 0.1, 0.2), (tq, 0.1, 0.4, 0.3)} be a neutrosophic on0.1, Y ∗ .0.5), The(qs, lower and upper of(sp, Y1 are as: (tq, 0.1, 0.4, 0.3)} be a neutroLet Y1 = {(pr,set 0.2, 0.1, 0.3, 0.3),approximations (rt, 0.2, 0.1, 0.4), 0.1,given 0.1, 0.2), ∗ sophic set on Y . The lower and upper approximations of Y1 are given as: SY1 = {( pr, 0.2, 0.1, 0.5), (qs, 0.1, 0.3, 0.3), (rt, 0.2, 0.1, 0.4), (sp, 0.1, 0.1, 0.2), (tq, 0.1, 0.4, 0.3)}, 1 = {(pr, 0.2, 0.1, 0.5), (qs, 0.1, 0.3, 0.3), (rt, 0.2, 0.1, 0.4), (sp, 0.1, 0.1, 0.2), (tq, 0.1, 0.4, 0.3)}, SYSY 1 = {( pr, 0.2, 0.2, 0.4), ( qs, 0.2, 0.4, 0.3), (rt, 0.2, 0.4, 0.4), ( sp, 0.2, 0.3, 0.2), ( tq, 0.2, 0.4, 0.3)}.

SY1 = {(pr, 0.2, 0.2, 0.4), (qs, 0.2, 0.4, 0.3), (rt, 0.2, 0.4, 0.4), (sp, 0.2, 0.3, 0.2), (tq, 0.2, 0.4, 0.3)}.

Thus, G = ( RX , SY ) and G = ( RX , SY ) are neutrosophic digraphs as shown in Figure 1.

Thus, G = (RX1 , SY1 )1 and1 G = (RX1 , SY11) are1 neutrosophic digraphs as shown in Fig. 1. p(0.4, 0.8, 0.2)

p(0.2, 0.1, 0.7)

b

1)

2) .3 , 0 .

0 .8 .6 ,

(0.2, 0

2) .1 , 0 .

(0.1, 0

t(0

, 0.

0 .5 .2 , t(0

, 0.

9)

.2 ,

G1 = (RX1 , SY1 )

0 .9

,0

.4 ,

0 .4

)

.2 (0

.4 ,0

,0

.3 )

b

,0

4)

.

, 0.

, 0.

r(0

.6 )

.1 4, 0

s(0

0 .9

(0.

2

b

.6 ,

)

0 .4 )

(0

0 .3

b

0 .2 ,

.4 )

.1 ,

, 0 .3

(0.2,

1, 0

b

,0

.6 )

, 0.

(0.2, 0.4, 0.3)

b

0 .5 )

0 .5

,0

0 .1 ,

.2 ,

b

0 .5

, (0.2

(0.

2

s(0

.3 ,

(0.1, 0.4, 0.3)

b

q (0

q(0

6)

b

b

.1 )

r(0

.7

.8 ,0

,0

.6 )

G1 = (RX1 , SY1 )

Figure 1: Neutrosophic Neutrosophicrough Rough Digraph G1( G=1 ,(G Figure 1. digraph G1 = G11),. G1 ) ∗ Example v, w, x,zy, a crisp set Rand R a neutrosophic tolerance on Xby given Example 2.2.

2. LetLet X ∗X=∗ {=u,{u, v, w, x, y, } z} be abecrisp set and a neutrosophic tolerance relation relation on X ∗ given by

R u v w x y z

u (1,1,0) (0.2,0.3,0.5) (0.5,0.6,0.9) (0.3,0.8,0.3) (0.3,0.2,0.1) (0.1,0.1,0.5)

v (0.2,0.3,0.5) (1,1,0) (0.9,0.5,0.6) (0.1,0.5,0.7) (0.8,0.9,0.1) (0.8,0.9,0.1)

w (0.5,0.6,0.9) (0.9,0.5,0.6) (1,1,0) (0.3,0.6,0.8) 5 (0.2,0.3,0.6) (0.7,0.6,0.6)

x (0.3,0.8,0.3) (0.1,0.5,0.7) (0.3,0.6,0.8) (1,1,0) (0.5,0.1,0.9) (0.8,0.7,0.2)

y (0.3,0.2,0.1) (0.8,0.9,0.1) (0.2,0.3,0.6) (0.5,0.1,0.9) (1,1,0) (0.6,0.5,0.9)

z (0.1,0.1,0.5) (0.8,0.9,0.1) (0.7,0.6,0.6) (0.8,0.7,0.2) (0.6,0.5,0.9) (1,1,0)

R 2018, 11, 59 u Algorithms

v w x y z 6 of 20 u (1,1,0) (0.2,0.3,0.5) (0.5,0.6,0.9) (0.3,0.8,0.3) (0.3,0.2,0.1) (0.1,0.1,0.5) v (0.2,0.3,0.5) (1,1,0) (0.9,0.5,0.6) (0.1,0.5,0.7) (0.8,0.9,0.1) (0.8,0.9,0.1) Let = {(u, 0.9, 0.3, 0.1 ), (v, 0.5, 0.6, 0.2(1,1,0) ), (w, 0.8, 0.5,(0.3,0.6,0.8) 0.3), ( x, 0.7, 0.6, 0.9), (y, 0.5, 0.2, 0.1), . w X (0.5,0.6,0.9) (0.9,0.5,0.6) (0.2,0.3,0.6) (0.7,0.6,0.6) (0.3,0.8,0.3) (0.1,0.5,0.7) (0.5,0.1,0.9) (z, 0.9, x0.7, 0.3 )} be a neutrosophic set on X ∗ .(0.3,0.6,0.8) Then the lower and(1,1,0) upper approximations of X are(0.8,0.7,0.2) given as follows: y (0.3,0.2,0.1) (0.8,0.9,0.1) (0.2,0.3,0.6) (0.5,0.1,0.9) (1,1,0) (0.6,0.5,0.9) (0.1,0.1,0.5) (0.8,0.9,0.1) (0.7,0.6,0.6) (0.8,0.7,0.2) (0.6,0.5,0.9) z RX = {(u, 0.5, 0.3, 0.3), (v, 0.5, 0.2, 0.3), (w, 0.6, 0.4, 0.3), ( x, 0.7, 0.3, 0.9), (y, 0.5, 0.2, 0.5), (z,(1,1,0) 0.5, 0.5, 0.8)},

{(u, 0.9, 0.6, 0.1)(v, , (v,0.5, 0.8,0.6, 0.7,0.2), 0.1),(w, (w,0.8, 0.8, 0.5, 0.6, 0.3 ), ((x, x, 0.8, ), (y, 0.1 ), (z, 0.9, 0.7, 0.2)}.

Let XRX == {(u, 0.9, 0.3, 0.1), 0.3), 0.7,0.7, 0.6,0.3 0.9), (y,0.6, 0.5,0.6, 0.2, 0.1), (z, 0.9, 0.7, 0.3)} be a neutrosophic set on X ∗ . Then the lower and upper approximations of X are given as ∗ ∗ ∗ ∗ follows: Let Y = {uv, vw, wx, xy, yz, zu, zw, vy} ⊆ X × X and S a neutrosophic tolerance relation on Y given as RX = {(u, 0.5, 0.3, 0.3), (v, 0.5, 0.2, 0.3), (w, 0.6, 0.4, 0.3), (x, 0.7, 0.3, 0.9), (y, 0.5, 0.2, 0.5), (z, 0.5, 0.5, 0.8)}, S uv vw wx xy yz zu zw vy RX = {(u, 0.9, 0.6, 0.1), (v, 0.8, 0.7, 0.1), (w, 0.8, 0.6, 0.3), (x, 0.8, 0.7, 0.3), (y, 0.6, 0.6, 0.1), (z, 0.9, 0.7, 0.2)}. uv

Let

(1,1,0) (0.2,0.3,0.6) (0.1,0.5,0.9) (0.3,0.8,0.3) (0.3,0.2,0.1) (0.1,0.1,0.5) (0.1,0.1,0.6) (o.2,0.3,0.5) (0.2,0.3,0.6) (0.3,0.5,0.8) ∗(0.1,0.3,0.7) (0.5,0.6,0.9) (0.8,0.9,0.1) (0.2,0.3,0.6) Y = {uv, vw, wx, xy, (1,1,0) yz, zu, zw, vy} ⊆ X × X ∗ and (0.7,0.6,0.6) S a neutrosophic tolerance relation on Y ∗ given wx (0.1,0.5,0.9) (0.3,0.5,0.8) (1,1,0) (0.3,0.1,0.9) (0.2,0.3,0.6) (0.3,0.6,0.6) (0.3,0.6,0.8) (0.5,0.1,0.9) S xy uv vw wx xy yz zu zw (0.1,0.5,0.7)vy (0.3,0.8,0.3) (0.1,0.3,0.7) (0.3,0.1,0.9) (1,1,0) (0.5,0.1,0.9) (0.3,0.2,0.2) (0.2,0.3,0.6) uv (0.3,0.8,0.3) (0.3,0.2,0.1) (0.1,0.1,0.5) (0.1,0.1,0.6) (o.2,0.3,0.5) yz (1,1,0) (0.3,0.2,0.1) (0.2,0.3,0.6) (0.7,0.6,0.6) (0.1,0.5,0.9) (0.2,0.3,0.6) (0.5,0.1,0.9) (1,1,0) (0.1,0.1,0.9) (0.6,0.5,0.9) (0.6,0.5,0.9) vw (0.2,0.3,0.6) (1,1,0) (0.3,0.5,0.8) (0.1,0.3,0.7) (0.7,0.6,0.6) (0.5,0.6,0.9) (0.8,0.9,0.1) (0.2,0.3,0.6) (0.1,0.1,0.5) (0.3,0.5,0.8) (0.5,0.6,0.9) (0.3,0.6,0.6) (0.3,0.2,0.2) (0.5,0.6,0.9) (0.3,0.3,0.1) (1,1,0) (0.3,0.1,0.9) (0.1,0.1,0.9) (0.2,0.3,0.6) (1,1,0) (0.3,0.6,0.6) (0.3,0.6,0.8) (0.5,0.1,0.9) wx zu (0.1,0.5,0.9) xy zw (0.3,0.8,0.3) (1,1,0) (0.5,0.1,0.9)(0.5,0.6,0.9) (0.3,0.2,0.2) (1,1,0) (0.2,0.3,0.6) (0.1,0.5,0.7) (0.1,0.1,0.6) (0.1,0.3,0.7) (0.8,0.9,0.1) (0.3,0.1,0.9) (0.3,0.6,0.8) (0.2,0.3,0.6) (0,6,0.5,0.9) (0.2,0.3,0.6) yz (0.3,0.2,0.1) (0.7,0.6,0.6) (0.2,0.3,0.6) (0.5,0.1,0.9) (1,1,0) (0.1,0.1,0.9) (0.6,0.5,0.9) (0.6,0.5,0.9) (0.2,0.3,0.5) (0.2,0.3,0.6) (0.5,0.1,0.9) (0.1,0.5,0.7) (0.6,0.5,0.9) (0.3,0.2,0.1) (0.2,0.3,0.6) (1,1,0) vy zu (0.1,0.1,0.5) (0.5,0.6,0.9) (0.3,0.6,0.6) (0.3,0.2,0.2) (0.1,0.1,0.9) (1,1,0) (0.5,0.6,0.9) (0.3,0.3,0.1) ∗ vw

zw vy

(0.1,0.1,0.6)

(0.8,0.9,0.1)

(0.3,0.6,0.8)

(0.2,0.3,0.6)

(0,6,0.5,0.9)

(0.5,0.6,0.9)

(1,1,0)

(0.2,0.3,0.6)

as

Let Y be a neutrosophic set on(0.5,0.1,0.9) Y ∗ defined as(0.1,0.5,0.7) Y = {(uv, 0.5, 0.2, 0.1), (vw, 0.5, 0.2, 0.3(0.2,0.3,0.6) ), (wx, 0.5, 0.3, (1,1,0) 0.3), (0.2,0.3,0.5) (0.2,0.3,0.6) (0.6,0.5,0.9) (0.3,0.2,0.1) ( xy, 0.5, 0.2, 0.3), (yz, 0.5, 0.2, 0.2), (zu, 0.5, 0.3, 0.2), (zw, 0.5, 0.4, 0.3), (vy, 0.5, 0.2, 0.1)}. Then the lower Let Y be a neutrosophic set on Y ∗ defined as Y = {(uv, 0.5, 0.2, 0.1), (vw, 0.5, 0.2, 0.3), (wx, 0.5, 0.3, 0.3), and 0.2, upper approximations Y are given (xy, 0.5, 0.3), (yz, 0.5, 0.2,of0.2), (zu, 0.5,as0.3, 0.2), (zw, 0.5, 0.4, 0.3), (vy, 0.5, 0.2, 0.1)}. Then the lower and up-

per approximations of Y are given as SY = {(uv, 0.5, 0.2, 0.3), (vw, 0.5, 0.2, 0.3), (wx, 0.5, 0.3, 0.3), ( xy, 0.5, 0.2, 0.3), (yz, 0.5, 0.2, 0.3), SY = {(uv, 0.5, 0.2, 0.3), (vw, 0.5, 0.2, 0.3), (wx, 0.5, 0.3 0.2,),0.3), (yz,0.2, 0.5,0.3 0.2, (zu, 0.5,0.5, 0.3, 0.3, 0.3)0.3), , (zw,(xy, 0.5, 0.2, (vy, 0.5, )}0.3), , (zu, 0.5, 0.3, 0.3), (zw, 0.5, 0.2, 0.3), (vy, 0.5, 0.2, 0.3)}, SY = {(uv, 0.5, 0.3, 0.1), (vw, 0.5, 0.4, 0.3), (wx, 0.5, 0.4, 0.3), ( xy, 0.5, 0.3, 0.3), (yz, 0.5, 0.4, 0.1),

SY = {(uv, 0.5, 0.3, 0.1), (vw, 0.5, 0.4, 0.3), (wx, 0.5, 0.4, 0.3), (xy, 0.5, 0.3, 0.3), (yz, 0.5, 0.4, 0.1), (zu, 0.5, 0.4, 0.1), (zw, 0.5, 0.4, 0.3), (vy, 0.5, 0.3, 0.1)}. (zu, 0.5, 0.4, 0.1), (zw, 0.5, 0.4, 0.3), (vy, 0.5, 0.3, 0.1)}.

Thus, G = ( RX, SY ) and G = ( RX, SY ) are the neutrosophic digraphs as shown in Figure 2.

Thus, G = (RX, SY ) and G = (RX, SY ) are the neutrosophic digraphs as shown in Fig. 2.

(0 b .6 ,0 .4 (0.5, ,0 0.3 .3 , 0. ) 3) b

b

.

5, 0 (0.

.

2, 0

3)

w

b

(0 b .8 ,0 (0. .6 5, 0 ,0 .4, .3 0.3 ) )

0.

6, 0.

6, 0.

1)

3)

1)

b

b

5, (0.

2) , 0.

0.3

y(

2, 0 .

4, 0 .

0.3) 0.4, , 5 .

(0

(0.5, 0.4, 0.3)

5, 0 .

0.

5, 0.

2, 0.

5)

(0.

.3) .2, 0 0 , 5 (0.

b

y(

(0.5, 0.2, 0.3)

w

b

8) 0.

5, 0. (0.5, 0.2, 0.3) 5, 0.

v

0.

5, .

(0

2) 0.

7, 0. (0.5, 0.4, 0.1) 9, 0.

.

1) 0. 0.5, 0 , ( 7 .

0 b 8, 0. ( v (0.5 , 0.

3.0 .1)

z(

u(0.9, 0.6, 0.1) b (0.5 .1) , 0.

0 , 3

z(

u(0.5, 0.3, 0.3) b (0.5 .3) , 0.

0 , ) 3, 0 2 .

3 .3) 0. 0.5, 0 , ( 2

x(0.8, 0.7, 0.3)

x(0.7, 0.3, 0.9)

G = (RX, SY )

G = (RX, SY )

Figure2:2. Neutrosophic Neutrosophic rough digraph G =G ( G,=G(G, ). G) Figure Rough Digraph

Now we discuss regular neutrosophic rough digraphs.

6 Definition 6. Let G = ( G, G ) be a neutrosophic rough digraph on a nonempty set X ∗ . The indegree of a vertex x ∈ G is the sum of membership degree, indeterminacy and falsity of all edges towards x from other vertices in G and G, respectively, represented by idG ( x ) and defined by idG ( x ) = idG ( x ) + idG ( x ),

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where idG ( x ) = ( ∑

TG (yx ),

idG ( x ) = ( ∑

TG (yx ),

x,y∈SY x,y∈SY

∑

IG (yx ),

∑

IG (yx ),

x,y∈SY x,y∈SY

∑

FG (yx )),

∑

FG (yx )).

x,y∈SY x,y∈SY

The outdegree of a vertex x ∈ G is the sum of membership degree, indeterminacy and falsity of all edges outward from x to other vertices in G and G, respectively, represented by odG ( x ) and defined by odG ( x ) = odG ( x ) + odG ( x ), where odG ( x ) = ( ∑

TG ( xy),

odG ( x ) = ( ∑

TG ( xy),

x,y∈SY x,y∈SY

∑

IG ( xy),

∑

IG ( xy),

x,y∈SY x,y∈SY

∑

FG ( xy)),

∑

FG ( xy)).

x,y∈SY x,y∈SY

dG ( x ) = idG ( x ) + odG ( x ) is called degree of vertex x.

Definition 7. A neutrosophic rough digraph is called a regular neutrosophic rough digraph of degree (m1 , m2 , m3 ) if dG ( x ) = (m1 , m2 , m3 ) ,∀ x ∈ X. Example 3. Let X ∗ = { p, q, r, s} be a nonempty set and R a neutrosophic tolerance relation on X ∗ is given as: R p q r s

p (1,1,0) (0.9,0.8,0.1) (0.7,0.5,0.8) (0.1,0.9,0.8)

q (0.1,0.9,0.8) (1,1,0) (0.1,0.9,0.8) (0.4,0.3,0.9)

r (0.7,0.5,0.8) (0.1,0.9,0.8) (1,1,0) (0.1,0.9,0.8)

s (0.1,0.9,0.8) (0.4,0.3,0.9) (0.1,0.9,0.8) (1,1,0)

Let X1 = {( p, 0.1, 0.4, 0.8), (q, 0.2, 0.3, 0.9), (r, 0.1, 0.6, 0.8), (s, 0.9, 0.6, 0.3)} be a neutrosophic set on X ∗ .

The lower and upper approximations of X1 are given as: RX1 = {( p, 0.1, 0.3, 0.8), (q, 0.2, 0.3, 0.9), (r, 0.1, 0.3, 0.8), (s, 0.8, 0.4, 0.4)}, RX1 = {( p, 0.1, 0.6, 0.8), (q, 0.4, 0.6, 0.8), (r, 0.1, 0.6, 0.8), (s, 0.9, 0.6, 0.3)}.

Let Y ∗ = { pq, qr, rs, sp} ⊆ X ∗ × X ∗ and S a neutrosophic tolerance relation on Y ∗ which is given as: S pq qr rs sp

pq (1,1,0) (0.1,0.9,0.8) (0.4,0.3,0.9) (0.1,0.9,0.8)

qr (0.1,0.9,0.8) (1,1,0) (0.1,0.9,0.8) (0.4,0.3,0.9)

rs (0.4,0.3,0.9) (0.1,0.9,0.8) (1,1,0) (0.1,0.9,0.8)

sp (0.1,0.9,0.8) (0.4,0.3,0.9) (0.1,0.9,0.8) (1,1,0)

Let Y1 = {( pq, 0.1, 0.3, 0.8), (qr, 0.1, 0.3, 0.3), (rs, 0.1, 0.3, 0.8), (sp, 0.1, 0.3, 0.8)} be a neutrosophic set on Y ∗ . The lower and upper approximations of Y1 are given as: SY1 = {( pq, 0.1, 0.3, 0.8), (qr, 0.1, 0.3, 0.3), (rs, 0.1, 0.3, 0.8), (sp, 0.1, 0.3, 0.8)}, SY1 = {( pq, 0.1, 0.3, 0.8), (qr, 0.1, 0.3, 0.3), (rs, 0.1, 0.3, 0.8), (sp, 0.1, 0.3, 0.8)}.

Thus, G1 = ( G1 , G1 ) is a regular neutrosophic rough digraph as shown in Figure 3.

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p(0.1, 0.6, 0.8)

p(0.1, 0.3, 0.8)

b

.

) (0.1 ,0 .3 , 0

.8 ) b

b

(0 .1

,0

.3 ,

0 .8

.

(0

)

1

,0

, .3

0.

8)

s(0.9, 0.6, 0.3)

,0

.8

.1 (0

,0

.

0 3,

.8

) (0.

1, 0 .3 ,0

.8 ) b

b

(0 .1

,0

.3 ,

0 .8

.

(0

)

1

. ,0

3,

0.

8)

q(0.4, 0.6, 0.8)

.1 (0

0 3,

q(0.2, 0.3, 0.9)

s(0.8, 0.4, 0.4)

b

b b

r(0.1, 0.6, 0.8)

r(0.1, 0.3, 0.8) G1 = (RX1 , SY1 )

G1 = (RX1 , SY1 )

Figure 3: Regular Neutrosophic DigraphG1G= Figure 3. Regular neutrosophicRough rough digraph ( G(G . 1) 1 = 11, )G 1, G SYLet {(pq, (qr,G0.1, 0.8), (sp, 0.1, 0.3, digraphs.

0.8)}, Definition 8. ( G0.3, ) and ( G0.3), ) be0.1, two0.3, neutrosophic rough Then the 1 =G 2 =0.3, 1 =0.1, 1 , G10.8), 2 , G2(rs, SY = {(pq, 0.1, 0.3, 0.8), (qr, 0.1, 0.3, 0.3), (rs, 0.1, 0.3, 0.8), (sp, 0.1, 0.3, 0.8)}.

1 direct sum of G1 and G2 is a neutrosophic rough digraph G = G1 ⊕ G2 = ( G1 ⊕ G2 , G1 ⊕ G2 ), G11 ,⊕G1G)2is a=regular ⊕ RX2 , SY1 ⊕ SY2 )digraph and Gas G2 in = Fig. ( RX ( RX1neutrosophic rough shown 3.1 ⊕ RX2 , SY1 ⊕ SY2 ) are Thus,where G1 = (G 1 ⊕ neutrosophic digraphs.

Definition 2.8. Let G1 = (G1 , G1 ) and G2 = (G2 , G2 ) be two neutrosophic rough digraphs. Then the direct sum of G1 and G2 is a neutrosophic rough digraph G = G1 ⊕ G2 = (G1 ⊕ G2 , G1 ⊕ G2 ), where G1 ⊕ G2 = (RX1 ⊕ RX2 , SY1 ⊕ SY2 ) and G1 ⊕ G2 = (RX1 ⊕ RX2 , SY1 ⊕ SY2 ) are neutrosophic digraphs.

(1) (1)

   TRX1 ( x ), TRX1 ⊕ RX2 ( x ) = TRX2 ( x ),    Tmax((x),  RX1 TRX1 ( x ), TRX2 ( x )), TRX1 ⊕RX2 (x) =  TRX2 (x),   IRX1 ( x ), (x), T  max(T RX1 RX2 (x)), IRX1 ⊕ RX2 ( x ) = IRX2 ( x ),   RX1 (x),  Imax ( IRX1 ( x ), IRX2 ( x )), IRX1 ⊕RX2 (x) =  IRX2 (x),   x ),1 (x), IRX2 (x)), FRX1 (RX  max(I  FRX1 ⊕ RX2 ( x ) = FFRX ( x ),  RX12 (x), ( (x), FRX1 ⊕RX2 (x) =  Fmin RX2 FRX1 ( x ), FRX2 ( x )),  min(FRX1 (x), FRX2 (x)),

(

if x ∈ RX1 − RX2 if x ∈ RX2 − RX1 −RX RX2 2 ifif x ∈ RX RX11 ∩ if x ∈ RX2 − RX1 ifif xx∈∈RX 2 1 −∩RX RX RX 1 2 if x ∈ RX2 − RX1 if x ∈ RX1 − RX2 if x ∈ RX ∩ RX if x ∈ RX12 − RX2 1 ifif x ∈ RX ∩ RX RX22 RX11 − if RX1 2 if xx ∈∈ RX RX21−−RX if RX if xx ∈∈ RX RX12∩−RX 21 if x ∈ RX1 ∩ RX2

y), if (x, y) ∈ SY TTSY1 (x, SY1 ( x, y ), if ( x, y ) ∈ SY11 TTSY2 (x, y), if (x, y) ∈ SY2 SY2 ( x, y), if ( x, y) ∈ SY2 ( ISY1 (x, y), if (x, y) ∈ SY1 ISY1 ⊕SY2 (x, y) = ISY1 (x, ( x, y), y), if if ((x, x, yy)) ∈ SY12 ∈ SY 2 ISY1 ⊕SY2 ( x, y) = ISY ISY2 ( x, y), if ( x, y) ∈ SY2 F (x, y), if (x, y) ∈ SY1 FSY1 ⊕SY2 (x, y) = ( SY1 FFSY (x, y), if (x, y) ∈ SY SY12 ( x, y ), if ( x, y ) ∈ SY12 FSY1 ⊕SY2 ( x, y) = FSY2 ( x, y), if ( x, y) ∈ SY2 TTSY1 ⊕SY2 (x, y) = SY1 ⊕SY2 ( x, y ) =

8

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(2)

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 if x ∈ RX1 − RX2 TRX1 (x), if x ∈ RX1 − RX2  TRX1 ( x ), (x), if x ∈ RX2 − RX1 TRX1 ⊕RX2 (x) = TRX TRX22 ( x ), if x ∈ RX2 − RX1 TRX1 ⊕ RX2 ( x )  =  max(TRX1 (x), TRX2 (x)), if x ∈ RX1 ∩ RX2  max( TRX1 ( x ), TRX2 ( x )), if x ∈ RX1 ∩ RX2 (x), if x ∈ RX1 − RX2 IRX if x ∈ RX1 − RX 2  IRX11 ( x ), (x) = IRX1 ⊕RX I (x), if x ∈ RX2 − RX1 if x ∈ RX2 − RX IRX22 ( x ), IRX1 ⊕2RX2 ( x )  = RX 1  if ∈xRX ∈ RX max(I 1 ∩ RX2 RX(2x(x)), )) , if x max(RX IRX11 (x), ( x ), IIRX ∩ RX 2 1 2  (x), if x ∈ RX RX2 FRX 1  if x ∈ RX1 − RX− 2  FRX11 ( x ), RX1 ∈2 RX FRX1F⊕RX2 (x)( x=) = FRX 2− ( x ), if xif∈ xRX FRX22(x), − RX 1 RX1 ⊕ RX2  min(F (x), F (x)), if x ∈ RX ∩ RX 12 2 (2x )), if x ∈ RX1 ∩ RX min( F RX1 ( x ), F RX RX1

RX2

(

T ( x, y), if ( x, y) ∈ SY1 TSY ⊕SY2 ( x, y) = T SY1 (x, y), if (x, y) ∈ SY1 1 1 TSY TSY1 ⊕SY2 (x, y) = SY ( x, y ), if ( x, y ) ∈ SY2 ( TSY22 (x, y), if (x, y) ∈ SY2 I ( x, y), if ( x, y) ∈ SY1 ISY ⊕SY2 ( x, y) = I SY1 (x, y), if (x, y) ∈ SY1 1 ISY ISY1 ⊕SY2 (x, y) = SY 1( x, y ), if ( x, y ) ∈ SY2 ( ISY22 (x, y), if (x, y) ∈ SY2 F ( x, y), if ( x, y) ∈ SY1 FSY ⊕SY2 ( x, y) = FSY1 (x, y), if (x, y) ∈ SY1 1 SY1( x, y ), if ( x, y ) ∈ SY2 FSY FSY1 ⊕SY2 (x, y) = FSY2 2 (x, y), if (x, y) ∈ SY2 Example 4. Let X ∗ = { p, q, r, s, t} be a set. Let G = ( G , G ) and G = ( G , G ) be two neutrosophic

2 2 1 1 1 2 Example 2.4. Let X ∗ = {p, r, s, t} be a set. Let G 1 = (G1 , G1 ) and G2 = (G2 , G2 ) be two neutrosophic ∗ asq,shown rough digraphs on X in Figures 1 and 4.

The direct sum of G and G is G = ( G ⊕ G , G ⊕ G ), 2 1 rough digraphs on X ∗ as shown in Fig.1 and Fig.4. The direct sum of G1 and G2 is 1G = 2(G11 ⊕ G2 2 , G1 ⊕ G2 ), where G1 ⊕ G2 = ( RX1 ⊕ RX2 , SY1 ⊕ SY2 ) and G1 ⊕ G2 = ( RX1 ⊕ RX2 , SY1 ⊕ SY2 ) are neutrosophic where G1 ⊕ G2 = (RX1 ⊕ RX2 , SY1 ⊕ SY2 ) and G1 ⊕ G2 = (RX1 ⊕ RX2 , SY1 ⊕ SY2 ) are neutrosophic digraphs digraphs as shown in Figure 5. as shown in Fig.5.

(p, 0.4, 0.6, 0.3)

(p, 0.2, 0.1, 0.3)

b

.4 ,

)

) b

0 .3

4,

,0

0 .2

0.

1,

(0.2, 0.3, 0.3) (r, 0.5, 0.6, 0.4) (s, 0.5, 0.6, 0.1)

1)

2)

0.

0.

(0 .2

.4 ,

4,

0.

(0.2, 0.1, 0.3) (r, 0.2, 0.4, 0.6) (s, 0.5, 0.1, 0.3) b

(0 b

,0

.

(0

4,

0 .3

b

(0 .4

.4 ,

)

0 .2 ,

.1 )

,0

0 .2

0 .2 ,

(0 .2

.1 ,

(t, 0.9, 0.6, 0.1)

,0

)

(0 .1

(q, 0.9, 0.6, 0.4)

.2 )

(q, 0.2, 0.4, 0.9)

(0 b

0 .2 ,

0 .1 ,

.

(0

(t, 0.4, 0.2, 0.9)

b

b

b

G2 = (RX2 , SY2 )

b

G2 = (RX2 , SY2 )

4. NeutrosophicRough rough digraph G =G ( G= 2 ).2 , G2 ) FigureFigure 4: Neutrosophic Digraph (G 2, G

Remark 2.1. The direct sum of two regular neutrosophic rough digraphs may not be regular neutrosophic rough digraph as it can be seen in the following example.

Example 2.5. Consider the two regular neutrosophic rough digraphs G1 = (G1 , G1 ) and G2 = (G2 , G2 ) as shown in Fig.3 and Fig.6, respectively, then the direct sum G = (G1 ⊕ G2 , G1 ⊕ G2 ) of G1 and G2 as shown in Fig.7 is not a regular neutrosophic rough digraph.

Remark 2.2. If G1 = (G1 , G1 ) and G2 = (G2 , G2 ) are two regular neutrosophic rough digraphs with degree (m1 , m2 , m3 ) and (n1 , n2 , n3 ) on X1∗ , X2∗ , respectively, and X1∗ ∩X2∗ = φ, then G1 ⊕ G2 is a regular neutrosophic rough digraph if and only if (m1 , m2 , m3 ) = (n1 , n2 , n3 ).

9

Algorithms 2018, 11, 59

10 of 20

p(0.2, 0.1, 0.3)

) .3, 0.2

1)

2)

, 0.

0 .8

.3 , 0 .

0 .2

)

) b

b

4)

0 .2

,

0 .3

)

0 .3 ) 0 .4 ,

, 0 .4

(0.2 ,

(0.2, 0

0 .3 )

0 .4 ,

(0.2, 0

.9 ,

3)

.4 , 0 .

(0.2 ,

1) , 0.

0 .8 .9 , t(0

3) .4 , 0 .

t(0

) .1, 0.2 (0.1, 0

2) .1 , 0 .

(0.2, 0

.6 ) .5 , 0 .4 , 0 t(0

.6 )

0 .4

.4 ,

0 .4 )

0 .1 )

.1b (0

.8 ,

0 .4 ,

0 .2 )

Figure 5: Neutrosophic Rough Digraph G = (G1 ⊕ G2 , G1 ⊕ G2 ) ) (0. ⊕ G ).

Figure 5.0.3Neutrosophic rough digraph G = ( G1 ⊕ G2 , G 2 12, , 0.

0 .4 1, 0 u(0, 0.2,00.7) .3 , u(0, 0.5, 0.7) , 0 , . .

4)

b

0 .7 ,

(0.2, 0.3, 0.3)

1)

,0

r(0 G1 ⊕ G2 = (RX1 ⊕ RX2 , SY1 ⊕ SY2 )

(0.4,

(0.1, 0

0.

.2 (0

(0 .4

)

(0.2, 0.4, 0.3)

b

9,

)

.4 ,0

.3 ,0

0 .4 )

t(0

b

0

.4, 0. . 2 , 0 .4

, 0 .9 ,

0.

, .2

,0

q (0.9

5,

0 .4 )

6)

0 .1 ,

(0.

2

b

1)

0 .2 ,

5, 0.

(0.4,

0 .5 )

G1 ⊕ G2 = (RX1 ⊕ RX2 , SY1 ⊕ SY2 )

(0

b

(0.2,

b

)

, 0 .9 ,

)

0 .2

0 .4 )

0 .6

3, 0.

.4 ,

0 .1 ,

0 .4 , r(0

.4 ,

p(0.4, 0.8, 0.2)

(0.

2

s( 0.

,0

0 .2 ,

)

b

(0.2, 0.1, 0.3)

.3 )

b

q (0.

,0

0 .2

(0 .4

)

(0.2,

.3

1 0.

(0.2, 0.4, 0.3)

0 .1 )

(0

0 .1 ,

2,

0 .4 ,

.4 )

.1 ,

(0

.2

. ,0

(0.4,

1, 0

6)

, 0.

(0 ) .1.3 ,0 ,0

(0.1, 0.4, 0.3) b

5, 0.

0 .5

b

b

, (0.2

.5 ,

3, 0.

, .1

0.

p(0.2, 0.1, 0.3)

(0.

2

2)

b

0 .5 )

0 .2 )

.5 , 0

)

0 .1 ,

0 .1 ,

.4 , 0

0 .2

, (0.2

(0.4,

.

(0 s(0

.1 ,

(0.1, 0.4, 0.3)

b

0 2,

b

,0

q (0.9

2

1

) .2

q (0.

.

(0

. ,0

,0

p(0.4, 0.8, 0.2) (0 .1

(0.2, 0

b

( Remark 1. The direct sum of two regular neutrosophic rough digraphss(may not be regular neutrosophic rough s( 0.

0 .5 5, ) ) 6 , digraph,0as shown in the following example.

0.

0.

0 .4 .5 , 9, ( 8, .4 , .

0 ) ( 0 0 0 0 .3 , ) (0.2, 0.1, 0.3) 0, , .7 .1 , 0. (0.2, 0.3, 0.3) .7 ) 0 2r(0.4 0 .7 2, , 0 = ( G) , G r(G , 0 the two .regular 2 , 0 Example 5. Consider neutrosophic rough digraphs G ) and G = ( 2 .

0 2 2 ) as 1 1 . .7) ⊕ RX , SY 2⊕, G 1 0 G1 ⊕ G2 ,=0 (RX1 ⊕ RX2 , SY1.7⊕ ⊕ G2 = (RX SY2 ) 1 2 1 ) SY2 ) 0, G , GG1⊕ shown in Figures(0 3 and 6, respectively, then the direct sum G = ( G1 (⊕ G ) of G and G as shown in 2 2 1 1 2 Figure 7 is not a( regular neutrosophic rough digraph.

Rough Digraph (G 0, Figure 5: Neutrosophic 0, = (G1 ⊕ G2 , G.71) ⊕ G2 ) 7) 0 .2 0 .2 0. ,0 , ,0 ,0 .2 .2 0 0 .7 ) . , 7)u(0, 00.5, , u(0, 0.2, 0.7) ( b 0.7) (0 b b

b

x(0.8, 0.5, 0.2)

b

b

b

b

b b

w(0, 0.5,)0.7)(0

.2 ,

G2 = (RX02., SY2 )

7)

b

b

(0 ,

,0

x(0.8, 0.5, 0.2)

0

7

,0 .7 .2 , ,0 2 .

G20= (RX2 , SY2 ) 0.7 ) , (0 b

b

Figure 6: Regular Neutrosophic Rough Digraph G2 = (G2 , G2 )

0 .2

,

0 2,

.7

)

(0 ,

0 .2

0 2,

.7

)

v(0.3, 0.5, 0.7)

, (0

0.

v(0.1, 0.2, 0.8)

x(0.7, 0.3, 0.3)

w(0, (0 0.7) ) 0.2,

, .2

b

v(0.3, 0.5, 0.7)

v(0.1, 0.2, 0.8)

x(0.7, 0.3, 0.3)

b

, 0 rough digraphs . .

Definition 2.9. Let0.G on crisp sets X1∗ 1 = (G,10, G1 ) and G2 = (G2 , G2 ) be two neutrosophic .7 ) 7 ,0 ) 0 ( and X2∗ respectively. The residue product of G1 and G2 is a neutrosophic rough(0digraph G = G1 ∗ G2 = (G1 ∗ G2 , G1 ∗ G2 ), where G1 ∗ G2 = (RX1 ∗ RX2 , SY1 ∗ SY2 ) and G1 ∗ G2 = (RX1 ∗ RX2 , SY1 ∗ SY2 ) are neutrosophic digraphs, respectively, such that b b (1) w(0, 0.2, 0.7)

w(0, 0.5, 0.7)

TRX1 ∗RX2 (x1 , x2 ) = max{TRX1 (x1 ), TRX2 (x2 )},

(RX G2 = (RX2 , SY2 ) 2) IRXG = max{I 2 =2 (x 1 , x22,) SY RX1 (x1 ), IRX2 (x2 )}, 1 ∗RX FRX1 ∗RX2 (x1 , x2 ) = min{FRX1 (x1 ), FRX2 (x2 )}, ∀(x1 , x2 ) ∈ RX1 × RX2 Figure Regular neutrosophic rough digraph G2 = ( G . 2 , G2 ) Figure 6:6.Regular Neutrosophic Rough Digraph G22, G=2 )(G Definition 2.9. Let G1 = (G1 , G1 ) and G2 = (G2 , G2 ) be two neutrosophic rough digraphs on crisp sets X1∗ 10 G2 is a neutrosophic rough digraph G = G1 ∗ G2 = and X2∗ respectively. The residue product of G1 and (G1 ∗ G2 , G1 ∗ G2 ), where G1 ∗ G2 = (RX1 ∗ RX2 , SY1 ∗ SY2 ) and G1 ∗ G2 = (RX1 ∗ RX2 , SY1 ∗ SY2 ) are neutrosophic digraphs, respectively, such that (1) TRX1 ∗RX2 (x1 , x2 ) = max{TRX1 (x1 ), TRX2 (x2 )},

IRX1 ∗RX2 (x1 , x2 ) = max{IRX1 (x1 ), IRX2 (x2 )}, FRX1 ∗RX2 (x1 , x2 ) = min{FRX1 (x1 ), FRX2 (x2 )}, ∀(x1 , x2 ) ∈ RX1 × RX2

Algorithms 2018, 11, 59

11 of 20

p(0.1, 0.3, 0.8)

p(0.1, 0.6, 0.8) b

0.2 ,0 w .7) (0 (0, 0.2 .1 ,0 ,0 .3 .7) ,0 .8 ) b

8) 7) 0. , 0.

2 2, . . ,0 , 0 .8) (0 0.1 0 ( 3, v 0. , .1 (0

b

s(0.9, 0.6, 0.3)

b

(0,

b

8) u (0 0. ) (0 .1, 0 ,0 .2 3, .3 .

0 .5 ,0 5, ,0 ,0 .8 .1 8, 0. ( ) ) .

0 0 7 7 , ( . .

0 ) 0 .2, 0 ( , x 2 0.7 . ) ,0 (0 b

b

b

(0 w( , 0.2, (0 0, 0 0.7) .1 . , 0 5, 0 . .3 , 0 7) .8 ) b

b

7) .7) 0.

0 2, .

5, 0. ,0 , 8) (0 3 0.

0.

3, v( . ,0 .1 (0

q(0.4, 0.6, 0.8)

b (0 8) u( .1, 0. ) , 0.

0 .3 ,0 3, .3 ,0 0. .2 ,0 3 1 8) . , .

0 0 0 , .7 ( ( ) 0 7 . ) ,0 7 .

0 .2, ,0 x( 0.7 .2 ) ,0 0 ( b b

q(0.2, 0.3, 0.9)

s(0.8, 0.4, 0.4)

b

b b

r(0.1, 0.3, 0.8)

r(0.1, 0.6, 0.8)

G1 ⊕ G2 = (RX1 ⊕ RX2 , SY1 ⊕ SY2 )

G1 ⊕ G2 = (RX1 ⊕ RX2 , SY1 ⊕ SY2 )

Figure 7: Neutrosophic Rough Digraph G = (G1 ⊕ G2 , G1 ⊕ G2 ) Figure 7. Neutrosophic rough digraph G = ( G1 ⊕ G2 , G1 ⊕ G2 ).

(x , x2 )(y y2()G=2 ,TGSY (x1 ,two y1 ),regular neutrosophic rough digraphs with degree ∗SY SY(1G G21 ,= Remark 2. If G1 T= 2 )1 are 1 , 2G1 )1 and ∗ ∗ (x3 1) , on x2 )(y ISY1 (x1 , y1 ),and X1∗ ∩ X2∗ = φ, then G1 ⊕ G2 is a regular (m1 , m2 , m3 ) andI(SY n1 ,∗SY n2 ,2 n X11 ,,y2X)2= , respectively, neutrosophic rough digraph if and only if (m1 , F mSY n2 , 1n,3y)1. ) ∈ SY1 , x1 6= y2 FSY y1(),n1 , ∀(x 2 , 1m(x 3 )1 ,= 1 ∗SY2 (x1 , x2 )(y1 , y2 ) = (2)

Definition 9. Let G1 = ( G1 , G1 ) and G2 = ( G2 , G2 ) be two neutrosophic rough digraphs on crisp sets X1∗ and X2∗ respectively. The residue product of G1 and G2 is a neutrosophic rough digraph G = G1 ∗ G2 = ( G1 ∗ G2 , G1 ∗ G2 ), where G1 ∗ G2 = ( RX1 ∗ RX2 , SY1 ∗ SY2 ) and TRX1 ∗RX2 (x1 , x2 ) = max{TRX1 (x1 ), TRX2 (x2 )}, G1 ∗ G2 = ( RX 1 ∗ RX2 , SY1 ∗ SY2 ) are neutrosophic digraphs, respectively, such that IRX1 ∗RX2 (x1 , x2 ) = max{IRX1 (x1 ), IRX2 (x2 )},

(1)

FRX1 ∗RX2 (x1 , x2 ) = min{FRX1 (x1 ), FRX2 (x2 )}, ∀(x1 , x2 ) ∈ RX1 × RX2 TRX1 ∗ RX2 ( x1 , x2 ) = max{ TRX1 ( x1 ), TRX2 ( x2 )}, TSY1 ∗SY2 (x1 , x2 )(y1 , y2 ) = TSY1 (x1 , y1 ), IRX ( x1 , x2 ) = max{ IRX1 ( x1 ), IRX2 ( x2 )}, 1 ∗ RX2 ISY1 ∗SY2 (x1 , x2 )(y1 , y2 ) = ISY1 (x1 , y1 ), FRX1 ∗ RX2 ( x1 , x2 ) = min{ FRX1 ( x1 ), FRX2 ( x2 )}, ∀( x1 , x2 ) ∈ RX1 × RX2 FSY1 ∗SY2 (x1 , x2 )(y1 , y2 ) = FSY1 (x1 , y1 ), ∀(x1 , y1 ) ∈ SY1 , x1 6= y2

and (G , GT2SY ) be rough digraphs on the two crisp sets Example 2.6. Let G1 = (G ( xtwo ), TSY11,∗G )(2y= 2 )2 = 1 , x2G 1, y 1 , y1neutrosophic SY12)( x 1 X1∗ = {p, q} and X2∗ = {u, v, w, x} as shown in Fig. 8 and Fig.9. Then the residue product of G1 and G2 is a I 1 ∗SY2 ( x1 , x2 )(y1 , y2 ) = ISY1 ( x1 , y1 ), neutrosophic rough digraphSYG = G1 ∗ G2 = (G1 ∗ G2 , G1 ∗ G2 ) where G1 ∗ G2 = (RX1 ∗ RX2 , SY1 ∗ SY2 ) and FSY , x2 )( y1 ,respective y2 ) = FSY1figures ( x1 , y1 )are , ∀(shown x1 , y1 )in∈ Fig.10.

SY1 , x1 6= y2 G1 ∗ G2 = (RX1 ∗ RX2 , SY ∗ SY and the 1 1 ∗SY22()x1 ∗ Theorem (2) 2.1. If G1 = (G1 , G1 ) and G2 = (G2 , G2 ) are two neutrosophic rough digraph such that |X2 | > 1, then their residue product is regular if and only if G1 is regular.

( x1 , x2 ) = max{ TRX1 (digraph. x1 ), TRX2 ( x2 )}, Proof. Let G1 ∗ G2 be TaRX regular 1 ∗ RX2 neutrosophic rough Then, for any two vertices (y1max , y2 ){ Iin X ∗ × X2∗ , ( x2 )}, 1, x IRX (x ( x2 1), and x2 ) = ∗ RX RX (1x1 ), IRX 2

1

1

2

dGmin 2 )x1 , x2 ) ∈ RX1 × RX2 1 , x(2x)1= G1 ∗G FRX1 ∗ RX2 ( x1 , x2 ) = FRX ), FdRX ( x22(y )}1,, y∀( 1 ∗G2{(x 2 1 ⇒ dG1 (x1 ) = dG1 (y1 ) This is true for all vertices in X1∗ . Hence G1 is a regular neutrosophic rough digraph.

TSY

( x1 , x2 )(y1 , y2 ) = TSY1 ( x1 , y1 ),

ISY

( x1 , x2 )(y1 , y2 ) = ISY111 ( x1 , y1 ),

FSY

( x1 , x2 )(y1 , y2 ) = FSY1 ( x1 , y1 ), ∀( x1 , y1 ) ∈ SY1 , x1 6= y2

1 ∗ SY2 1 ∗ SY2 1 ∗ SY2

Algorithms 2018, 11, 59

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Example 6. Let G1 = ( G1 , G1 ) and G2 = ( G2 , G2 ) be two neutrosophic rough digraphs on the two crisp sets X1∗ = { p, q} and X2∗ = {u, v, w, x } as shown in Figures 8 and 9. Then the residue product of G1 and G2 is a neutrosophic rough digraph G = G1 ∗ G2 = ( G1 ∗ G2 , G1 ∗ G2 ) where G1 ∗ G2 = ( RX1 ∗ RX2 , SY1 ∗ SY2 ) and G1 ∗ G2 = ( RX1 ∗ RX2 , SY1 ∗ SY2 ) and the respective figures are shown in Figure 10.

(0.1,0.3, 0.3,0.1) 0.1) (0.1,

0.2, 0.7, 0.8) (p,(p, 0.2, 0.7, 0.8)

(q, 0.2, 0.2, 0.4, 0.6) (q, 0.6) bb

b

b

(0.1, 0.3,, SY 0.1)) = (RX GG 1 1= (RX1 ,1 SY11)

(p, 0.2, 0.7, 0.8) b

0.5, 0.9, 0.2) (p,(p, 0.5, 0.9, 0.2)

(q, 0.2, 0.4, 0.6) b

G1 = (RX1 , SY1 ) (0.1,0.3, 0.3,0.1) 0.1) (0.1,

(q, 0.5, 0.5, 0.4, 0.4, 0.1) (q, 0.1)

b

b

(p, 0.5, 0.9, 0.2) b

b

b

(0.1, 0.3, 0.1) G = (RX , SY ) G1 1= (RX1 ,1SY11)

(q, 0.5, 0.4, 0.1) b

G1 = (RXRough Figure DigraphG1G= 1 , SY1 ) 1 = Figure8:8. Neutrosophic Neutrosophic rough digraph ( G(G ).1 ).

1 ,1G 1, G Figure 8: Neutrosophic Rough Digraph G 1 = (G1 , G1 ). (u, 0.7, 0.7,, G 0.2)). (u,8:0.3, 0.2, 0.8) Figure Neutrosophic Rough Digraph (u, G10.7, = (G 1 1 0.2) 0.7, (u, 0.3, 0.2, 0.8) b

(0 ( .3 0. (0 , 0 3 , .3 .2 0 . , 0 , 0 2 , .2 .3 0 , 0 ) .3 .3 ) )

)

(0 (0 .2 .2 ( , 0 , 00. .2 .22, , 0 , 00. .4 .24, ) )0.

4

.

(0

0. ) 1, .4 .4 ) 0. ,0 ,0 1, .1 .1 ,0 ,0 .1 .1 (0 (0

b

b

b

b

4)

b

b

b

(w, 0.5, 0.7, 0.1)

(w, 0.3, 0.2, 0.7) b

b

b

b

(w,= 0.5, 0.7,, 0.1) (w,G0.3, 0.2, 0.7) G (RX 2 = (RX2 , SY2 ) 2 2 SY2 ) (w, 0.5, 0.7, 0.1) (w, 0.3, 0.2, 0.7) G2 = (RX2 , SY2 ) G2 = (RX2 , SY2 ) 9: Neutrosophic Rough Digraph 2 = (G 2 , G2 ).

GFigure G2 =G(RX 2 = (RX2 , SY2 ) 2 , SY2 ) Figure((q, 9: u), Neutrosophic Rough DigraphGG G ).

2 ,G 9.0.3, Neutrosophic rough digraph == (u), G(G 2 ,0.9, 2 ((p, 2 ). 20.2) ((p, u), 0.3, 0.7, 0.8) Figure 20.7, 0.4, 0.6) (0 (0 Figure 9: Neutrosophic Rough Digraph G2 = (G , G .12 ).

2 .1 , ((p, x), 0.7, 0.9, 0.2) ((q, x), 0.7, 0.7, 0.5) ((p, x), 0.7, 0.5) 0.7, 0.7, 0.2)0.9, 0.2) ((q, x), ((q, ((p,0.9, x), 0.7, x), 0.7, 0.7, 0.5)

(0 ( .10.1 ,0 , (0 .30. .1 ,0 , 03, .3 .10. ,0 ) 1) .1 )

((q, x), 0.2, 0.4, 0.6)

((p, x), 0.2, 0.7, 0.8)

b

((q, v), 0.6, 0.6, 0.1)

12 12

b

(0.1, 0.3, 0.1) (0 0.1) (0.1, 0.3, b .1 ,0 .3 ,0 .1 b b ) (0 (0 .1, ((p, w), 0.5, 0.9, 0.1) ((q, w), 0.5, 0.7, 0.1) .1 0. , 0 3, .3 0 , 0 2 ,.1SY G1 ∗ G2 b = (RX1 ∗ RX ∗ SY2 ) b .1 b ) 1 b ) ((p, w), 0.5, 0.9, 0.1) ((q, w), 0.5, 0.7, 0.1) ((p, w), 0.5, 0.9, 0.1) ((q, w), 0.5, 0.7, 0.1) = (G1 ∗ G2 , G1 ∗ G2 ).

G1 ∗ G2 = (RX1 ∗ RX2 , SY1 ∗ SY2 ) G1 ∗ G2 = (RX1 ∗ RX2 , SY1 ∗ SY2 )

Figure 10: Neutrosophic Rough Rough DigraphGG==(G (G1∗ ∗GG,2G , G ∗ G).2 ).

Figure 10: 10.

Neutrosophic Figure Neutrosophic roughDigraph digraph G ( G11 ∗ G22, G1 1∗1∗GG 2 )2.

12

b

((p, v), 0.6, 0.9, 0.1)

(0.1, 0.3, 0.1)

((q, v), 0.6,((q, 0.6,v), 0.1) 0.9, 0.1) ((p, v), 0.6, 0.6, 0.6, 0.1) ((p, v), 0.6, 0.9, 0.1)

Rough Digraph G

b

b

(0.1, 0.3, 0.1)

b

((q, w), 0.3, 0.4, 0.6) ((p, w), 0.3, 0.7, 0.7) ((q, w), 0.3, 0.4, 0.6) ((p, w), 0.3, 0.7, 0.7) Figure 10: Neutrosophic G1 ∗ G2 = (RX1 ∗ RX2 , SY1 ∗ SY2 ) G1 ∗ G2 = (RX1 ∗ RX2 , SY1 ∗ SY2 )

(0.1, 0.3, 0.1) (0.1, 0.3, 0.1) (0.1, 0.3, 0.1) (0.1, 0.3, 0.1)

b

) .1 ) ) ,0 .1 .1 .3 ,0 ,0 .3 1, 0 .3 . ,0 ,0 .1 (0 .1 (0 (0

G1 b∗ G2 = (RX1 ∗ RXb2 , SY1 ∗ SY2 )

b

) ) .1 .1 ) ,0 , 0 .1 .3 .3 , 0 ,0 , 0 .3 .1 .1 , 0 (0 (0 0.1 (

(0.1, 0.3, 0.1)

0.1) 0.3, 0.1) (0.1, 0.3, (0.1, ) .1 ,0 ) ) .3 .1 .1 ,0 ,0,0 .1 .3 .3 (0 ,0 ,0 .1 .1 (0 (0

b

((p, w), 0.3, 0.7, 0.7)

1) 0.

3, . ,0 .1 1) (0b 0).

30,.1 . , 0, .10.3 (.10b , (0 b

b

0 ((p, u), ((q, u), 0.7, 0.7, 0.1) ) 0.7, 0.9,.0.2) 3, .1 ( 0 b 0 b , 0 0.7, 0.9, ((p,3u), .10.2) .1) ((q, u), 0.7, 0.7, 0.1) . , 0 ( 0. b 0 b ) .

1, 3 1 .

1 ,0 . , 0 ( 0. .1 , 0) 3, ) .3.1 0. ,30, 0 b 1 1 ) . 0.

0 (0.1, 0.3, 0.1) (, .1 (0 0.1) 0.3, 0.1) (0.1, 0.3, (0.1, (0.1, 0.3, 0.1)

(

((p, v), 0.3, 0.7, 0.8) ((q, v), 0.3, 0.4, 0.6) ((p,((p, v), v), 0.3,0.3, 0.7,0.7, v),v), 0.3, 0.4, 0.8) 0.6) ((q, 0.3, 0.4, 0.8) ((q, 0.6)

(0.1, 0.1)0.1) 0.3,0.3, (0.1,

(0.1, 0.3, 0.1)

) .1 ,0 .3 ,0 .1 (0

b

((q, u), 0.7, 0.7, 0.1)

b

,

((q, w), 0.3, 0.4, 0.6)

b

b

b

0 0. 0.6) ((p, u), 0.3, ) 0.7, 0.8) .3, ((q, u), 0.3, 0.4, 1, . 1 b (0 0.

0.

00.3, b 1 , ((p, u), 0.7, 0.8) .

3, ((q, u), 0.3, 0.4, 0.6) 1 ) 3 . , 0. ( 0 ( 0.

0. b 0 b , 1) .1 3, 1 1) 1 . , ,0 0 ( 0 (0, 0.

0 .1 .3 . . ) 3 1 3 1 ,0 ) ,0 ,0 0. , 0. .1 .3 .1) b, b ) 0.3, 0.1) ,0 .1 0.3 (0.1, 0 .1 ( , 1) .

1 ) . ,0 (0 .3 b ,0 b 1) 1 .

(0.1, 0.3, 0.1) . ) 0 b b 1 (0 3, 0.

(0.1, 0.3, 0.1) 0.

3, .1) .

1, .

0 (0.1, 0.3, 0.1) ,0 , (0b .1) b .1 .3 0 (01, 0 3, 0.1) . .

0 , 0 ( , .1 .3 (0.1, 0.3, 0.1) (0b .1, 0 b 0 (0.1, 0.3, 0.1) (b b

) 1) .10. , 03, .30. , 01, .10.

(0 (

0.8) ((q,((q, 0.7, 0.2, ((p, 0.6)0.6) 0.4,0.4, 0.2,0.2, 0.8) x),x), 0.7, 0.2, x),x), ((p,

b

b

(0 (0 .1 (0, 0 .1 (0 ,0 .1.3 (0 .1 .3 , 0, 0 .1 (,00 ,0 ,0 .3.1 ..13 .1 , 0) .3 ,,00 ) ,0 .1 . .

3 1 ) .1 ,)0 ) .1 )

(0 (0 .1 .1 , 0 (,00 .2 .1.2, , 0 ,00 .4 .2.4, ) )0

b

b

(v, 0.6, 0.6, 0.1)

) .3 3 ) , 0 0 . 0 .3 ) .2 2 , , 0 0 . 0 .2 , .3 , (0 (0.3 .3, (0 (x,(x, 0.7,0.7, 0.7,0.7, 0.5)0.5) (x, 0.7, 0.7, 0.5)

(x, 0.1, 0.3, 0.9)

(x, 0.1, (x, 0.1, 0.3,0.3, 0.9)0.9)

b

b

b

b

(0 .3,(0 (0 0..23, .3, , 00.2 0.2 .1), 0 , 0 .1 ) .1 )

.4

(u, 0.7, 0.7, 0.2)

(u, 0.3, 0.2, 0.8) b

b

(v, 0.6,(v, 0.6, 0.1) 0.6, 0.6, 0.1) ) ) .1 .1 ) , 0 , 0 0 .1 .2 .2 , , 0 , 0 0 .2 .3 .3 , (0 (0 0.3 (

)

b

(v, 0.3, 0.2, 0.9) (v, 0.3, 0.2,0.2, 0.9)0.9) (v, 0.3, ) .4 , 0 4) ) .1 0 . 0 .4 , 0 1, , .2 0 . .1 (0 .2, .2, 0 (0 (0

b

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Theorem 1. If G1 = ( G1 , G1 ) and G2 = ( G2 , G2 ) are two neutrosophic rough digraph such that | X2∗ | > 1, then their residue product is regular if and only if G1 is regular.

Proof. Let G1 ∗ G2 be a regular neutrosophic rough digraph.

Then, for any two vertices ( x1 , x2 ) and (y1 , y2 ) in X1∗ × X2∗ , dG1 ∗G2 ( x1 , x2 ) = dG1 ∗G2 (y1 , y2 ) ⇒ dG1 ( x1 ) = dG1 (y1 ). ∗ This is true for all vertices in X1 . Hence G1 is a regular neutrosophic rough digraph.

Conversely, suppose that G1 = ( G1 , G1 ) is a (m1 , m2 , m3 )-regular neutrosophic rough digraph and G2 = ( G2 , G2 ) is any neutrosophic rough digraph with | X2∗ | > 1. If | X2∗ | > 1, then dG1 ∗G2 ( x1 , x2 ) = dG1 ( x1 ) = (m1 , m2 , m3 ). This is a constant ordered-triplet for all vertices in X1∗ × X2∗ . Hence G1 ∗ G2 is a regular neutrosophic rough digraph.

3. Applications to Decision-Making In this section, we present some real life applications of neutrosophic rough digraphs in decision making. In decision-making, the selection is facilitated by evaluating each choice on the set of criteria.

The criteria must be measurable and their outcomes must be measured for every decision alternative.

3.1. Online Reviews and Ratings Customer reviews are increasingly available online for a wide range of products and services.

As customers search online for product information and to evaluate product alternatives, they often have access to dozens or hundreds of product reviews from other customers. These reviews are very helpful in product selection. However, only considering the good reviews about a product is not very helpful in decision-making. The customer should keep in mind bad and neutral reviews as well. We use percentages of good reviews, bad reviews and neutral reviews of a product as truth membership, false membership and indeterminacy respectively.

Mrs. Sadia wants to purchase a refrigerator. For this purpose she visits web sites of different refrigerator companies. The refrigerator companies and their ratings by other customers are shown in Table 1.

Table 1. Companies and their ratings.

X∗

Good Reviews

Neutral

Bad Reviews

PEL Dawlance Haier Waves Orient

45% 52% 51% 47% 51%

29% 25% 43% 41% 35%

37% 49% 45% 38% 48%

Here X ∗ = {Pel(P),Dawlance(D),Haier(H),Waves(W),Orient(O)} and the neutrosophic set on X ∗ according to the reviews will be X = {( P, 0.45, 0.29, 0.37), ( D, 0.52, 0.25, 0.49), ( H, 0.51, 0.43, 0.45), (W, 0.47, 0.41, 0.38)(O, 0.51, 0.35, 0.48)}. The neutrosophic tolerance relation on X ∗ is given below R P D H W O

P (1,1,0) (0.5,0.6,0.9) (0.2,0.3,0.6) (0.1,0.2,0.3) (0.4,0.6,0.8)

D (0.5,0.6,0.9) (1,1,0) (0.1,0.6,0.9) (0.4,0.5,0.9) (0.9,0.8,0.2)

H (0.2,0.3,0.6) (0.1,0.6,0.9) (1,1,0) (0.2,0.9,0.6) (0.1,0.9,0.7)

W (0.1,0.2,0.3) (0.4,0.5,0.9) (0.2,0.9,0.6) (1,1,0) (0.2,0.5,0.9)

O (0.4,0.6,0.8) (0.9,0.8,0.2) (0.1,0.9,0.7) (0.2,0.5,0.9) (1,1,0)

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The lower and upper approximations of X are as follows: RX = {( P, 0.45, 0.29, 0.49), ( D, 0.51, 0.25, 0.49), ( H, 0.51, 0.35, 0.45),

(W, 0.45, 0.41, 0.40), (O, 0.51, , 0.25, 0.49)},

RX = {( P, 0.50, 0.35, 0.37), ( D, 0.52, 0.43, 0.48), ( H, 0.51, 0.43, 0.45),

(W, 0.47, 0.43, 0.37), (O, 0.52, 0.43, 0.48)}.

Let Y ∗ = {( P, D ), ( P, H ), ( D, H ), ( D, W ), ( H, W ), ( H, O), (W, P), (W, O), (O, P), (O, D )} be the subset of X ∗ × X ∗ and the tolerance relation S on Y ∗ is given as follows: S (P,D) (P,H) (D,H) (D,W) (H,W) (H,O) (W,P) (W,O) (O,P) (O,D)

(P,D) (1,1,0) (0.1,0.6,0.9) (0.1,0.6,0.9) (0.4,.5,.9) (0.2,0.3,0.9) (0.2,0.3,0.6) (0.1,0.2,0.9) (0.1,0.2,0.3) (0.4,0.6,0.9) (0.4,0.6,0.8)

(P,H) (0.1,0.6,0.9) (1,1,0) (0.5,0.6,0.9) (0.2,0.6,0.9) (0.2,0.3,0.6) (0.1,0.3,0.7) (0.1,0.2,0.6) (0.1,0.2,0.7) (0.2,0.3,0.8) (0.1,0.6,0.9)

(D,H) (0.1,0.6,0.9) (0.5,0.6,0.9) (1,1,0) (0.2,0.6,0.9) (0.1,0.6,0.9) (0.1,0.6,0.9) (0.2,0.3,0.9) (0.1,0.5,0.9) (0.2,0.3,0.6) (0.1,0.6,0.9)

(D,W) (0.4,0.5,0.9) (0.2,0.6,0.9) (0.2,0.9,0.6) (1,1,0) (0.1,0.6,.9) (0.1,0.5,0.9) (0.1,0.2,0.9) (0.2,0.5,0.9) (0.1,0.2,0.3) (0.4,0.5,0.9)

(H,W) (0.2,0.3,0.9) (0.2,0.3,0.6) (0.1,0.6,0.9) (0.1,0.6,0.9) (1,1,0) (0.2,0.5,0.9) (0.1,0.2,0.6) (0.2,0.5,0.9) (0.1,0.2,0.7) (0.1,0.5,0.9)

S (P,D) (P,H) (D,H) (D,W) (H,W) (H,O) (W,P) (W,O) (O,P) (O,D)

(H,O) (0.2,0.3,0.6) (0.1,0.3,0.7) (0.2,0.3,0.9) (0.1,0.2,0.9) (0.1,0.2,0.6) (1,1,0) (0.2,0.6,0.8) (0.2,0.9,0.6) (0.1,0.6,0.8) (0.1,0.8,0.7)

(W,P) (0.1,0.2,0.9) (0.1,0.2,0.6) (0.1,0.5,0.9) (0.2,0.5,0.9) (0.2,0.5,0.9) (0.2,0.6,0.8) (1,1,0) (0.4,0.6,0.8) (0.2,0.5,0.9) (0.2,0.5,0.9)

(W,O) (0.1,0.2,0.3) (0.1,0.2,0.7) (0.2,0.3,0.6) (0.1,0.2,0.3) (0.1,0.2,0.7) (0.2,0.9,0.6) (0.4,0.6,0.8) (1,1,0) (0.2,0.5,0.9) (0.2,0.5,0.9)

(O,P) (0.4,0.6,0.9) (0.2,0.3,0.8) (0.1,0.6,0.9) (0.4,0.5,0.9) (0.1,0.5,0.9) (0.1,0.6,0.8) (0.2,0.5,0.9) (0.2,0.5,0.9) (1,1,0) (0.5,0.6,0.9)

(O,D) (0.4,0.6,0.8) (0.1,0.6,0.9) (0.1,0.6,0.9) (0.1,0.5,0.9) (0.2,0.5,0.9) (0.1,0.8,0.7) (0.2,0.5,0.9) (0.2,0.5,0.9) (0.5,0.6,0.9) (1,1,0)

Thus, the lower and upper approximations of Y are calculated as follows: SY = {(( P, D ), 0.42, 0.23, 0.47), (( P, H ), 0.45, 0.28, 0.45), (( D, H ), 0.50, 0.21, 0.45),

(( D, W ), 0.43, 0.22, 0.45), (( H, W ), 0.41, 0.30, 0.44), (( H, O), 0.51, 0.22, 0.46), ((W, P), 0.42, 0.26, 0.40), ((W, O), 0.42, 0.23, 0.44), ((O, P), 0.43, 0.25, 0.48),

((O, D ), 0.50, 0.22, 0.48)} SY = {(( P, D ), 0.42, 0.30, 0.44), (( P, H ), 0.50, 0.30, 0.41), (( D, H ), 0.50, 0.30, 0.45),

(( D, W ), 0.43, 0.30, 0.45), (( H, W ), 0.41, 0.30, 0.44), (( H, O), 0.51, 0.30, 0.46), ((W, P), 0.42, 0.26, 0.37), ((W, O), 0.45, 0.30, 0.44), ((O, P), 0.50, 0.28, 0.45),

((O, D ), 0.50, 0.30, 0.47)}.

Thus, G = ( RX, SY ) and G = ( RX, SY ) are the neutrosophic digraphs as shown in Figure 11.

To find the best company, we use the following formula: S ( vi ) = ∑ vi

where

∈X∗

( TRX (vi )× TRX (vi ))+( IRX (vi )× IRX (vi ))−( FRX (vi )× FRX (vi )) 1−{ T (vi v j )+ I (vi v j )− F (vi v j )}

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T (vi v j ) = max∗ TSY (vi v j ) × max∗ TSY (vi v j ), vj ∈X

vj ∈X

vj ∈X

vj ∈X

vj ∈X

vj ∈X

I (vi v j ) = max∗ ISY (vi v j ) × max∗ ISY (vi v j ), F (vi v j ) = min∗ FSY (vi v j ) × min∗ FSY (vi v j ).

By direct calculations we have S( P) = 0.167, S( D ) = 0.156, S( H ) = 0.268, S(W ) = 0.272, S(O) = 0.155.

From the above calculations it is clear that Waves is the best company for refrigerator.

P (0.50, 0.35, 0.37)

P (0.45, 0.29, 0.49)

b b

(0.5 0, 0 .30, 0.45 )

37)

(0.42, 0.

26, 0.

(0 .5 2, 0.

43 ,0 .4 8) O

45)

21, 0.

(0.5 0, 0 .

40)

(0.42, 0.26 , 0.

(0 .5 1, 0.

25 ,0 .4 9) O

G = (RX, SY )

8) .4 ,0 43 0.

2, .5 (0

b

1) .4 ,0 30 0.

0, .5 (0

9) .4 ,0 25 0.

1, .5 (0

5) .4 ,0 28 0.

5, .4 (0

(0.42 , 0.23 , 0.47 ) . ) 8 0 .4 , ) ,0 2 .2 3 0 , .45 4 .50 b 0 . ( ,0 (0 .22 0 , (0.5 43 (0.

(0 1, 0.22 , 0.4 .4 6) 2, 0.

23 ,0 .4 4) ) , 0.44 , 0.30 W( b b (0.41 0.4 45) 5, 0 , 0. .41 .35 0 , , 0.

1 40) 0.5 H(

5) (0.42 .4 , 0.30 , 0.44 ,0 8 ) 2 .

0 .47) 0 , , 0 .3 0 5) 0 , 0 b .5 (0.5 0.4 0 ( 0, 0.3 (0.5 , (0 1, 0.3 .43 0, 0.

(0 .4 46) 5, 0.

30 ,0 .4 4) .44) 30, 0 W( b 1, 0. b ) 0.4 (0.4 .45 7, 0 3, 0 .43 0.4 , 0. , 1 37) 0.5 H(

D

D

8) .4 ,0 25

b

G = (RX, SY ) Figure G= Figure 11.11: G= ( G,(G, G )G) .

3.2. Context of Recruitment

T (vi vj ) = max∗ TSY (vi vj ) × max∗ TSY (vi vj ), vj ∈X

vj ∈X

vj ∈X

vj ∈X

(visomething vj ), I(vfor = max ISY (viavailable vj ) × max∗isISY i vj ) the Choosing the right candidate not that should be left to vj position ∈X ∗ vj ∈X F (vi vj ) = min∗ FSY (vi vj ) × min∗ FSY (vi vj ). chance or guesswork.

How choose the right Bytodirect calculations wecandidate? have S(P ) =process 0.167, S(D) 0.156, S(H) = 0.268, S(W ) =is0.272, S(O) =against 0.155. the suitability In any recruitment the =ability of the candidate weighed of the candidate.

Pakistan Telecommunication Company Limited (PTCL) wants to recruit From above calculations it is clear that Waves is the best company for refrigerator. a person for the post of administrator.

To keep the procedure simple, the company wants appoint of their employee on the basis of education (Edu) and experience (Exp).

3.2 toContext Recruitment Let Choosing X ∗ = {(the C1,right Edu)candidate , (C1, Exp ), the (C2,position Edu), (available C2, Expis ), not (C3,something Edu), (C3, Exp )} bebethe of candidates for that should leftset to chance or guesswhowork. applied for the post and their corresponding attributes. Let R be a neutrosophic tolerance on X ∗ How to choose the right candidate. given as follows: In any recruitment process the ability of the candidate is weighed up against the suitability of the candidate.

Pakistan Telecommunication Company Limited(PTCL) wants to recruit a person for the post of administrator.

To keep the simple the (C1,Exp) company wants to appoint their employee on the basis of education(Edu) and R procedure (C1,Edu) (C2,Edu) (C2,Exp) (C3,Edu) (C3,Exp) ∗ experience(Exp).

Let X = {(C1, Edu), (C1, Exp), (C2, Edu), (C2, Exp), (C3, Edu), (C3, Exp)} be the set of (C1,Edu) (1,1,0) (0.3,0.6,0.1) (0.6,0.7,0.2) (0.6,0.5,0.8) (0.3,0.2,0.1) (0.9,0.1,0.1) candidates who applied for the post and their corresponding attributes. Let R be a neutrosophic tolerance on (C1,Exp) (0.3,0.6,0.1) (1,1,0) (0.9,0.9,0.3) (0.8,0.7,0.6) (0.4,0.5,0.9) (0.3,0.1,0.1) X ∗ given as follows: (C2,Edu) (0.6,0.7,0.2) (0.9,0.9,0.3) (1,1,0) (0.6,0.5,0.1) (0.3,0.2,0.1) (0.4,0.8,0.7) R (C2,Exp) (C1,Edu) (0.6,0.5,0.8) (C1,Exp) (0.8,0.7,0.6) (C2,Edu) (0.6,0.5,0.1) (C2,Exp) (1,1,0) (C3,Edu) (0.1,0.1,0.2) (C3,Exp) (0.5,0.6,0.7) (C1,Edu) (C3,Edu) (1,1,0) (0.3,0.2,0.1)(0.3,0.6,0.1) (0.4,0.5,0.9)(0.6,0.7,0.2) (0.3,0.2,0.1)(0.6,0.5,0.8) (0.1,0.1,0.2)(0.3,0.2,0.1) (1,1,0) (0.9,0.1,0.1) (0.2,0.1,0.2) (C1,Exp) (C3,Exp)(0.3,0.6,0.1) (0.9,0.1,0.1) (1,1,0) (0.3,0.1,0.1)(0.9,0.9,0.3) (0.4,0.8,0.7)(0.8,0.7,0.6) (0.5,0.6,0.7)(0.4,0.5,0.9) (0.2,0.1,0.2)(0.3,0.1,0.1) (1,1,0) (1,1,0) (0.6,0.5,0.1) (0.3,0.2,0.1) (0.4,0.8,0.7) .

(C2,Edu) (0.6,0.7,0.2) (0.9,0.9,0.3) (C2,Exp) (0.6,0.5,0.8) (0.8,0.7,0.6) (0.6,0.5,0.1) (1,1,0) (0.1,0.1,0.2) (0.5,0.6,0.7) (C3,Edu) (0.3,0.2,0.1) (0.4,0.5,0.9) (0.3,0.2,0.1) (0.1,0.1,0.2) (1,1,0) (0.2,0.1,0.2) (1,1,0) (C3,Exp) (0.9,0.1,0.1) (0.3,0.1,0.1) (0.4,0.8,0.7) (0.5,0.6,0.7) (0.2,0.1,0.2) Let X = {((C1, Edu), 0.9, 0.1, 0.5), ((C1, Exp), 0.2, 0.6, 0.5), ((C2, Edu), 0.7, 0.2, 0.3), ((C2, Exp), 0.1, 0.3, 0.9), ((C3, Edu), 0.4, 0.6, 0.8), ((C3, Exp), 0.8, 0.1, 0.2)} be a neutrosophic set define on X ∗ . Then the

15

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Let X = {((C1, Edu), 0.9, 0.1, 0.5), ((C1, Exp), 0.2, 0.6, 0.5), ((C2, Edu), 0.7, 0.2, 0.3), ((C2, Exp), 0.1, 0.3, 0.9), ((C3, Edu), 0.4, 0.6, 0.8), ((C3, Exp), 0.8, 0.1, 0.2)} be a neutrosophic set define on X ∗ . Then the lower and upper approximations of X are given as: RX = {((C1, Edu), 0.2, 0.1, 0.6), ((C1, Exp), 0.2, 0.2, 0.8), ((C2, Edu), 0.1, 0.2, 0.6),

((C2, Exp), 0.1, 0.3, 0.9), ((C3, Edu), 0.2, 0.6, 0.8), ((C3, Exp), 0.2, 0.1, 0.5)},

RX = {((C1, Edu), 0.9, 0.6, 0.2), ((C1, Exp), 0.7, 0.6, 0.2), ((C2, Edu), 0.7, 0.6, 0.3),

((C2, Exp), 0.6, 0.6, 0.3), ((C3, Edu), 0.4, 0.6, 0.2), ((C3, Exp, 0.9, 0.3, 0.2)}.

Let Y ∗ = {(C1, Edu)(C1, Exp), (C1, Exp)(C2, Edu), (C1, Edu)(C3, Exp), (C3, Exp)(C1, Exp), (C1, Exp)(C2, Exp), (C2, Exp)(C2, Edu), (C3, Exp)(C3, Edu), (C3, Edu)(C2, Exp), (C3, Exp)(C2, Exp)} ⊆ X ∗ × X ∗ and S be a neutrosophic tolerance relation on Y ∗ given as follows: S (C1,Edu)(C1,Exp) (C1,Exp)(C2,Edu) (C1,Edu)(C3,Exp) (C3,Exp)(C1,Exp) (C1,Exp)(C2,Exp) (C2,Exp)(C2,Edu) (C3,Exp)(C2,Exp) (C3,Exp)(C3,Edu) (C3,Edu)(C2,Exp)

(C1,Edu)(C1,Exp) (1,1,0) (0.3,0.6,0.3) (0.3,0.1,0.1) (0.9,0.1,0.1) (0.3,0.6,0.6) (0.6,0.5,0.8) (0.8,0.1,0.6) (0.4,0.1,0.9) (0.3,0.2,0.6)

S (C1,Edu)(C1,Exp) (C1,Exp)(C2,Edu) (C1,Edu)(C3,Exp) (C3,Exp)(C1,Exp) (C1,Exp)(C2,Exp) (C2,Exp)(C2,Edu) (C3,Exp)(C2,Exp) (C3,Exp)(C3,Edu) (C3,Edu)(C2,Exp)

(C1,Exp)(C2,Edu) (0.3,0.6,0.3) (1,1,0) (0.3,0.6,0.7) (0.3,0.1,0.3) (0.6,0.5,0.1) (0.8,0.7,0.6) (0.3,0.1,0.1) (0.3,0.1,0.1) (0.4,0.5,0.9)

(C2,Exp)(C2,Edu) (0.6,0.5,0.8) (0.8,0.7,0.6) (0.4,0.5,0.8) (0.5,0.6,0.7) (0.6,0.5,0.6) ( 1,1,0) (0.5,0.5,0.7) (0.3,0.2,0.7) (0.1,0.1,0.2)

(C1,Edu)(C3,Exp) (0.3,0.1,0.1) (0.3,0.6,0.7) (1,1,0) (0.3,0.1,0.1) (0.3,0.6,0.7) (0.4,0.5,0.8) (0.5,0.1,.7) (0.2,0.1,.2) (0.3,.2,.7)

(C3,Exp)(C2,Exp) (0.8,0.1,0.6) (0.3,0.1,0.1) (0.5,0.1,0.7) (0.8,0.7,0.6) (0.3,0.1,0.1) (0.5,0.5,0.7) (1,1,0) (0.1,0.1,0.2) (0.2,0.1,0.2)

(C3,Exp)(C1,Exp) (0.9,0.1,0.1) (0.3,0.1,0.3) (0.3,0.1,0.1) (1,1,0) (0.3,0.1,0.6) (0.5,0.6,0.7) (0.8,0.7,0.6) (0.4,0.5,0.9) (0.2,0.1,0.6)

(C3,Exp)(C3,Edu) (0.4,0.1,0.9) (0.3,0.1,0.1) (0.2,0.1,0.2) (0.4,0.5,0.9) (0.1,0.1,0.2) (0.3,0.2,0.7) (0.1,0.1,0.2) ( 1,1,0) (0.1,0.1,0.2)

(C1,Exp)(C2,Exp) (0.3,0.6,0.6) (0.6,0.5,0.1) (0.3,0.6,0.7) (0.3,0.1,0.6) (1,1,0) (0.6,0.5,0.6) (0.3,0.1,0.1) (0.1,0.1,0.2) (0.4,0.5,0.9)

(C3,Edu)(C2,Exp) (0.3,0.2,0.6) (0.4,0.5,0.9) (0.3,0.2,0.7) (0.2,0.1,0.6) (0.4,0.5,0.9) (0.1,0.1,0.2) (0.2,0.1,0.2) (0.1,0.1,0.2) (1,1,0)

Let Y = {((C1, Edu)(C1, Exp), 0.2, 0.1, 0.1), ((C1, Exp)(C2, Edu), 0.1, 0.1, 0.3), ((C1, Edu)(C3, Exp), 0.2, 0.1, 0.2), ((C3, Exp)(C1, Exp), 0.2, 0.1, 0.2), ((C1, Exp)(C2, Exp), 0.1, 0.2, 0.3), ((C2, Exp)(C2, Edu), 0.1, 0.2, 0.3)), ((C3, Exp)(C2, Exp), 0.1, 0.1, 0.3), ((C3, Exp)(C3, Edu), 0.2, 0.1, 0.2), ((C3, Edu)(C2, Exp), 0.1, 0.3, 0.3)} be neutrosophic rough set on Y ∗ . Then the lower and upper approximations of Y are given as follows: SY = {((C1, Edu)(C1, Exp), 0.2, 0.1, 0.3), ((C1, Exp)(C2, Edu), 0.1, 0.1, 0.3),

((C1, Edu)(C3, Exp), 0.2, 0.1, 0.3), ((C3, Exp)(C1, Exp), 0.2, 0.1, 0.3),

((C1, Exp)(C2, Exp), 0.1, 0.2, 0.3), ((C2, Exp)(C2, Edu, 0.1, 0.2, 0.3)), ((C3, Exp)(C2, Exp), 0.1, 0.1, 0.3), ((C3, Exp)(C3, Edu), 0.1, 0.1, 0.3), ((C3, Edu)(C2, Exp), 0.1, 0.3, 0.3)}, SY = {((C1, Edu)(C1, Exp), 0.2, 0.2, 0.1), ((C1, Exp)(C2, Edu), 0.2, 0.3, 0.2),

((C1, Edu)(C3, Exp), 0.2, 0.2, 0.1), ((C3, Exp)(C1, Exp), 0.2, 0.2, 0.1),

((C1, Exp)(C2, Exp), 0.2, 0.2, 0.1), ((C2, Exp)(C2, Edu, 0.2, 0.2, 0.3)), ((C3, Exp)(C2, Exp), 0.2, 0.2, 0.2), ((C3, Exp)(C3, Edu), 0.2, 0.2, 0.2), ((C3, Edu)(C2, Exp), 0.2, 0.3, 0.2)}

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b

(0 (0 .2 .2 , ,0 0 .1 .1 ,0 ,0 .3 .3 ) )

) .3 .3 ) ,0 0 .2 .2 , ,0 0 .1 .1 , (0 (0

) ) .3 .3 ,0 ,0 .1 .1 ,0 ,0 .1 .1 (0 (0 0.5)0.5) , , .1 .1 ,0 ,0 ) ) .2 .2 .3 .3 , 0 , 0 , 0 , 0 p) p) .1 .1 x x ,0 ,0 E ,E .2 .2 3 , 3 (0 (0 ((C ((C

b

(( C (( 2, E C 2, du E ), 0 ((C1, Exp), 0.2, 0.2, 0.8) du b .1 b ), , 0 ((C1, Exp), 0.2, 0.2, 0.8) 0 (0.2, 0.1, 0.3) (0.1, 0.1, 0.3) .2 b b .1 ,0 ,0 (0.2, 0.1, 0.3) (0.1, 0.1, 0.3) .2 .6 ) ,0 .6 )

(0.1, 0.1, 0.3) (0.1, 0.1, 0.3)

b

b

(( (( C C 2, 2, E E xp x ), p), (0 ( 0 . 0 . .1 0 .1 1, 1 , 0 , 0 . , 0 .2 0 .2 3 , .3 , 0 , 0 . , 0 .3 0 .3 9 ) .9 ) ) )

(C (C 1, 1, E E d du u ), ), 0 0 . .2 2, , 0 0 . .1 1, , 0 0 . .6 6) )

Thus, G = ( RX, SY ) and G = ( RX, SY ) are the neutrosophic digraphs as shown in Figures 12 and 13.

b

b

3) 0.

3 , .3 ) . ,0 0 .1 .3 , (0 1, 0 . b (0 ((C3, Edu), 0.2, 0.6, 0.8) b ((C3, Edu), 0.2, 0.6, 0.8)

Figure 12. Neutrosophic Digraph G = ( RX, SY ) (C (C 1, 1, E E d du u ), ), 0 0 . .9 9, , 0 0 . .6 6, , 0 . 0 .2 2) )

Figure 12: Neutrosophic Digraph G = (RX, SY ) Figure 12: Neutrosophic Digraph G = (RX, SY ) (( C (( 2, C E 2, d E u) du , 0 ((C1, Exp), 0.7, 0.6, 0.2) ), b .7 b ((C1, Exp), 0.7, 0.6, 0.2) 0. , 0 (0.2, 0.2, 0.1) (0.2, 0.3, 0.2) b b 7 , 0 .6 , .6 0 .3 (0.2, 0.2, 0.1) (0.2, 0.3, 0.2) ,0 ) .3 )

b

b

b

) .2 .2 ) ,0 0 .3 .3 , ,0 0 .2 .2 , (0 (0

(0 (0 .2 .2 , ,0 0 .2 .2 ,0 ,0 .1 .1 ) )

(( (( CC 2, 2, EE xpx ),p), 0. 0 (0.(0 6 , .6 2 , .2 0., 0 0., 0 6 , .6 2 , .2 0., 0 0., 0 3 ) .3 3 ) .3 ) )

) ) .2 .2 ,0,0 .2 .2 ,0,0 .2 .2 (0 (0 .2).2) 0 0 3, 3, 0. 0.

9 , .9 , ) ) .1 .1 0., 0 , 0 , 0 ),p) .2 .2 xp x ,0 ,0 E E .2 .2 3 , 3 , (0 (0 ((C((C b

(0.2, 0.2, 0.2) (0.2, 0.2, 0.2)

b

b

b

) .2 , 00.2) 3 . , 0 3, .2 0 .

(00.2, (

b ((C3, Edu), 0.4, 0.6, 0.2) ((C3, Edu), 0.4, 0.6, 0.2)

G = (RX, SY ) G = (RX, SY ) Figure13.

13: Neutrosophic Digraph GG==(RX, SY ) ) Figure DigraphG ( RX, Figure 13:Neutrosophic Neutrosophic Digraph = (RX, SYSY ) max{I

(C1), I

(C2), I

(C3)} = max{0.8, 0.65, 0.65} = 0.8

RY (C2), IRY (C3)} = max{0.8, 0.65, 0.65} = 0.8 max{IRY (C1), Ithe To find the best employee using calculations, we have RY RY following RY

Thus, C1 is best employee for the post under consideration. So PTCL can hire C1 for the post of administrator.

Thus, C1 is best employee for the post under consideration.

So PTCL can IRY (C1,Edu )+ IRY (C1,Exp ) 0.9+hire 0.7 C1 for the post of administrator.

IRY (C1) =

IRY (C2) = IRY (C3) =

2 IRY (C2,Edu)+ IRY (C2,Exp) 2 IRY (C3,Edu)+ IRY (C3,Exp) 2

=

= =

2 0.7+0.6 2 0.4+0.9 2

= 0.8

= 0.65 = 0.65

max{ IRY (C1), IRY (C2), IRY (C3)} = max{0.8, 0.65, 0.65} = 0.8 17 17

Thus, C1 is the best employee for the post under consideration. So, PTCL can hire C1 for the post of administrator.

4. Comparative Analysis of Rough Neutrosophic Digraphs and Neutrosophic Rough Digraphs Rough neutrosophic digraphs and neutrosophic rough digraphs are two different notions on the basis of their construction and approach. In rough neutrosophic digraphs, the relation defined on the universe of discourse is a crisp equivalence relation that only classifies the elements which are related. On the other hand, in neutrosophic rough digraphs the relation defined on the set is

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neutrosophic tolerance relation. The neutrosophic tolerance relation not only classifies the elements of the set which are related but also expresses their relation in terms of three components, namely truth membership (T), Indeterminacy (I) and falsity (F). This approach leaves room for indeterminacy and incompleteness. Below, we apply the method of rough neutrosophic digraphs to the above presented application (online reviews and ratings).

Here X ∗ = { Pel ( P), Dawlance( D ), Haier ( H ), Waves(W ), Orient(O)} and the neutrosophic set ∗ on X according to the reviews will be X = {( P, 0.45, 0.29, 0.37), ( D, 0.52, 0.25, 0.49), ( H, 0.51, 0.43, 0.45), (W, 0.47, 0.41, 0.38), (O, 0.51, 0.35, 0.48)}. The equivalence relation on X ∗ is given below R P D H W O

P 1 0 1 0 1

D 0 1 0 0 0

H 1 0 1 0 1

W 0 0 0 1 0

O 1 0 1 0 1

The lower and upper approximations of X are as follows: RX = {( P, 0.45, 0.29, 0.48), ( D, 0.52, 0.25, 0.49), ( H, 0.45, 0.29, 0.48),

(W, 0.47, 0.41, 0.38), (O, 0.45, , 0.29, 0.48)},

RX = {( P, 0.51, 0.43, 0.37), ( D, 0.52, 0.25, 0.49), ( H, 0.51, 0.43, 0.37),

(W, 0.47, 0.41, 0.38), (O, 0.51, 0.43, 0.37)}.

Let Y ∗ = {( P, D ), ( P, H ), ( D, H ), ( D, W ), ( H, W ), ( H, O), (W, P), (W, O), (O, P), (O, D )} be the subset of X ∗ × X ∗ and the equivalence relation S on Y ∗ is given as follows: S (P,D) (P,H) (D,H) (D,W) (H,W) (H,O) (W,P) (W,O) (O,P) (O,D)

(P,D) 1 0 0 0 0 0 0 0 0 0

(P,H) 0 1 0 0 0 1 0 0 1 1

(D,H) 0 0 1 0 0 0 0 0 0 0

(D,W) 0 0 0 1 0 0 0 0 0 0

(H,W) 0 0 0 0 1 0 0 0 0 0

(H,O) 0 1 0 0 0 1 0 0 1 1

(W,P) 0 0 0 0 0 0 1 1 0 0

(W,O) 0 0 0 0 0 0 1 1 0 0

(O,P) 0 1 0 0 0 1 0 0 1 1

(O,D) 0 1 0 0 0 1 0 0 1 1

Thus, the lower and upper approximations of Y are calculated as follows: SY = {(( P, D ), 0.45, 0.25, 0.48), (( P, H ), 0.42, 0.24, 0.37), (( D, H ), 0.45, 0.25, 0.47),

(( D, W ), 0.45, 0.24, 0.48), (( H, W ), 0.45, 0.29, 0.38), (( H, O), 0.42, 0.24, 0.37), ((W, P), 0.42, 0.22, 0.37), ((W, O), 0.42, 0.22, 0.37), ((O, P), 0.42, 0.24, 0.37),

((O, D ), 0.42, 0.24, 0.37)} SY = {(( P, D ), 0.45, 0.25, 0.48), (( P, H ), 0.45, 0.29, 0.37), (( D, H ), 0.45, 0.25, 0.47),

(( D, W ), 0.45, 0.24, 0.48), (( H, W ), 0.45, 0.29, 0.38), (( H, O), 0.45, 0.29, 0.37), ((W, P), 0.45, 0.29, 0.35), ((W, O), 0.45, 0.29, 0.35), ((O, P), 0.45, 0.29, 0.37),

((O, D ), 0.45, 0.29, 0.37)}.

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To find the best company ratings, we use the following formula: S ( vi ) = ∑

vi ∈ X ∗

( TRX (vi )× TRX (vi ))+( IRX (vi )× IRX (vi ))−( FRX (vi )× FRX (vi )) 1−{ T (vi v j )+ I (vi v j )− F (vi v j )}

where T (vi v j ) = max∗ TSY (vi v j ) × max∗ TSY (vi v j ), vj ∈X

vj ∈X

vj ∈X

vj ∈X

vj ∈X

vj ∈X

I (vi v j ) = max∗ ISY (vi v j ) × max∗ ISY (vi v j ), F (vi v j ) = min∗ FSY (vi v j ) × min∗ FSY (vi v j ).

By direct calculations, we have S( P) = 0.20, S( D ) = 0.0971, S( H ) = 0.2077, S(W ) = 0.2790, S(O) = 0.2011.

From the above calculations, we have Waves as the best choice and Dawlance as the least choice for refrigerator. This is because the relation applied in this method is crisp equivalence relation which does not consider the uncertainty between the companies of the same equivalence class. Whereas in our proposed method, least choice for refrigerator is different. So, the results may vary when we apply the method of rough neutrosophic digraphs and neutrosophic rough digraphs on the same application. This means that rough neutrosophic digraphs and neutrosophic rough digraphs have a different approach.

5. Conclusions Neutrosophic set and rough set are two different theories to deal with uncertainty and imprecise and incomplete information. Due to the limitation of human knowledge to understand the complex problems, it is very difficult to apply only a single type of uncertainty method to deal with such problems. Therefore, it is necessary to develop hybrid models by incorporating the advantages of many other different mathematical models dealing with uncertainty. Thus, by combining these two mathematical tools, we have presented a new hybrid model, namely, neutrosophic rough digraphs.

We have escribed regular neutrosophic rough digraphs and we have presented novel applications of our proposed hybrid in decision-making. We have given a comparison of both models, rough neutrosophic digraphs and neutrosophic rough digraphs. We plan to extend our research work to (1) Neutrosophic rough hypergraphs; (2) Bipolar neutrosophic rough hypergraphs; (3) Soft rough neutrosophic graphs; (4) Decision support systems based on neutrosophic rough information.

Author Contributions: M.A., N.I. and S.S. conceived and designed the experiments; F.S. analyzed the data; N.I. and S.S. wrote the paper.

Acknowledgments: The authors are very thankful to the Editor and referees for their valuable comments and suggestions for improving the paper.

Conflicts of Interest: The authors declare that they have no conflict of interest regarding the publication of the research article.

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