Annalsof Fuzzy Mathematicsand Informatics
Volume14,No.1,(July2017),pp.1–27
ISSN:2093–9310(printversion)
ISSN:2287–6235(electronicversion) http://www.afmi.or.kr
@FMI
c KyungMoonSaCo. http://www.kyungmoon.com
Applicationofintuitionisticneutrosophicgraph structuresindecision-making
MuhammadAkram,MuzzamalSitara
Received9January2017; Revised8February2017; Accepted16February2017
Abstract. Inthisresearchstudy,wepresentconceptofintuitionisticneutrosophicgraphstructures.Weintroducethecertainoperationson intuitionisticneutrosophicgraphstructuresandelaboratethemwithsuitableexamples.Further,weinvestigatesomeremarkablepropertiesofthese operators.Moreover,wediscussahighlyworthwhilereal-lifeapplicationof intuitionisticneutrosophicgraphstructuresindecision-making.Lastly,we elaborategeneralprocedureofourapplicationbydesigninganalgorithm.
2010AMSClassification: 03E72,05C72,05C78,05C99
Keywords: Intuitionisticneutrosophicgraphstructures,Decisionmaking
CorrespondingAuthor: MuhammadAkram(m.akram@pucit.edu.pk)
1. Introduction
G
raphicalmodelsareextensivelyusefultoolsforsolvingcombinatorialproblemsofdifferentfieldsincludingoptimization,algebra,computerscience,topology andoperationsresearchetc.Fuzzygraphicalmodelsarecomparativelymorecloseto nature,becauseinnaturevaguenessandambiguityoccurs.Therearemanycomplex phenomenaandprocessesinscienceandtechnologyhavingincompleteinformation. Todealsuchcasesweneededatheorydifferentfromclassicalmathematics.Graph structuresasgeneralizedsimplegraphsarewidelyusedforstudyofedgecolored andedgesignedgraphs,alsohelpfulandcopiouslyusedforstudyinglargedomains ofcomputerscience.Initiallyin1965,Zadeh[29]proposedthenotionoffuzzy setstohandleuncertaintyinalotofrealapplications.Fuzzysettheoryisfinding largenumberofapplicationsinrealtimesystems,whereinformation inherentin systemshasvariouslevelsofprecision.Afterwards,Turksen[26]proposedtheidea ofinterval-valuedfuzzyset.Butinvarioussystems,therearemembershipandnonmembershipvalues,membershipvalueisinfavorofaneventandnon-membership valueisagainstofthatevent.Atanassov[8]proposedthenotionofintuitionistic
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
fuzzysetin1986.Theintuitionisticfuzzysetsaremorepracticalandapplicablein real-lifesituations.Intuitionisticfuzzysetdealwithincompleteinformation,that is,degreeofmembershipfunction,non-membershipfunctionbutnotindeterminate andinconsistentinformationthatexistsdefinitelyinmanysystems, includingbelief system,decision-supportsystemsetc.In1998,Smarandache[24]proposedanother notionofimprecisedatanamedasneutrosophicsets.“Neutrosophicsetisapart ofneutrosophywhichstudiestheorigin,natureandscopeofneutralities,aswell astheirinteractionswithdifferentideationalspectra”.Neutrosophicsetisrecently proposedpowerfulformalframework.Forconvenientusageof neutrosophicsetsin real-lifesituations,Wangetal.[27]proposedsingle-valuedneutrosophicsetasageneralizationofintuitionisticfuzzyset[8].Aneutrosophicsethasthreeindependent componentshavingvaluesinunitinterval[0,1].Ontheotherhand,Bhowmikand Pal[10, 11]introducedthenotionsofintuitionisticneutrosophicsetsandrelations. Kauffman[16]definedfuzzygraphonthebasisofZadeh’sfuzzyrelations[30].Rosenfeld[21]investigatedfuzzyanalogueofvariousgraph-theoreticideasin1975.Later on,Bhattacharyagavesomeremarksonfuzzygraphin1987.BhutaniandRosenfeld discussedM-strongfuzzygraphswiththeirpropertiesin[12].In2011,Dineshand Ramakrishnan[15]putforwardfuzzygraphstructuresandinvestigateditsproperties.In2016,AkramandAkmal[1]proposedthenotionofbipolarfuzzygraph structures.Broumietal.[13]portrayedsingle-valuedneutrosophicgraphs.Akram andShahzadi[2]introducedthenotionofneutrosophicsoftgraphswithapplications. AkramandShahzadi[4]highlightedsomeflawsinthedefinitionsofBroumietal. [13]andShah-Hussain[22].Akrametal.[5]alsointroducedthesingle-valuedneutrosophichypergraphs.Representationofgraphsusingintuitionisticneutrosophic softsetswasdiscussedin[3].Inthispaper,wepresentconceptofintuitionistic neutrosophicgraphstructures.Weintroducethecertainoperationsonintuitionistic neutrosophicgraphstructuresandelaboratethemwithsuitableexamples.Further, weinvestigatesomeremarkablepropertiesoftheseoperators.Moreover,wediscuss ahighlyworthwhilereal-lifeapplicationofintuitionisticneutrosophicgraphstructuresindecision-making.Lastly,weelaborategeneralprocedure ofourapplication bydesigninganalgorithm.
Wehaveusedstandarddefinitionsandterminologiesinthispaper.Forothernotations,terminologiesandapplicationsnotmentionedinthepaper,thereadersare referredto[3,6,7,9,13,14,17,18,20,22,23,25,28,30].
2. IntuitionisticNeutrosophicGraphStructures
Definition2.1. ([23]).Let G1 =(P,P1,P2,...,Pr )and G2 =(P ′,P ′ 1,P ′ 2,...,P ′ r ) betwoGSs,Cartesianproductof ˇ G1 and ˇ G1 isdefinedas: G1 × G2 =(P × P ′,P1 × P ′ 1,P2 × P ′ 2,...,Pr × P ′ r ), where Ph × P ′ h = {(k1l)(k2l) | l ∈ P ′,k1k2 ∈ Ph }∪{(kl1)(kl2) | k ∈ p,l1l2 ∈ P ′ h }, h =(1, 2,...,r).
Definition2.2. ([23]).Let G1 =(P,P1,P2,...,Pn )and G2 =(P ′,P ′ 1,P ′ 2,...,P ′ r ) betwoGSs,crossproductof ˇ G1 and ˇ G2 isdefinedas: G1 ∗ G2 =(P ∗ P ′,P1 ∗ P ′ 1,P2 ∗ P ′ 2,...,Pr ∗ P ′ r ), 2
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where Ph ∗ P ′ h = {(k1l1)(k2l2) | k1k2 ∈ Ph,l1l2 ∈ P ′ h }, h =(1, 2,...,r).
Definition2.3. ([23]).Let ˇ G1 =(P,P1,P2,...,Pr )and ˇ G2 =(P ′,P ′ 1,P ′ 2,...,P ′ r ) betwoGSs,lexicographicproductof G1 and G2 isdefinedas: ˇ G1 • ˇ G2 =(P • P ′,P1 • P ′ 1,P2 • P ′ 2,...,Pr • P ′ r ), where Ph •P ′ h = {(kl1)(kl2) | k ∈ P,l1l2 ∈ P ′ h}∪{(k1l1)(k2l2) | k1k2 ∈ Ph,l1l2 ∈ P ′ h }, h =(1, 2,...,r).
Definition2.4. ([23]).Let G1 =(P,P1,P2,...,Pr )and G2 =(P ′,P ′ 1,P ′ 2,...,P ′ r ) betwoGSs,strongproductof ˇ G1 and ˇ G2 isdefinedas: G1 ⊠ G2 =(P ⊠ P ′,P1 ⊠ P ′ 1,P2 ⊠ P ′ 2,...,Pr ⊠ P ′ r ), where Ph ⊠ P ′ h = {(k1l)(k2l) | l ∈ P ′,k1k2 ∈ Ph}∪{(kl1)(kl2) | k ∈ P,l1l2 ∈ P ′ h}∪ {(k1l1)(k2l2) | k1k2 ∈ Ph,l1l2 ∈ P ′ h }, h =(1, 2,...,r).
Definition2.5. ([23]).Let G1 =(P,P1,P2,...,Pr )and G2 =(P ′,P ′ 1,P ′ 2,...,P ′ n) betwoGSs,compositionof ˇ G1 and ˇ G2 isdefinedas: G1 ◦ G2 =(P ◦ P ′,P1 ◦ P ′ 1,P2 ◦ P ′ 2,...,Pr ◦ P ′ r ), where Ph ◦ P ′ h = {(k1l)(k2l) | l ∈ P ′,k1k2 ∈ Ph }∪{(kl1)(kl2) | k ∈ P,l1l2 ∈ P ′ h}∪ {(k1l1)(k2l2) | k1k2 ∈ Ph,l1,l2 ∈ P ′ suchthat l1 = l2}, h =(1, 2,...,r).
Definition2.6. ([23]).Let ˇ G1 =(P,P1,P2,...,Pr )and ˇ G2 =(P ′,P ′ 1,P ′ 2,...,P ′ r ) betwoGSs,unionof G1 and G2 isdefinedas: ˇ G1 ∪ ˇ G2 =(P ∪ P ′,P1 ∪ P ′ 1,P2 ∪ P ′ 2,...,Pr ∪ P ′ r ) Definition2.7. ([23]).Let G1 =(P,P1,P2,...,Pr )and G2 =(P ′,P ′ 1,P ′ 2,...,P ′ r ) betwoGSs,joinof G1 and G2 isdefinedas: ˇ G1 + ˇ G2 =(P + P ′,P1 + P ′ 1,P2 + P ′ 2,...,Pr + P ′ r ), where P + P ′ = P ∪ P ′ , Ph + P ′ h = Ph ∪ P ′ h ∪ P ′′ h for h =(1, 2,...,r). P ′′ h consists ofallthoseedgeswhichjointheverticesof P and P ′ Definition2.8. ([19]).Let V beafixedset.Ageneralizedintuitionisticfuzzyset I of V isanobjecthavingtheform I={(v,µI (v),νI (v))|v ∈ V },wherethefunctions µI : V → [0, 1]and νI : V → [0, 1]definethedegreeofmembershipanddegreeof nonmembershipofanelement v ∈ V ,respectively,suchthat min{µI (v),νI (v)}≤ 0 5,forall v ∈ V
Thisconditioniscalledthegeneralizedintuitionisticcondition. Definition2.9. ([10, 11]).Aset I = {TI (v),II (v),FI (v): v ∈ V } issaidtobean intuitionisticneutrosophic(IN)set,if (i) {TI (v) ∧ II (v)}≤ 0.5, {II (v) ∧ FI (v)}≤ 0.5, {FI (v) ∧ TI (v)}≤ 0.5, (ii)0 ≤ TI (v)+ II (v)+ FI (v) ≤ 2.
Definition2.10. Anintuitionisticneutrosophicgraphisapair G =(A,B)with underlyingset V ,where TA, FA, IA : V → [0, 1]denotethetruth,falsityand indeterminacymembershipvaluesoftheverticesin V and TB , FB , IB : E ⊆ V × V → [0, 1]denotethetruth,falsityandindeterminacymembershipvaluesoftheedges kl ∈ E suchthat
3
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(i) TB (kl) ≤ TA(k) ∧ TA(l), FB (kl) ≤ FA(k) ∨ FA(l),IB (kl) ≤ IA(k) ∧ IA(l), (ii) TB (kl) ∧ IB (kl) ≤ 0.5,TB (kl) ∧ FB (kl) ≤ 0.5,IB (kl) ∧ FB (lk) ≤ 0.5, (iii)0 ≤ TB (kl)+ FB (kl)+ IB (kl) ≤ 2, ∀ k,l ∈ V
Definition2.11. ˇ Gi =(O,O1 ,O2,...,Or )issaidtobeanintuitionisticneutrosophicgraphstructure(INGS)ofgraphstructure ˇ G =(P,P1,P2,...,Pr ),if O = <k,T (k),I(k),F (k) > and Oh = <kl,Th(kl),Ih(kl),Fh(kl) > aretheintuitionistic neutrosophic(IN)setsonthesets P and Ph,respectivelysuchthat
(i) Th(kl) ≤ T (k) ∧ T (l), Ih (kl) ≤ I(k) ∧ I(l), Fh(kl) ≤ F (k) ∨ F (l), (ii) Th(kl) ∧ Ih (kl) ≤ 0.5,Th(kl) ∧ Fh(kl) ≤ 0.5,Ih (kl) ∧ Fh(kl) ≤ 0.5, (iii)0 ≤ Th(kl)+ Ih(kl)+ Fh (kl) ≤ 2, forallkl ∈ Oh , h ∈{1, 2,...,r}, where, O and Oh areunderlyingvertexandh-edgesetsofINGS ˇ Gi, h ∈{1, 2,...,r}
O1(0 2, 0 4, 0 6)
k1(0.3,0.4,0. 5) O1(0. 5, 0. 2, 0. 6) O2(0 .2 ,0 .3 ,0 .4)
k6(0 .2, 04, 0.6) O1(0 2, 0 2, 0 5) O3(0 2, 0 1, 0 3)
O3(0 .1 ,0 .4 ,0 .5)O2(0 5, 0 4, 0 6)
k7(0 7, 0 2, 0 5) k 2 (0 2 , 0 5 , 0 4) k8(0 5, 0 4, 0 6)
O2(0. 1, 0. 3, 0. 5) O1(0. 1, 0. 2, 0. 5) O3(0 .1 ,0 .1 ,0 .4)
Example2.12. Anintuitionisticneutrosophicgraphstructureisrepresentedin Fig. 1. k3(0 .5 , 04 , 0 .3)
O3(0 .2 ,0 .3 ,0 .6) O2(0. 2, 0. 1, 0. 6) k4(0. 2, 0. 2, 0. 5)
k 5 (0 6 , 0 4 , 0 . 5)
Figure1. Anintuitionisticneutrosophicgraphstructure
NowwedefinetheoperationsonINGSs. Definition2.13. Let ˇ Gi1 =(O1,O11,O12,...,O1r )and ˇ Gi2 =(O2,O21,O22,...,Q2r ) beINGSsofGSs G1 =(P1,P11,P12,...,P1r )and G2 =(P2,P21,P22,...,P2r ),respectively. Cartesianproductof Gi1 and Gi2,denotedby ˇ Gi1 × ˇ Gi2 =(O1 × O2,O11 × O21,O12 × O22,...,O1r × O2r ), isdefinedas: (i)
T(O1 ×O2 )(kl)=(TO1 × TO2 )(kl)= TO1 (k) ∧ TO2 (l) I(O1 ×O2 )(kl)=(IO1 × IO2 )(kl)= IO1 (k) ∧ IO2 (l) F(O1 ×O2 )(kl)=(FO1 × FO2 )(kl)= FO1 (k) ∨ FO2 (l) forall kl ∈ P1 × P2, (ii)
T(O1h ×O2h )(kl1)(kl2)=(TO1h × TO2h )(kl1)(kl2)= TO1 (k) ∧ TO2h (l1l2) 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) forall k ∈ P1 , l1l2 ∈ P2h , 4
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(iii)
T(O1h ×O2h )(k1l)(k2l)=(TO1h × TO2h )(k1l)(k2l)= TO2 (l) ∧ TO1h (k1k2) 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) forall l ∈ P2 , k1k2 ∈ P1h
k1(0 5, 0 2, 0 6)
k3(0 4, 0 3, 0 4)
O12(0 4, 0 3, 0 6) O11(0 5, 0 2, 0 8)
Example2.14. Consider ˇ Gi1 =(O1,O11,O12)and ˇ Gi2 =(O2,O21,O22)aretwo INGSsofGSs G1 =(P1,P11,P12)and G2 =(P2,P21,P22)respectively,asrepresented inFig. 2,where P11 = {k1k2}, P12 = {k3k4}, P21 = {l1l2}, P22 = {l2l3} l3(0 5, 0 4, 0 5) l1(0 2, 0 2, 0 3) l2(0 3, 0 3, 0 4)
k4(0 5, 0 3, 0 6) k2(0 5, 0 3, 0 8)
Gi1 = (O1,O11,O12) Figure2. TwoINGSs ˇ Gi1 and ˇ Gi2
O22(0. 3, 0. 3, 0. 5) O 21 (0 2 , 0 2 , 0 4) Gi2 = (O2,O21,O22)
Cartesianproductof ˇ Gi1 and ˇ Gi2 definedas ˇ Gi1 × ˇ Gi2 = {O1 ×O2,O11 ×O21,O12 × O22} isrepresentedinFig. 3
k1 l2 (0 3, 0 2, 0 6)
k2 l2(0 3, 0 3, 0 8)
k1l1(0 .2 ,0 .2 ,06)
O11 × O21 (0 3, 0 2, 0 8) O12 × O22(0 . 3, 0. 2, 0. 6) O11×O21(0. 2, 0. 2, 0. 6) 5
O 11 × O 21(0 .2 , 0 2 , 0 . 8)
k2l1(02, 0. 2, 08) k1l3(0 .5 , 0 .2 , 0 .6)
O12 ×O22(0 .3 , 03 ,08)
O11 × O21 (0 5, 0 2, 0 8) O11 × O21 (0 2, 0 2, 0 8)
k2l3(0. 5, 0. 3, 0. 8)
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k3 l2 (0 3, 0 2, 0 4)
O12 × O22(0 . 3, 0. 2, 05) O11×O21(0. 2, 0. 2, 0. 4)
k3l1(0 .2 ,0 .2 ,04)
k4 l2(0 3, 0 3, 0 6)
O12 × O22 (0 3, 0 3, 0 6)
k3l3(0 .4 , 0 .3 , 05)
O12 × O22 (0 2, 0 2, 0 6)
O 11 × O 21(0 2 , 0 2 , 0 . 6)
O12 ×O22(0 .3 , 0 .3 ,0 .6)
k4l3(0. 5, 0. 3, 06) k4l1(02, 0. 2, 06)
O12 × O22 (0 4, 0 3, 0 6)
Figure3. ˇ Gi1 × ˇ Gi2
Theorem2.15. Cartesianproduct ˇ Gi1× ˇ Gi2 = (O1×O2,O11×O21,O12×O22,...,O1r × O2r ) oftwoINGSsofGSs G1 and G2 isanINGSof G1 × G2 Proof. Weconsidertwocases:
Case1: For k ∈ P1, l1l2 ∈ P2h T(O1h ×O2h )((kl1)(kl2))= TO1 (k) ∧ TO2h (l1l2) ≤ TO1 (k) ∧ [TO2 (l1) ∧ TO2 (l2)] =[TO1 (k) ∧ TO2 (l1)] ∧ [TO1 (k) ∧ TO2 (l2)] = T(O1 ×O2 )(kl1) ∧ T(O1 ×O2 )(kl2), I(O1h ×O2h )((kl1)(kl2))= IO1 (k) ∧ IO2h (l1l2)
≤ IO1 (k) ∧ [IO2 (l1) ∧ IO2 (l2)] =[IO1 (k) ∧ IO2 (l1)] ∧ [IO1 (k) ∧ IO2 (l2)]
= I(O1 ×O2 )(kl1) ∧ I(O1 ×O2 )(kl2),
F(O1h ×O2h )((kl1)(kl2))= FO1 (k) ∨ FO2h (l1l2)
≤ FO1 (k) ∨ [FO2 (l1) ∨ FO2 (l2)] =[FO1 (k) ∨ FO2 (l1)] ∨ [FO1 (k) ∨ FO2 (l2)]
= F(O1 ×O2 )(kl1) ∨ F(O1 ×O2 )(kl2),
for kl1,kl2 ∈ P1 × P2
Case2: For k ∈ P2, l1l2 ∈ P1h
T(O1h ×O2h )((l1k)(l2k))= TO2 (k) ∧ TO1h (l1l2)
≤ TO2 (k) ∧ [TO1 (l1) ∧ TO1 (l2)]
=[TO2 (k) ∧ TO1 (l1)] ∧ [TO2 (k) ∧ TO1 (l2)]
= T(O1 ×O2 )(l1k) ∧ T(O1 ×O2 )(l2k), 6
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I(O1h ×O2h )((l1k)(l2k))= IO2 (k) ∧ IO1h (l1l2) ≤ IO2 (k) ∧ [IO1 (l1) ∧ IO1 (l2)] =[IO2 (k) ∧ IO1 (l1)] ∧ [IO2 (k) ∧ IO1 (l2)]
= I(O1 ×O2 )(l1k) ∧ I(O1 ×O2 )(l2k),
F(O1h ×O2h )((l1k)(l2k))= FO2 (k) ∨ FO1h (l1l2) ≤ FO2 (k) ∨ [FO1 (l1) ∨ FO1 (l2)] =[FO2 (k) ∨ FO1 (l1)] ∨ [FO2 (k) ∨ FO1 (l2)] = F(O1 ×O2 )(l1k) ∨ F(O1 ×O2 )(l2k), for l1k,l2k ∈ P1 × P2 Bothcasesexists ∀h ∈{1, 2,...,r}.Thiscompletestheproof.
Definition2.16. Let Gi1 =(O1,O11,O12,...,Q1r )and Gi2 =(O2,O21,O22,...,Q2r ) beINGSsofGSs ˇ G1 =(P1,P11,P12,...,P1r )and ˇ G2 =(P2,P21,P22,...,P2r ),respectively.Crossproductof Gi1 and Gi2,denotedby ˇ Gi1 ∗ ˇ Gi2 =(O1 ∗ O2,O11 ∗ O21,O12 ∗ O22,...,O1r ∗ O2r ), isdefinedas: (i)
T(O1 ∗O2 )(kl)=(TO1 ∗ TO2 )(kl)= TO1 (k) ∧ TO2 (l) I(O1 ∗O2 )(kl)=(IO1 ∗ IO2 )(kl)= IO1 (k) ∧ IO2 (l) F(O1 ∗O2 )(kl)=(FO1 ∗ FO2 )(kl)= FO1 (k) ∨ FO2 (l) forall kl ∈ P1 × P2, (ii)
T(O1h ∗O2h )(k1l1)(k2 l2)=(TO1h ∗ TO2h )(k1l1)(k2l2)= TO1h (k1k2) ∧ TO2h (l1l2) I(O1h ∗O2h )(k1l1)(k2 l2)=(IO1h ∗ IO2h )(k1l1)(k2l2)= IO1h (k1k2) ∧ IO2h (l1l2) F(O1h ∗O2h )(k1l1)(k2l2)=(FO1h ∗ FO2h )(k1l1)(k2l2)= FO1h (k1k2) ∨ FO2h (l1l2) forall k1k2 ∈ P1h , l1l2 ∈ P2h .
Example2.17. CrossproductofINGSs ˇ Gi1 and ˇ Gi2 showninFig. 2 isdefinedas ˇ Gi1 ∗ ˇ Gi2 = {O1 ∗ O2,O11 ∗ O21,O12 ∗ O22} andisrepresentedinFig. 4
k4 l3 (0 5, 0 3, 0 6)
k1 l3 (0 5, 0 2, 0 6)
O11∗O21(0 .2 ,02 ,0 .8)
k2 l1 (0 2, 0 2, 0 8) k3 l2 (0 3, 0 3, 0 4)
k1 l2 (0 3, 0 2, 0 6)
O11 ∗O21(0 . 2, 0. 2, 0. 8) O12 ∗O22(0 . 3, 0. 3, 0. 6) k4 l1 (0 2, 0 2, 0 6)
k2 l3 (0 5, 0 3, 0 8)
k2 l2 (0 3, 0 3, 0 8) k3 l3 (0 4, 0 3, 0 5) k3 l1 (0 2, 0 2, 0 4) k4 l2 (0 3, 0 3, 0 6) k1 l1 (0 2, 0 2, 0 6)
O12∗O22(0 .3 ,0 .3 ,0 . 6)
Figure4. Gi1 ∗ Gi2
Theorem2.18. Crossproduct ˇ Gi1∗ ˇ Gi2 = (O1∗O2,O11∗O21,O12∗O22,...,O1r ∗O2r ) oftwoINGSsofGSs G1 and G2 isanINGSof G1 ∗ G2 7
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Proof. Forall k1l1,k2l2 ∈ P1 ∗ P2
T(O1h ∗O2h )((k1l1)(k2l2))= TO1h (k1k2) ∧ TO2h (l1l2)
≤ [TO1 (k1) ∧ TO1 (k2)] ∧ [TO2 (l1) ∧ TO2 (l2)]
=[TO1 (k1) ∧ TO2 (l1)] ∧ [TO1 (k2) ∧ TO2 (l2)]
= T(O1 ∗O2 )(k1l1) ∧ T(O1 ∗O2 )(k2l2),
I(O1h ∗O2h )((k1l1)(k2l2))= IO1h (k1k2) ∧ IO2h l1l2)
≤ [IO1 (k1) ∧ IO1 (k2)] ∧ [IO2 (l1) ∧ IO2 (l2)] =[IO1 (k1) ∧ IO2 (l1)] ∧ [IO1 (k2) ∧ IO2 (l2)]
= I(O1 ∗O2 )(k1l1) ∧ I(O1 ∗O2 )(k2l2),
F(O1h ∗O2h )((k1l1)(k2l2))= FO1h (k1k2) ∨ FO2h (l1l2) ≤ [FO1 (k1) ∨ FO1 (k2)] ∨ [FO2 (l1) ∨ FO2 (l2)] =[FO1 (k1) ∨ FO2 (l1)] ∨ [FO1 (k2) ∨ FO2 (l2)] = F(O1 ∗O2 )(k1l1) ∨ F(O1 ∗O2 )(k2l2), for h ∈{1, 2,...,r}.Thiscompletestheproof.
Definition2.19. Let Gi1 =(O1,O11,O12,...,O1r )and Gi2 =(O2,O21,O22,...,O2r ) beINGSsofGSs ˇ G1 =(P1,P11,P12,...,P1r )and ˇ G2 =(P2,P21,P22,...,P2r ),respectively.Lexicographicproductof Gi1 and Gi2,denotedby ˇ Gi1 • ˇ Gi2 =(O1 • O2,O11 • O21,O12 • O22,...,O1r • O2r ), isdefinedas:
(i)
T(O1 •O2 )(kl)=(TO1 • TO2 )(kl)= TO1 (k) ∧ TO2 (l) I(O1 •O2 )(kl)=(IO1 • IO2 )(kl)= IO1 (k) ∧ IO2 (l) F(O1 •O2 )(kl)=(FO1 • FO2 )(kl)= FO1 (k) ∨ FO2 (l) forall kl ∈ P1 × P2
(ii)
T(O1h •O2h )(kl1)(kl2)=(TO1h • TO2h )(kl1)(kl2)= TO1 (k) ∧ TO2h (l1l2) 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) forall k ∈ P1 , l1l2 ∈ P2h , (iii)
T(O1h •O2h )(k1l1)(k2 l2)=(TO1h • TO2h )(k1l1)(k2l2)= TO1h (k1k2) ∧ TO2h (l1l2) I(O1h •O2h )(k1l1)(k2 l2)=(IO1h • IO2h )(k1l1)(k2l2)= IO1h (k1k2) ∧ IO2h (l1l2) F(O1h •O2h )(k1l1)(k2l2)=(FO1h • FO2h )(k1l1)(k2l2)= FO1h (k1k2) ∨ FO2h (l1l2) forall k1k2 ∈ P1h , l1l2 ∈ P2h .
Example2.20. LexicographicproductofINGSs Gi1 and Gi2 showninFig. 2 is definedas Gi1 • Gi2 = {O1 • O2,O11 • O21,O12 • O22} andisrepresentedinFig. 5 8
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
k1 l1 (0 2, 0 2, 0 6)
k2 l1 (0 2, 0 2, 0 8) k2 l3 (0 5, 0 3, 0 8)
O11 •O21(0. 2, 0. 2, 08) O 11 • O 21 (0 2, 0 2, 0 6)
k1 l2 (0 3, 0 2, 0 6)
k4 l1 (0 2, 0 2, 0 6)
O 11 • O 21 (0 2 , 0 2 , 0 8)
O11•O21(0 .2 ,0 .2 ,0 .8)
O12 •O22(0. 3, 0. 3, 08)
k2 l2 (0 3, 0 3, 0 8)
O12 • O22 (0 3, 0 2, 0 6)
k1 l3 (0 5, 0 2, 0 6)
O11 • O21 (0 2, 0 2, 0 6) O12•O22(0 .3 ,0 .3 ,0 .6)
k3 l2 (0 3, 0 2, 0 6) k4 l2 (0 3, 0 3, 0 6)
O12 •O22(0. 3, 0. 3, 0. 6)
k3 l1 (0 2, 0 2, 0 4)
k3 l3 (0 4, 0 3, 0 5)
k4 l3 (0 5, 0 3, 0 6)
O 12 • O 22 (0 3 , 0 3 , 0 6) O11 •O21(0. 2, 0. 2, 0. 4) O 12 • O 22 (0 3, 0 3, 0 5) Figure5. ˇ Gi1 • ˇ Gi2 Theorem2.21. Lexicographicproduct ˇ Gi1 • ˇ Gi2 = (O1•O2,O11•O21,O12•O22,...,O1r • O2r ) oftwoINGSsoftheGSs G1 and G2 isanINGSof G1 • G2.
Proof. Weconsidertwocases:
Case1: For k ∈ P1, l1l2 ∈ P2h
T(O1h •O2h )((kl1)(kl2))= TO1 (k) ∧ TO2h (l1l2) ≤ TO1 (k) ∧ [TO2 (l1) ∧ TO2 (l2)] =[TO1 (k) ∧ TO2 (l1)] ∧ [TO1 (k) ∧ TO2 (l2)]
= T(O1 •O2 )(kl1) ∧ T(O1 •O2 )(kl2), I(O1h •O2i )((kl1)(kl2))= IO1 (k) ∧ IO2h (l1l2)
≤ IO1 (k) ∧ [IO2 (l1) ∧ IO2 (l2)] =[IO1 (k) ∧ IO2 (l1)] ∧ [IO1 (k) ∧ IO2 (l2)]
= I(O1 •O2 )(kl1) ∧ I(O1 •O2 )(kl2),
F(O1h •O2i )((kl1)(kl2))= FO1 (k) ∨ FO2h (l1l2)
≤ FO1 (k) ∨ [FO2 (l1) ∨ FO2 (l2)] =[FO1 (k) ∨ FO2 (l1)] ∨ [FO1 (k) ∨ FO2 (l2)]
= F(O1 •O2 )(kl1) ∨ F(O1 •O2 )(kl2), for kl1,kl2 ∈ P1 • P2 9
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
Case2: For k1k2 ∈ P1h ,l1l2 ∈ P2h
T(O1h •O2h )((k1l1)(k2l2))= TO1h (k1k2) ∧ TO2h (l1l2) ≤ [TO1 (k1) ∧ TO1 (k2)] ∧ [TO2 (l1) ∧ TO2 (l2)]
=[TO1 (k1) ∧ TO2 (l1)] ∧ [TO1 (k2) ∧ TO2 (l2)]
= T(O1 •O2 )(k1l1) ∧ T(O1 •O2 )(k2l2),
I(O1h •O2h )((k1l1)(k2l2))= IO1h (k1k2) ∧ IO2h (l1l2)
≤ [IO1 (k1) ∧ IO1 (k2)] ∧ [IO2 (l1) ∧ IO2 (l2)] =[IO1 (k1) ∧ IO2 (l1)] ∧ [IO1 (k2) ∧ IO2 (l2)]
= I(O1 •O2 )(k1l1) ∧ I(O1 •O2 )(k2l2),
F(O1h •O2h )((k1l1)(k2l2))= FO1h (k1k2) ∨ FO2h (l1l2) ≤ [FO1 (k1) ∨ FO1 (k2)] ∨ [FO2 (l1) ∨ FO2 (l2)]
=[FO1 (k1) ∨ FO2 (l1)] ∨ [FO1 (k2) ∨ FO2 (l2)] = F(O1 •O2 )(k1l1) ∨ F(O1 •O2 )(k2l2),
for k1l1,k2l2 ∈ P1 • P2 Bothcasesholdfor h ∈{1, 2,...,r}.Thiscompletestheproof.
Definition2.22. Let ˇ Gi1 =(O1,O11,O12,...,O1r )and ˇ Gi2 =(O2,O21,O22,...,O2r ) beINGSsofGSs G1 =(P1,P11,P12,...,P1r )and G2 =(P2,P21,P22,...,P2r ),respectively.Strongproductof Gi1 and Gi2,denotedby ˇ Gi1 ⊠ ˇ Gi2 =(O1 ⊠ O2,O11 ⊠ O21,O12 ⊠ O22,...,O1r ⊠ O2r ), isdefinedas:
(i)
T(O1 ⊠O2 )(kl)=(TO1 ⊠ TO2 )(kl)= TO1 (k) ∧ TO2 (l) I(O1 ⊠O2 )(kl)=(IO1 ⊠ IO2 )(kl)= IO1 (k) ∧ IO2 (l) F(O1 ⊠O2 )(kl)=(FO1 ⊠ FO2 )(kl)= FO1 (k) ∨ FO2 (l) forall kl ∈ P1 × P2, (ii)
T(O1h ⊠O2h )(kl1)(kl2)=(TO1h ⊠ TO2h )(kl1)(kl2)= TO1 (k) ∧ TO2h (l1l2)
I(O1h ⊠O2h )(kl1)(kl2)=(IO1h ⊠ IO2h )(kl1)(kl2)= IO1 (k) ∧ IO2h (l1l2)
T(O1h ⊠O2h )(k1l)(k2l)=(TO1h ⊠ TO2h )(k1l)(k2l)= TO2 (l) ∧ TO1h (k1k2)
I(O1h ⊠O2h )(k1l)(k2l)=(IO1h ⊠ IO2h )(k1l)(k2l)= IO2 (l) ∧ IO2h (k1k2)
F(O1h ⊠O2h )(kl1)(kl2)=(FO1h ⊠ FO2h )(kl1)(kl2)= FO1 (k) ∨ FO2h (l1l2) forall k ∈ P1 , l1l2 ∈ P2h , (iii)
F(O1h ⊠O2h )(k1l)(k2l)=(FO1h ⊠ FO2h )(k1l)(k2l)= FO2 (l) ∨ FO2h (k1k2) forall l ∈ P2 , k1k2 ∈ P1h, (iv)
T(O1h ⊠O2h )(k1l1)(k2l2)=(TO1h ⊠ TO2h )(k1l1)(k2l2)= TO1h (k1k2) ∧ TO2h (l1l2)
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) forall k1k2 ∈ P1h , l1l2 ∈ P2h
Example2.23. StrongproductofINGSs Gi1 and Gi2 showninFig. 2 isdefined as Gi1 ⊠ Gi2 = {O1 ⊠ O2,O11 ⊠ O21,O12 ⊠ O22} andisrepresentedinFig. 6 10
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
k1 l1 (0 2, 0 2, 0 6)
O 11 ⊠ O 21 (0 2 , 0 2 , 0 8)
k2 l1 (0 2, 0 2, 0 8)
O11 ⊠ O21 (0 2, 0 2, 0 6)
O11 ⊠O21(0. 2, 0. 2, 0. 8)
O11 ⊠ O21 (0 2, 0 2, 0 8)
O11⊠O21(0 .2 ,02 ,0 .8)
k4 l1 (0 2, 0 2, 0 6)
k3 l2 (0 3, 0 3, 0 4)
k3 l1 (0 2, 0 2, 0 4)
k1 l2 (0 3, 0 2, 0 6) k2 l3 (0 5, 0 3, 0 8)
O12 ⊠O22(0. 3, 0. 3, 0. 8)
O12⊠O22(0 .3 ,0 .2 ,0 .6) O 11 ⊠ O 21 (0 3 , 0 2 , 0 8) O 11 ⊠ O 21 (0 5 , 0 2 , 0 8)
k2 l2 (0 3, 0 3, 0 8)
k1 l3 (0 5, 0 2, 0 6)
O11 ⊠O21(0. 2, 0. 2, 0. 4) O12 ⊠ O22 (0 3, 0 3, 0 5) O12 ⊠O22(0. 3, 0. 3, 0. 6)
O12⊠O22(0 .3 ,0 .3 ,0 .6) O 12 ⊠ O 22 (0 3 , 0 . 3 , 0 6) O 12 ⊠ O 22 (0 4 , 0 3 , 0 6) O 12 ⊠ O 22 (0 2 , 0 2 , 0 6)
k3 l3 (0 4, 0 3, 0 5)
O11⊠O21(0 .2 ,0 .2 ,0 .6)
k4 l2 (0 3, 0 3, 0 6) O12 ⊠ O22 (0 3, 0 3, 0 6)
Figure6. ˇ Gi1 ⊠ ˇ Gi2
k4 l3 (0 5, 0 3, 0 6)
Theorem2.24. Strongproduct Gi1 ⊠Gi2 = (O1 ⊠O2,O11 ⊠O21,O12 ⊠O22,...,O1r ⊠ O2r ) oftwoINGSsoftheGSs ˇ G1 and ˇ G2 isanINGSof ˇ G1 ⊠ ˇ G2 Proof. Therearethreecases: Case1: For k ∈ P1, l1l2 ∈ P2h T(O1h ⊠O2h )((kl1)(kl2))= TO1 (k) ∧ TO2h (l1l2) ≤ TO1 (k) ∧ [TO2 (l1) ∧ TO2 (l2)] =[TO1 (k) ∧ TO2 (l1)] ∧ [TO1 (k) ∧ TO2 (l2)] = T(O1 ⊠O2 )(kl1) ∧ T(O1 ⊠O2 )(kl2), I(O1h ⊠O2h )((kl1)(kl2))= IO1 (k) ∧ IO2h (l1l2) ≤ IO1 (k) ∧ [IO2 (l1) ∧ IO2 (l2)] =[IO1 (k) ∧ IO2 (l1)] ∧ [IO1 (k) ∧ IO2 (l2)] = I(O1 ⊠O2 )(kl1) ∧ I(O1 ⊠O2 )(kl2), 11
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
F(O1h ⊠O2h )((kl1)(kl2))= FO1 (k) ∨ FO2h (l1l2)
≤ FO1 (k) ∨ [FO2 (l1) ∨ FO2 (l2)] =[FO1 (k) ∨ FO2 (l1)] ∨ [FO1 (k) ∨ FO2 (l2)]
= F(O1 ⊠O2 )(kl1) ∨ F(O1 ⊠O2 )(kl2),
for kl1,kl2 ∈ P1 ⊠ P2 Case2: For k ∈ P2, l1l2 ∈ P1h
T(O1h ⊠O2h )((l1k)(l2k))= TO2 (k) ∧ TO1h (l1l2)
≤ TO2 (k) ∧ [TO1 (l1) ∧ TO1 (l2)] =[TO2 (k) ∧ TO1 (l1)] ∧ [TO2 (k) ∧ TO1 (l2)]
= T(O1 ⊠O2 )(l1k) ∧ T(O1 ⊠O2 )(l2k),
I(O1h ⊠O2h )((l1k)(l2k))= IO2 (k) ∧ IO1h (l1l2)
≤ IO2 (k) ∧ [IO1 (l1) ∧ IO1 (l2)] =[IO2 (k) ∧ IO1 (l1)] ∧ [IO2 (k) ∧ IO1 (l2)]
= I(O1 ⊠O2 )(l1k) ∧ I(O1 ⊠O2 )(l2k),
F(O1h ⊠O2h )((l1k)(l2k))= FO2 (k) ∨ FO1h (l1l2)
≤ FO2 (k) ∨ [FO1 (l1) ∨ FO1 (l2)] =[FO2 (k) ∨ FO1 (l1)] ∨ [FO2 (k) ∨ FO1 (l2)]
= F(O1 ⊠O2 )(l1k) ∨ F(O1 ⊠O2 )(l2k),
for l1k,l2k ∈ P1 ⊠ P2 Case3: Forevery k1k2 ∈ P1h,l1l2 ∈ P2h
T(O1h ⊠O2h )((k1l1)(k2l2))= TO1h (k1k2) ∧ TO2h (l1l2) ≤ [TO1 (k1) ∧ TO1 (k2)] ∧ [TO2 (l1) ∧ TO2 (l2)]
=[TO1 (k1) ∧ TO2 (l1)] ∧ [TO1 (k2) ∧ TO2 (l2)]
= T(O1 ⊠O2 )(k1l1) ∧ T(O1 ⊠O2 )(k2l2),
I(O1h ⊠O2h )((k1l1)(k2l2))= IO1h (k1k2) ∧ IO2h (l1l2)
≤ [IO1 (k1) ∧ IO1 (k2)] ∧ [IO2 (l1) ∧ IO2 (l2)] =[IO1 (k1) ∧ IO2 (l1)] ∧ [IO1 (k2) ∧ IO2 (l2)]
= I(O1 ⊠O2 )(k1l1) ∧ I(O1 ⊠O2 )(k2l2),
F(O1h ⊠O2h )((k1l1)(k2l2))= FO1h (k1k2) ∨ FO2h (l1l2)
≤ [FO1 (k1) ∨ FO1 (k2)] ∨ [FO2 (l1) ∨ FO2 (l2)] =[FO1 (k1) ∨ FO2 (l1)] ∨ [FO1 (k2) ∨ FO2 (l2)]
= F(O1 ⊠O2 )(k1l1) ∨ F(O1 ⊠O2 )(k2l2),
for k1l1,k2l2 ∈ P1 ⊠ P2, and h =1, 2,...,r Thiscompletestheproof. 12
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
Definition2.25. Let ˇ Gi1 =(O1,O11,O12,...,O1r )and ˇ Gi2 =(O2,O21,O22,...,O2r ) beINGSsofGSs ˇ G1 =(P1,P11,P12,...,P1r )and ˇ G2 =(P2,P21,P22,...,P2r ),respectively.Thecompositionof ˇ Gi1 and ˇ Gi2,denotedby ˇ Gi1 ◦ ˇ Gi2 =(O1 ◦ O2,O11 ◦ O21,O12 ◦ O22,...,O1r ◦ O2r ), isdefinedas:
(i)
T(O1 ◦O2 )(kl)=(TO1 ◦ TO2 )(kl)= TO1 (k) ∧ TO2 (l) I(O1 ◦O2 )(kl)=(IO1 ◦ IO2 )(kl)= IO1 (k) ∧ IO2 (l) F(O1 ◦O2 )(kl)=(FO1 ◦ FO2 )(kl)= FO1 (k) ∨ FO2 (l) forall kl ∈ P1 × P2, (ii)
T(O1h ◦O2h )(kl1)(kl2)=(TO1h ◦ TO2h )(kl1)(kl2)= TO1 (k) ∧ TO2h (l1l2) I(O1h ◦O2h )(kl1)(kl2)=(IO1h ◦ IO2h )(kl1)(kl2)= IO1 (k) ∧ IO2h (l1l2)
T(O1h ◦O2h )(k1l)(k2l)=(TO1h ◦ TO2h )(k1l)(k2l)= TO2 (l) ∧ TO1h (k1k2)
I(O1h ◦O2h )(k1l)(k2l)=(IO1h ◦ IO2h )(k1l)(k2l)= IO2 (l) ∧ IO2h (k1k2)
F(O1h ◦O2h )(kl1)(kl2)=(FO1h ◦ FO2h )(kl1)(kl2)= FO1 (k) ∨ FO2h (l1l2) forall k ∈ P1 , l1l2 ∈ P2h , (iii)
F(O1h ◦O2h )(k1l)(k2l)=(FO1h ◦ FO2h )(k1l)(k2l)= FO2 (l) ∨ FO2h (k1k2) forall l ∈ P2 , k1k2 ∈ P1h, (iv)
T(O1h ◦O2h )(k1l1)(k2 l2)=(TO1h ◦ TO2h )(k1l1)(k2l2)= TO1h (k1k2) ∧ TO2 (l1) ∧ TO2 (l2) I(O1h ◦O2h )(k1l1)(k2 l2)=(IO1h ◦ IO2h )(k1l1)(k2l2)= IO1h (k1k2) ∧ IO2 (l1) ∧ IO2 (l2) F(O1h ◦O2h )(k1l1)(k2l2)=(FO1h ◦ FO2h )(k1l1)(k2l2)= FO1h (k1k2) ∨ FO2 (l1) ∨ FO2 (l2) forall k1k2 ∈ P1h , l1l2 ∈ P2h suchthat l1 = l2
Example2.26. ThecompositionofINGSs ˇ Gi1 and ˇ Gi2 showninFig. 2 isdefined as: ˇ Gi1 ◦ ˇ Gi2 = {O1 ◦ O2,O11 ◦ O21,O12 ◦ O22}
k1 l1 (0 2, 0 2, 0 6) k2 l2 (0 3, 0 3, 0 8)
O11 ◦ O21 (0 2, 0 2, 0 6) 13
O11 ◦O21(0. 2, 0. 2, 0. 8) O12 ◦ O22 (0 3, 0 2, 0 6) O11 ◦O21(03, 0. 2, 0. 8) O11◦O21(0 .2 ,0 .2 ,0 .8)
andisrepresentedinFig. 7 k1 l2 (0 3, 0 2, 0 6) k2 l3 (0 5, 0 3, 0 8)
O11◦O21(0 .3 ,0 .2 ,0 .8) O 11 ◦ O 22 (0 3 , 0 2 , 0 8) O 11 ◦ O 21 (0 5 , 0 2 , 0 8) O 11 ◦ O 21 (0 2 , 0 2 , 0 8) O11 ◦O21(0 .2 , 02 , 0 .8)
O12 ◦ O22 (0 3, 0 3, 0 8)
k1 l3 (0 5, 0 2, 0 6) k2 l1(0 2, 0 2, 0 8)
O11◦O21(02,02,08) O11 ◦ O21 (0 2, 0 2, 0 8)
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
k4 l1(0 2, 0 2, 0 6)
k3 l2 (0 3, 0 3, 0 4)
O12 ◦ O22 (0 2, 0 2, 0 6)
k4 l3 (0 5, 0 3, 0 6)
O11 ◦O21(02, 0. 2, 0. 4) O12 ◦ O22 (0 3, 0 3, 0 6) O12 ◦O22(0. 3, 0. 3, 0. 6)
O12 ◦O22(0.2,0.2,0.6)
O12◦O22(0 .3 ,0 .3 ,0 .5) O 12 ◦ O 22 (0 3 , 0 3 , 0 6) O 12 ◦ O 22 (0 4 , 0 3 , 0 6) O 12 ◦ O 22 (0 2 , 0 2 , 0 6) O12 ◦O22(02 , 0.2 , 06)
O12 ◦ O22 (0 2, 0 2, 0 6)
k3 l1(0 2, 0 2, 0 4)
O11◦O21(0 .2 ,0 .2 ,0 .6)
O12 ◦ O22 (0 3, 0 3, 0 6)
k4 l2 (0 3, 0 3, 0 6)
Figure7. Gi1 ◦ Gi2
k3 l3 (0 4, 0 3, 0 5)
Theorem2.27. Thecomposition ˇ Gi1 ◦ ˇ Gi2 = (O1 ◦O2,O11 ◦O21,O12 ◦O22,...,O1r ◦ O2r ) oftwoINGSsofGSs G1 and G2 isanINGSof G1 ◦ G2 Proof. Weconsiderthreecases:
Case1: For k ∈ P1, l1l2 ∈ P2h T(O1h ◦O2h )((kl1)(kl2))= TO1 (k) ∧ TO2h (l1l2)
≤ TO1 (k) ∧ [TO2 (l1) ∧ TO2 (l2)] =[TO1 (k) ∧ TO2 (l1)] ∧ [TO1 (k) ∧ TO2 (l2)]
= T(O1 ◦O2 )(kl1) ∧ T(O1 ◦O2 )(kl2),
I(O1h ◦O2h )((kl1)(kl2))= IO1 (k) ∧ IO2h (l1l2)
≤ IO1 (k) ∧ [IO2 (l1) ∧ IO2 (l2)] =[IO1 (k) ∧ IO2 (l1)] ∧ [IO1 (k) ∧ IO2 (l2)]
= I(O1 ◦O2 )(kl1) ∧ I(O1 ◦O2 )(kl2),
F(O1h ◦O2h )((kl1)(kl2))= FO1 (k) ∨ FO2h (l1l2) ≤ FO1 (k) ∨ [FO2 (l1) ∨ FO2 (l2)] =[FO1 (k) ∨ FO2 (l1)] ∨ [FO1 (k) ∨ FO2 (l2)]
= F(O1 ◦O2 )(kl1) ∨ F(O1 ◦O2 )(kl2),
for kl1,kl2 ∈ P1 ◦ P2
Case2: For k ∈ P2, l1l2 ∈ P1h
T(O1h ◦O2h )((l1k)(l2k))= TO2 (k) ∧ TO1h (l1l2)
≤ TO2 (k) ∧ [TO1 (l1) ∧ TO1 (l2)] =[TO2 (k) ∧ TO1 (l1)] ∧ [TO2 (k) ∧ TO1 (l2)]
= T(O1 ◦O2 )(l1k) ∧ T(O1 ◦O2 )(l2k), 14
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
I(O1h ◦O2h )((l1k)(l2k))= IO2 (k) ∧ IO1h (l1l2) ≤ IO2 (k) ∧ [IO1 (l1) ∧ IO1 (l2)] =[IO2 (k) ∧ IO1 (l1)] ∧ [IO2 (k) ∧ IO1 (l2)]
= I(O1 ◦O2 )(l1k) ∧ I(O1 ◦O2 )(l2k),
F(O1h ◦O2h )((l1k)(l2k))= FO2 (k) ∨ FO1h (l1l2)
≤ FO2 (k) ∨ [FO1 (l1) ∨ FO1 (l2)] =[FO2 (k) ∨ FO1 (l1)] ∨ [FO2 (k) ∨ FO1 (l2)]
= F(O1 ◦O2 )(l1k) ∨ F(O1 ◦O2 )(l2k),
for l1k,l2k ∈ P1 ◦ P2
Case3: For k1k2 ∈ P1h ,l1,l2 ∈ P2 suchthat l1 = l2
T(O1h ◦O2h )((k1l1)(k2l2))= TO1h (k1k2) ∧ TO2 (l1) ∧ TO2 (l2) ≤ [TO1 (k1) ∧ TO1 (k2)] ∧ TO2 (l1) ∧ TO2 (l2) =[TO1 (k1) ∧ TO2 (l1)] ∧ [TO1 (k2) ∧ TO2 (l2)] = T(O1 ◦O2 )(k1l1) ∧ T(O1 ◦O2 )(k2l2),
I(O1h ◦O2h )((k1l1)(k2l2))= IO1h (k1k2) ∧ IO2 (l1) ∧ IO2 (l2) ≤ [IO1 (k1) ∧ IO1 (k2)] ∧ [IO2 (l1) ∧ IO2 (l2)] =[IO1 (k1) ∧ IO2 (l1)] ∧ [IO1 (k2) ∧ IO2 (l2)]
= I(O1 ◦O2 )(k1l1) ∧ I(O1 ◦O2 )(k2l2),
F(O1h ◦O2h )((k1l1)(k2l2))= FO1h (k1k2) ∨ FO2 (l1) ∨ FO2 (l2) ≤ [FO1 (k1) ∨ FO1 (k2)] ∨ [FO2 (l1) ∨ FO2 (l2)] =[FO1 (k1) ∨ FO2 (l1)] ∨ [FO1 (k2) ∨ FO2 (l2)] = F(O1 ◦O2 )(k1l1) ∨ F(O1 ◦O2 )(k2l2), for k1l1,k2l2 ∈ P1 ◦ P2 Allcasesholdsfor h =1, 2,...,r.Thiscompletestheproof.
Definition2.28. Let ˇ Gi1 =(O1,O11,O12,...,O1r )and ˇ Gi2 =(O2,O21,O22,...,O2r ) beINGSsofGSs ˇ G1 =(P1,P11,P12,...,P1r )and ˇ G2 =(P2,P21,P22,...,P2r ),respectively.Theunionof Gi1 and Gi2,denotedby Gi1 ∪ Gi2 =(O1 ∪ O2,O11 ∪ O21,O12 ∪ O22,...,O1r ∪ O2r ), isdefinedas: (i)
T(O1 ∪O2 )(k)=(TO1 ∪ TO2 )(k)= TO1 (k) ∨ TO2 (k) I(O1 ∪O2 )(k)=(IO1 ∪ IO2 )(k)= IO1 (k) ∨ IO2 (k) F(O1 ∪O2 )(k)=(FO1 ∪ FO2 )(k)= FO1 (k) ∧ FO2 (k) forall k ∈ P1 ∪ P2, (ii)
T(O1h ∪O2h )(kl)=(TO1h ∪ TO2h )(kl)= TO1h (kl) ∨ TO2h (kl) I(O1h ∪O2h )(kl)=(IO1h ∪ IO2h )(kl)= IO1h (kl) ∨ IO2h (kl) F(O1h ∪O2h )(kl)=(FO1h ∪ FO2h )(kl)= FO1h (kl) ∧ FO2h (kl) forall kl ∈ P1h ∪ P2h 15
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Example2.29. TheunionoftwoINGSs ˇ Gi1 and ˇ Gi2 showninFig. 2 isdefinedas Gi1 ∪ Gi2 = {O1 ∪ O2,O11 ∪ O21,O12 ∪ O22} andisrepresentedinFig. 8. l1(0 2, 0 2, 0 3) l3(0 5, 0 4, 0 5) l2(0 3, 0 3, 0 4) k3(0 4, 0 3, 0 4) k2(0 5, 0 3, 0 8)
k4(0 5, 0 3, 0 6) k1(0 5, 0 2, 0 6) O11 ∪ O21(0 2, 0 2, 0 4) O12 ∪ O22(0 3, 0 3, 0 5) O 12 ∪ O 22 (0 4 , 0 3 , 0 6) O 11 ∪ O 21 (0 5 , 0 2 , 0 . 8)
Figure8. ˇ Gi1 ∪ ˇ Gi2
Theorem2.30. Theunion Gi1 ∪Gi2 = (O1 ∪O2,O11 ∪O21,O12 ∪O22,...,O1r ∪O2r ) oftwoINGSsoftheGSs ˇ G1 and ˇ G2 isanINGSof ˇ G1 ∪ ˇ G2
Proof. Let k1k2 ∈ P1h ∪ P2h.Therearetwocases:
Case1: For k1,k2 ∈ P1,bydefinition 2.28, TO2 (k1)= TO2 (k2)= TO2h (k1k2)= 0, IO2 (k1)= IO2 (k2)= IO2h (k1k2)=0, FO2 (k1)= FO2 (k2)= FO2h (k1k2)= 1.Thus,
T(O1h ∪O2h )(k1k2)= TO1h (k1k2) ∨ TO2h (k1k2) = TO1h (k1k2) ∨ 0 ≤ [TO1 (k1) ∧ TO1 (k2)] ∨ 0 =[TO1 (k1) ∨ 0] ∧ [TO1 (k2) ∨ 0] =[TO1 (k1) ∨ TO2 (k1)] ∧ [TO1 (k2) ∨ TO2 (k2)] = T(O1 ∪O2 )(k1) ∧ T(O1 ∪O2 )(k2),
I(O1h ∪O2h )(k1k2)= IO1h (k1k2) ∨ IQ2h (k1k2)
= IO1h (k1k2) ∨ 0 ≤ [IO1 (k1) ∧ IO1 (k2)] ∨ 0 =[IO1 (k1) ∨ 0] ∧ [IO1 (k2) ∨ 0] =[IO1 (k1) ∨ IO2 (k1)] ∧ [IO1 (k2) ∨ IO2 (k2)] = I(O1 ∪O2 )(k1) ∧ I(O1 ∪O2 )(k2), 16
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F(O1h ∪O2h )(k1k2)= FO1h (k1k2) ∧ FO2h (k1k2) = FO1i (k1k2) ∧ 1 ≤ [FO1 (k1) ∨ FO1 (k2)] ∧ 1 =[FO1 (k1) ∧ 1] ∨ [FO1 (k2) ∧ 1] =[FO1 (k1) ∧ FO2 (k1)] ∨ [FO1 (k2) ∧ FO2 (k2)] = F(O1 ∪O2 )(k1) ∨ F(O1 ∪O2 )(k2),
for k1,k2 ∈ P1 ∪ P2 Case2: For k1,k2 ∈ P2,bydefinition 2.28, TO1 (k1)= TO1 (k2)= TO1h (k1k2)= 0, IO1 (k1)= IO1 (k2)= IO1h (k1k2)=0, FO1 (k1)= FO1 (q2)= FO1h (k1k2)= 1,so
T(O1h ∪O2h )(k1k2)= TO1h (k1k2) ∨ TO2i (k1k2) = TO2i (k1k2) ∨ 0 ≤ [TO2 (k1) ∧ TO2 (k2)] ∨ 0 =[TO2 (k1) ∨ 0] ∧ [TO2 (k2) ∨ 0] =[TO1 (k1) ∨ TO2 (k1)] ∧ [TO1 (k2) ∨ TO2 (k2)] = T(O1 ∪O2 )(k1) ∧ T(O1 ∪O2 )(k2),
I(O1h ∪O2h )(q1k2)= IO1h (k1k2) ∨ IO2h (k1k2) = IO2h (k1k2) ∨ 0 ≤ [IO2 (k1) ∧ IO2 (k2)] ∨ 0 =[IO2 (k1) ∨ 0] ∧ [IO2 (k2) ∨ 0] =[IO1 (k1) ∨ IO2 (k1)] ∧ [IO1 (k2) ∨ IO2 (k2)] = I(O1 ∪O2 )(k1) ∧ I(O1 ∪O2 )(k2),
F(O1h ∪O2h )(k1k2)= FO1h (k1k2) ∧ FO2h (k1k2) = FO2h (k1k2) ∧ 1 ≤ [FO2 (k1) ∨ FO2 (k2)] ∧ 1 =[FO2 (k1) ∧ 1] ∨ [FO2 (k2) ∧ 1] =[FO1 (k1) ∧ FO2 (k1)] ∨ [FO1 (k2) ∧ FO2 (k2)] = F(O1 ∪O2 )(k1) ∨ F(O1 ∪O2 )(k2), for k1,k2 ∈ P1 ∪ P2 Bothcaseshold ∀h ∈{1, 2,...,r}.Thiscompletestheproof.
Theorem2.31. Let ˇ G = (P1 ∪ P2,P11 ∪ P21,P12 ∪ P22,...,P1r ∪ P2r ) betheunion oftwoGSs G1 = (P1,P11,P12,...,P1r ) and G2 = (P2,P21,P22,...,P2r ).Thenevery INGS ˇ Gi = (O,O1,O2,...,Or ) of ˇ G isunionofthetwoINGSs ˇ Gi1 and ˇ Gi2 ofGSs G1 and G2,respectively.
17
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Proof. Firstly,wedefine O1,O2,O1h and O2h for h ∈{1, 2,...,r} as: TO1 (k)= TO (k),IO1 (k)= IO (k),FO1 (k)= FO (k), if k ∈ P1, TO2 (k)= TO (k),IO2 (k)= IO (k),FO2 (k)= FO (k), if k ∈ P2, TO1h (k1k2)= TOh (k1k2),IO1h (k1k2)= IOh (k1k2),FO1h (k1k2)= FOh (k1k2), if k1k2 ∈ P1h ,
TO2h (k1k2)= TOh (k1k2),IO2h (k1k2)= IOh (k1k2),FO2h (k1k2)= FOh (k1k2), if k1k2 ∈ P2h Then O = O1 ∪ O2 and Oh = O1h ∪ O2h, h ∈{1, 2,...,r} Nowfor k1k2 ∈ Plh, l =1, 2, h =1, 2,...,r,
TOlh (k1k2)= TOh (k1k2) ≤ TO (k1) ∧ TO (k2)= TOl (k1) ∧ TOl (k2), IOlh (k1k2)= IOh (k1k2) ≤ IO (k1) ∧ IO (k2)= IOl (k1) ∧ IOl (k2), FOlh (k1k2)= FOh (k1k2) ≤ FO (k1) ∨ FO (k2)= FOl (k1) ∨ FOl (k2),i.e., ˇ Gil =(Ol ,Ol1,Ol2,...,Olr )isanINGSof ˇ Gl ,l=1,2. Thus Gi =(O,O1 ,O2,...,Or ),anINGSof G = G1 ∪ G2,istheunionofthetwo INGSs Gi1 and Gi2
Definition2.32. Let Gi1 =(O1,O11,O12,...,O1r )and Gi2 =(O2,O21,O22,...,O2r ) beINGSsofGSs ˇ G1 =(P1,P11,P12,...,P1r )and ˇ G2 =(P2,P21,P22,...,P2r ),respectivelyandlet P1 ∩ P2 = ∅ Join of Gi1 and Gi2,denotedby ˇ Gi1 + ˇ Gi2 =(O1 + O2,O11 + O21,O12 + O22,...,O1r + O2r ),
isdefinedas:
(i)
T(O1 +O2 )(k)= T(O1 ∪O2 )(k) I(O1 +O2 )(k)= I(O1 ∪O2 )(k) F(O1 +O2 )(k)= F(O1 ∪O2 )(k) forall k ∈ P1 ∪ P2, (ii)
T(O1h +O2h )(kl)= T(O1h ∪O2h )(kl) I(O1h +O2h )(kl)= I(O1h ∪O2h )(kl) F(O1h +O2h )(kl)= F(O1h ∪O2h )(kl) forall kl ∈ P1h ∪ P2h , (iii)
T(O1h +O2h )(kl)=(TO1h + TO2h )(kl)= TO1 (k) ∧ TO2 (l) I(O1h +O2h )(kl)=(IO1h + IO2h )(kl)= IO1 (k) ∧ IO2 (l) F(O1h +O2h )(kl)=(FO1h + FO2h )(kl)= FO1 (k) ∨ FO2 (l) forall k ∈ P1 , l ∈ P2
Example2.33. ThejoinoftwoINGSs ˇ Gi1 and ˇ Gi2 showninFig. 2 isdefinedas ˇ Gi1 + ˇ Gi2 = {O1 + O2,O11 + O21,O12 + O22} andisrepresentedintheFig. 9
Theorem2.34. Thejoin Gi1 +Gi2 = (O1 +O2,O11 +O21,O12 +O22,...,O1r +O2r ) oftwoINGSsofGSs ˇ G1 and ˇ G2 isINGSof ˇ G1 + ˇ G2
Proof. Let k1k2 ∈ P1h + P2h.Therearethreecases:
Case1: For k1,k2 ∈ P1,bydefinition 2.32, TO2 (k1)= TO2 (k2)= TO2h (k1k2)= 0, IO2 (k1)= IO2 (k2)= IO2h (k1k2)=0, FO2 (k1)= FO2 (k2)= FO2h (k1k2)= 18
l 3 (0 5 , 0 4 , 0 5)
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
l2 (0 3, 0 3, 0 4)
1,so,
(0.5,0.2,0.6) (0.5,0.3,0.8)
(0.3,0.2,0.6) (0.3,0.3,0.8)
k4(0 .5 ,0 .3 ,0 .6) k1(0. 5, 0. 2, 0. 6)
(0 2, 0 2, 0 6) (05,0 3,0 6) (0.2,0.2,0.8) (0.4,0.3,0.5)
(0.3,0.3,0.6) (0.3,0.3,0.4) (0.2,0.2,0.6) (0.2,0.2,0.4)
O11+O21(0 .2 ,0 .2 ,0 .4) O12+O22(03, 0. 3, 0. 5) O 12 + O 22 (0 4 , 0 3 , 0 6) O 11 + O 21 (0 5 , 0 2 , 0 . 8)
k3 (0 4, 0 3, 0 4) k2 (0 5, 0 3, 0 8)
Figure9. ˇ Gi1 + ˇ Gi2
T(O1h +O2h )(k1k2)= TO1h (k1k2) ∨ TO2h (k1k2) = TO1h (k1k2) ∨ 0 ≤ [TO1 (k1) ∧ TO1 (k2)] ∨ 0 =[TO1 (k1) ∨ 0] ∧ [TO1 (q2) ∨ 0] =[TO1 (k1) ∨ TO2 (k1)] ∧ [TO1 (k2) ∨ TO2 (k2)] = T(O1 +O2 )(k1) ∧ T(O1 +O2 )(k2),
I(O1h +O2h )(k1k2)= IO1h (k1k2) ∨ IO2h (k1k2) = IO1h (k1k2) ∨ 0 ≤ [IO1 (k1) ∧ IO1 (k2)] ∨ 0 =[IO1 (k1) ∨ 0] ∧ [IO1 (k2) ∨ 0] =[IO1 (k1) ∨ IO2 (k1)] ∧ [IO1 (k2) ∨ IO2 (k2)] = I(O1 +O2 )(k1) ∧ I(O1 +O2 )(k2),
F(O1h +O2h )(k1k2)= FO1h (k1k2) ∧ FO2h (k1k2) = FO1h (k1k2) ∧ 1 ≤ [FO1 (k1) ∨ FO1 (k2)] ∧ 1 =[FO1 (k1) ∧ 1] ∨ [FO1 (k2) ∧ 1] =[FO1 (k1) ∧ FO2 (k1)] ∨ [FO1 (k2) ∧ FO2 (k2)] = F(O1 +O2 )(k1) ∨ F(O1 +O2 )(k2), for k1,k2 ∈ P1 + P2 19
l 1 (0 2 , 0 2 , 0 3)
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Case2: For k1,k2 ∈ P2,bydefinition 2.32, TO1 (k1)= TO1 (k2)= TO1h (k1k2)= 0, IO1 (k1)= IO1 (k2)= IO1h (k1k2)=0, FO1 (k1)= FO1 (k2)= FO1h (k1k2)= 1,so
T(O1h +O2h )(k1k2)= TO1i (k1k2) ∨ TO2i (k1k2) = TO2i (k1k2) ∨ 0 ≤ [TO2 (k1) ∧ TO2 (k2)] ∨ 0 =[TO2 (k1) ∨ 0] ∧ [TO2 (k2) ∨ 0] =[TO1 (k1) ∨ TO2 (k1)] ∧ [TO1 (k2) ∨ TO2 (k2)] = T(O1 +O2 )(k1) ∧ T(O1 +O2 )(k2),
I(O1h +O2h )(k1k2)= IO1h (k1k2) ∨ IO2h (k1k2) = IO2h (k1k2) ∨ 0 ≤ [IO2 (k1) ∧ IO2 (k2)] ∨ 0 =[IO2 (k1) ∨ 0] ∧ [IO2 (k2) ∨ 0] =[IO1 (k1) ∨ IO2 (k1)] ∧ [IO1 (k2) ∨ IO2 (k2)] = I(O1 +O2 )(k1) ∧ I(O1 +O2 )(k2),
F(O1h +O2h )(k1k2)= FO1h (k1k2) ∧ FO2h (k1k2) = FO2h (k1k2) ∧ 1 ≤ [FO2 (k1) ∨ FO2 (k2)] ∧ 1 =[FO2 (k1) ∧ 1] ∨ [FO2 (k2) ∧ 1] =[FO1 (k1) ∧ FO2 (k1)] ∨ [FO1 (k2) ∧ FO2 (k2)] = F(O1 +O2 )(k1) ∨ F(O1 +O2 )(k2),
for q1,q2 ∈ P1 + P2 Case3: For k1 ∈ P1, k2 ∈ P2,bydefinition 2.32, TO1 (k2)= TO2 (k1)=0, IO1 (k2)= IO2 (k1)=0, FO1 (k2)= FO2 (k1)=1,so T(O1h +O2h )(k1k2)= TO1 (q1) ∧ TO2 (k2) =[TO1 (k1) ∨ 0] ∧ [TO2 (k2) ∨ 0] =[TO1 (k1) ∨ TO2 (k1)] ∧ [TO2 (k2) ∨ TO1 (k2)] = T(O1 +O2 )(k1) ∧ T(O1 +O2 )(k2),
I(O1h +O2h )(k1k2)= IO1 (k1) ∧ IO2 (k2) =[IO1 (k1) ∨ 0] ∧ [IO2 (k2) ∨ 0] =[IO1 (k1) ∨ IO2 (k1)] ∧ [IO2 (k2) ∨ IO1 (k2)] = I(O1 +O2 )(k1) ∧ I(O1 +O2 )(k2), 20
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F(O1h +O2h )(k1k2)= FO1 (k1) ∨ FO2 (k2) =[FO1 (k1) ∧ 1] ∨ [FO2 (k2) ∧ 1] =[FO1 (k1) ∧ FO2 (k1)] ∨ [FO2 (k2) ∧ FO1 (k2)] = F(O1 +O2 )(k1) ∨ F(O1 +O2 )(k2), for k1,k2 ∈ P1 + P2 Allthesecaseshold ∀h ∈{1, 2,...,r}.Thiscompletestheproof.
Theorem2.35. If G = (P1 + P2,P11 + P21,P12 + P22,...,P1r + P2r ) isthejoinof thetwoGSs ˇ G1 = (P1,P11,P12,...,P1r ) and ˇ G2 = (P2,P21,P22,...,P2r ).Theneach strongINGS Gi = (O,O1 ,O2,...,Or ) of G,isjoinofthetwostrongINGSs Gi1 and ˇ Gi2 ofGSs ˇ G1 and ˇ G2,respectively.
Proof. Wedefine Ol and Olh for l =1, 2and h =1, 2,...,r as: TOl (k)= TO (k),IOk (k)= IO (k),FOl (k)= FO (k), if k ∈ Pl , TOlh (k1k2)= TOh (k1k2),IOlh (k1k2)= IOh (k1k2),FOlh (k1k2)= FOh (k1k2),if k1k2 ∈ Plh Nowfor k1k2 ∈ Plh, l =1, 2, h =1, 2,...,r, TOlh (k1k2)= TOh (k1k2)= TO (k1) ∧ TO (k2)= TOl (k1) ∧ TOl (k2), IOlh (k1k2)= IOh (k1k2)= IO (k1) ∧ IO (k2)= IOl (k1) ∧ IOl (k2), FOlh (k1k2)= FOh (k1k2)= FO (k1) ∨ FO (k2)= FOl (k1) ∨ FOl (k2),i.e., Gil =(Ol ,Ol1,Ol2,...,Olr )isstrongINGSof Gl , l =1, 2 Moreover, ˇ Gi isthejoinof ˇ Gi1 and ˇ Gi2 asshown: Accordingtothedefinitions 2.28 and 2.32, O = O1 ∪ O2 = O1 + O2 and Oh = O1h ∪ O2h = O1h + O2h , ∀k1k2 ∈ P1h ∪ P2h. When k1k2 ∈ P1h + P2h (P1h ∪ P2h),i.e., k1 ∈ P1 and k2 ∈ P2, TOh (k1k2)= TO (k1) ∧ TO (k2)= TOl (k1) ∧ TOl (k2)= T(O1h +O2h )(k1k2), IOh (k1k2)= IO (k1)∧IO (k2)= IOl (k1)∧IOl (k2)= I(O1h +O2h )(k1k2), FOh (k1k2)= FO (k1) ∨ FO (k2)= FOl (k1) ∨ FOl (k2)= F(O1h +O2h )(k1k2), when k1 ∈ P2, k2 ∈ P1,wegetsimilarcalculations.It’struefor h =1, 2,...,r.This completestheproof.
3. Application
AccordingtoIMFdata,1 75billionpeoplearelivinginpoverty,theirlivingis estimatedtobelessthantwodollarsaday.Povertychangesbyregion,forexamplein Europeitis3%,andintheSub-SaharanAfricaitisupto65%.Werankthecountries oftheWorldaspoororrich,usingtheirGDPpercapitaasscale.Poor countriesare tryingtocatchupwithrichordevelopedcountries.Butthisratioisverysmall,that’s whytradeofpoorcountriesamongthemselvesisveryimportant.Therearedifferent typesoftradeamongpoorcountries,forexample:agriculturalorfooditems,raw minerals,medicines,textilematerials,industrialsgoodsetc.UsingINGS,wecan estimatebetweenanytwopoorcountrieswhichtradeiscomparativelystrongerthan others.Moreover,wecandecide(judge)whichcountryhaslarge numberofresources forparticulartypeofgoodsandbettercircumstancesforitstrade.Wecanfigureout, forwhichtrade,anexternalinvestorcaninvesthismoneyinthese poorcountries. Further,itwillbeeasytojudgethatinwhichfieldthesepoorcountriesaretryingto 21
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Table1. INsetOofninepoorcountriesonglobe
PoorCountry T I F
Congo 0.5 0.3 0.2 Liberia 0.4 0.4 0.3 Burundi 0.4 0.4 0.4 Tanzania 0.5 0.5 0.4 Uganda 0.4 0.4 0.5 SierraLeone 0.5 0.4 0.4 Zimbabwe 0.3 0.4 0.4 Kenya 0.5 0.3 0.3 Zambia 0.4 0.4 0.4
bebetter,andcanbehelped.Itwillalsohelpindecidingthatinwhichtradethey areweak,andshouldbefacilitated,sothattheycanbeindependentandimprove theirlivingstandards. WeconsiderasetofninepoorcountriesintheWorld:
Let O betheINseton P ,asdefinedinTable 1.InTable 1,symbol T demonstrates thepositiveaspectsofthatpoorcountry,symbol I indicatesitsnegativeaspects, whereas F denotesthepercentageofambiguityofitsproblemsfortheWorld. Let weusefollowingalphabetsforcountrynames: CO=Congo,L=Liberia,B=Burundi,T=Tanzania,U=Uganda,SL=Sierra Leone,ZI=Zimbabwe,K=Kenya,ZA=Zambia.Foreverypairofpoor countries inset P ,differenttradeswiththeir T , I and F valuesaredemonstratedinTables 2 8,where T , F and I indicatesthepercentageofoccurrence,non-occurrenceand uncertainty,respectivelyofaparticulartradebetweenthosetwopoorcountries
Table2. INsetofdifferenttypesoftradebetweenCongoand otherpoorcountriesin P
Typeoftrade (CO,L) (CO,B) (CO,T) (CO,U) (CO,K)
Fooditems (0.1,0.2,0.3) (0.4,0.2,0.1) (0.2,0.1,0.4) (0.4,0.3,0.5) (0.2,0.1,0.3) Chemicals (0.2,0.4,0.3) (0.1,0.2,0.1) (0.1,0.2,0.4) (0.3,0.2,0.4) (0.5,0.1,0.1) Oil (0.4,0.2,0.1) (0.4,0.3,0.2) (0.5,0.1,0.2) (0.4,0.2,0.2) (0.5,0.3,0.1)
Rawminerals (0.3,0.1,0.1) (0.4,0.3,0.3) (0.4,0.2,0.2) (0.4,0.1,0.2) (0.5,0.1,0.1)
Textileproducts (0.2,0.3,0.3) (0.1,0.3,0.4) (0.1,0.2,0.4) (0.1,0.3,0.2) (0.2,0.1,0.3)
Goldanddiamonds (0.4,0.1,0.1) (0.4,0.2,0.2) (0.2,0.2,0.4) (0.2,0.2,0.4) (0.1,0.3,0.3)
P = {Congo, Liberia, Burundi, Tanzania, Ugenda, SierriaLeone, Zimbabwe, Kenya, Zambia}.22
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
Table3. INsetofdifferenttypesoftradebetweenLiberiaand otherpoorcountriesin P
Typeoftrade (L,B) (L,T) (L,U) (L,SL) (L,ZI) Fooditems (0.4,0.2,0.2) (0.4,0.3,0.2) (0.3,0.3,0.4) (0.3,0.3,0.3) (0.2,0.3,0.3) Chemicals (0.2,0.2,0.4) (0.1,0.4,0.3) (0.3,0.3,0.3) (0.2,0.2,0.4) (0.1,0.3,0.3) Oil (0.1,0.1,0.4) (0.2,0.3,0.3) (0.1,0.1,0.4) (0.2,0.4,0.3) (0.2,0.2,0.3) Rawminerals (0.3,0.1,0.3) (0.2,0.2,0.3) (0.2,0.1,0.4) (0.3,0.2,0.3) (0.2,0.1,0.3) Textileproducts (0.1,0.3,0.4) (0.1,0.3,0.3) (0.2,0.1,0.3) (0.1,0.2,0.3) (0.2,0.2,0.3) Goldanddiamonds (0.2,0.1,0.4) (0.2,0.1,0.3) (0.3,0.1,0.3) (0.4,0.1,0.1) (0.3,0.1,0.1)
Table4. INsetofdifferenttypesoftradebetweenBurundiand otherpoorcountriesin P
Typeoftrade (B,T) (B,U) (B,SL) (B,ZI) (B,K) Fooditems (0.3,0.2,0.2) (0.4,0.1,0.2) (0.3,0.3,0.1) (0.3,0.3,0.2) (0.3,0.2,0.2) Chemicals (0.1,0.2,0.3) (0.2,0.1,0.3) (0.2,0.4,0.3) (0.3,0.4,0.3) (0.3,0.3,0.1) Oil (0.1,0.1,0.4) (0.2,0.3,0.4) (0.2,0.4,0.3) (0.2,0.2,0.5) (0.1,0.3,0.4) Rawminerals (0.2,0.1,0.3) (0.4,0.2,0.3) (0.4,0.2,0.4) (0.3,0.2,0.2) (0.4,0.2,0.2) Textileproducts (0.3,0.1,0.1) (0.2,0.4,0.3) (0.3,0.2,0.2) (0.3,0.2,0.1) (0.4,0.1,0.2)
Goldanddiamonds (0.3,0.2,0.3) (0.3,0.4,0.3) (0.1,0.4,0.2) (0.2,0.4,0.2) (0.2,0.3,0.4)
Table5. INsetofdifferenttypesoftradebetweenTanzaniaand otherpoorcountriesin P
Typeoftrade (T,U) (T,SL) (T,ZI) (T,K) (T,ZA) Fooditems (0.4,0.2,0.1) (0.5,0.1,0.1) (0.3,0.1,0.2) (0.4,0.3,0.2) (0.3,0.2,0.2) Chemicals (0.2,0.3,0.3) (0.2,0.3,0.4) (0.2,0.3,0.3) (0.4,0.1,0.4) (0.3,0.4,0.4) Oil (0.1,0.3,0.3) (0.4,0.1,0.3) (0.3,0.4,0.2) (0.2,0.3,0.3) (0.1,0.3,0.3) Rawminerals (0.3,0.3,0.4) (0.4,0.3,0.3) (0.3,0.2,0.1) (0.4,0.2,0.3) (0.3,0.2,0.3)
Textileproducts (0.2,0.4,0.3) (0.2,0.4,0.4) (0.1,0.3,0.4) (0.2,0.3,0.2) (0.4,0.1,0.2) Goldanddiamonds (0.3,0.4,0.3) (0.4,0.3,0.4) (0.3,0.1,0.1) (0.2,0.2,0.2) (0.4,0.3,0.3)
Table6. INsetofdifferenttypesoftradebetweenSierraLeone andotherpoorcountriesin P
Typeoftrade (SL,ZI) (SL,K) (SL,ZA) (SL,CO) (L,K) Fooditems (0.3,0.3,0.2) (0.4,0.2,0.1) (0.2,0.4,0.3) (0.5,0.1,0.1) (0.4,0.1,0.2) Chemicals (0.2,0.3,0.4) (0.3,0.2,0.2) (0.2,0.4,0.4) (0.2,0.2,0.3) (0.2,0.3,0.3) Oil (0.1,0.3,0.4) (0.2,0.2,0.3) (0.3,0.4,0.2) (0.5,0.2,0.1) (0.3,0.3,0.3)
Rawminerals (0.3,0.2,0.2) (0.5,0.2,0.1) (0.3,0.1,0.1) (0.3,0.3,0.3) (0.4,0.1,0.2)
Textileproducts (0.2,0.4,0.2) (0.3,0.2,0.3) (0.2,0.2,0.4) (0.2,0.2,0.3) (0.3,0.3,0.2)
Goldanddiamonds (0.3,0.1,0.1) (0.1,0.2,0.4) (0.2,0.3,0.3) (0.4,0.1,0.2) (0.3,0.2,0.3)
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MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27
Table7. INsetofdifferenttypesoftradebetweenZimbabweand otherpoorcountriesin P
Typeoftrade (ZI,K) (ZI,ZA) (ZI,U) (ZI,CO) Fooditems (0.3,0.2,0.2) (0.3,0.1,0.1) (0.3,0.1,0.1) (0.2,0.1,0.1) Chemicals (0.3,0.3,0.2) (0.2,0.4,0.3) (0.3,0.2,0.2) (0.2,0.1,0.2) Oil (0.1,0.3,0.3) (0.1,0.4,0.4) (0.3,0.2,0.1) (0.3,0.1,0.1) Rawminerals (0.3,0.1,0.2) (0.3,0.2,0.1) (0.3,0.2,0.3) (0.2,0.3,0.1) Textileproducts (0.2,0.2,0.2) (0.2,0.4,0.3) (0.2,0.3,0.3) (0.2,0.3,0.1) Goldanddiamonds (0.3,0.3,0.1) (0.3,0.2,0.1) (0.3,0.2,0.2) (0.3,0.2,0.1)
Table8. INsetofdifferenttypesoftradebetweenZambiaand otherpoorcountriesin P
Typeoftrade (ZA,CO) (ZA,L) (ZA,B) (ZA,K) Fooditems (0.3,0.1,0.2) (0.3,0.1,0.2) (0.4,0.2,0.1) (0.3,0.1,0.3) Chemicals (0.2,0.2,0.2) (0.2,0.2,0.1) (0.3,0.2,0.2) (0.3,0.1,0.1) Oil (0.4,0.1,0.1) (0.2,0.1,0.1) (0.3,0.2,0.1) (0.3,0.2,0.2) Rawminerals (0.3,0.1,0.1) (0.4,0.1,0.1) (0.4,0.2,0.2) (0.4,0.1,0.1) Textileproducts (0.2,0.2,0.2) (0.2,0.2,0.3) (0.2,0.3,0.2) (0.3,0.1,0.2) Goldanddiamonds (0.1,0.2,0.4) (0.4,0.3,0.2) (0.2,0.3,0.2) (0.3,0.2,0.1)
Manyrelationscanbedefinedontheset P ,wedefinefollowingrelationsonset P as:
P1 =Fooditems, P2 =Chemicals, P3 =Oil, P4 =Rawminerals, P5 =Textile products, P6 =Goldanddiamonds,suchthat(P,P1 ,P2,P3,P4,P5,P6)isaGS. Anyelementofarelationdemonstratesaparticulartradebetween thosetwopoor countries.As(P,P1 ,P2,P3,P4,P5,P6)isGS,that’swhyanyelementcanappearin onlyonerelation.Therefore,anyelementwillbeconsideredinthatrelation,whose valueofTishigh,andvaluesofI,Farecomparativelylow,usingdataofabove tables.
WritedownT,IandFvaluesoftheelementsinrelationsaccordingtoabovedata, suchthat O1, O2, O3, O4, O5, O6 areINsetsonrelations P1, P2, P3, P4, P5, P6, respectively.
Let P1 = {(Burundi, Congo), (SierraLeone, Congo), (Burundi, Zambia)}, P2 = {(Kenya,Congo)}, P3 = {(Congo, Zambia), (Congo, Tanzania), (Zimbabwe, Congo)}, P4 = {(Congo, Uganda), (SierraLeone, Kenya), (Zambia, Kenya)}, P5 = {(Burundi, Zimbabwe), (Tanzania, Burundi)}, P6 = {(SierraLeone, Liberia), (Uganda, SierraLeone), (Zimbabwe, SierraLeone)}
Let O1 = {((B,CO), 0 4, 0 2, 0 1), ((SL,CO), 0 5, 0 1, 0 1), ((B,ZA), 0 4, 0 2, 0 1)}, O2 = {((K,CO), 0.5, 0.1, 0.1)}, O3 = {((CO,ZA), 0.4, 0.1, 0.1), ((CO,T ), 0.5, 0.1, 0.2), ((ZI,CO), 0.3, 0.1, 0.1)}, O4 = {((CO,U ), 0.4, 0.1, 0.2), ((SL,K), 0.5, 0.2, 0.1), ((ZA,K), 0.4, 0.1, 0.1)}, O5 = {((B,ZI), 0.3, 0.2, 0.1), ((T,B), 0.3, 0.1, 0.1)}, O6 = {((SL,L), 0.4, 0.1, 0.1), ((U,SL), 0.4, 0.2, 0.1), ((ZI,SL), 0 3, 0 1, 0 1)}.Obviously,(O, O1, O2, O3, O4, O5, O6)isanINGSas showninFig. 10 24
(0 3, 0 1, 0 1)
Oil
(0 4, 0 1, 0 1) (0 5, 0 1, 0 1) (0. 3, 0. 1, 0. 1) (0 .4 ,0 .1 ,0 . 1) (0 4, 0 2, 0 1)
Rawminerals
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27 Liberia Uganda
Goldanddiamonds
(0 4, 0 2, 0 1) (0 .5 , 01 , 0 .2)
Tanzania Zimbabwe (0 4, 0 2, 0 1)
Leone Zambia
Chemicals Oil Oil
(0 5, 0 1, 0 1)
Goldanddiamonds Goldanddiamonds (0 4, 0 1, 0 1)
Sierra Kenya Congo Burundi
Fooditems Fooditems
(0 4, 0 1, 0 2)
Fooditems
Rawminerals Textileproducts Textileproducts
Rawminerals
(0 5, 0 2, 0 1) (0 3, 0 1, 0 1) (0 3, 0 2, 0 1)
Figure10. INGSindicatingeminenttradebetweenanytwopoorcountries
EveryedgeofthisINGSdemonstratestheprominenttradebetweentwopoorcountries,forexampleprominenttradebetweenCongoandZambiaisOil,itsT,Fand Ivaluesare0.4,0.1and0.1,respectively.Accordingtothesevalues,despiteof poverty,circumstancesofCongoandZambiaare40%favorableforoiltrade,10% areunfavorable,and10%areuncertain,thatis,sometimestheymaybefavorable andsometimesunfavorable.WecanobservethatCongoisvertexwithhighestvertexdegreeforrelationoilandSierraLeoneisvertexwithhighestvertexdegreefor relationgoldanddiamonds.Thatis,amongtheseninepoorcountries,Congois mostfavorableforoiltrade,andSierraLeoneismostfavorablefortradeofgoldand diamonds.ThisINGSwillbeusefulforthoseinvestors,whoareinterestedtoinvest intheseninepoorcountries.Forexampleaninvestorcaninvestinoil inCongo. Andifsomeonewantstoinvestingoldanddiamonds,thisINGSwillhelphimthat SierraLeoneismostfavorable.
AbigadvantageofthisINGSisthatUnitedNations,IMF,WorldBank, andrich countriescanbeawareofthefactthatinwhichfieldsoftrade,thesepoorcountries aretryingtobebetterandcanbehelpedtomaketheireconomicconditionsbetter. Moreover,INGSofpoorcountriescanbeverybeneficialforthem,itmayincrease tradeaswellasforeignaidandeconomichelpfromtheWorld,andcan presenttheir 25
MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,1–27 betteraspectsbeforetheWorld.
Wenowexplaingeneralprocedureofthisapplicationbyfollowingalgorithm. Algorithm:
1.Inputavertexset P = {C1,C2,...,Cn} andaINset O definedonset P
2.InputINsetoftradeofanyvertexwithallotherverticesandcalculate T , F ,and I ofeachpairofverticesusing, T (CiCj ) ≤ min(T (Ci),T (Cj )), F (Ci Cj ) ≤ max(F (Ci ),F (Cj )), I(CiCj ) ≤ min(I(Ci),I(Cj )).
3.RepeatStep2foreachvertexinset P .
4.Definerelations P1,P2,...,Pn ontheset P suchthat(P,P1 ,P2,...,Pn)is aGS.
5.Consideranelementofthatrelation,forwhichitsvalueof T iscomparatively high,anditsvaluesof F and I arelowthanotherrelations.
6.Writedownallelementsinrelationswith T , F and I values,corresponding relations O1,O2,...,On areINsetson P1,P2,P3,...,Pn,respectivelyand (O,O1 ,O2,...,On)isanINGS.
4. Conclusions
Fuzzygraphicalmodelsarehighlyutilizedinapplicationsofcomputerscience. Especiallyindatabasetheory,clusteranalysis,imagecapturing,datamining,control theory,neuralnetworks,expertsystemsandartificialintelligence.Inthisresearch paper,wehaveintroducedcertainoperationsonintuitionisticneutrosophicgraph structures.Wehavediscussedanovelandworthwhilereal-lifeapplicationofintuitionisticneutrosophicgraphstructureindecision-making.Wehave intensionsto generalizeourconceptsto(1)ApplicationsofINsoftGSsindecision-making(2) ApplicationsofINroughfuzzyGSsindecision-making,(3)ApplicationsofINfuzzy softGSsindecision-making,and(4)ApplicationsofINroughfuzzysoftGSsin decision-making.
Acknowledgment: TheauthorsarethankfultoEditor-in-Chiefandthereferees fortheirvaluablecommentsandsuggestions.
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MuhammadAkram (m.akram@pucit.edu.pk,makrammath@yahoo.com)
DepartmentofMathematics,UniversityofthePunjab,NewCampus,Lahore, Pakistan
MuzzamalSitara (muzzamalsitara@gmail.com)
DepartmentofMathematics,UniversityofthePunjab,NewCampus,Lahore, Pakistan
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