Fuzzy Critical Path with Various Measures
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1 Interntonl Journl of themtcs Trens n Technology IJTT Volume 5 umer 5- Ferury 08 Fuzzy Crtcl Pth wth Vrous esures V.nusuy n P.lsownr. ssocte Professor PG n eserch Deprtment of themtcs Seethlkshm mswm college Truchrpll- eserch Scholr PG n eserch Deprtment of themtcs Seethlkshm mswm college Truchrpll- strct In opertons reserch networks ply n mportnt role s qute often the prolem of etermnng n optmum soluton cn e looke upon s the prolem of selectng the est sequence of opertons out of fnte numer of vlle lterntves tht cn e represente s network. etwork grm plys vtl role to etermne proect completon tme. etwork Scheulng s technque use for plnnng n scheulng lrge proects n the vrous fels such s constructon frcton purchsng etc. etwork nlyss s technque whch etermnes the vrous sequences of ctvtes concernng proect n the proect completon tme. The populr metho of ths technque s wely use s the crtcl pth metho. In ths pper we fn the fuzzy crtcl pth n cyclc proect network usng mgntue mesure to entfy the fuzzy crtcl pth from type- trpezol fuzzy numers. n llustrtve exmple s lso nclue to emonstrte our propose pproch. Keywors: Fuzzy crtcl pth type- trpezol fuzzy numers cyclc proect network mgntue mesure centro mesure. S themtcs Suect Clssfcton 00: 05C7 I.Introucton The purpose of crtcl pth methocp s to n the plnnng n control of lrge complex proects. The pproch requres the conserton of :. Wht ctvtes re to e one. The sequence n whch they wll e performe. The resources requre n. The tme requre for ech ctvty. The crtcl pth metho technque proves for the network crtcl pth whch conssts of the sequence of proect ctvtes tht etermne the mnmum requre proect tme. Ths pper nlyze the crtcl pth n generl proect network wth fuzzy ctvty tmes. The fuzzy mesures were ntrouce y sugeno[9]. We propose mgntue n centro mesure[0] for fuzzy numers to crtcl pth metho n fuzzy proect network where the urton tme of ech ctvty n fuzzy proect network s represente y type- trpezol fuzzy numer. The structure of ths pper s s follows: In secton we hve some sc concepts. Secton contns some propertes regrng clculton of the totl slck fuzzy tme. Secton gves the network termnology. Secton 5 gves n lgorthm to fn the crtcl pth comne wth type- trpezol fuzzy numer usng mgntue n centro mesure metho. To llustrte the propose lgorthm the numercl exmple s solve n secton 6. II. sc concepts.. Type- fuzzy numer[8]: et X e type- fuzzy set efne n the unverse of scourse f the followng contons re stsfe then X s clle type- fuzzy numer.. X s norml.. X s convex set.. The support of X s close n oune. ISS: Pge 66
2 Interntonl Journl of themtcs Trens n Technology IJTT Volume 5 umer 5- Ferury 08 ISS: Pge 67.. orml type- fuzzy numer type- fuzzy numertfs X s s to e norml f ts Foot of ncertnty FO s norml ntervl type- fuzzy numer ITFS n t hs prmry memershp functon.. ton on type- fuzzy numers et = forll FO = et = forll FO = e two norml type- fuzzy numers. y usng extenson prncple we hve let = forll forll FO FO =.5 gntue esure et = c:λ e trpezol fuzzy numer such tht <<c<. It s converte to trngulr fuzzy numer c such tht < <. The mgntue mesure of the trngulr fuzzy numer wth prmetrc form s gven y g.6. Centro esure et e α-cut ntervl numer then the centro mesure of x xx.7.ottons: t = The ctvty etween noe n. ESF =The erlest strtng fuzzy tme of noe. FF = The ltest fnshng fuzzy tme of noe.
3 Interntonl Journl of themtcs Trens n Technology IJTT Volume 5 umer 5- Ferury 08 T S F = The totl slck fuzzy tme of t. p n = the n th fuzzy pth. P = the set of ll fuzzy pths n proect network Fp n = The totl slck fuzzy tme of pth p n n proect network. III.Propertes[56]: Property :. Forwr pss clculton To clculte the erlest strtng fuzzy tme n the proect network set the ntl noe to zero for strtng e ESF ESF mx { ESF TSF } =numer of preceng noes. ESF =The erlest strtng fuzzy tme of noe. nkng vlue s utlze to entfy the mxmum vlue. Erlest fnshng fuzzy tme = Erlest strtng fuzzy tme + Fuzzy ctvty tme. Property.. ckwr pss clculton To clculte the ltest fnshng tme n the proect network set FF mn { FF SET } n FF n ESFn. = numer of succeeng noes. nkng vlue s utlze to entfy the mnmum vlue. test strtng fuzzy tme= test fnshng Fuzzy tme - Fuzzy ctvty tme. Property.. For the ctvty t < Totl fuzzy slck: SFT Property.. FF ESF SFT or FF SFT ESF n; F p S F T n n p k to the estnton noe k= to m. p k P p n s the possle pths n fuzzy cyclc proect network from source noe IV. etwork termnology[]: Conser recte cyclc proect network GVE consstng of fnte set of noes V={..n} n set of m recte eges E V V. Ech ege s enote y n orere pr where. V n.. In ths network we specfy two noes enote y s n t whch re the source noe n the estnton noe respectvely. We efne pth p s sequence p =[=. - l =] of lterntng noes n eges. The exstence of t lest one pth p s n GVE s ssume for every noe V {S }. enotes trpezol type fuzzy numer ssocte wth the ege corresponng to the length necessry to trnsverse from to. V. lgorthm 5. lgorthm for fnng crtcl pth[78]: Step : Estmte the fuzzy ctvty tme wth respect to ech ctvty. Step : et ESF n clculte E S F =. n y usng property. Step : et F Fn E S F n clculte F F n =n- n-... y usng property. Step : Clculte SFT wth respect to ech ctvty n proect network y usng property. Step 5 : Clculte mgntue mesure n centro mesure for ech ctvty usng efnton.5. Step 6 : If mgntue mesure n centro mesure re zero those ctvtes re clle s fuzzy crtcl ctvty n the corresponng pth s entfe s fuzzy crtcl pth. ISS: Pge 68
4 Interntonl Journl of themtcs Trens n Technology IJTT Volume 5 umer 5- Ferury 08 VI. umercl exmple []: Exmple 6.: The prolem s to fn the crtcl pth n crtcl pth length etween source noe to estnton noe n the fuzzy cyclc proect network hvng 6 vertces n 7 eges wth type- fuzzy numer. Soluton : The ege lengths re P Q S T V P S 6 Q 5 T V Fg.6. Tle: ctvtes fuzzy urtons n totl slck fuzzy tme for ech ctvty ctvty< Fuzzy ctvty tme Defuzzfe ctvty tme converte n to trngulr fuzzy numer Defuzzfe ctvty tme Totl flot ISS: Pge 69
5 Interntonl Journl of themtcs Trens n Technology IJTT Volume 5 umer 5- Ferury Tle: ctvtes fuzzy urtons n totl slck fuzzy tme for ech ctvty ctvty -< - Fuzzy ctvty tme Fuzzy ctvty tmeα-cut ntervl numer α=0.5 Totl flot Centro mesure ISS: Pge 70
6 Interntonl Journl of themtcs Trens n Technology IJTT Volume 5 umer 5- Ferury 08 From the tle we oserve tht. The expecte tme n terms of trpezol fuzzy numers re efuzzfe usng efnton.6 n.7 for ll ctvtes n the gven cyclc proect network.. y usng property. ll possle pths P={ } re foun.. Fuzzy crtcl pth s entfe wth the help of mgntue n centro mesure.. The pth s the fuzzy crtcl pth n gven cyclc fuzzy proect network y usng oth the mesures. VII. Concluson: In ths work mgntue mesure n centro mesure re use to fn the fuzzy crtcl pth n cyclc proect network. we foun tht the crtcl pth whch s otne y oth the mesures re sme. Hence we conclue tht the otne crtcl pth s unque one. eferences []. V.nusuy n.sthy Type- fuzzy shortest pth Interntonl Journl of Fuzzy themtcl rchve []. V.nusuy n P. lsownr Fuzzy crtcl pth wth type- trpezolfuzzy numers tonl conference on Grph Theory Fuzzy Grph Theory nther pplctons Jml cemc eserch Journl: n nterscplnry SpeclIssueFe-06 pp [].G. ortoln. n.degn revew of some methos for rnkng fuzzysusets fuzzy sets n systems Vol pp.-9 []. S.Chns n P.Zelnsk Crtcl pth nlyss n the network wth fuzzy ctvtytmes Fuzzy Sets n Systems [5]. S.Elzeth n.suth Fuzzy crtcl pth prolem for proect networkinterntonl ournl of pure n pple mthemtcs [6]. G.S.ng n T.-Chenhn Fuzzy crtcl pth proect network Informton nngement scences [7]. S.H.suton Fuzzy crtcl pth metho IEEE Trns. System n Cyernetcs [8]. J.J.O ren CP n Constructon ngement ew York c-grw-hll 99. [9].Sugeno. Fuzzy mesures n fuzzy ntegrls- survey In Gupt srs ngnes 977pp [0]. S.Yo n FT.n Fuzzy crtcl pth metho se on sgne stnce rnkngof fuzzy memers IEEE Trnsctons on Systems n n Cyernetcs-Prt :Systems n Humns []...Zeh Fuzzy sets Informton n Control []...Zeh The concepts of lngustc vrle n ts pplcton topproxmte resonng prt Informton Scences ; ISS: Pge 7
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Today. CS 188: Artificial Intelligence Fall Recap: Search. Example: Pancake Problem. Example: Pancake Problem. General Tree Search.
CS 88: Artificil Intelligence Fll 00 Lecture : A* Serch 9//00 A* Serch rph Serch Tody Heuristic Design Dn Klein UC Berkeley Multiple slides from Sturt Russell or Andrew Moore Recp: Serch Exmple: Pncke
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