Multi-Attribute Group Decision Making based on Bidirectional Projection Measure

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1 IJSRD - Ieraioal Joural for Scieific Research & Develope Vol. 4, Issue 10, 2016 ISSN (olie): Muli-Aribue Group Decisio Makig based o Bidirecioal Projecio Measure S. Sala 1 Dr. V. Diwakar Reddy 2 Dr. G. Krishaiah 3 1 PG Scholar 2 Professor 3 Red.Professor 1,2,3 Depare of Mechaical Egieerig 1,2,3 Sri Vekaeswara Uiversiy, Tirupai, Idia Absrac This paper develops a ehod for solvig he uliple aribue group decisio akig probles wih he ierval valued eurosophic iforaio. Iiially, propose a o-liear equaio usig Euclidea disace easures based o axiizig deviaio ehod. The o-liear equaios solved by Lagrage fucio o obai aribue weighs. Furher, ha rakig he aleraives based o bidirecioal projecio easures fro Ideal Ierval Neurosophic Esiaes Reliabiliy Soluio (IINERS) ad Ideal Ierval Neurosophic Esiaes U-reliabiliy Soluio (IINEURS) o selec he bes aleraive. Fially, a illusraive exaple deosraes he applicaio of he proposed ehod. The effeciveess ad advaages of he proposed ehod are show by he coparaive aalysis wih exisig relaive ehods. Key words: Ierval Valued Neurosophic Ses (IVNS), Euclidea Disace Measure, Maxiizig Deviaio Mehod, Bidirecioal Projecio Measures, Group Decisio Makig I. INTRODUCTION The Neurosophic Ses was developed by Saradache 1 o oly o deal wih he decisio iforaio which is ofe icoplee, ideeriae, ad icosise bu also o iclude he ruh ebership degree, he falsiy ebership degree, ad he ideeriacy ebership degree. For sipliciy ad pracical applicaio, Wag proposed he sigle-valued NS (SVNS) ad he ierval-valued NS (IVNS) which are he isaces of NS ad gave soe operaios o hese ses 2, 3. Sice is appearace, ay fruiful resuls have appeared 4, 5. O oe had, ay researchers have proposed soe aggregaio operaors of SVNS ad INS ad applied he o MADM probles O he oher had, soe researchers have also proposed eropy ad siilariy easure of he SVNS ad IVNS ad applied he o MADM probles 11, 12.The above probles ha are relaed o he aribue weighs are copleely kow. However, wih he develope of he iforaio sociey ad iere echology, he socioecooic eviroe ges ore coplex i ay decisio areas, such as capial ivese decisio akig, edical diagosis, ad persoel exaiaio. Oly oe decisio aker cao deal wih he coplex probles. Accordigly, i is ecessary o gaher uliple decisio akers wih differe kowledge srucures ad experieces o coduc a group decisio akig. I soe circusaces, i is difficul for he decisio akers o give he iforaio of he aribue weighs correcly, which akes he aribue weighs icopleely kow or copleely ukow. How o derive he aribue weighs fro he give eurosophic iforaio is a ipora opic. I iuiioisic fuzzy eviroes, ay researchers have proposed soe progra odels o obai he icopleely kow aribue weighs or he copleely ukow aribue weighs, such as Xu proposed he deviaio-based ehod 13, he ideal soluio-based ehod 14, ad he group cosesus-based ehod 15 ad Li e al. proposed he cosisecy-based ehod 16. Uder he eurosophic eviroe, Sahi ad Liu proposed he axiizig deviaio ehod 17, 21. I his paper, we deal wih MAGDM proble which he iforaio expressed by IVNS, ad he aribue weighs are copleely ukow. Our ai is: 1) To deerie aribue weighs based o axiizig deviaio ehod usig Euclidea disace easure. 2) Aggregae hese weighs based o Ideal Ierval Neurosophic Esiaes Reliabiliy Soluio (IINERS) ad Ideal Ierval Neurosophic Esiaes Ureliabiliy Soluio (IINEURS). 3) Rakig he aleraives based o Bidirecioal Projecio Measures. The res of he paper is orgaized as follows. Secio 2 briefly describes soe basic coceps of NS, SVNS ad IVNS. Secio3 proposes a bidirecioal projecio easure based uliple aribue group decisio akig ehodology. I sec. 4, a illusraive exaple is preseed o deosrae he applicaio of he proposed ehod, Sec.5 he effeciveess ad advaages of he proposed ehod are deosraed by he coparaive aalysis wih exisig relaive ehods. Fially, Sec. 6coais coclusios ad fuure work. A. Neurosophic Se (NS) II. PRELIMINARIES Le X be a space of pois (objecs) ad x X. A eurosophic se A i X is defied by a ruh ebership fucio (x), a ideeriacy ebership fucio I(x) ad a falsiy ebership fucio FA(x). (x), I(x) ad FA(x) are real sadard or real osadard subses of 0,1. Tha is TA(x): X 0,1, IA(x): X 0,1 ad FA(x):X 0,1. There is o resricio o he su of (x), I(x) ad FA(x), so 0 supta(x) supia(x) supfa(x) 3. B. Coplie of NS The coplee of a eurosophic se A is deoed by A c ad is defied as TA c (x) = {1} (x), IA c (x) = {1} IA(x) ad FA c (x) = {1} FA(x) for all x X. C. Sigle Valued Neurosophic Ses (SVNS) Le X be a uiverse of discourse. A sigle valued eurosophic se A over X is a objec havig he for A={ x, ua(x), wa(x), va(x) :x X}where ua(x):x 0,1, wa(x):x 0,1 ad va(x):x 0,1 wih 0 ua(x)wa(x) va(x) 3 for all x X. The iervals (x), (x) ad (x) deoe he ruh ebership degree, he ideeriacy ebership All righs reserved by 325

2 degree ad he falsiy ebership degree of x o A, respecively. D. Coplee of SVNS The coplee of a SVNS A is deoed by A c ad is defied as (x) = (x), (x) = 1(x), ad va c (x) = u(x) for all x X. Tha is, A c = { x, (x),1wa(x), ua(x) : x X}. E. Ierval Neurosophic Ses (INS) Le ua (x) = ua (x), ua ( x), wa (x) = wa (x), wa (x) ad va (x) = va (x), va (x), he A ={ x, ua (x),ua (x), wa (x),wa (x), va (x), va (x) : x X} wih he codiio, 0 supua (x) supwa (x) supva (x) 3 for all x X. Here, we oly cosider he sub-uiary ierval of 0, 1. Therefore, a INS is clearly eurosophic se. F. Coplie of INS The coplee of a INS A is deoed by A c ad is defied as ua c(x) = v(x), (wa ) c (x)=1wa (x), (wa )(x)= 1wA (x) ad va c(x)=u(x) for all x X. Tha is, A c={ x, va (x),va (x),1wa (x),1wa (x),ua (x),ua (x) :x X}. G. INS Rakig Suppose ha A 1 = (a 1, b 1, c 1, d 1, e 1, f 1) ad A 2 = (a 2, b 2, c 2, d 2, e 2, f 2) are wo ierval valued eurosophic ses The we defie he rakig ehod as follows: If L (A 1) > L (A 2), he A 1>A 2. If L (A 1) = L (A 2) ad N (A 1) > N (A 2), he A 1>A 2. H. INS Disace Measurig Fucios (Ye 2014a)Le x = (T L 1,T U 1,I L U 1, I 1,F L U 1, F 1 ), ad y = (T L 2,T U 2,I L 2, I U 2,F L 2, F U 2 ) be wo INVs, he The Haig disace bewee x ad y is defied as follows d H (x, y) = 1 ( T 6 1 L T L 2 T U 1 T U 2 I L 1 I L 2 I U 1 I U 2 F L 1 F L 2 F U 1 F U 2 ) (1) The Euclidia disace bewee x ad is defied as follows. d E (x, y) = 1 6 ((T 1 L T 2 L ) 2 (T 1 U T 2 U ) 2 (I 1 L I 2 L ) 2 (I 1 U I 2 U ) 2 (F 1 L F 2 L ) 2 (F 1 U F 2 U ) 2 ) (2) 1) Lagrage Fucio L(w, ) =k=1 λ k 1 w 6 j=1 i=1 s=1 j (( upv ) E) ( w 12 j=1 j 2 1) We copue Parial derivaives of L as follows, L = λ w k=1 k i=1 s=1 (( upv ) E w j = 0 ) j of DM L = w j=1 j 2 1 = 0 Where, is Lagrage uliplier ad λ k is weigh d k ( upv ) E = ((u 1 L u 2 L ) 2 (u 1 U u 2 U ) 2 (p 1 L p 2 L ) 2 (p 1 U p 2 U ) 2 (v 1 L v 2 L ) 2 (v 1 U v 2 U ) 2 A siple ad exac forula for deeriig he aribue weighs as follows: w k=1 λ k i=1 s=1 (( upv ) E ) j = ( λ k (( upv ) E ) j=1 k=1 i=1 s=1 I. IRNPIS Le α ad β be he collecio of beefi aribues ad cos aribues, respecively. R s is he Ierval Relaive Neurosophic Posiive Ideal Soluio (IRNPIS). R s = r s1, r s2,,r s is defied as a soluio i which every copoe r sj = < T j, I j, F j > is characerized by T j = {(ax{t ij }) i-h aribue α, (i{t ij }) i-h aribue β} J. IRNNIS The Ierval Relaive Neurosophic Negaive Ideal Soluio (IRNNIS). R s = r s1, r s2,,r s is defied as a soluio i which every copoe = < T j, I j, F j > is r sj characerized by T j = {(i{t ij }) i-h aribue α, (ax{t ij }) i-h aribue β} I j = {(ax{i ij }) i-h aribue α, (i{i ij }) i-h aribue β}, F j = {(ax{f ij }) i-h aribue α, (i{f ij }) i-h aribue β}, I he eurosophic decisio arix D S = < T ij, I ij, F ij > for i=1,2,, ad j=1,2,, K. The ideal ierval eurosophic esiaes reliabiliy soluio (IINERS) ad he ideal ierval eurosophic esiaes u-reliabiliy soluio (IINEURS) for ierval eurosophic decisio arix For a ierval eurosophic decisio akig arix D S = < T ij, I ij, F ij > for i=1,2,, ad j=1,2,,. T ij,i ij,f ij are he degrees of ebership, degree of ideeriacy ad degree of o-ebership of he aleraive A i saisfyig he aribue C j. The ierval eurosophic esiae reliabiliy soluio (see defiiio 2.13, ad 2.14) ca be deeried fro he cocep of SVNS cube 22. The ideal ierval eurosophic esiaes reliabiliy soluio (IINERS) is preseed as follows Y S = y S1,y S2,y S3,y S4 = ax i {U i1 }, i i {P i1 }, i i {V i1 }, ax i {U i2 }, i i {P i2 }, i i {V i2 }, ax i {U i3 }, i i {P i3 }, i i {V i3 }, ax i {U i4 }, i i {P i4 }, i i {V i4 }, The ideal ierval eurosophic esiaes ureliabiliy soluio (IINEURS) is preseed as follows Y S = y S1,y S2,y S3,y S4 = i i {U i1 }, ax i {P i1 }, ax i {V i1 }, i i {U i2 }, ax i {P i2 }, ax i {V i2 }, i i {U i3 }, ax i {P i3 }, ax i {V i3 }, i i {U i4 }, ax i {P i4 }, ax i {V i4 }, III. PROPOSED METHOD Sep 1: Le X k or A (k) = (a (k) ij ) x be a ierval eurosophic decisio arix, where a (k) ij =(u (k) ij, u (k) ij, p (k) ij, p (k) ij, v (k) ij, v (k) ij ) is a aribue value, give by he decisio aker d k, for he aleraive A I wih respec o he aribue C j. Sep 2: Usig he aribue values give by decisio akers, deeried he aribue weighs by give forula: w k=1 λ k i=1 s=1 (( upv ) E ) j = j=1(k=1 λ k i=1 s=1(( upv ) E ) Where, λ k = weigh of DM d k ( upv )E = ((u 1 L 1) 2 (u 1 U 1) 2 (p 1 L ) 2 (p 1 U ) 2 (v 1 L ) 2 (v 1 U ) 2 All righs reserved by 326

3 Here Euclidea disace easured fro each aleraive A i wih respecive o he aribue C j o Posiive Ideal Soluio (PIS) assuig {1,1, 0, 0,0, 0} Sep 3: A weighed aleraive decisio arix is obaied by calculaig y i kj= y li kj, y ui kj = w jx li kj, w jx lu kj (k = 1, 2,, ; j = 1, 2,, ; i = 1, 2,, )for X i (i = 1, 2,, ), Sep 4: Ideifyhe Ideal Ierval Neurosophic Esiaes Reliabiliy Soluio (IINERS) accordig o followig equaio: Y S = y S1,y S2,y S3,y S4 = ax i {U i1 }, i i {P i1 }, i i {V i1 }, ax i {U i2 }, i i {P i2 }, i i {V i2 }, ax i {U i3 }, i i {P i3 }, i i {V i3 }, ax i {U i4 }, i i {P i4 }, i i {V i4 }, Ad he Ideal Ierval Neurosophic Esiaes Ureliabiliy Soluio (IINEURS) is as follows: Y S = y S1,y S2,y S3,y S4 = i i {U i1 }, ax i {P i1 }, ax i {V i1 }, i i {U i2 }, ax i {P i2 }, ax i {V i2 }, i i {U i3 }, ax i {P i3 }, ax i {V i3 }, i i {U i4 }, ax i {P i4 }, ax i {V i4 }, Sep 5: The bidirecioal projecio easure bewee each weighed aleraive decisio arix Y i (i = 1, 2,, ) ad IINERSY IINEURSY - ca be calculaed by BProj (Y i,y ) = Where Y i Y Y i Y Y i Y Y i.y, Y i = v k (y li kj ) 2 (y ui kj ) 2 k=1 j=1 Y = v k (y l kj ) 2 (y u kj ) 2 k=1 j=1 Y i.y = k=1 v k j=1y li kj y l kj y ui kj y u kj Ad also, bidirecioal projecio easures fro IINEURS BProj (Y i,y - ) = Where Y i Y Y i Y Y i Y Y i.y, Y i = v k (y li kj ) 2 (y ui kj ) 2 k=1 j=1 Y = v k (y l kj ) 2 (y u kj ) 2 k=1 j=1 Y i.y - = k=1 v k j=1y li kj y l kj y ui kj y u kj Sep 6: Deerie he Relaive Correlaio Coefficies based o bidirecioal projecio easures by followig equaio: RCC i = BProj (Y i,y ) BProj (Y i,y )BProj (Y i,y ) Sep 7: The, he aleraives are raked i a descedig order accordig o he values of RCC i. The greaer value of RCC i eas he beer aleraive. A. Sep 1 IV. ILLUSTRATIVE EXAMPLE The decisio akig proble is adaped fro 25. Suppose ha a orgaizaio plas o iplee ERP syse. The firs sep is o fora projec ea ha cosiss of CIO ad wo seior represeaives fro user depares. By collecig all iforaio abou ERP vedors ad syses, projec ea chooses four poeial ERP syses Ai (i = 1, 2, 3, 4) as cadidaes. The copay eploys soe exeral professioal orgaizaios (expers) o aid his decisio akig. The projec ea selecs four aribues o evaluae he aleraives: (1) C1: fucio ad echology, (2) C2: sraegic fiess, (3) C3: vedors abiliy, ad (4) C4: vedor s repuaio. Suppose ha here are hree decisio akers, deoed by D1, D2, D3, whose correspodig weigh vecor is λ = (1/3, 1/3, 1/3). The four possible aleraives are o be evaluaed uder hese four aribues ad are i he for of IVNNs for each decisio aker, as show i he followig: 1) Ierval valued eurosophic decisio arix: D1= {{0.4, 0.5, 0.2, 0.3, 0.3, 0.5} {0.3, 0.4, 0.3, 0.6, 0.2, 0.4} {0.2, 0.5, 0.2, 0.6, 0.3, 0.5} {0.5, 0.6, 0.3, 0.5, 0.2, 0.5}} {{0.6, 0.7, 0.1, 0.2, 0.2, 0.3} {0.1, 0.3, 0.1, 0.4, 0.2, 0.5} {0.4, 0.5, 0.2, 0.5, 0.3, 0.7} {0.2, 0.4, 0.1, 0.4, 0.3, 0.3}} {{0.3, 0.4, 0.2, 0.3, 0.3, 0.4} {0.3, 0.6, 0.2, 0.3, 0.2, 0.5} {0.2, 0.7, 0.2, 0.4, 0.3, 0.6} {0.2, 0.6, 0.4, 0.7, 0.2, 0.7}} {{0.2, 0.6, 0.1, 0.2, 0.1, 0.2} {0.2, 0.5, 0.4, 0.5, 0.1, 0.6} {0.3, 0.5, 0.1, 0.3, 0.2, 0.2} {0.4, 0.4, 0.1, 0.6, 0.1, 0.5}} D2= {{0.4, 0.6, 0.1, 0.3, 0.2, 0.4} {0.3, 0.5, 0.1, 0.4, 0.3, 0.4} {0.4, 0.5, 0.2, 0.4, 0.1, 0.3} {0.3, 0.6, 0.3, 0.6, 0.3, 0.6}} {{0.3, 0.5, 0.1, 0.2, 0.2, 0.3} {0.3, 0.4, 0.2, 0.2, 0.1, 0.3} {0.2, 0.7, 0.3, 0.5, 0.3, 0.6} {0.2, 0.5, 0.2, 0.7, 0.1, 0.2}} {{0.5, 0.6, 0.2, 0.3, 0.3, 0.4} {0.1, 0.4, 0.1, 0.3, 0.3, 0.5} {0.5, 0.5, 0.4, 0.6, 0.3, 0.4} {0.1, 0.2, 0.1, 0.4, 0.5, 0.6}} {{0.3, 0.4, 0.1, 0.2, 0.1, 0.3} {0.3, 0.3, 0.1, 0.5, 0.2, 0.4} {0.2, 0.3, 0.4, 0.5, 0.5, 0.6} {0.3, 0.3, 0.2, 0.3, 0.1, 0.4}} D3= {{0.1,0.3, 0.2,0.3, 0.4,0.5} {0.3,0.3, 0.1,0.3, 0.3,0.4} {0.2,0.6, 0.3,0.5, 0.3,0.5} {0.4,0.6, 0.3,0.4, 0.2,0.3}} {{0.3,0.6, 0.3,0.5, 0.3,0.5} {0.3,0.4, 0.3,0.4, 0.3,0.5} {0.3,0.5, 0.2,0.4, 0.1,0.5} {0.1,0.2, 0.3,0.5, 0.3,0.4}} {{0.4,0.5, 0.2,0.4, 0.2,0.4} {0.2,0.3, 0.1,0.1, 0.3,0.4} {0.1,0.4, 0.2,0.6, 0.3,0.6} {0.4,0.5, 0.2,0.6, 0.1,0.3}} {{0.2,0.4, 0.3,0.4, 0.1,0.3} {0.1,0.4, 0.2,0.5, 0.1,0.5} {0.3,0.6, 0.2,0.4, 0.2,0.2} {0.2,0.4, 0.3,0.3, 0.2,0.6}} W 4= B. Sep2 Aribue weighs are obaied fro Ma lab usig W 1= , W 2=0.2593, W 3=0.2560, W 4= C. Sep 3 axiizig deviaio ehod wih Euclidea disace easures as follows. {{ , , , , , } { } {{0.0997, , ,0.7751, , } { } {{0.1040, , , , , } { } {{ , , ,0.7505, , } { } { } { }} { } { }} { } { }} { } { }} All righs reserved by 327

4 D. Sep 4 Ideal Ierval Neurosophic Esiaes Reliabiliy Soluio (IINERS) for isace firs aribue is as follows {0.1040, , , , , } Ad he Ideal Ierval Neurosophic Esiaes Ureliabiliy Soluio (IINEURS) for isace firs aribue is as follows { , , } Sep 5: The bidirecioal projecio easure bewee each weighed aleraive decisio arix Y i (i = 1, 2,, ) ad IINERS Y &IINEURSY - Aleraive Y Y - A A A A Sep 6: Relaive Correlaio Coefficies based o bidirecioal projecio easures for aleraives A i A 1= A 2= A 3= A 4= Sep 7: The, he aleraives are raked i a descedig order accordig o he values of RCC i. The greaer value of RCC i eas he beer aleraive. Here, A 2> A 4> A 1> A 3 V. COMPARATIVE ANALYSIS AND DISCUSSION The resuls obai fro wo exaples wih parially kow ad copleely ukow weighs are copared o Sahi ad Liu 24 ad Liu ad Luo 25 ehods. 1) Sahi ad Liu 24 developed score ad accuracy discriiaio fucios for MCGDM proble afer proposig wo aggregaio operaors. The ukow weighs of aribues are deeried by cosrucig liear equaio based o axiizig deviaio ehod. The aribue weighs are obaied by solvig liear equaio usig Lagrage echique. The idividual decisio arixes are grouped wih aid of geoeric weighed aggregaio operaor. For each aleraive weighed aggregaed eurosophic values are calculaed usig obaied aribue weighs o aggregaed group decisio arix. Therefore he rakig of each aleraive is based o score ad accuracy fucios applied o aleraive weighed aggregaed eurosophic values. 2) Liu ad Luo 25 proposed weighed disace fro posiive ideal soluio o each aleraive based liear equaio for deeriig ukow weighs of aribues afer observig soe drawback for MAGDM uder SVNS. The liear fucio ais o iiize overall weighed disace fro PIS where aribue weighs are ukow. The parially kow or ukow codiios are subjeced o proposed liear equaio ad solved usig ay liear prograig echique resuls weighs of aribues. The rakig of aleraives give based o weighed haig disace fro PIS. The proposed odel also exeded o IVNS. Table 1: Weighed aleraive decisio arix 3) Proposed ehod aied o ehace resuls accuracy, flexible o operae ad effeciveess. I able 2 exaple is evaluaed wih copleely ukow weighs. The ukow weighs of aribues are deeried by cosrucig liear equaio based o axiizig deviaio ehod usig Euclidea disace easure. The aribue weighs are obaied by solvig liear equaio usig Lagrage echique. Ad rakig he aleraives based o bidirecioal projecio easures. The he proposed ehod give siilar resuls o 24 ad 25. Therefore he proposed ehod is accurae, flexible ad effecive. Type of Proble Copleely Ukow weighs Sachi ad Liu 24 A2 A4 A1 A3 Liu ad Luo 25 A2 A4 A1 A3 Table 2: Coparisos of Mehods VI. CONCLUSION Proposed Mehod A 2> A 4> A 1> A 3 Therefore, prese ehod for solvig he uliple aribue group decisio akig probles wih he ierval valued eurosophic iforaio was developed. Iiially, a oliear equaio usig Euclidea disace easures based o axiizig deviaio ehod developed. The he oliear equaios solved by Lagrage fucio o obai aribue weighs. Furher, ha rak he aleraives based o bidirecioal projecio easures fro IINERS ad IINEURS o selec he bes aleraive. Fially, a illusraive exaple deosraed he applicaio of he developed ehod, ad he he effeciveess ad raioaliy of he developed ehod are deosraed by he coparaive aalysis wih exisig relaive ehods. I he fuure work, we shall exed he bidirecioal projecio ehod o oher decisio daa, such as refied eurosophic ubers ad eurosophic ses, ad develop he applicaios such as paer recogiio ad edical diagosis. REFERENCES 1 F. Saradache, Neurosophy. Neurosophic Probabiliy, Se, ad Logic, ProQues Iforaio ad Learig, A Arbor, Mich, USA, H. Wag, F. Saradache, Y.-Q. Zhag, ad R. Suderraa, Ierval Neurosophic Ses ad Logic: Theory ad Applicaios i Copuig, Hexis, Phoeix, Ariz, USA, H. Wag, F. Saradache, Y. Q. Zhag, ad R. Suderraa, Sigle valued eurosophic ses, Muli space ad Muli srucure, o. 4, pp , S. Broui, M. Talea, A. Bakali, ad F. Saradache, Sigle valued eurosophic graphs, Joural of New Theory, vol. 10,pp , S. Broui,M. Talea,A. Bakali, ad F. Saradache, O bipolar sigle valued eurosophic graphs, Joural of New Theory, o. 11, pp , P. Liu ad Y. Wag, Ierval eurosophic prioriized OWA operaor ad is applicaio o uliple aribue decisio akig, Joural of Syses Sciece ad Coplexiy, pp. 1 17, All righs reserved by 328

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