[Lkshmi, 5): Februry, 0] ISSN: 77-955 IOR), Publiction Imct Fctor: 785 IJESRT INTERNTIONL JOURNL OF ENGINEERING SCIENCES & RESERCH TECHNOLOGY SUB -TRIDENT FORM THROUGH FUZZY SUB -TRINGULR FORM Prveen Prksh, MGeethLkshmi * Hed o the Dertment, Mthemtics Dertment, Hindustn University,Chenni, Indi * ssistnt Proessor, Mthemtics Dertment,KCG college o Technology,Chenni,Indi BSTRCT This er dels with the new concet to ind Shortest Pth nd the Otimum Solution with the hel o Fuzzy Numbers Here the Fuzzy Sub-Tngulr Form is obtined rom the Pscl s Tngle Grded Men long with the hel o uzzy numbers nd this orm is gin converted to Sub-Tdent Form The Minimum vlue o Sub-Tdent Form gives the shortest Pth nd lso the Otimum Solution with suitble numecl exmle KEYWORDS: Grded Men, Fuzzy Numbers, Otimum Solution, Pscl s Tngle, nd Sub-Tdent Form INTRODUCTION One o the most imortnt roblem in trnsorttion network is the shortest th roblem The distnce o shortest route is clculted using this shortest th roblem rom source node to the destintion node Dubois nd Prde introduced the uzzy shortest th roblem with the hel o Floyd s lgothm in the yer 980 []Lter in the yer 99, Okd nd Gen [] introduced the sme roblem with the hel o Dijkstr s lgothm LotiZdeh introduced the uzzy set theory in the yer 95[]Then Chen nd Hsieh introduced the Fuzzy Grded Men Integrtion Reresenttion [] nd [7] ter tht the reresenttion nd liction o uzzy number is given by SHilern in the yer 997 []In the yer 000, Okd nd Soer concentrted on shortest th roblem on network in which uzzy number, insted o rel number is ssigned to ech rc length [5] In this roosed method the Sub-Tdent Form through Fuzzy Sub-Tdent Form by using trezoidl uzzy numbers with the hel o Pscl s Tngle Grded Men which gives the Shortest Pth nd the Otiml Solution Here this er consists o Sections: Bsic deinitions nd nottions in the irst section, the roosed method in the second section, the Working Rule or the lgothm in the third section, identiying the shortest th nd obtining the otiml solution by giving suitble numecl exmle in the ourth section nd inlly the ith section gives the conclusion bsed on our study PRELIMINRIES In this section, some bsic deinition o uzzy set theory nd uzzy number re discussed [] Deinition : uzzy set x in X is chrctezed by membershi unction ) reresents grde o membershi o ) More generl reresenttion or uzzy set is given by x, x x) / x X Deinition : The cut o uzzy set x X x, where 0, x o the Universe o discourse X is deined s htt: // wwwijesrtcom Interntionl Journl o Engineeng Sciences & Reserch Technology [8]
[Lkshmi, 5): Februry, 0] ISSN: 77-955 IOR), Publiction Imct Fctor: 785 Deinition uzzy set deined on the set o rel numbers is sid to be uzzy number i its membershi unction : [0, ] hs the ollowing chrctestics ) b) c) is convex i x ) x) min{ x ), x)} x, x X, is norml i there exists n x ) is iecewise continuous x such tht i Reresenttion o Generlized Trezoidl) Fuzzy Number mx x) In generl, generlized uzzy number is descbed t ny uzzy subset o the rel line R, whose membershi unction stisies the ollowing conditions: is continuous ming rom R to [0,], x)=0, x c, x) L x) is stctly incresing on [c,] x) w, x b, x) R x) is stctly decresing on [b,d], x) 0, d x Where 0 w nd, b, c nd d re rel numbers We denote this tye o generlized uzzy number s c,, b, d; w) LR When w=, we denote this tye o generlized uzzy number s c,, b, d) LR When Lx) nd Rx) re stght line, then is Trezoidl uzzy number, we denote it s c,, b, d) Grded Men Integrtion Reresenttion In 998, Chen nd Hsieh [] nd [7] roosed grded men integrtion reresenttion or reresenting generlized uzzy number Suose L, R re inverse unctions o L nd R resectively, nd the grded men h- level vlue o generlized uzzy number c,, b, d; w) LR is h[ L h) R h)]/ Then the grded men integrtion reresenttion o generlized uzzy number bsed on the integrl vlue o grded men h-level is ) w 0 L h h) R w 0 hdh h) dh c b d 0, Where h is between 0 nd w, 0<w ; Pscl s Tngle Grded Men roch The Grded Men Integrtion Reresenttion or generlized uzzy number by Chen nd Hsieh [] - [8]Lter SkKdhr Bbu nd BRjesh nnd introduces Pscl s Tngle Grded Men in Sttisticl Otimiztion [0]But the resent roch is very simle ws o nlyzing uzzy vbles to get the otimum shortest th This htt: // wwwijesrtcom Interntionl Journl o Engineeng Sciences & Reserch Technology [85]
[Lkshmi, 5): Februry, 0] ISSN: 77-955 IOR), Publiction Imct Fctor: 785 rocedure is tken rom the ollowing Pscl s tngle We tke the coeicients o uzzy vbles s Pscl s tngle numbers Then we just dd nd divide by the totl o Pscl s number nd we cll it s Pscl s Tngle Grded Men roch Figure: Pscl s Tngle The ollowing re the Pscl s tngulr roch: Let,,, ) nd B b, b, b, ) re two trezoidl uzzy numbers then we cn tke the b coeicient o uzzy numbers rom Pscl s tngles nd ly the roch we get the ollowing ormul: b b b b ) ; ; 8 8,, b, b, b b The coeicients o, nd re,,, This roch cn be extended ir n-dimensionl Pscl s Tngulr uzzy order lso, PROPOSED METHOD Fuzzy Sub-Tngulr Form o Pscl s Tngle Tngulr Fuzzy Numbers: The Pscl s Tngle or Tngulr Fuzzy Number is given in igure: nd the Sub -Tngles or Tngulr Fuzzy Number is given in Figure: Set: I) s ollows: Figure: Figure: Sub Tngle: i) Sub-Tngle: ii) Sub-Tngle: iii) Set: I htt: // wwwijesrtcom Interntionl Journl o Engineeng Sciences & Reserch Technology [8]
[Lkshmi, 5): Februry, 0] ISSN: 77-955 IOR), Publiction Imct Fctor: 785 P = ) P, q = ) P B, r = ) P C P = ) P, q = ) P B, r = ) P C P = ) P, q = ) P B, r = ) P C The Fuzzy Sub-Tngulr Form or Tngulr Fuzzy Number is given by FST, q, r ), where q r r, rr q q q r r B Trezoidl Fuzzy Numbers: The Pscl s Tngle or Trezoidl Fuzzy Number is given in igure: nd the Sub -Tngles or Trezoidl Fuzzy Number is given in Figure: 5Set: II) s ollows: Figure: Figure: 5 Sub Tngle: i) Sub-Tngle: ii) Sub-Tngle: iii) Set: II P = P = P = ) ) ), q =, q =, q =, r = 7, r =, r = 7 The Fuzzy Sub-Tngulr Form or Trezoidl Fuzzy Number is given by FST, q, r ), where q r htt: // wwwijesrtcom Interntionl Journl o Engineeng Sciences & Reserch Technology [87] q q q r r r, rr C Pentgonl Fuzzy Numbers: The Pscl s Tngle or Pentgonl Fuzzy Number is given in igure: nd the Sub -Tngles or Pentgonl Fuzzy Number is given in Figure: 7Set: III) s ollows:
[Lkshmi, 5): Februry, 0] ISSN: 77-955 IOR), Publiction Imct Fctor: 785 Figure: Figure: 7 Sub Tngle: i) Sub-Tngle: ii) Sub-Tngle: iii) Set: III P = P = P = ) ) ), q =, q = 0, q = 8, r = 5, r = 5 htt: // wwwijesrtcom Interntionl Journl o Engineeng Sciences & Reserch Technology [88], r = The Fuzzy Sub-Tngulr Form or Pentgonl Fuzzy Number is given by FST, q, r ), where q r Similrly we cn extend this to dierent uzzy numbers Sub-Tdent Form, 0 q q q r r r, rr The Sub-Tdent Form o Fuzzy Number is given by ST qq rr, where rr re the Grded Mens o the Pscl s Tngle rom the Fuzzy Tngulr Form LGORITHM The Working Rule or the Sub-Tdent Form to ind the shortest th nd the otimum solution is given by the ollowing lgothm: Ste: Inut the uzzy number s edge weight Ste: ind uzzy sub-tngulr orm FST ) o uzzy number using Pscl s Tngle Grded Men tken in three sides o Pscl s Tngle Ste: converting uzzy sub-tngulr orm FST ) to Sub-Tdent Form ST )
[Lkshmi, 5): Februry, 0] ISSN: 77-955 IOR), Publiction Imct Fctor: 785 Ste: ind the minimum vlue o the Sub-Tdent Form ST ) Ste: 5 Reet Ste: or ll the djcent edges nd the minimum o ll djcent edges rve t the shortest th Ste: Otimum Solution is obtined by otsol min ST 00 ILLUSTRTIVE EXMPLE In order to illustrte the bove rocedure consider smll network shown in igure: 8 where ech rc length is reresented s trezoidl uzzy number [9]: Figure: 8 0, 0, 08, ) 0, 0, 0, 08) 5 0, 0, 05, 0) 0, 05, 0, 07) 7 0, 0, 05, 07) 05, 07, 08, 09) 0, 07, 08, 09) 0, 0, 0, 0) 0, 0, 07, 08) llustrtive Exmle Tble: Pth Tble: Fuzzy Sub-Tngulr Form Sub-Tdent Form FST, q, r ) q r ST ) Minimum ST ),),),) 0,0,078) 07,0,078) 0,0,089) 07 055 0587 0587,) 0555,055,05) 087 087,7) 07,0700,089) 08879 08879 TOTL Minimum 85 ST ) min ST The Minimum Vlue is obtined using the Sub-Tdent Form in the th, ) The only djcent edge to the th, ) is, ) nd, 7) Thereore the Shortest Pth is 7 Suose the node is gin divided into two edges then gin we hve to use the sme Sub-Tdent Form or the ths nd choose the minimum vlue The Otimum Solution s the minimum cost is given by otsol min ST 00 8500 htt: // wwwijesrtcom Interntionl Journl o Engineeng Sciences & Reserch Technology [89]
[Lkshmi, 5): Februry, 0] ISSN: 77-955 IOR), Publiction Imct Fctor: 785 85 CONCLUSION The im o this er is to ind the otimum shortest th nd the minimum otimum solution by using the simlest orm clled the Sub-Tdent Form through Fuzzy Sub-Tngulr Form using trezoidl uzzy numbers This method is very simle while comng to ll the existing methods REFERENCES [] LZdeh, Fuzzy Sets, Inormtion nd Control, Vol8, 8-5,95 [] D Dubois nd HPrde, Fuzzy sets nd Systems : cdemic ress, New York 980 [] SOkd nd MGen, Fuzzy shortest th roblem, Comuters nd Industl Engineeng, Vol 7,No -, 5-8,99 [] SHilern, Reresenttion nd liction o uzzy numbers, Fuzzy Sets nd Systems, Vol9, No, 59-8, 997 [5] OkdS nd SoerT, Shortest Pth Problem on network with Fuzzy rc Lengths, Fuzzy Sets nd Systems : vol09, No, 9-0,000 [] Shn-Huo Chen, Oertions on Fuzzy Numbers with Functions Pncile, Tmkng JMngement Sci,Vol), -5 [7] Shn-Huo Chen nd Chin Hsun Hseih, Grded Men Integrtion Reresenttion o Generlized Fuzzy Number, Journl o Chinese Fuzzy System ssocition: Tiwn, vol5, no, -7,000 [8] Shn-Huo Chen nd Chin Hsun Hseih, Reresenttion, Rnking, Distnce nd Similty o L-R tye uzzy number nd lictions, ustrlin Journl o Intelligent Inormtion Processing Systems: ustrli, vol, n0, 7-9,000 [9] ThH, Oertion Reserch Introduction, Prentice Hll o Indi Pvt Ltd, New Delhi,00 [0] SKKhdr Bbu nd Rjesh nndb, Sttisticl otimiztion or Generlised Fuzzy Number, Interntionl Journl o Modern Engineeng Reserch: Vol, 7-5, 0 [] Felix nd Victor Devdoss, New Decgonl Fuzzy Number under Uncertin Linguistic Environment, Interntionl Journl o Mthemtics nd its liction: Vol, 89-97, 05 htt: // wwwijesrtcom Interntionl Journl o Engineeng Sciences & Reserch Technology [90]