Application of intuitionistic neutrosophic graph structures in decision-making

2017·0.27 MB·English

by  Muhammad Akram, Muzzamal Sitara;

Annals of Fuzzy Mathematics and Informatics
FMI
@
Volume14, No. 1, (July 2017), pp. 1–27
ISSN:2093–9310 (print version) (cid:13)c KyungMoon Sa Co.
ISSN:2287–6235 (electronic version) http://www.kyungmoon.com
http://www.afmi.or.kr
Application of intuitionistic neutrosophic graph
structures in decision-making
Muhammad Akram, Muzzamal Sitara
Received 9 January 2017; Revised 8 February2017; Accepted 16 February 2017
Abstract. In this research study, we present concept of intuitionis-
ticneutrosophicgraph structures. Weintroducethecertain operations on
intuitionistic neutrosophicgraph structuresand elaborate them with suit-
ableexamples. Further,weinvestigatesomeremarkablepropertiesofthese
operators. Moreover,wediscussahighlyworthwhilereal-lifeapplicationof
intuitionisticneutrosophicgraphstructuresindecision-making. Lastly,we
elaborate general procedure of our application bydesigning an algorithm.
2010 AMS Classification: 03E72, 05C72,05C78, 05C99
Keywords: Intuitionistic neutrosophic graph structures, Decision making
Corresponding Author: Muhammad Akram ([email protected])
1. Introduction
G
raphical models are extensively useful tools for solving combinatorial prob-
lems of different fields including optimization, algebra, computer science, topology
andoperationsresearchetc. Fuzzygraphicalmodelsarecomparativelymorecloseto
nature,becauseinnaturevaguenessandambiguityoccurs. Therearemanycomplex
phenomena andprocessesin science and technologyhavingincomplete information.
To deal such cases we needed a theory different from classical mathematics. Graph
structures as generalized simple graphs are widely used for study of edge colored
and edge signed graphs, also helpful and copiously used for studying large domains
of computer science. Initially in 1965, Zadeh [29] proposed the notion of fuzzy
sets to handle uncertainty in a lot of real applications. Fuzzy set theory is finding
large number of applications in real time systems, where information inherent in
systems has various levels of precision. Afterwards, Turksen [26] proposed the idea
ofinterval-valuedfuzzy set. Butin varioussystems,there aremembershipandnon-
membership values, membership value is in favor of an event and non-membership
value is against of that event. Atanassov [8] proposed the notion of intuitionistic
MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,1–27
fuzzy set in 1986. The intuitionistic fuzzy sets are more practical and applicable in
real-life situations. Intuitionistic fuzzy set deal with incomplete information, that
is, degree of membership function, non-membership function but not indeterminate
andinconsistentinformationthatexists definitely in many systems, including belief
system, decision-support systems etc. In 1998, Smarandache [24] proposed another
notion of imprecise data named as neutrosophic sets. “Neutrosophic set is a part
of neutrosophy which studies the origin, nature and scope of neutralities, as well
as their interactions with different ideational spectra”. Neutrosophic set is recently
proposed powerful formal framework. For convenient usage of neutrosophic sets in
real-lifesituations,Wangetal. [27]proposedsingle-valuedneutrosophicsetasagen-
eralization of intuitionistic fuzzy set[8]. A neutrosophic set has three independent
components having values in unit interval [0, 1]. On the other hand, Bhowmik and
Pal [10, 11] introduced the notions of intuitionistic neutrosophic sets and relations.
Kauffman[16]definedfuzzygraphonthebasisofZadeh’sfuzzyrelations[30]. Rosen-
feld [21] investigated fuzzy analogue of various graph-theoretic ideas in 1975. Later
on,Bhattacharyagavesomeremarksonfuzzygraphin1987. BhutaniandRosenfeld
discussed M-strong fuzzy graphs with their properties in [12]. In 2011, Dinesh and
Ramakrishnan [15] put forward fuzzy graph structures and investigated its prop-
erties. In 2016, Akram and Akmal [1] proposed the notion of bipolar fuzzy graph
structures. Broumi et al. [13] portrayed single-valued neutrosophic graphs. Akram
andShahzadi[2]introducedthenotionofneutrosophicsoftgraphswithapplications.
Akram and Shahzadi [4] highlighted some flaws in the definitions of Broumi et al.
[13] and Shah-Hussain [22]. Akram et al. [5] also introduced the single-valued neu-
trosophic hypergraphs. Representation of graphs using intuitionistic neutrosophic
soft sets was discussed in [3]. In this paper, we present concept of intuitionistic
neutrosophicgraphstructures. Weintroducethecertainoperationsonintuitionistic
neutrosophic graphstructures and elaborate them with suitable examples. Further,
we investigate some remarkable properties of these operators. Moreover,we discuss
a highly worthwhile real-life application of intuitionistic neutrosophic graph struc-
tures in decision-making. Lastly, we elaborate general procedure of our application
by designing an algorithm.
We have used standard definitions and terminologies in this paper. For other no-
tations, terminologies and applications not mentioned in the paper, the readers are
referred to [3,6, 7, 9, 13, 14, 17, 18, 20, 22, 23, 25, 28, 30].
2. Intuitionistic Neutrosophic Graph Structures
Definition 2.1. ([23]). Let Gˇ = (P,P ,P ,...,P ) and Gˇ = (P′,P′,P′,...,P′)
1 1 2 r 2 1 2 r
be two GSs, Cartesian product of Gˇ and Gˇ is defined as:
1 1
Gˇ ×Gˇ =(P ×P′,P ×P′,P ×P′,...,P ×P′),
1 2 1 1 2 2 r r
where P ×P′ = {(k l)(k l) | l ∈ P′, k k ∈ P }∪{(kl )(kl ) | k ∈ p,l l ∈ P′},
h h 1 2 1 2 h 1 2 1 2 h
h=(1,2,...,r).
Definition 2.2. ([23]). Let Gˇ = (P,P ,P ,...,P ) and Gˇ = (P′,P′,P′,...,P′)
1 1 2 n 2 1 2 r
be two GSs, cross product of Gˇ and Gˇ is defined as:
1 2
Gˇ ∗Gˇ =(P ∗P′,P ∗P′,P ∗P′,...,P ∗P′),
1 2 1 1 2 2 r r
2
MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,1–27
where P ∗P′ = {(k l )(k l )|k k ∈P , l l ∈P′}, h=(1,2,...,r).
h h 1 1 2 2 1 2 h 1 2 h
Definition 2.3. ([23]). Let Gˇ = (P,P ,P ,...,P ) and Gˇ = (P′,P′,P′,...,P′)
1 1 2 r 2 1 2 r
be two GSs, lexicographic product of Gˇ and Gˇ is defined as:
1 2
Gˇ •Gˇ =(P •P′,P •P′,P •P′,...,P •P′),
1 2 1 1 2 2 r r
whereP •P′ ={(kl )(kl )|k ∈P,l l ∈P′}∪{(k l )(k l )|k k ∈P ,l l ∈P′},
h h 1 2 1 2 h 1 1 2 2 1 2 h 1 2 h
h=(1,2,...,r).
Definition 2.4. ([23]). Let Gˇ = (P,P ,P ,...,P ) and Gˇ = (P′,P′,P′,...,P′)
1 1 2 r 2 1 2 r
be two GSs, strong product of Gˇ and Gˇ is defined as:
1 2
Gˇ ⊠Gˇ =(P ⊠P′,P ⊠P′,P ⊠P′,...,P ⊠P′),
1 2 1 1 2 2 r r
where P ⊠P′ = {(k l)(k l)|l ∈P′, k k ∈P }∪{(kl )(kl )|k ∈P,l l ∈P′} ∪
h h 1 2 1 2 h 1 2 1 2 h
{(k l )(k l )|k k ∈P , l l ∈P′}, h=(1,2,...,r).
1 1 2 2 1 2 h 1 2 h
Definition 2.5. ([23]). Let Gˇ = (P,P ,P ,...,P ) and Gˇ = (P′,P′,P′,...,P′)
1 1 2 r 2 1 2 n
be two GSs, composition of Gˇ and Gˇ is defined as:
1 2
Gˇ ◦Gˇ =(P ◦P′,P ◦P′,P ◦P′,...,P ◦P′),
1 2 1 1 2 2 r r
where P ◦P′ = {(k l)(k l)| l ∈P′, k k ∈P }∪{(kl )(kl ) | k ∈P,l l ∈ P′} ∪
h h 1 2 1 2 h 1 2 1 2 h
{(k l )(k l )|k k ∈P , l ,l ∈P′ such that l 6=l }, h=(1,2,...,r).
1 1 2 2 1 2 h 1 2 1 2
Definition 2.6. ([23]). Let Gˇ = (P,P ,P ,...,P ) and Gˇ = (P′,P′,P′,...,P′)
1 1 2 r 2 1 2 r
be two GSs, union of Gˇ and Gˇ is defined as:
1 2
Gˇ ∪Gˇ =(P ∪P′,P ∪P′,P ∪P′,...,P ∪P′).
1 2 1 1 2 2 r r
Definition 2.7. ([23]). Let Gˇ = (P,P ,P ,...,P ) and Gˇ = (P′,P′,P′,...,P′)
1 1 2 r 2 1 2 r
be two GSs, join of Gˇ and Gˇ is defined as:
1 2
Gˇ +Gˇ =(P +P′,P +P′,P +P′,...,P +P′),
1 2 1 1 2 2 r r
where P +P′ = P ∪P′, P +P′ = P ∪P′ ∪P′′ for h = (1,2,...,r). P′′ consists
h h h h h h
of all those edges which join the vertices of P and P′.
Definition 2.8. ([19]). LetV beafixedset. Ageneralizedintuitionistic fuzzysetI
of V is an object having the form I={(v,µ (v),ν (v))|v ∈ V}, where the functions
I I
µ : V → [0,1] and ν : V → [0,1] define the degree of membership and degree of
I I
nonmembership of an element v ∈V, respectively, such that
min{µ (v),ν (v)}≤0.5, for all v ∈V.
I I
This condition is called the generalized intuitionistic condition.
Definition 2.9. ([10, 11]). A set I ={T (v),I (v),F (v) : v ∈V} is said to be an
I I I
intuitionistic neutrosophic (IN)set, if
(i) {T (v)∧I (v)}≤0.5, {I (v)∧F (v)}≤0.5, {F (v)∧T (v)}≤0.5,
I I I I I I
(ii) 0≤T (v)+I (v)+F (v)≤2.
I I I
Definition 2.10. An intuitionistic neutrosophic graph is a pair G = (A,B) with
underlying set V, where T , F , I : V → [0,1] denote the truth, falsity and
A A A
indeterminacy membershipvalues of the vertices in V andT , F , I : E ⊆V ×V
B B B
→[0,1]denote the truth, falsity andindeterminacy membershipvalues ofthe edges
kl∈E such that
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MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,1–27
(i) T (kl)≤T (k)∧T (l), F (kl)≤F (k)∨F (l), I (kl)≤I (k)∧I (l),
B A A B A A B A A
(ii) T (kl)∧I (kl)≤0.5, T (kl)∧F (kl)≤0.5, I (kl)∧F (lk)≤0.5,
B B B B B B
(iii) 0≤T (kl)+F (kl)+I (kl)≤2, ∀ k,l∈V.
B B B
Definition 2.11. Gˇ = (O,O ,O ,...,O ) is said to be an intuitionistic neutro-
i 1 2 r
sophic graph structure(INGS) of graph structure Gˇ = (P,P ,P ,...,P ), if O =
1 2 r
<k,T(k),I(k),F(k)>andO =<kl,T (kl),I (kl),F (kl)>aretheintuitionistic
h h h h
neutrosophic(IN) sets on the sets P and P , respectively such that
h
(i) T (kl)≤T(k)∧T(l), I (kl)≤I(k)∧I(l), F (kl)≤F(k)∨F(l),
h h h
(ii) T (kl)∧I (kl)≤0.5, T (kl)∧F (kl)≤0.5, I (kl)∧F (kl)≤0.5,
h h h h h h
(iii) 0≤T (kl)+I (kl)+F (kl)≤2, for all kl∈O , h∈{1,2,...,r},
h h h h
where,OandO areunderlyingvertexandh-edgesetsofINGSGˇ ,h∈{1,2,...,r}.
h i
Example 2.12. An intuitionistic neutrosophic graph structure is represented in
Fig. 1.
k.,.(0,2.0504)2b OOk2O17(((20(00.02...27,1,0,,.0003...,k322,01,,.0(4.000)5...)553,))O03O.b(420,(.0b01..,2O5,0).011.(,1,00..04.2)6,)0O.4O3(,10(0.01..6,5k,0)6.04.(,02,b0.2.05.b,)60)Ok.O48O2,3((01(0(00.0.6...525)1,,,,0000...3.424,,,0,00..06.5.)6)6)) bk.,.,(0.60405)5
k3(0.5,0.4,b0O.33)(0.2,0.1,0.3k)4(0.b2,0.2,0.5)
Figure 1. An intuitionistic neutrosophic graph structure
Now we define the operations on INGSs.
Definition2.13. LetGˇ =(O ,O ,O ,...,O )andGˇ =(O ,O ,O ,...,Q )
i1 1 11 12 1r i2 2 21 22 2r
be INGSs of GSs Gˇ = (P ,P ,P ,...,P ) and Gˇ = (P ,P ,P ,...,P ), re-
1 1 11 12 1r 2 2 21 22 2r
spectively.
Cartesian product of Gˇ and Gˇ , denoted by
i1 i2
Gˇ ×Gˇ = (O ×O ,O ×O ,O ×O ,...,O ×O ),
i1 i2 1 2 11 21 12 22 1r 2r
is defined as:
 T(O1×O2)(kl)=(TO1 ×TO2)(kl)=TO1(k)∧TO2(l)
(i)  I(O1×O2)(kl)=(IO1 ×IO2)(kl)=IO1(k)∧IO2(l)
F(O1×O2)(kl)=(FO1 ×FO2)(kl)=FO1(k)∨FO2(l)
for all kl∈P ×P ,
1 2
 T(O1h×O2h)(kl1)(kl2)=(TO1h ×TO2h)(kl1)(kl2)=TO1(k)∧TO2h(l1l2)
(ii)  I(O1h×O2h)(kl1)(kl2)=(IO1h ×IO2h)(kl1)(kl2)=IO1(k)∧IO2h(l1l2)
F(O1h×O2h)(kl1)(kl2)=(FO1h ×FO2h)(kl1)(kl2)=FO1(k)∨FO2h(l1l2)
for all k ∈P , l l ∈P ,
1 1 2 2h
4
MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,1–27
 T(O1h×O2h)(k1l)(k2l)=(TO1h ×TO2h)(k1l)(k2l)=TO2(l)∧TO1h(k1k2)
(iii)  I(O1h×O2h)(k1l)(k2l)=(IO1h ×IO2h)(k1l)(k2l)=IO2(l)∧IO2h(k1k2)
F(O1h×O2h)(k1l)(q2l)=(FO1h ×FO2h)(k1l)(k2l)=FO2(l)∨FO2h(k1k2)
for all l∈P , k k ∈P .
2 1 2 1h
Example 2.14. Consider Gˇ = (O ,O ,O ) and Gˇ = (O ,O ,O ) are two
i1 1 11 12 i2 2 21 22
INGSsofGSsGˇ =(P ,P ,P )andGˇ =(P ,P ,P )respectively,asrepresented
1 1 11 12 2 2 21 22
in Fig. 2, where P ={k k }, P ={k k }, P ={l l }, P ={l l }.
11 1 2 12 3 4 21 1 2 22 2 3
k1(0.5,0.2,0.6) k3(0.4,0.3,0.4) l1(0.2,0b.2,0.3) l3(0.5,0b.4,0.5)
b b
)4
OO1121((00..45,,00..32,,00..68)) .,.,.O02020(12 O22(0.3,0.3,0.5)
k4(0.5,0b .3,0.6) k2(0.5,b0.3,0.8) l2(0.3,b0.3,0.4)
Gˇi1=(O1,O11,O12) Gˇi2=(O2,O21,O22)
Figure 2. Two INGSs Gˇ and Gˇ
i1 i2
CartesianproductofGˇ andGˇ definedasGˇ ×Gˇ ={O ×O ,O ×O ,O ×
i1 i2 i1 i2 1 2 11 21 12
O } is represented in Fig. 3.
22
k1l2(0.3,0.2,0.6) k2l2(0.3,0.3,0.8)
bO11×O21(0.3,0.2,0.8) b
k1l1(0.2b,0.2,O0.161)× O21(0.2,0.2,0.6O)12×k1Obl223(0.(30,0..52,,0.06).2,0.6) k2l1(0.2,0.2O1,1×O021(0..2,08.2b,0).8) O12×O22(0.3,0.3,0.8)k2l3(0.5,b0.3,0.8)
O11×O21(0.2,0.2,0.8) O11×O21(0.5,0.2,0.8)
5
MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,1–27
k3l2(0.3,0.2,0.4) k4l2(0.3,0.3,0.6)
bO12×O22(0.3,0.3,0.6) b
k3l1(0.2b,0.2,O0.141)× O21(0.2,0.2,0.4O)12×k3Obl223(0.(3,00..42,0,.50).3,0.5) k4l1(0.2,0.2O,11×O021(0..2,06.2b,0).6) O12×O22(0.3,0.3,0.6)k4l3(0.5,b0.3,0.6)
O12×O22(0.2,0.2,0.6) O12×O22(0.4,0.3,0.6)
Figure 3. Gˇ ×Gˇ
i1 i2
Theorem2.15. CartesianproductGˇ ×Gˇ =(O ×O ,O ×O ,O ×O ,...,O ×
i1 i2 1 2 11 21 12 22 1r
O ) of two INGSs of GSs Gˇ and Gˇ is an INGS of Gˇ ×Gˇ .
2r 1 2 1 2
Proof. We consider two cases:
Case 1: For k ∈P , l l ∈P
1 1 2 2h
T(O1h×O2h)((kl1)(kl2))=TO1(k)∧TO2h(l1l2)
≤T (k)∧[T (l )∧T (l )]
O1 O2 1 O2 2
=[T (k)∧T (l )]∧[T (k)∧T (l )]
O1 O2 1 O1 O2 2
=T(O1×O2)(kl1)∧T(O1×O2)(kl2),
I(O1h×O2h)((kl1)(kl2))=IO1(k)∧IO2h(l1l2)
≤I (k)∧[I (l )∧I (l )]
O1 O2 1 O2 2
=[I (k)∧I (l )]∧[I (k)∧I (l )]
O1 O2 1 O1 O2 2
=I(O1×O2)(kl1)∧I(O1×O2)(kl2),
F(O1h×O2h)((kl1)(kl2))=FO1(k)∨FO2h(l1l2)
≤F (k)∨[F (l )∨F (l )]
O1 O2 1 O2 2
=[F (k)∨F (l )]∨[F (k)∨F (l )]
O1 O2 1 O1 O2 2
=F(O1×O2)(kl1)∨F(O1×O2)(kl2),
for kl ,kl ∈P ×P .
1 2 1 2
Case 2: For k ∈P , l l ∈P
2 1 2 1h
T(O1h×O2h)((l1k)(l2k))=TO2(k)∧TO1h(l1l2)
≤T (k)∧[T (l )∧T (l )]
O2 O1 1 O1 2
=[T (k)∧T (l )]∧[T (k)∧T (l )]
O2 O1 1 O2 O1 2
=T(O1×O2)(l1k)∧T(O1×O2)(l2k),
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MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,1–27
I(O1h×O2h)((l1k)(l2k))=IO2(k)∧IO1h(l1l2)
≤I (k)∧[I (l )∧I (l )]
O2 O1 1 O1 2
=[I (k)∧I (l )]∧[I (k)∧I (l )]
O2 O1 1 O2 O1 2
=I(O1×O2)(l1k)∧I(O1×O2)(l2k),
F(O1h×O2h)((l1k)(l2k))=FO2(k)∨FO1h(l1l2)
≤F (k)∨[F (l )∨F (l )]
O2 O1 1 O1 2
=[F (k)∨F (l )]∨[F (k)∨F (l )]
O2 O1 1 O2 O1 2
=F(O1×O2)(l1k)∨F(O1×O2)(l2k),
for l k,l k ∈P ×P .
1 2 1 2
Both cases exists ∀h∈{1,2,...,r}. This completes the proof. (cid:3)
Definition2.16. LetGˇ =(O ,O ,O ,...,Q )andGˇ =(O ,O ,O ,...,Q )
i1 1 11 12 1r i2 2 21 22 2r
be INGSs of GSs Gˇ = (P ,P ,P ,...,P ) and Gˇ = (P ,P ,P ,...,P ), re-
1 1 11 12 1r 2 2 21 22 2r
spectively. Cross product of Gˇ and Gˇ , denoted by
i1 i2
Gˇ ∗Gˇ = (O ∗O ,O ∗O ,O ∗O ,...,O ∗O ),
i1 i2 1 2 11 21 12 22 1r 2r
is defined as:
 T(O1∗O2)(kl)=(TO1 ∗TO2)(kl)=TO1(k)∧TO2(l)
(i)  I(O1∗O2)(kl)=(IO1 ∗IO2)(kl)=IO1(k)∧IO2(l)
F(O1∗O2)(kl)=(FO1 ∗FO2)(kl)=FO1(k)∨FO2(l)
for all kl∈P ×P ,
1 2
 T(O1h∗O2h)(k1l1)(k2l2)=(TO1h ∗TO2h)(k1l1)(k2l2)=TO1h(k1k2)∧TO2h(l1l2)
(ii)  I(O1h∗O2h)(k1l1)(k2l2)=(IO1h ∗IO2h)(k1l1)(k2l2)=IO1h(k1k2)∧IO2h(l1l2)
F(O1h∗O2h)(k1l1)(k2l2)=(FO1h ∗FO2h)(k1l1)(k2l2)=FO1h(k1k2)∨FO2h(l1l2)
for all k k ∈P , l l ∈P .
1 2 1h 1 2 2h
Example 2.17. Cross product of INGSs Gˇ and Gˇ shown in Fig. 2 is defined as
i1 i2
Gˇ ∗Gˇ = {O ∗O ,O ∗O ,O ∗O } and is represented in Fig. 4.
i1 i2 1 2 11 21 12 22
k2l1(0.2,0.2,0.8) k4l3(0.5,0.3,0.6)k3l3(0.4,0.3,0.5) k2l2(0.3,0.3,0.8)
b O11∗O21(0.2,0.2,k01.8l)3(0.5,0b.2,0.6) O12∗O22(0.3,0.3,0.6) b b O12∗O22(0.3,0.3k,04.6l1)(0.2,0b.2,0.6O)11∗O21(0.2,0.2,0.8) b
k2l3(0.5,0.3,0.8) k3l1(0.2,0.2,0.4)
k1l2(0.3,0.2,0.b6)kb3l2(0.3,0.3,0.4) b b k4l2(0.3,0.3,0.6b) bk1l1(0.2,0.2,0.6)
Figure 4. Gˇ ∗Gˇ
i1 i2
Theorem2.18. CrossproductGˇ ∗Gˇ =(O ∗O ,O ∗O ,O ∗O ,...,O ∗O )
i1 i2 1 2 11 21 12 22 1r 2r
of two INGSs of GSs Gˇ and Gˇ is an INGS of Gˇ ∗Gˇ .
1 2 1 2
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MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,1–27
Proof. For all k l ,k l ∈P ∗P
1 1 2 2 1 2
T(O1h∗O2h)((k1l1)(k2l2))=TO1h(k1k2)∧TO2h(l1l2)
≤[T (k )∧T (k )]∧[T (l )∧T (l )]
O1 1 O1 2 O2 1 O2 2
=[T (k )∧T (l )]∧[T (k )∧T (l )]
O1 1 O2 1 O1 2 O2 2
=T(O1∗O2)(k1l1)∧T(O1∗O2)(k2l2),
I(O1h∗O2h)((k1l1)(k2l2))=IO1h(k1k2)∧IO2hl1l2)
≤[I (k )∧I (k )]∧[I (l )∧I (l )]
O1 1 O1 2 O2 1 O2 2
=[I (k )∧I (l )]∧[I (k )∧I (l )]
O1 1 O2 1 O1 2 O2 2
=I(O1∗O2)(k1l1)∧I(O1∗O2)(k2l2),
F(O1h∗O2h)((k1l1)(k2l2))=FO1h(k1k2)∨FO2h(l1l2)
≤[F (k )∨F (k )]∨[F (l )∨F (l )]
O1 1 O1 2 O2 1 O2 2
=[F (k )∨F (l )]∨[F (k )∨F (l )]
O1 1 O2 1 O1 2 O2 2
=F(O1∗O2)(k1l1)∨F(O1∗O2)(k2l2),
for h∈{1,2,...,r}. This completes the proof. (cid:3)
Definition2.19. LetGˇ =(O ,O ,O ,...,O )andGˇ =(O ,O ,O ,...,O )
i1 1 11 12 1r i2 2 21 22 2r
be INGSs of GSs Gˇ = (P ,P ,P ,...,P ) and Gˇ = (P ,P ,P ,...,P ), re-
1 1 11 12 1r 2 2 21 22 2r
spectively. Lexicographic product of Gˇ and Gˇ , denoted by
i1 i2
Gˇ •Gˇ = (O •O ,O •O ,O •O ,...,O •O ),
i1 i2 1 2 11 21 12 22 1r 2r
is defined as:
 T(O1•O2)(kl)=(TO1 •TO2)(kl)=TO1(k)∧TO2(l)
(i)  I(O1•O2)(kl)=(IO1 •IO2)(kl)=IO1(k)∧IO2(l)
F(O1•O2)(kl)=(FO1 •FO2)(kl)=FO1(k)∨FO2(l)
for all kl∈P ×P
1 2
 T(O1h•O2h)(kl1)(kl2)=(TO1h •TO2h)(kl1)(kl2)=TO1(k)∧TO2h(l1l2)
(ii)  I(O1h•O2h)(kl1)(kl2)=(IO1h •IO2h)(kl1)(kl2)=IO1(k)∧IO2h(l1l2)
F(O1h•O2h)(kl1)(kl2)=(FO1h •FO2h)(kl1)(kl2)=FO1(k)∨FO2h(l1l2)
for all k ∈P , l l ∈P ,
1 1 2 2h
 T(O1h•O2h)(k1l1)(k2l2)=(TO1h •TO2h)(k1l1)(k2l2)=TO1h(k1k2)∧TO2h(l1l2)
(iii)  I(O1h•O2h)(k1l1)(k2l2)=(IO1h •IO2h)(k1l1)(k2l2)=IO1h(k1k2)∧IO2h(l1l2)
F(O1h•O2h)(k1l1)(k2l2)=(FO1h •FO2h)(k1l1)(k2l2)=FO1h(k1k2)∨FO2h(l1l2)
for all k k ∈P , l l ∈P .
1 2 1h 1 2 2h
Example 2.20. Lexicographic product of INGSs Gˇ and Gˇ shown in Fig. 2 is
i1 i2
defined as Gˇ •Gˇ = {O •O ,O •O ,O •O } and is represented in Fig. 5.
i1 i2 1 2 11 21 12 22
8
MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,1–27
k1l1(0.2,0.2,0.6) k2l1(0.2,0.2,0.8) k2l3(0.5,0.3,0.8)
•OO.,(0.121,0.22016)bb O1O1•11O•2O1(201.(20,.20.,20,.20.,b08b)80)20.2,0(..,.O12O118•) O12• O22(0.3,0.3,0.8) bb
k1l2(0.3,0.2,0.6) O12•O22(0k.23l,20(0.2.3,0,0.6.3),0.8) k1l3(0.5,0.2,0.6)
O11•O21(0.2,0.2,0.6)
kk34ll11((00..22,,00..22,,00..46))bb O11• O21(0.2,0.2,0.4) )50k3.0,3.0,(.2O22O1•3lkb3b3(l02.(40,.03.,30,.02.,50).6O)O121•2•OO222(20(.03.,3,0.03.,3,0.06bb.6))603030(.,.,.OO2221)•kk44ll32((00..53,,00..33,,00..66))
Figure 5. Gˇ •Gˇ
i1 i2
Theorem2.21. LexicographicproductGˇ •Gˇ =(O •O ,O •O ,O •O ,...,O •
i1 i2 1 2 11 21 12 22 1r
O ) of two INGSs of the GSs Gˇ and Gˇ is an INGS of Gˇ •Gˇ .
2r 1 2 1 2
Proof. We consider two cases:
Case 1: For k ∈P , l l ∈P
1 1 2 2h
T(O1h•O2h)((kl1)(kl2))=TO1(k)∧TO2h(l1l2)
≤T (k)∧[T (l )∧T (l )]
O1 O2 1 O2 2
=[T (k)∧T (l )]∧[T (k)∧T (l )]
O1 O2 1 O1 O2 2
=T(O1•O2)(kl1)∧T(O1•O2)(kl2),
I(O1h•O2i)((kl1)(kl2))=IO1(k)∧IO2h(l1l2)
≤I (k)∧[I (l )∧I (l )]
O1 O2 1 O2 2
=[I (k)∧I (l )]∧[I (k)∧I (l )]
O1 O2 1 O1 O2 2
=I(O1•O2)(kl1)∧I(O1•O2)(kl2),
F(O1h•O2i)((kl1)(kl2))=FO1(k)∨FO2h(l1l2)
≤F (k)∨[F (l )∨F (l )]
O1 O2 1 O2 2
=[F (k)∨F (l )]∨[F (k)∨F (l )]
O1 O2 1 O1 O2 2
=F(O1•O2)(kl1)∨F(O1•O2)(kl2),
for kl ,kl ∈P •P .
1 2 1 2
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MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,1–27
Case 2: For k k ∈P ,l l ∈P
1 2 1h 1 2 2h
T(O1h•O2h)((k1l1)(k2l2))=TO1h(k1k2)∧TO2h(l1l2)
≤[T (k )∧T (k )]∧[T (l )∧T (l )]
O1 1 O1 2 O2 1 O2 2
=[T (k )∧T (l )]∧[T (k )∧T (l )]
O1 1 O2 1 O1 2 O2 2
=T(O1•O2)(k1l1)∧T(O1•O2)(k2l2),
I(O1h•O2h)((k1l1)(k2l2))=IO1h(k1k2)∧IO2h(l1l2)
≤[I (k )∧I (k )]∧[I (l )∧I (l )]
O1 1 O1 2 O2 1 O2 2
=[I (k )∧I (l )]∧[I (k )∧I (l )]
O1 1 O2 1 O1 2 O2 2
=I(O1•O2)(k1l1)∧I(O1•O2)(k2l2),
F(O1h•O2h)((k1l1)(k2l2))=FO1h(k1k2)∨FO2h(l1l2)
≤[F (k )∨F (k )]∨[F (l )∨F (l )]
O1 1 O1 2 O2 1 O2 2
=[F (k )∨F (l )]∨[F (k )∨F (l )]
O1 1 O2 1 O1 2 O2 2
=F(O1•O2)(k1l1)∨F(O1•O2)(k2l2),
for k l ,k l ∈P •P .
1 1 2 2 1 2
Both cases hold for h∈{1,2,...,r}. This completes the proof.
(cid:3)
Definition2.22. LetGˇ =(O ,O ,O ,...,O )andGˇ =(O ,O ,O ,...,O )
i1 1 11 12 1r i2 2 21 22 2r
be INGSs of GSs Gˇ = (P ,P ,P ,...,P ) and Gˇ = (P ,P ,P ,...,P ), re-
1 1 11 12 1r 2 2 21 22 2r
spectively. Strong product of Gˇ and Gˇ , denoted by
i1 i2
Gˇ ⊠Gˇ = (O ⊠O ,O ⊠O ,O ⊠O ,...,O ⊠O ),
i1 i2 1 2 11 21 12 22 1r 2r
is defined as:
(i)  IT((OO11⊠⊠OO22))((kkll))==((ITOO11⊠⊠ITOO2)2()k(kl)l)==IOTO1(1k()k)∧∧IOT2O(2l)(l)
F(O1⊠O2)(kl)=(FO1 ⊠FO2)(kl)=FO1(k)∨FO2(l)
for all kl∈P ×P ,
1 2
(ii)  IT((OO11hh⊠⊠OO22hh))((kkll11))((kkll22))==((ITOO11hh⊠⊠ITOO2h2h)()k(kl1l1)()k(kl2l2))==IOTO1(1k()k)∧∧IOT2Oh2(hl1(ll12l)2)
F(O1h⊠O2h)(kl1)(kl2)=(FO1h ⊠FO2h)(kl1)(kl2)=FO1(k)∨FO2h(l1l2)
for all k ∈P , l l ∈P ,
1 1 2 2h
(iii)  IT((OO11hh⊠⊠OO22hh))((kk11ll))((kk22ll))==((ITOO11hh⊠⊠ITOO2h2h)()k(k11l)l()k(k22l)l)==ITOO2(2l()l)∧∧IOT2Oh1(hk(1kk12k)2)
F(O1h⊠O2h)(k1l)(k2l)=(FO1h ⊠FO2h)(k1l)(k2l)=FO2(l)∨FO2h(k1k2)
for all l∈P , k k ∈P ,
2 1 2 1h
(iv)  TI((OO11hh⊠⊠OO22hh))((kk11ll11))((kk22ll22))==((ITOO11hh⊠⊠ITOO2h2h)()k(k11l1l1)()k(k22l2l2))==ITOO1h1(hk(1kk12k)2)∧∧IOT2Oh2(hl1(ll12l)2)
F(O1h⊠O2h)(k1l1)(k2l2)=(FO1h ⊠FO2h)(k1l1)(k2l2)=FO1h(k1k2)∨FO2h(l1l2)
for all k k ∈P , l l ∈P .
1 2 1h 1 2 2h
Example 2.23. Strong product of INGSs Gˇ and Gˇ shown in Fig. 2 is defined
i1 i2
as Gˇ ⊠Gˇ = {O ⊠O ,O ⊠O ,O ⊠O } and is represented in Fig. 6.
i1 i2 1 2 11 21 12 22
10