CRITICAL PATH ANALYSIS IN A PROJECT NETWORK USING RANKING METHOD IN INTUITIONISTIC FUZZY ENVIRONMENT R. Sophia Porchelvi 1, G.

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1 Interntonl Journl of dvnce eserch IJO.or ISSN Interntonl Journl of dvnce eserch IJO.or Volume Issue Mrch 05 Onlne: ISSN 0-94 CITICL PTH NLYSIS IN POJECT NETWOK USING NKING METHOD IN INTUITIONISTIC FUZZY ENVIONMENT. Soph Porchelv G. Sudh Deprtment of Mthemtcs.D.M Collee for women ( utonomous NpttnmInd. E-ml: sophporchelv@ml.com Deprtment of Mthemtcs.V.C. Collee ( utonomous Mnnmpndl Ind. E-ml: venkt_sudh@yhoo.n KeyWords Crtcl Pth Method (CPM Intutonstc fuzzy project network nkn method Trnulr Intutonstc Fuzzy number (TIFN. STCT In ths pper n lorthm s presented to perform crtcl pth nlyss n n ntutonstc fuzzy envronment. Trnulr Intutonstc fuzzy numbers re used to represent ctvty tmes n the project network. nkn procedures re ppled on ntutonstc fuzzy numbers to fnd the crtcl pth. numercl emple llustrtes the method. IJO 05

2 Interntonl Journl of dvnce eserch IJO.or ISSN Introducton The crtcl pth method s vtl tool for plnnn nd control of comple projects. ccordn to crtcl pth method the decson mker cn control the tme nd cost of the project. CPM hs been used n busness mnement fctory producton etc. Here new pproch for fndn the crtcl pth s ntroduced. The bse de behnd ths method s the use of fuzzy vlues under mprecse condtons. The concept of fuzzy ws ntroduced by Zdeh [9] n 965. Vrous pplctons of fuzzy sets hve been studed by reserchers n dfferent felds. T.J.oss [] publshed n nterestn book on fuzzy sets theory nd ts pplctons n 005. In ths pper method for fndn crtcl pth n n Intutonstc fuzzy project network s presented s dscussed by []. rnkn procedure on Intutonstc fuzzy numbers s ppled for ettn the crtcl pths (efer []. Ths pper s ornzed s follows. In secton prelmnry concepts nd defntons n ntutonstc fuzzy set theory nd the procedure for fndn crtcl pth usn TIFN re provded. n lorthm s presented n secton. n llustrtve emple to fnd the crtcl pth s eplned n secton 4. The lst secton drws some concludn remrks.. Prelmnry concepts. Intutonstc Fuzzy Set: n Intutonstc fuzzy set ~ n X s ven by set of ordered trples: ~ ~ ( v ~ ( / X where ~ ~ : X [0 ] re functons such tht 0 ~ ( v ~ ( v for ll X. For ech the numbers ~ ndv ( represent the deree of membershp nd deree of non-membershp of the element X to X respectvely. ( ~. Intutonstc Fuzzy Number (IFN: n ntutonstc fuzzy subset ~ = ~ ( v ~ ( / X of rel lne s clled n Intutonstc Fuzzy Number (IFN f the follown holds: ( There ests m ~ ( m ndv ~ ( m 0 ( s contnuous mppn from to the closed ntervl [0] nd for ll the relton 0 ~ v ( holds. ( ~ The membershp nd non-membershp functon of ~ s of the follown form: 0 m f m m ~ m ( h m m 0 m where f ( nd h ( re strctly ncresn nd decresn functon m mndm m respectvely. m f m m;0 f f v ~ ( 0 m h m m ;0 h h 0 m Here m s the men vlue of ~. re clled left nd rht spreds of membershp functon ~ ( respectvely. represents left nd rht spreds of non membershp functon v ~ ( respectvely. = IJO 05

3 Interntonl Journl of dvnce eserch IJO.or ISSN IJO 05 Trnulr Intutonstc Fuzzy Number (TIFN (TIFN ~ s n ntutonstc fuzzy set n wth the follown ( ( ~ ~ ndv otherwse 0 ( ~ otherwse v ( ~ where nd ( ~ ~ v.4 Chen nd Chen Metrc Dstnce nkn Procedure Chen nd Chen proposed metrc dstnce method ( see [ ] to rnk fuzzy numbers. Let nd be two fuzzy numbers defned s follows: L m m

4 Interntonl Journl of dvnce eserch IJO.or ISSN L m m where m nd m re the men of nd. The metrc dstnce between nd cn be clculted s follows: L L D( = h ( y h ( y dy h ( y h ( y ( dy 0 0 where h L h h L nd h re the nverse functons of L L nd respectvely. In order to rnk fuzzy numbers Chen nd Chen ( see [ ] let the fuzzy number =0 then the metrc dstnce between nd 0 s clculted s follows: L D(0 = h ( y dy h ( y ( dy 0 0 the lrer vlue of D(0 s the better rnkn of ccordn to Chen nd Chen the membershp functon of s defned s follows: f f f where µ nd σ re clculted s follows: 4 y z p q r where becomes TIFN = + p ; =y + q ; = z + r The nverse functons h L nd h of L respectvely re shown s follows: h L (y = (µ - σ + σ y h (y = (µ + σ - σ y. Proposed Method for fndn the crtcl pth n ntutonstc fuzzy sense Nottons: N : The set of ll nodes n project network j : The trnulr ntutonstc fuzzy erlest fuzzy tme of j TIFLF j : The trnulr ntutonstc fuzzy ltest fuzzy tme of j TIFTF : The trnulr ntutonstc fuzzy totl flot of the ctvty -j TIFt : The trnulr ntutonstc fuzzy ctvty tme of nodes nd j TIFP : The trnulr ntutonstc fuzzy th pth TIFP(j : The set of ll nodes connected to ll predecessor ctvtes of node j TIFS(j : The set of ll nodes connected to ll successor ctvtes of node j TIFCP: The trnulr ntutonstc fuzzy completon tme of pth lorthm: Step : Clculte j = m =TIFLF = 0 TIFt / TIFP( j j j N nd IJO 05

5 Interntonl Journl of dvnce eserch IJO.or ISSN Step : Clculte TIFLF = mn TIFLF j TIFt / j TIFS ( j j n j N nd TIFLF n = n Step : Clculte TIFTF = TIFLF j TIFt ; j N Step 4 : Fnd ll the possble pths nd clculte TIFCP n project network Step 5 : Fnd the rnkn vlue of TIFCP(P = 4 nd compute the crtcl pth. 4. Numercl Emple Consder network wth the trnulr ntutonstc fuzzy rc lenths s shown below: P 4 4 P P 46 P 47 * ** 6 P 68 8 P 6 P P 78 P 5 P Fure- Trnulr Intutonstc Fuzzy Project Network The possble pths of n trnulr ntutonstc fuzzy project network re TIFP : ; TIFP : ; TIFP : ; TIFP 4 : Step To clculte the trnulr ntutonstc fuzzy erlest strt: Set = Clculte j j = by usn j = m TIFt / TIFP( j j j TIF( N nd = TIFLF = 0 = TIFt = ; ; ; ctvty TIFt Intutonstc Fuzzy ctvty Tme TIFP : TIFP : TIFP 4 : TIFP 5 : TIFP 6 : TIFP 46 : TIFP 47 : TIFP 57 : TIFP 68 : TIFP 78 : IJO 05

6 Interntonl Journl of dvnce eserch IJO.or ISSN Step To clculte the trnulr ntutonstc fuzzy ltest fnsh: 96 8 Set TIFLF 8 = Clculte TIFLF j j = 7654 by usn TIFLF = mn TIFLF TIFt / j TIFS ( j j n j TIF( N nd TIFLF n = n j TIFLF 7 =TIFLF 8 TIFt 78 = ; TIFLF TIFLF ; TIFLF TIFLF ; TIFLF TIFLF = Step To clculte the trnulr ntutonstc fuzzy totl flot: Clculte TIFTF wth respect to ech ctvty by usn TIFTF = TIFLF j TIFTF = TIFTF 4 = TIFTF 6 = TIFTF 47 = TIFTF 68 = ; TIFTF = TIFTF 5 = 44 ; ; TIFTF 46 = ; TIFTF 57 = 44 TIFTF 78 = ; TIFt Step 4 To et the possble pths s below : Fnd ll the possble pths nd clculte TIFCP n project network. P = { (478 (468 (68 (578} TIFP = (478 then TIFCP(P = TIFTF TIFTF 4 TIFTF 47 TIFTF 78 = TIFP = (468 then TIFCP(P = TIFTF TIFTF 4 TIFTF 46 TIFTF6 8 = 444 TIFP = (68 then TIFCP(P = TIFTF TIFTF 6 TIFTF 68 = TIFP 4 = (578 then TIFCP(P 4 = TIFTF TIFTF 5 TIFTF 57 TIFTF 78 = 059 Step 5 To obtn the crtcl pth usn rnkn procedure : : 4 P 4.5 L (y =.5 + y nd (y = y (TIFCP(P = dy 0.5 y dy 5.5 y 0 = 6.4 Smlrly for fndn (TIFCP(P =.8 ; (TIFCP(P = 5.6 ; (TIFCP(P 4 =.67 Snce (TIFCP(P < (TIFCP(P < (TIFCP(P 4 < (TIFCP(P Intutonstc fuzzy crtcl pth s IJO 05

7 Interntonl Journl of dvnce eserch IJO.or ISSN Concluson In ths pper smple pproch s provded to fnd the Trnulr Intutonstc fuzzy totl durton tmes nd the crtcl pths when the ctvty tmes re TIFNs. The proposed method ves trnulr solutons nd nformtons for mkn project mnement decsons. eferences [] Chen LS Chen CH (005 Selectn IS personnel usn rnkn fuzzy number by metrc dstnce method Europen Journl of Opertons eserch 60 ( : *+ T.N. Chun nd J.Y. Kun (005 The fuzzy shortest pth lenth nd the correspondn shortest pth n network Computers nd Opertons eserchvol.no.6pp *+ P.K. De nd mt hnchr (0 Fuzzy Crtcl Pth nlyss by nkn Method eserch Ind Publctons vol.7 no. pp.5-4. [4] D. Dubos nd H. Prde (980 Fuzzy Sets nd Systems: Theory nd pplctons cdemc Press New York. *5+ C.M. Klen (99 Fuzzy shortest pths Fuzzy Sets nd Systemsvol.9no.pp.7-4. *6+. Krn Ydv. njt sws (009 Fndn Shortest Pth usn n Intellent Technque Interntonl Journl of Enneern nd Technoloy vol.no *7+ K.C. Ln nd M.S. Chern(99 The fuzzy shortest pth problem nd ts most vtl rcs Fuzzy Sets nd Systemsvol.58no.pp.4-5. [8]. Noor Gn nd M. Mohmed Jbrull (00 On Serchn Intutonstc Fuzzy Shortest Pth n Net work ppled Mthemtcl Scences No. 69 pp *9+ S. Okd nd T. Soper (000 shortest pth problem on network wth fuzzy rc lenths Fuzzy Sets nd Systemsvol.09 no. pp *0+ P. Pndn nd P. jendrn (00 new lorthm for mnmum pth n network ppled Mthemtcl Scences vol.4 no. 54 pp [] N. v Shnkr V. Sreesh nd P. Phn ushn o (0 Crtcl pth nlyss n the fuzzy project network eserch Ind publctons vol 7 No. pp [] oss T.J (005 Fuzzy Loc wth Enneern pplctons Mc Grw Hll. *+. Soph Porchelv nd G. Sudh (0 modfed lorthm for solvn shortest pth problem wth Intutonstc fuzzy rc lenths Interntonl Journl of Scentfc nd Enneern eserch vol.4 no.0pp *4+. Soph Porchelv nd G. Sudh Intutonstc Fuzzy Crtcl pth n network Interntonl conference on Mthemtcl Methods nd Computtons proc Feb 04. *5+. Soph Porchelv nd G. Sudh (04 Computton of shortest pth n fuzzy network usn Trnulr Intutonstc Fuzzy Number Interntonl Journl of Scentfc nd Enneern eserch vol.5 no.pp *6+. Soph Porchelv nd G. Sudh (04 new pproch for fndn Mnmum Pth n network usn Trnulr Intutonstc Fuzzy Number Interntonl Journl of Current eserch Vol.6 Issue- 08 pp *7+. Soph Porchelv nd G. Sudh (04 Modfed pproch on Shortest Pth n Intutonstc Fuzzy Envronment Indn Journl of ppled eserchvol.6 Issue- 08 pp *8+ Tkhsh M.T. Ymkn. (005 On fuzzy Shortest Pth Problem wth fuzzy prmeter n lorthm mercn pproch Proceedns of the nnul Meetn of the North Fuzzy Informton Processn Socety pp [9] ] Zdeh L (965 Fuzzy Sets Informt. Control IJO 05