Right Skewed Distribution of Activity Times in PERT

Transcription

1 N.Rv Shnkr et l. / Interntonl Journl of Engneerng Scence nd Technolog (IJEST) Rght Skewed Dstrbuton of Actvt Tmes n PERT N.Rv Shnkr, S. Suresh Bbu, Y.L.P.Thorn, D.Rghurm Dept.of Appled Mthemtcs, GIS, GITAM Unverst, Vskhptnm, Ind Abstrct A usul supposton n project mngement s tht the dstrbuton for most ctvtes n project network s rght skewed. The prme objectve of ths pper s to fnd new pth flot n Progrm Evluton nd Revew Technque (PERT) for rght skewed dstrbuton of ctvt tmes n project network. The new pth flot concept wll brng useful plnnng nformton to the decson mngers nd the plnners n the project constructon. Our new pth flot n PERT re compred wth norml, lognorml pproxmtons wth two prmeters nd lso wth bet pproxmtons wth three prmeters. The comprson revels tht bet pproxmtons wth three prmeters performs better thn norml nd lognorml pproxmtons suggested. Kewords: PERT, Estmtng Men, Vrnce, Project durton, Rght skewed dstrbutons, Pth flot. 1.Introducton The PERT s the smplest method nd elds the most optmstc results, whle Monte Crlo smulton produce the most conservtve result. In PERT project network, the ctvt tmes re ssumed to be bet dstrbuton nd three prmeters (optmstc tme (), most lkel tme (m) nd pessmstc tme (b)) re used to estmte the mens nd vrnces of the ctvt tmes. The bet dstrbuton s wdel used to model probblt dstrbutons of vrbles or project prmeters n mn res of opertons reserch. The resons wh the bet dstrbuton s so wdel used s tht t s extremel verstle, n tht vret of uncertntes cn be usefull modeled b t. For exmple, t cn ccommodte vret of skewnesses, both postve nd negtve, nd thus, when skewness s n mportnt fctor, the bet dstrbuton s often used [1]. Rsk nlss s hvng mjor nfluence on mngement decsons nvolvng nvestments. Morgn nd Henron [] note tht the flexblt of the bet dstrbuton encourges ts emprcl use n wde rnge of pplctons. These pplctons nclude the smulton of sstems n engneerng, prtculrl where the rndom vrble to be modeled s rto. Therefore, n vret of cses, there s often need to estmte the dstrbuton. Further, the bet dstrbuton cn be estmted reltvel esl from dt on just the optmstc, pessmstc nd most lkel vlues, nd snce mngers nd plnners fnd t eser to estmte these three ponts thn other sttstcl prmeters, the bet dstrbuton hs been ppled n mn prctcl problems. Cottrell [3] suggested n pproxmton of the norml dstrbuton to determne expected tme nd vrnce usng two tme estmtes. Mohn et l [4], suggested two prmeter lognorml pproxmton for estmtng ctvt tmes n PERT nd lso showed tht t works well when the dstrbutons re hghl rght skewed. A usul supposton n project mngement s tht the dstrbuton for most ctvtes s rght skewed [4]. An ctvt n the project s sd to be smmetrc, rght nd left skewed respectvel when b-m = m-, b-m b m > m-, nd b-m < m-. For rght skewed ctvt, the skewness k = >1. m Rsk nlss s hvng mjor nfluence on mngement decsons nvolvng nvestments. The bet dstrbuton s seen s sutble model n rsk nlss becuse t provdes wde vret of dstrbutonl shpes over fnte ntervl [5]. A gret del of reserch hs been crred out on methodologes for estmtng project tme dstrbutons. The methods cn be brodl grouped nto four mn pproches: exct nlss, nltcl pproxmton, nltcl boundng, nd smulton. Yousr H. Abdelkder [6] developed the moment s method for fndng project completon tme when ctvtes re Webull dstrbuted. Further, the bet dstrbuton cn be estmted reltvel esl from dt on just the optmstc, pessmstc nd most lkel vlues, nd snce mngers nd plnners fnd t eser to estmte these three ponts thn other sttstcl prmeters, the bet dstrbuton hs been ppled n mn prctcl problems. ISSN : Vol. 3 No. 4 Apr

2 N.Rv Shnkr et l. / Interntonl Journl of Engneerng Scence nd Technolog (IJEST) Jn Kozłowsk et l.[9] ponted out tht speces bod sze dstrbutons re rght-skewed, smmetrc or left-skewed, but rght-skewness strongl prevls. Mohn et l [4] developed method for determnng expected ctvt tmes wth onl two tme estmtes (,m) or (b,m) usng lognorml dstrbuton. He ponted out tht lognorml men pproxmton s better thn Trdtonl PERT [10], Premchndr[8], Cottrell[3], nd works exceptonll when dstrbutons re hghl rght skewed. The purpose of ths pper s to show the effectveness of new pth flot when the ctvtes re rght skewed. In secton we dscuss some relevnt PERT formule to estmte men nd vrnce. In secton 3 we propose method to fnd new totl flot b usng the ctvt men nd vrnce of rel world projects. In secton 4 exmple nlss s presented. In secton 5 we drw the conclusons.. PERT pproxmtons.1 Tme estmtes wth three prmeters The followng re some ctvt durton estmtes wth three prmeters, m nd b usng bet dstrbuton..1.1 Tme estmtes of the men nd vrnce[ 7] The cretors of Trdtonl PERT [7,10] worked out the bsc concepts of the PERT nlss, nd suggested the estmtes of the men nd vrnce vlues 1 µ = ( + 4m + b) (1) 6 1 = ( b ) () 36 Subject to the ssumpton tht the denst dstrbuton of the ctvt tme s α 1 β 1 Γ( α + β ) ( ) ( b ) f Y ( ) =, < < b, α, β > 0. (3) α + β 1 Γ( α) Γ( β ) ( b ).1. Gnzburg [11] tme estmtes Generlzng the ssumpton p + q 4 Gnzburg proposed the estmtes + 9m + b µ = (4) 13 ( b ) m m = (5) 168 b b for the generl bet dstrbuton. + b B ssumng tht p =1, q = nd m =, he further mproved these estmtes estmtes when the 3 estmted mode of the ctvt tme s locted n the tl of the dstrbuton s follows 8 + 5b µ = (6) 13 10( b ) = (b-) /3 (7) Rv shnkr nd Sreesh [1] tme estmtes We proposed PERT pproxmtons[1] for ctvt durton dstrbuton s follows m + 5b µ = (8) 7 ISSN : Vol. 3 No. 4 Apr

3 N.Rv Shnkr et l. / Interntonl Journl of Engneerng Scence nd Technolog (IJEST) ( b ) = (9) 35. Tme estmtes wth two prmeters The followng re some ctvt durton estmtes wth two prmeters (, m ) or (b,m )...1 Cottrell [3] tme estmtes Cottrell [3] determned expected ctvt tmes wth onl two tme estmtes (b,m) usng norml dstrbuton to develop smplfed verson of the PERT for project plnnng. PERT pproxmtons usng tme estmtes re µ = m b m = z where z = (10).. Mohn et l [4] tme estmtes Mohn et l [4] developed n lterntve method for determnng expected ctvt tmes wth onl two tme estmtes (,m) or (b,m) usng lognorml dstrbuton. PERT pproxmtons usng tme estmtes (,m) re 1 * z z = + log( m) 4 * * µ = log[ ] + z (11) PERT pproxmtons usng tme estmtes (b,m) re 1 * z z = + + log( b m) 4 (1) * * µ = log[ b] z * * where nd µ re the men nd stndrd devton of the underlng norml dstrbuton nd z = Proposed method to fnd new totl flot n PERT network for rght skewed project 3.1 Clculton of pth flot for rght skewed project For the current defnton of the PERT, the expected durton of n pth n the network s T, = 1,,,n nd ts vrnce s, = 1,,,n. T nd represent, respectvel, the sum of ll ctvtes durton nd ther ctvt durton vrnce on the pth. Accordng to the centrl lmt theorem, the expected fnsh tme of n T T pth n the network s rndom vrble wth norml dstrbuton N (T, ). Then, the λ = wll be of the stndrd norml dstrbuton N(0,1). In project plnnng, the requred probblt level P of totl fnsh tme s usull gven to determne the crtcl pth nd expected totl tme. Then λ wll be obtned from ISSN : Vol. 3 No. 4 Apr

4 N.Rv Shnkr et l. / Interntonl Journl of Engneerng Scence nd Technolog (IJEST) the stndrd norml dstrbuton tble. The expected fnsh tme for n pth n the network cn be expressed s follows ccordng to norml dstrbuton : Ts = T + λ (9) The crtcl pth cn be found b the followng T = mx { T } = mx{ T + } = T (10) s s λ + λ where the pth s the crtcl pth wth durton T s T + λ = under probblt level P. All the other pths wll be consdered s non crtcl pths. Accordng to the current pth flot concept, the flot of non-crtcl pth s defned s the dfference between the crtcl pth expected durton nd tht of pth nd cn be expressed s follows : TF = T T (11) 3. Modfed pproch of pth flot for rght skewed project The pth flot on the noncrtcl pth s defned s the followng for the PERT network plnnng : MTF = T s T s (1) when the noncrtcl pth flot wll be used to cope wth uncertntes on ste, the followng equton must be stsfed to mntn tht the crtcl pth remns unchnged or the noncrtcl pth does not become the crtcl pth. MTF = TF - λ ( ) (13) 4. Exmple nlss The current concept of pth flot s bsed on trdtonl crtcl pth method( CPM). The ctvt durton n CPM s ssumed s determnstc, nd the expected durton fnsh tme for n pth s constnt. The non crtcl pth flot completel usng cn onl chnge the expected tme on the noncrtcl pth wthn the crtcl pth durton T n rght skewed network. Usng PERT, both the ctvt durton nd pth expected tme wll become non determnstc. In rel plnnng nd constructon projects, requred durton s gven from the Ts T expert. λ = on n pth wll be constructed to the stndrd norml dstrbuton where T s s totl durton. Then the probblt P relte to the cn be found from the norml dstrbuton tble. The smllest P wll be the crtcl pth. The crtcl pth cn be determned s follows : Ts T λ = mn (14) Fg. 1 : Project Network ISSN : Vol. 3 No. 4 Apr

5 N.Rv Shnkr et l. / Interntonl Journl of Engneerng Scence nd Technolog (IJEST) Tble I : Actvt tmes nd ther skewness Actvt Actvt Nme m b b m k = m 1- Shell : Loft I.B. Structure : L out Mch Fdn. Loft I.B. Ppng Lout Shell : Assemble I.B. Structure : Fb Mch Fdn. Fbrcte I.B. Ppng Fb I.B. Structure : Assemb I.B. Structure : Instll Erect I.B Erect Foundton Engne : Instll Complete #rd DK Engne : Fnsh Boler: Instll Boler : Test FINAL Test Fg. 1 shows n 18-ctvt project network n the shp buldng domn. The uncertnt n ech ctvt durton could be elcted through expert judgment v optmstc tme, most lkel tme m, pessmstc tme b s descrbed n Tble I. Modern-d shp producton s mnufcturng domn n whch nnovtve desgn nd buld strteges requre specl ttenton to rsk fctors tht m mpct cost nd delver tme. Two mjor rsk res re the mpct of ECO s nd crne unvlblt. Engneerng chnges m come from vret of sources- such s owner-requested chnges, ndequte desgn specfctons, nterfce problems for vendor-furnshed equpment, etc. Crnes re used to lft lrge prefbrcted unts nd ther unvlblt due to outges m result n substntl project dels. Generll the project dels re due to the del of ctvtes n the project. The ctvt dels cn be found b fndng ther erlest nd ltest tmes whch n turn depends up on the ctvt men tme. Tkng the network n Fg. 1 s the exmple, the dfferent pth flot results wll be compred usng dfferent tme estmtes nd presented n Tble II to Tble VI for λ = 1.65.For the dfferent λ, the modfed totl flot for trdtonl [10] pproxmtons on ll pths s clculted nd presented n Tble VII. Tble II : Totl flot for Gnzberg [ 11 ] pproxmtons when λ =1.65 Pth T TS = T - λ TF = MTF = T -T TS -TS ISSN : Vol. 3 No. 4 Apr

6 N.Rv Shnkr et l. / Interntonl Journl of Engneerng Scence nd Technolog (IJEST) Tble III : Totl flot for Trdtonl [10] pproxmtons when λ = 1.65 Pth T TS = TF = MTF = T - λ T -T TS -TS Tble IV : Totl flot for Rv Shnkr & Sreesh [ 1 ] pproxmtons when λ =1.65 Pth T TS = T - λ TF = MTF = T -T TS -TS Tble V : Totl flot for Mohn et l [ 4 ] pproxmtons when λ = 1.65 Pth T TS = T - λ TF = MTF = T -T TS -TS Tble VI : Totl flot for Cottrell [ 3 ] pproxmtons when λ = 1.65 Pth T TS = T - λ TF = T -T MTF = TS -TS ISSN : Vol. 3 No. 4 Apr

7 N.Rv Shnkr et l. / Interntonl Journl of Engneerng Scence nd Technolog (IJEST) Tble VII Pth flots wth dfferent λ for Trdtonl [10] pproxmtons Pth Pth flot wth dfferent λ Conclusons The PERT method s technque tht llows us to mnge the schedulng of project. The mn ssumpton n PERT s tht the ctvt durtons n project cn be estmted precsel nd tht the re sttstcll ndependent. The estmtes of the men project durton nd ts totl costs belong to the most mportnt ctvtes n mngng constructon project. Ther mportnce s especll promnent n project plnnng nd contrctng. In such stutons correct estmtes of project durton nd costs enbles mngers to mnmze possble losses. Estmtng men nd vrnce s ver chllengng feld, n whch reserchers dedcte ther knowledge nd effort to hndle nterestng problems whch hve drect relton wth socet, now nd n the future. In ths pper new pth flot n PERT for rght skewed dstrbuton of ctvt tmes n project network hs proposed. The new pth flot concept wll brng useful plnnng nformton to the decson mngers nd the plnners n the project constructon. Our new pth flot n PERT hs compred wth norml, lognorml pproxmtons wth two prmeters nd lso wth bet pproxmtons wth three prmeters. The comprson revels tht bet pproxmtons wth three prmeters performs better thn norml nd lognorml pproxmtons suggested. References [1] Soumo D. Motr, Skewness nd the Bet Dstrbuton, Journl of opertonl reserch socet. Vol 41, No. 10, pp ,1990. [] Morgn nd Henron, A Gude to Delng wth Uncertnt n Quntttve Rsk nd Polc Anlss Publshed b the Press Sndcte of the Unverst of Cmbrdge. Reprnt 003. [3] Cottrell W Smplfed progrm evluton nd revew technque (PERT). Journl of constructon Engneerng nd Mngement 15/1:16-,1999. [4] S Mohn, M Goplkrshnn, H Blsubrmnn nd A Chndrshekhr, A lognorml pproxmton of ctvt durton n PERT usng two tme estmtes, Journl of the Opertonl Reserch Socet 58, , 007. [5] Dvd Johnson, The Trngulr Dstrbuton s Prox for the Bet Dstrbuton n Rsk Anlss, The Sttstcn 46, No.3, pp ,1997. [6] Yousr H. Abdelkder, Evlutng project completon tmes when ctvt tmes re Webull dstrbuted, Europen Journl of Opertons Reserch 157, , 004. [7] D.G.Mlcolm, J.H.Roseboom, C.E.Clrk nd W.Fzr (1959) Applcton of technque for reserch nd development progrm evluton. Opertons Reserch 7, [8] Premchndr I.M., An pproxmton of the ctvt durton dstrbuton n PERT, Computers nd Opertons Reserch 8, ,001. [9] Jn Kozłowsk nd Adm. T. Gwelczk, Wh re speces' bod sze dstrbutons usull skewed to the rght? Functonl Ecolog, 00,16, C.E.Clrk (196) The PERT model for the dstrbuton of n ctvt. Opertons Reserch 10, [11] Dmtr Golenko-Gnzburg, On the Dstrbuton of Actvt Tme n PERT Journl of Opertonl Reserch Socet, Vol.39, No.8, pp , [1] N.Rv Shnkr, V.Sreesh, An Approxmton for the Actvt Durton Dstrbuton, Supportng Orgnl PERT, Appled Mthemtcl Scences, Vol.3, 009, no.57, ISSN : Vol. 3 No. 4 Apr