KEYWORDS: Triangular Fuzzy Number - Fully Fuzzy Time Cost Trade Off problem Goal Programming.

Transcription

1 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. IJESR INERNAIONAL JOURNAL OF ENGINEERING SIENES & RESEARH EHNOLOGY SOLUION PROEURE FOR FUZZY PROJE RASHING PROBLEM HROUGH GOAL PROGRAMMING EHNIQUE M.Evagele Jebaeel*. Paul hayabara * PG a Reearch epartmet of Mathematc Bhop Heber ollege ruchrappall-0 07 aml Nau Ia. PG a Reearch epartmet of Mathematc Bhop Heber ollege ruchrappall-0 07 aml Nau Ia. ABSRA Proect maagemet oe of the mot mportat fel bue a utry. Every tak a orgazato ca be take to accout a a proect. me ot rae Off problem oe of the ma apect of proect cheulg. I th paper we have preete a ew algorthm for olvg Fully Fuzzy me ot rae Off problem through Goal Programmg techque. Ug th techque the proect maager wll be able to eterme the mmum otal cot of the proect a mmum urato of the proect ealy. A llutrato prove to emotrate the effcecy of the metho. EYWORS: ragular Fuzzy Number - Fully Fuzzy me ot rae Off problem Goal Programmg. INROUION Proect Maagemet a very mportat fel employe for cheulg actvte a motorg the progre compettve a fluctuatg evromet. he feable urato requre to perform a pecfc proect eterme ug crtcal path metho. However becaue of compettve prorte tme mportat a the completo tme of a proect eterme ug crtcal path metho houl be reuce to meet a eale requete. I cheulg a proect t geerally coere to expete the urato of ome actvte through expag extra buget orer to compre the proect completo tme. h proceure ca be coere uer ether ome fxe avalable buget or a threhol of proect completo tme. h problem kow a tme cot trae off problem or proect crahg problem the proect maagemet lterature. he ma obectve of thee k of problem to eterme the optmum urato a cot houl be age to the actvte uch that the overall cot mmze. he proect urato ca be hortee by the accelerato of the crtcal actvty tme. he accelerato of the actvty tme ca be acheve ug more reource (ug more prouctve equpmet materal or hrg more worker whch mea hgher cot. Proect crahg problem aalyze how to mofy proect actvte o a to acheve the traeoff betwee the proect cot a the completo tme. IME OS RAE OFF PROBLEMS IN IFFEREN NAURE By revewg the lterature t oberve that there are everal tue vetgate a aalyze the proect maagemet problem. Mathematcal a heurtc metho are the two maor approache ue to olve the tme cot trae off problem proect cheulg. Mathematcal metho covert the proect tme cot trae off problem to mathematcal moel a utlze lear programmg teger programmg yamc programmg goal programmg or mult-obectve lear programmg to olve the problem. However formulatg the obectve fucto a well a the requre cotrat tme-coumg a proe to error. Heurtc metho prove a way to obta goo oluto but o ot guaratee optmalty. However they requre le computatoal effort tha mathematcal metho. http: // Iteratoal Joural of Egeerg Scece & Reearch echology [8]

2 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. he problem of proect tme cot trae off wa frt trouce by elly []. By aumg that rect cot of a actvty chage wth tme mathematcal programmg moel were evelope to mmze the proect rect cot []. hereafter may reearcher have evelope mathematcal programmg moel for thee k of problem. he tme cot trae off problem have bee extevely vetgate. May moel have bee propoe a they ca be categorze to two type: etermtc cheulg a o-etermtc cheulg. Recetly Yag [] propoe a chace-cotrae programmg moel to aalyze the tme cot trae off problem where fug varablty coere. Yag [] took buget ucertaty to accout o proect tme cot trae off a chacecotrae programmg moel. A hybr tellget algorthm tegratg mulato a geetc algorthm wa ege for olvg the propoe moel. he above metoe tme cot trae off moel maly bae o probablty theory. A geerally kow t requre a pror prectable regularty or a poteror frequecy trbuto to cotruct the probablty trbuto of actvty tme. However real worl applcato ome actvty tme mut be forecate ubectvely; for example we have to ue huma ugmet tea of tochatc aumpto to eterme actvty tme. A alteratve way to eal wth mprece ata to employ the cocept of fuzze whereby the vague actvty tme ca be repreete by fuzzy et. he ma avatage of methoologe bae o fuzzy theory are that they o ot requre pror prectable regularte or poteror frequecy trbuto a they ca eal wth mprece put formato cotag feelg a emoto quatfe bae o the eco-maker ubectve ugmet. I the lterature there are everal tue that have vetgate the proect maagemet problem wth fuzzy parameter. Lu et al. [8] propoe a Fuzzy Optmal cotructo me ot rae Off metho. I ther tuy the actvty urato accepte a fuzzy umber. A acceptable rk level efe a the mmum cocept of the fuzzy et theory; fuzzy urato are traforme to crp et. he the geetc algorthm techque are ue to f the optmal or ear optmal oluto. Arka a Gugor [] apple Fuzzy Goal Programmg to the me ot rae Off problem wth two obectve whch are mmum completo tme a crahg cot. he aprato level of the obectve are efe a fuzzy umber. he goal programmg olve ug max-m approach. Wag a Lag [9] olve proect maagemet eco problem wth multple fuzzy goal ther tuy. he goal of the problem are efe ug lear memberhp fucto a the multple Fuzzy Goal Programmg problem olve after traformg to t crp equvalet ug Bellma a Zaeh fuzzy eco cocept. Ehtehara et al. [] preete a ew approach for the oluto of me ot rae Off problem wth ucerta cot. A approprate geetc algorthm ue to f the oluto of the Mult Obectve Fuzzy me ot Problem. L [] propoe a approach to olve proect crahg problem wth ucerta actvty tme a crah cot. he cofece-terval etmate a the prevou tattcal ata are ue to olve the fuzzy proect crahg problem. I th tuy level (- fuzzy umber were erve from cofece terval etmate of the tattcal ata wth a tace rakg whch ue to efe the fuzzy orerg. he actvte executo tme a cot the aly cot are accepte a tragular fuzzy umber. he propoe approach explctly embe the fuzzy et theory to the optmzato proceure a the a mult obectve geetc algorthm ue to olve the cotuou a mult obectve fuzzy tme cot moel. Maa a e [] olve me ot rae Off problem wth fuzzy actvty urato tme. he fuzzy actvty urato tme efe a fuzzy varable bae o elf-ual creblty meaure. he the obtae me ot rae Off problem olve wth a hybr tellget algorthm tegratg fuzzy mulato a geetc algorthm. Ghazafar et al. [0 ] propoe a mathematcal moel to eal wth Fuzzy me ot rae Off problem. he ormal a crah urato of actvte are coere a tragular fuzzy umber. For the oluto of the fuzzy problem a rakg fuzzy umber metho ue. Lag [9] propoe a pobltc lear programmg approach for the oluto of Fuzzy Mult Obectve proect maagemet eco problem. he fuzzy parameter are efe ug the tragular poblty trbuto. I the propoe pobltc lear programmg approach the fuzzy obectve a the fuzzy cotrat are traforme to ther crp equvalet. he the obtae mult obectve lear programmg problem traforme to a equvalet lear programmg problem ug Zmmerma fuzzy eco cocept a the mmum operator. he a a [] propoe a ew approach to olve me ot rae Off problem. h approach ecrbe the mmum total crah cot of a proect etwork va a memberhp fucto whch completely coerve all the fuzze of parameter a the correpog optmal actvty tme for each actvty uer fferet poblty level are obtae. oluay Gocke [] propoe a oluto proce for the Fuzzy Mult Obectve Proect rahg problem. http: // Iteratoal Joural of Egeerg Scece & Reearch echology [9]

3 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. I th paper we have preete a ew algorthm for olvg Fully Fuzzy me ot rae Off problem through Goal Programmg techque. Ug th techque the proect maager wll be able to eterme the mmum total cot of the proect a mmum urato of the proect ealy. h oe of the eaet metho to olve Fuzzy me a ot optmzato problem occurrg real lfe tuato. A llutrato prove to emotrate the effcecy of the propoe metho. PRELIMINARIES I th ecto ome bac efto of fuzzy theory efe by aufma Gupta a Zmmerma are preete. efto he charactertc fucto ca be geeralze to a fucto : X A of a crp et A X ag a value ether 0 or to each member X. h fucto A uch that the value age to the elemet of the uveral et X fall wth a [0 pecfe rage.e. ]. he age value cate the memberhp grae of the elemet the et A. he fucto efe by efto A A ( x A A A {( A A( x : x X calle the memberhp fucto a the et } for each x X calle a fuzzy et. A fuzzy et efe o the et of real umber R a to be a fuzzy umber f t memberhp fucto ha the followg charactertc:. ( x : R [0 ] cotuou. A. ( x 0 for all ( a ] [ c. A. ( x trctly creag o [a b] a trctly ecreag o [b c]. A for all x b where a b c.. ( x A efto ragular fuzzy umber a fuzzy umber repreete wth three pot a follow: A = (a a a th repreetato terprete a memberhp fucto. A 0 f x a a x a x a ( x f a x a a a a x f a x a a a efto A tragular fuzzy umber A ( a a a B ( b b a to be a o-egatve tragular fuzzy umber f a oly f a 0 a b. he et of all thee tragular fuzzy umber eote byf (R. http: // Iteratoal Joural of Egeerg Scece & Reearch echology [0]

4 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. Fully Fuzzy Lear Programmg problem (FFLP Lear programmg oe of the mot frequetly apple operato reearch techque. We aume that all parameter a varable are real umber. But real worl evromet o ot have prece formato. So the fuzzy umber a fuzzy varable houl be ue Lear Programmg problem. he taar form FFLP problem wth m fuzzy equalty cotrat a fuzzy varable a follow: Maxmze (or Mmze ( Subect to A X b X ( X a o-egatve fuzzy umber. Where [ c ] [ ] [ X x A a ] m b [ b ] m c x a b F( R where =... m a =.... * * * a * Remark []: x ( x x x a to be a exact optmal oluto of problem ( f t atfe the followg tatemet: x [ * where x F(... * * x ]... Ax * b x ( x { x x S x / Ax b [ * ] x x where x F( } * problem x x we have that x Remark []: Let be a optmal oluto of problem ( a there ext a exact optmal oluto of problem ( calle a alteratve exact optmal oluto. x * x * x ( cae mmzato x * the x alo a PROBLEM ESRIPION Wth the progre of the proect proect maager alway ee to make traeoff betwee the cot a the completo tme. Sometme maager may make eco orer to fh the proect ooer wth proect cot augmet by acceleratg the proect cheule whch alo ame a proect crahg proect maagemet. I other cae motvate by reucg the proect cot maager may be cocrpte to acrfce wth prologg the proect completo tme. herefore t aturally erable for maager to f a cheule to complete a proect wth the balace of the cot a completo tme. he total cot fucto of a proect ha two compoet: rect a rect cot. rect cot are curre becaue of the performace of proect actvte whle rect cot clue thoe tem that are ot rectly relate to vual proect actvte a thu ca be aee for the etre proect. I geeral rect cot creae almot learly wth the creae of proect urato a uually aume a a percetage of proect rect cot. he proect tme cot trae off problem thu reuce to eterme proect cot agat proect urato. A poble way to olve tme cot trae off problem to ue a mathematcal programmg moel whoe obectve fucto cotructe o that proect rect cot mmze a the mpoe cotrat guaratee a ere proect eale whle the preceece requremet of the etwork are matae. Parameter a eco varable of moel are a follow: Parameter Number of actual actvte rah tme for actvty N Normal tme for actvty N ot of og actvty ormal tme. http: // Iteratoal Joural of Egeerg Scece & Reearch echology []

5 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. Slope cot for actvty eco Varable: otal cot of tme a qualty; Startg tme of oe ; Plae tme of the actvty Proect completo tme. I th paper tme parameter tartg tme varable a cot of the proect are coere tragular fuzzy umber. A proect ca be repreete by a actvty-o-arc etwork G = (V A where V = { } the et of oe repreetg the mletoe a A the et of arc repreetg the actvte. I the etwork oe a repreet the tart a e of the proect repectvely. I th paper the ormal a crah actvty urato a ormal a crah cot are aume to be ucerta varable. he omplete Fuzzy Mult Obectve Moel for Fully Fuzzy me ot rae Off problem preete a follow: M Subect to 0 x 0 ( N x x N ( P I * M. ( GOAL PROGRAMMING [] he Goal Programmg (GP a mportat techque for eco Maker to olve Mult Obectve eco Makg (MOM problem fg a et of atfyg oluto. Goal Programmg to mmze the evato betwee the achevemet of goal a ther aprato level. he mmzato proce ca be accomplhe wth varou type of metho uch a thoe of Lexcographc Goal Programmg (LGP Weghte Goal Programmg (WGP a MINMAX (hebyhev Goal Programmg. he Mathematcal formulato of Weghte Goal Programmg expree a follow: M f ( X 0 ( g X F ( F a feable et ( Where a are the repectve potve weght attache to thee evato the achevemet fucto. http: // Iteratoal Joural of Egeerg Scece & Reearch echology []

6 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. max 0 f ( X g a max 0 g f ( X th goal a (X the lear fucto of the th goal. f are repectvely over a uer achevemet of the ALGORIHM O SOLVE FULLY FUZZY PROJE RASHING PROBLEMS he followg a ew algorthm to f the optmal oluto of me ot rae Off problem ug Goal Programmg techque. he tep of propoe algorthm are gve below: Step : Set up the mathematcal formulato of the Fuzzy me ot rae Off problem a gve (. Step : he formulate Mult Obectve Lear Mathematcal moel for problem ( ca be wrtte a gve below: M Subect to ( ( ( ( ( 0 ( ( ( ( ( ( ( N N N ( P ( I * ( M ( {( N N } ( 0 N ( http: // Iteratoal Joural of Egeerg Scece & Reearch echology []

7 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. Step : he above Mult Obectve Lear Mathematcal Moel for problem ( may be wrtte a gve below. M Subect 0 ( { N to N { N I I I } * M * * 0 ( 0 N } 0 ( P 0 { N N ( 0 } 0 Step : Poblem ( coverte to the Mult Obectve Lear Programmg problem wth three crp fucto a gve below: M 0 M Subect to 0 M M M 0 M { N N 0 I * I * I * } { N N 0 0 { N N } ( ( P 0 } http: // Iteratoal Joural of Egeerg Scece & Reearch echology []

8 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. http: // Iteratoal Joural of Egeerg Scece & Reearch echology [] Step : Form the Weghte Goal Programmg moel gve ecto for the moel (. he followg the WGP for moel (: (7 R Q P to Subect M * ( * ( *( ( } { } { } { I I I P N N N N N N Where PQ a R eote the goal ettg for cot a a the goal ettg for tme a α a β are the repectve potve weght attache to the evato the achevemet fucto.

9 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. Step (: Solve the Goal Programmg Moel (7 gve tep ( the total fuzzy cot of the proect a optmal fuzzy urato of the proect obtae by ubttute the obtae value ( a (. NUMERIAL EXAMPLE Lt of actvte for creatg a catee a factory gve below wth other relevat etal. Job A mut precee all other whle ob G mut follow a other ob ca ru cocurretly. able repreet the ecrpto of the proect. I th proect tme parameter a cot of the proect are coere tragular fuzzy umber form. Irect cot of the proect per ay ( Actvte formato gve able. Actvty Normal tme ( N able ecrpto of the proect Actvty ecrpto (A Pla approval a te preparato (B Layg fouato ( Rag bulg wall ( le proofg (E Itall electrcty (F Itall Plumbg (G oect ervce to fh able he Fuzzy ata of Proect rah Normal cot tme ( ( Slope cot (per ay A( ( (90 ( ( ( B( (7 9 ( 7 9 ( (7 8 9 ( ( ( 8 ( E( (8 8 8 ( F( (7 9 ( 7 G( (0 (8 0 ( ( ( ( ( ( ( ( ( ( ( ( http: // Iteratoal Joural of Egeerg Scece & Reearch echology []

10 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. he oluto proceure ug the propoe metho gve the prevou ecto ecrbe a follow: Frt formulate the Fuzzy Mult Obectve Lear Mathematcal Moel for the gve proect accorg to tep (. he formulate FMOLP moel ca be wrtte the followg form: M ( M ( 0 Subect to { N { N { N N N 0 } I I I } } * * * 0 ( P &... N (8 0 Seco ecompoe the Mult Obectve Lear Programmg moel (8 wth x crp obectve fucto for the proect accorg to tep (. Obta the goal for each obectve the followg way: Moel ( ca be reuce to x epeet Lear Programmg moel. Solvg thee x moel by commo approach for Lear Programmg x goal value for each obectve fucto obtae. he obtae goal value from the above proce are R. 000 for R for a R for a 8 ay for ay for a ay for. Fally th proect formulate a Goal Programmg moel a gve below: http: // Iteratoal Joural of Egeerg Scece & Reearch echology [7]

11 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. M 0. * 0. * Subect 0 0. * 0.8* 0.8* to { N { N 0.8* N N } I I I * * * 0. * 0. * 0 } 0.* ( P * * 0 { N N 0 (9 } Ma Obectve of th moel to reuce the total proect cot a the urato of the proect. So we elect the weght value for potve a egatve evato are 0. a Solve the above Goal Programmg Moel (9 ug LINGO oftware package. he value of mmum fuzzy total cot a plae fuzzy urato of the proect have bee eterme ug LINGO olver. A computer package calle LINGO (LINGO 000 [7] ue o a peroal computer to olve the mathematcal moel of the example proect. LINGO a commercal package ug the power of lear a o-lear optmzato to formulate large problem cocely olve them a aalyze the oluto. I all tete ru the lear mathematcal moel of the example proect requre le tha oe eco o LINGO to obta the optmal oluto. he ample proect olve ealy ug Goal Programmg techque a the computatoal reult are tabulate. http: // Iteratoal Joural of Egeerg Scece & Reearch echology [8]

12 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. able rahg cot for each actvty Actvty Proect urato rahg ot A ( (0 0 0 B ( 7 9 ( (7 8 9 (0 0 0 ( 8 (0 0 0 E (8 8 8 (000 F (7 9 (0 0 0 G (8 0 ( he optmum crah cot for each actvty of the ample proect preete able. he mmum proect cot ( a the mmum urato ( 9 ay. he obtae reult for the ample proect have bee compare wth the metho preete by Evagele [7 8]. ONLUSION I th paper a ew algorthm ha bee propoe to olve the Fully Fuzzy me ot rae Off problem. Mult Obectve Lear Programmg Problem formulate the form Goal Programmg a fou the oluto ug extg proceure. By a mple example the obtae reult of preete algorthm ha bee compare wth the metho preete by Evagele [7 8]. It coclue that the reult coce wth each other. REFERENES [] Ammar M. Optmzato of Proect me ot rae Off Problem wth coute ah Flow J. otr. Eg. Maage. Vol. 7 No. ISSN 07-9/0/ [] Arka F. Gugor Z. A applcato of fuzzy goal programmg to a mult obectve proect etwork problem. Fuzzy Set a Sytem [] he S. a M. me-ot rae-off Aaly of Proect Network Fuzzy Evromet. Europea Joural of Operato Reearch ISSN [] hg hag Mult-hoce Goal Programmg. Omega [] Ehtehara E. Afhar A. a Abbaa R. Fuzzy-bae MOGA approach to tochatc tme cot traeoff problem. Automato otructo Vol. 8 No. pp [] R. Ezzat A ew algorthm to olve fully fuzzy lear programmg problem ug the MOLP problem. Apple Mathematcal Moellg. o:0.0/.apm [7] Evagele M. Jebaeel. Paul hayabara a I. hrtopher Voth A New Soluto Proceure to Fully Fuzzy me ot rae Off Problem Jamal Acaemc Reearch Joural February 0 ISBN No: pp [8] Evagele M. Jebaeel. Paul hayabara A Algorthm to Solve Fully Fuzzy me ot rae Off Problem Iteratoal Joural of Egeerg Scece a Iovatve echology ISSN No: 9-97 Vol. Iue 0. [9] Feg. Lu L. a Bur S. Ug geetc algorthm to olve cotructo tme cot trae off problem J. omput v. Eg. ( [0] Ghazafar M. A.Youefl M.S. JabalAmel A. Bozorg-Amr A ew approach to olve tme- cot trae off problem wth fuzzy eco varable. It J Mauf ech ol : [] Ghazafar M. Shahaagh a A.Youefl A applcato of Poblty Goal Programmg to the me ot rae off Problem. Frt Jot ogre o Fuzzy a Itellget Sytem Ferow Uverty of Mahha 007. [] Hegazy. Optmzato of cotructo tme cot trae off aaly ug geetc algorthm. a. J. v. Eg [] J.R. elly rtcal Path Plag a Scheulg: Mathematcal ba Operato Reearch. vol.9 pp [] e H. Maa W. N Y. Optmzato Moel a a GA Bae Algorthm for Stochatc me-ot rae Off. Apple Mathematc a omputato [] L H. a Love P. Ug Improve Geetc Algorthm to Facltate me-ot Optmzato. J. otr. Eg. Maage. ( http: // Iteratoal Joural of Egeerg Scece & Reearch echology [9]

13 [Jebaeell (: May 0] ISSN: 77-9 (IOR Publcato Impact Factor:.78 (ISRA Impact Factor:. [] L F.. Fuzzy rahg Problem o Proect maagemet Bae o ofece-iterval Etmate. Eghth Iteratoal oferece o Itellget Sytem eg a Applcato. OI: 0.09/ISA [7] LINGO 000. LINGO uer maual LINO Sytem Ic. hcago. [8] Lu L. Bur S. a Feg. otructo me-ot rae-off Aaly ug LP/IP Hybr Metho. J. otr. Eg. Maage. ( [9].F. Lag E.J. Wag a.y. g A Stuy o Proect crahg eco wth multple fuzzy goal. Joural of the hee Ittute of Iutral Egeer Vol. 0 o. pp [0].F. Lag Applyg fuzzy goal programmg to proect maagemet eco wth multple goal ucerta evromet. Expert Sytem wth Applcato: A Iteratoal Joural Vol. 7 Iue Page: [] aregha H. a aher S. O the crete me ot a Qualty rae Off Problem. Appl. Math. omput [] oluay Gocke Soluto of Fuzzy Mult-Obectve Proect rahg Problem. Neural omput & Applc : 7-7. OI: 0.007/ z 0. [] Yag I. Ug Elct Partcle Swarm Optmzato to Facltate Bcrtero me-ot rae Off aaly. J. otr. Eg. Maage. ( http: // Iteratoal Joural of Egeerg Scece & Reearch echology [70]