Buffer Management Method for Multiple Projects in the CCPM-MPL Representation

Transcription

1 Industra ngnrng & Managmnt Systms Vo No 4 Dcmbr 22 pp ISSN ISSN KII Buffr Managmnt Mthod for Mutp Projcts n th CCPM-MP Rprsntaton Nguyn h Ngoc ruc* Dpartmnt of Informaton Scnc and Contro ngnrng Nagaoka Unvrsty of chnoogy Nagaoka Japan Yoshnor ak Dpartmnt of ctrca ngnrng Nagaoka Unvrsty of chnoogy Nagaoka Japan Hroyuk Goto Dpartmnt of Industra and Systms ngnrng Hos Unvrsty Kogan Japan Hrotaka akahash Dpartmnt of Humants Yamanash wa Cog Kofu Japan (Rcvd: August 9 22 / Rvsd: Novmbr / Accptd: Novmbr 26 22) ABSRAC hs rsarch proposs a framwork of buffr managmnt for mut-projct systms n th crtca chan projct managmnt (CCPM) mthod xprssd n th form of max-pus nar (MP) rprsntaton. Snc tm buffrs ar nsrtd n th projcts for absorbng uncrtants n task duratons and protctng th compton tms th proposd mthod provds a procdur for frqunty survyng th rats of consumd buffrs and th rat of apsd tms. hr raton xprsss th prformanc of th projcts whch s pottd on a chart through th comptd procsss. h chart prsnts th currnt prformanc of th projcts and thr ntracton whch arts managrs to mak ncssary dcsons at th rght tm for managng ach projct and th ntr mut-projct systm. h proposd framwork can anayz th compx systm rady and t nabs managrs to mak an ffctv dcson on schdung. h ffctvnss of th framwork s dmonstratd through a numrca xamp. Kywords: Max-Pus nar Systm Crtca Chan Projct Managmnt m Buffrs Mutp Projcts Buffr Managmnt Fvr Chart * Corrspondng Author -ma: s79@stn.nagaokaut.ac.jp. INRODUCION h crtca chan projct managmnt (CCPM) s a w-known mthod for pannng and managng projcts. h concpt of th CCPM s an outgrowth of th thory of constrans (OC) frst dvopd by Godratt (99) whch has drawn much attnton for many rsarchrs (Hrron t a. 22; Cohn t a. 24; ach 25; uk t a. 26). hs rsarchs hav many focusd on carfyng th concpts of th CCPM mthod dscussng svra ratd ssus such as rsourc confct buffr s sz and buffr managmnt. In projct managmnt nta schdu s frqunty changd du to unprdctab rasons such as xtrna uncrtants procssng changs and rsourc confcts. hus th CCPM provds a mthod for nsrtng tm buffrs whch pay a ro for absorbng uncrtants n th task duratons to protct th compton tm. Morovr th CCPM mthod has achvd succsss not ony for a sng projct but for arg-sca mut-projcts (Cohn t a. 24; ach 25). On th othr hand th max-pus nar (MP) rprsntaton s known as an ffcnt too for dscrbng a cass of dscrt vnt systms (DSs) (Hdrgott t a.

2 ruc ak Goto and akahash: Industra ngnrng & Managmnt Systms Vo No 4 Dcmbr 22 pp KII ; Goto 2). h typca and sgnfcant fatur of DSs s that vnts occur at dscrt tm nstants and th vaus of th ntrna stats chang non-contnuousy. hs knd of systms typcay appars n manufacturng systms transportaton systms projct managmnt and so on. Snc unprdctab changs n th xcuton tms may nfunc th compton tm svra rsarchs that consdr uncrtants hav bn carrd out (Hdrgott 26). Howvr du to th nonnarty n th stats of th systm t s dffcut to hand argsca systms. Furthrmor an appcaton of th CCPM framwork to an MP rprsntaton was studd n our prvous papr (akahash t a. 29; ruc t a. 2). hrn a mthod for dtrmnng and nsrtng tm buffrs to absorb uncrtants s consdrd usng an MP rprsntaton. W frst dntfd a crtca path whch s th ongst path of succssv tasks wth zro foat tm. hn w rducd th procssng tm and nstad projct and fdng buffrs. A projct buffr s nsrtd btwn th nd of th crtca path and ts succdng output. Fdng buffrs ar nsrtd whrvr a non-crtca path jons nto a crtca path. hs buffrs wr frqunty montord and controd wth a buffr managmnt pocy (Kasahara t a. 29) n whch th bhavor of a projct s faturd by th rats of buffr consumpton and procssng tm. Howvr ths works wr dsgnd ony for a sng-projct systm. Rcnty a CCPM-MP framwork for dtrmnng tm buffrs has bn proposd for mut-projct systms (ruc t a. 22) an xtndd and mprovd vrson of a sng-projct cas. h projct and fdng buffrs ar dtrmnd for ach projct n arg-sca systms; th buffrs ar usd to absorb uncrtants n th ntrna task duratons of ach projct. Morovr th constrants btwn th projcts ar dtrmnd to nsrt capacty buffrs; thy pay a ro for absorbng uncrtants n th prcdng projct and protct th subsqunt projcts. hs rsarch consdrs a framwork for managng ths buffrs n a mut-projct systm. Partcuary nstad of mnmzng th duraton of ach projct as th cas of a sng projct th mut-projct schdung ams to maxmz th throughput of th ntr systm by sharng th sam rsourcs. hus t s dffcut for managrs to comprhnd th ratonshp btwn ths projcts wth mutua dpndnc by appyng th sngprojct framwork (Kasahara t a. 29) on by on makng th contro of th who systm tough. h proposd framwork hrn xprsss th prformanc for a projcts and thr ratonshp n th compx systm wth mutp nput tms whch can art managrs to mak xtra actons at th propr tm to avod tardnss both for ndvdua projct and for th ntr systm. In ths papr th ffct of cost and rsourc confcts s not consdrd for th sak of smpcty. For furthr dscusson of rsourc confcts rfr to Yoshda t a. (2). h rmanng contnts ar prsntd n th foowng four sctons. Scton 2 provds a brf rvw of th thortca background for ths papr: max-pus agbra MP rprsntaton and CCPM-MP framwork. Scton 3 drvs a concpt of buffr managmnt n th CCPM mthod and proposs a framwork of buffr managmnt n th form of th MP rprsntaton. h proposd framwork s confrmd through a numrca xamp n Scton 4. Fnay concusons and rcommndatons ar gvn n Scton HORICA BACKGROUND 2. Max-Pus Agbra Max-pus agbra s known as th schdu agbra for dscrbng a crtan cass of DSs. For a st D R { } whr R s th ntr ra st oprators for addton and mutpcaton ar dfnd as: x y max( x y) x y x y () h symbo corrsponds to mutpcaton n convntona agbra and s oftn omttd whn no confuson s ky to ars. For xamp w smpy wrt xy for xprssng x y. hs oprators hod th commutatv assocatv and dstrbutv aws. h zro and unt mnts for ths ar gvn by ε ( ) and ( ) rspctvy. hs hod x ε ε x ε and x ε ε x ε for an arbtrary ε ( ). Morovr w dfn ε ( ) and assum a proprty ε ε ε ε ε. Furthrmor th foowng two oprators ar dfnd for subsqunt dscussons: An oprator for th powr of x y mn( x y) x y x y (2) a R s dfnd as: a x x a. (3) Oprators for mutp numbrs ar as foows. If m n x max( x x x ) (4) x mn( x x x ). (5) n k m k m m n n k m k m m n For a matrx X D m n [ X ] j xprsss th ( j) th mnt of X and X s th transpos matrx of X. For X Y D m n ( ) ( ) [ X Y] max [ X] j [ Y ] j j (6) [ X Y] mn [ X] j [ Y ] j j (7) If m X D Y D p

3 Buffr Managmnt Mthod for Mutp Projcts n th CCPM-MP Rprsntaton Vo No 4 Dcmbr 22 pp KII 399 ( ) ( ) [ X Y] k [ X] j [ Y ] j j (8) [ X Y] k [ X] j [ Y ] j j (9) h prorty of oprators and ar hghr than that of oprators and whr s w-dfnd f X has at ast on non- ε ntry n vry row. h zro and unt mnts for matrcs ar; ε mn s a matrx whos a mnts ar ε n D m n and s a matrx whos dagona mnts ar and off-dagona mnts ar ε. For a vctor v D dag( v ) D rpr- m m n snts a matrx whos dagona mnts ar [ v ] and offdagona mnts ar ε. 2.2 Max-Pus nar Rprsntaton W brfy rvw th MP rprsntaton for a crtan cass of dscrt vnt systms dvopd by Yoshda t a. (2). W assum th foowng constrants mposd on th focusd systm: h numbr of procsss xtrna nputs and xtrna outputs ar n p and q rspctvy. A procsss ar usd ony onc for a sng job. h subsqunt job cannot start procssng wh th currnt job s at work. Procsss that hav prcdnc constrants cannot start procssng unt thy hav rcvd a th rqurd parts from th prcdng procsss. Procsss wth xtrna nputs cannot start unt a th rqurd matras hav arrvd. h procssng starts as soon as a th condtons abov ar satsfd. For th k th job n procss ( n ) t d ( k )( ) ( ) x k and ( ) x k b th procssng tm procssng start tm and procssng compton tm rspc- tvy. W gv th nta condton by x() ε n. Morovr [ u ( k) ] rprsnts th matra arrva tm from xtrna nput ( p ) and [ y ( k) ] rprsnts th output tm of th fnshd product to xtrna output ( q ). h foowng matrcs Pk F B and C ar ntroducd for rprsntng th structur of th systm: [ P k] { d( k):f j ε :othrws} j [ F ] j [ B ] j [ C ] j : f procss has a prcdng procss j ε : f procss dos not hav a prcdng procss j : f procss has an xtrrna nput j ε : f procss dos not hav an xtrna nput j : f procss has an xtrrna output j ε : f procss dos not hav an xtrna output j whr F B and C ar rfrrd to as th adjacncy nput and output matrcs rspctvy. h arst compton tm s dfnd as th mnmum vau wth whch th corrspondng procss can compt procssng. Spcfcay th arst compton tms of a procsss ar cacuatd by: x ( k) ( PF) P[ x ( k) Bu ( k)] () * k k * whr ( PF k ) n PF k ( PF k ) and an nstanc ( n ) dpnds on th prcdnc-ratonshps of th systm. h corrspondng output tms ar gvn by: y ( k) C x ( k) () h arst startng tms of a procsss ar gvn by: x ( k) P x ( k). (2) k Furthrmor th atst startng tms ar dfnd as th maxmum vau by whch th sam output tm basd on th arst tm can b accompshd. hus th atst startng tms of a procsss and th atst fdng tms ar gvn by: * x ( ) ( ) ( ) k PkF Pk C y k (3) u ( k) B x ( k) (4) A crtca path s undrstood as a srs of procsss wth zro tota foat. ota foat s th maxmum tm whch can b movd backward wthout changng th compton tm. hs s dfnd as th dffrnc btwn th atst and arst startng tms. Spcfcay th tota foats ω ( k) of a procsss ar obtand as: ( x ) ω ( k) dag ( k) x ( k) (5) Morovr th crtca path s dntfd by th st of procss numbrs that satsfy: { [ ω ( k) ] } (6) 2.3 CCPM-MP Framwork for Dtrmnng m Buffrs n Mutp Projcts W brfy rvw th framwork for dtrmnng tm buffrs n a mut-projct systm (ruc t a. 22). Consdrng a gnra systm wth m procsss (tasks) and n projcts n vctors 2 n of m-dmnson ar ntroducd wth th foowng proprts: : f procss bongs to projct j j (7) ε :othrws for a ( m ) and j ( j n ) and j xprsss whch procsss bong to projct j. Not that procss s mpcty dsgnd to bong a crtan projct basd

4 ruc ak Goto and akahash: Industra ngnrng & Managmnt Systms Vo No 4 Dcmbr 22 pp KII 4 on th prorty of th projcts and th procsss. hn w dfn a matrx D m n that rprsnts th ayr of projcts as foows: [ 2 n ] (8) Morovr w rduc th procssng tm of tasks to /3 of th orgna tm and dfn th sz of tm buffrs as /3 of th corrspondng path duraton (akahash t a. 29). h poston and sz of th buffrs ar dtrmnd as bow: Projct buffr (PB) Projct buffrs ar nsrtd btwn th nd of crtca paths and thr succdng xtrna outputs to protct th crtca paths. h postons ar dtrmnd by nspctng th mnts of th output matrx C f [ C ] ε and j a PB s nsrtd bhnd procss. j h szs of th PBs ar obtand by: (dag( p ) ) 3 b x y (9) whr matrx s dfnd from th st of crtca procsss ( ) dtrmnd n q. (6) as: :f and [ ] [ ] j j (2) ε :othrws whr Fββ dag( v) F dag( v) Fββ rprsnts th transtons from non-crtca procsss to non-crtca ons. Capacty buffr (CB) Capacty buffrs ar nsrtd at th transtons btwn projcts to absorb uncrtants n th prcdng projct and protct th subsqunt projcts. W frst dtrmn transtons for ach par of th projcts. Gnray transtons from projct A to projct B(A B) ar dfnd through two vctors h and h wth th foowng proprts as: :f τ w F β v (24) ε :othrws whr ϑ s th st of procsss n projct A wth a succssor bongng to projct B. FAB dag( B) F dag ( A) F AB rprsnts th transtons from procsss n projct A to procsss n projct B. :f ϕ [ ] hw FAB A (25) ε :othrws whr ϕ s th st of procsss n projct B wth a prdcssor bongng to projct A. akng ths togthr CBs ar nsrtd at transtons btwn procsss ϑ and ϕ. h szs of th CBs from projct A to B ar cacuatd by: Fdng buffr (FB) Fdng buffrs ar nsrtd whr a non-crtca path jons nto a crtca path to protct th crtca paths from day whch may occur n th non-crtca paths. o dfn th postons of FBs w frst ntroduc two vctors v and w wth th proprts as [ v ] { : β ε: β} and [ w ] { : ε: } whr β s th st of non-crtca procsss th postons ar dntfd usng two vctors v and w wth th foowng proprts: :f γ v Fβ w (2) ε :othrws whr γ s th subst of β havng a succssor cassfd as. F dag( w β ) F dag( v ) F β rprsnts th transtons from non-crtca procsss to crtca ons. :f τ w Fβ v (22) ε :othrws whr τ s th subst of havng a prdcssor cassfd as β. hn FBs ar nstad at transtons btwn procsss γ and τ. h szs of th FBs ar stmatd as: * b (dag( f Pk v ) ( Fββ Pk) v ) 3 (23) (dag( ) ( ) * ) 3 bc Pk h FAAPk A (26) whr FAA dag( A) F dag( A). F AA rprsnts th transtons btwn procsss n projct A. Smary w can fnd th postons and szs of CBs from projct B to A or btwn any par of th projcts. Rdfnd systm wth th CCPM appd h rprsntng matrcs ntroducd n Scton 2.2 ar rdfnd for asy comprhndng th structur of th systm aftr th CCPM appd. Aftr th nsrton of th PB th output matrx C γ s rformuatd n th foowng mannr: C b C γ dag( p) (27) whr th subscrpt γ xprsss that th matrx was rdfnd. Aftr th nsrton of th FBs and CBs th adjacncy matrx F γ s rformuatd as: Fγ F Fβ F c (28) whr Fβ Fβ dag( bf ) : F β and b f xprsss th poston and szs of th FBs rspctvy and ( ( )) n c xy c xy xy x y F F dag ( b ) (29)

5 Buffr Managmnt Mthod for Mutp Projcts n th CCPM-MP Rprsntaton Vo No 4 Dcmbr 22 pp KII 4 whr n s th numbr of projcts. h subscrpt c xprsss th CBs. F xy prsnts th transtons of procsss from projct x to y and ( b c) xyxprsss th sz of th corrspondng CBs. Not that F c ε mn f n. /3 Morovr matrcs P P and B rman unchangd. γ k 3. BUFFR MANAGMN 3. Concpt of th CCPM In th CCPM for managng and obsrvng th projcts prformanc th buffrs ar frqunty montord to protct th crtca path as w as th projct s compton tm. hs mchansm s cad Buffr Managmnt whch can dtct a potnta probm and ras a warnng sgna to managrs. As th projcts procd f a task apss a ongr tm than xpctd th task consums th buffr on th corrspondng path. hus th buffr managmnt frqunty compars two paramtrs: th consumpton rat of th PB and th progrss rat of th corrspondng crtca path for xprssng th currnt prformanc of th projct. In th mut-projct systm ths paramtrs ar frqunty chckd for a projcts and pottd on a fvr chart as shown n Fgur. hs chart shows th ratonshp btwn buffr usd (%) vrsus tm usd (%) through th comptd procsss. h chart s dvdd nto thr zons: grn yow and rd whch ar dtrmnd mprcay by thrshods sttngs. ypcay w bas on th thrshods sttngs proposd by ach (25) as std n ab. Usng th fvr chart th projcts status s artd to projct managrs and th dcson v for an xtra acton s suggstd. Spcfcay f th currnt status s n th grn zon th projct s gong w and th managrs nd not tak an acton. If th status s n th yow zon th projct asssss a probm and nds a rcovry pan to avod furthr buffr roson. If th status s n th rd zon th projct w possby b at and th managrs shoud ntat th acton. 3.2 Proposd Mthod W dvop a framwork of buffr managmnt for a mut-projct systm whch focuss on th systm aftr th nsrton of th tm buffrs. h ovra procdur for buffr managmnt s summarzd as th fowchart n Fgur 2. A squncng dscrpton for th procdur s xprssd by usng th MP rprsntaton as foows: Cacuat PB usd (%) Start Cassfy crtca procsss Survy currnt progrss tm q.(3) qs.(3)-(33) Cacuat m usd (%) Frqunty updat qs.(34)-(36) qs.(37)-(39) Pot on a chart (a projcts) Fg. Anayz th currnt status nd Fgur 2. Fowchart of buffr managmnt. m usd (%) Fgur. An xamp of a fvr chart. ab. Buffr thrshods Buffr usd (%) Grn to yow Yow to rd transton (%) transton (%) W agan consdr th mut-projct systm wth n projcts and m procsss mntond n Scton 2.3. Usng n vctors ( 2 n ) whch rprsnt th procsss bongng to n projcts drvd n q. (7) w dfn n vctors 2 n that satsfy: : and [ ] [ ] f x x (3) ε :othrws whr s th st of crtca procsss dtrmnd by q. (6) and xprsss whch crtca procsss b- x

6 ruc ak Goto and akahash: Industra ngnrng & Managmnt Systms Vo No 4 Dcmbr 22 pp KII 42 ong to projct x( x n ). hus th managrs can montor ach projct n ony on procdur for th ntr systm nstad of consdrng thm ndpndnty. Nxt w ntroduc a vctor z whch rprsnts th actua compton tms of th comptd procsss at th montorng stag. hus th managrs can survy th vau of ths vctor frqunty whnvr montorng th projct s xcuton. Vctor z has th foowng proprty: : [ ] t η z F β v (3) ε :othrws whr t s dfnd as actua compton tm of procss and η rprsnts th st of comptd procss numbrs at th montorng stag and ε ( ) s mntond n Scton 2.. In ordr to manag th buffrs for ach projct w dcompos vctor z to n vctors z z z n whch rprsnt n projcts by: z dag( z) (32) x whr x { n } and z has th foowng proprts: x : [ ] t η z F β v (33) ε :othrws In ordr to know th projcts prformanc at th montorng stag w cacuat th rats of th PBs consumpton and th apsd tms. hs vaus ar dtrmnd for ach projct as th foowng procdur: Rato of th PBs consumpton (buffr usd %): Frst w cassfy th orgna PB for ach projct (x): b px p x b C (34) whr b p s th sz of th PBs obtand from q. (9). hn w dfn th PB consumd by th comptd procsss for projct (x) usng th foowng vctor: ( b ) dag( xγ ) z (35) x usd x whr x γ R prsnts th arst compton tms of th systm aftr th CCPM appcaton drvd n Scton 2.3. Fnay th consumpton rat of th PB at th montorng stag s cacuatd through vctor r as: b px bx x usd r ( b ). (36) Rato of th progrss tm ovr th crtca path duraton (tm apsd %) Frst w cacuat th duraton tm of th crtca path for ach projct (x): d (dag( u ) y ) C (37) x γ γ x b whr u γ x γ uγ xprsss th atst fdng tms on th crtca paths and x γ and y γ ar th atst startng tms and th output tms of th systm aftr th CCPM appcaton drvd n Scton 2.3. hn w dfn th compton tm of procsss at th montorng stag wthout consdrng th ffct of nput tms as: z ( B u ) z. (38) x x γ x whr u γ xprsss th atst fdng tms whch s drvd n Scton 2.3. Fnay th rato of th progrss tm ovr th stmatd compton tm s obtand through vctor r as: tx r z. (39) tx x dx h rato of buffr consumpton (%) vrsus th corrspondng progrss tm (%) s pottd rspctvy for a projcts through th comptd procsss on th sam fvr chart (Fgur ). h chart xprsss th currnt status of a projcts and thr ntracton. hus managrs can mak a propr dcson for protctng ach projct and th ntr systm. 4. NUMRICA XAMP A numrca xamp for a smp systm s prsntd to factat bttr undrstandng of th proposd framwork. Fgur 3 shows a smp systm wth two projcts consstng of thr nputs two outputs and ght procsss. Not that th dashd arrow from procss (3) to (6) s consdrd as th prcdnc constrant btwn th two projcts. Projct conssts of procsss () (2) (3) and (5). Projct 2 conssts of procsss (4) (6) (7) and (8). u 3 u 4 Procsss d 23 () (3) (5) d 3 d 39 d 59 (2) u 5 d 73 Output (4) (6) (8) d 46 d 69 d 83 (7) Projct Output Projct 2 Fgur 3. A smp systm wth two projcts ( and 2). 4. Dtrmnaton of m Buffrs Usng th MP rprsntaton ntroducd n Scton 2.2 th rprsntaton matrcs for th ntr systm ar gvn as foows:

7 Buffr Managmnt Mthod for Mutp Projcts n th CCPM-MP Rprsntaton Vo No 4 Dcmbr 22 pp KII 43 εεεεεεεε εεεεεεε εεεεεεε εεεεεεεε εεεε εεε F C εεεεεε εεεεεεε εεεεεε εεε εεεε εεεεεε εεεεεεε Pk dag( ) B εεε εεεε. εεεεεεεε Assumng that th nta condton s x() ε 8 and th nput tms u [ 345] th output tms and th crtca path ar dntfd as y (8 22) and { }. rspctvy. Morovr w appy th MP rprsntaton prsntd n Scton 2.3 to th systm. Frst th procssng /3 tms ar gvn by ( ). From q. (7) w dfn two vctors as: ( εεεε) 2 ( εεε ε ). h matrx gvn n q. (8) s: εεεε εεε ε Projct buffr Snc th crtca path s dntfd as { } and [ C] 5 and [ C] 28 th postons of th PBs ar bhnd procsss (5) and (8). From qs. (9) and (2) w obtan th szs of th PBs as b (7 6). Fdng buffr From qs. (2) (23) w obtan th postons of th FBs nsrtd at th transtons of procss (2) (5) and procss (7) (8). h szs of th FBs ar dtrmnd p as b ( εεεεε ε). f Capacty buffr From qs. (24) (26) a CB s nsrtd at th transton of procss (3) (6). Morovr th sz of th CB at th transton from projct to 2 s ( b c) 2 ( εε4 εε εεε). Fgur 4 shows th systm wth th tm buffrs nsrtd for th mut-projct systm n Fgur 3. h systm wth th CCPM appd h matrcs for rprsntng th structur of th attr systm ar rdfnd as foows. Usng qs. (27) (29) w obtan th adjacncy and output matrcs as: εεεεεεεε εεεεεεε εεεεεεε εεεεεεεε F ε εεεεε εε4 εεεε εεε εεεε εεεεε ε C γ dag(3233) Pk B εεεε7 εεε εεεεεεε6 εεεεεεε εεε εεεε εεεεεεεε Usng qs. ()-(6) th arst compton tms and th corrspondng output tms ar cacuatd as x γ ( ) and y γ ( 6) rspctvy. h atst fdng tms ar obtand as u γ ( 34 7). Morovr th crtca path s dntfd as { }. 4.2 Buffr Managmnt Aftr th nsrton of th tm buffrs th mutprojct systm s montord and controd through th Procsss (2) FB u 3 d 2 FB () (3) (5) d d 3 3 d 5 3 PB PB7 Output Projct CB u 4 u 5 CB4 (4) (6) (8) d 4 2 d 6 3 d 8 (7) FB d 7 FB PB PB6 Output Projct 2 Fgur 4. h mut-projct systm aftr th appcaton of crtca chan projct managmnt mthod.

8 ruc ak Goto and akahash: Industra ngnrng & Managmnt Systms Vo No 4 Dcmbr 22 pp KII 44 buffr managmnt proposd n Scton 3.2. From q. (3) w obtan two vctors as: [ ε ε εεε] [ εεε ε ε] 2. W assum hr that th st of th comptd procsss η and th vctor of th actua compton tms z hav bn coctd at th motorng stag as: η { } z (267ε 8 ε) Usng q. (32) w dcompos z nto: z dag( z) [ ε6 εεεεε] z dag( z) [ εεε7 ε ε ε] 2 2 Rato of th PB s consumpton - Cassfy th orgna PB of ach projct: b b C p p 7 p2 p 2 6 b b C - Dfn th PBs consumd by th comptd procsss on ach projct (buffr usd): ( b) dag( x ) z usd γ [ 2 ε 5 ε ε ε ε ε] ( b2) usd dag( xγ ) z2 [ εεε ε2 εε] Rato of th corrspondng progrss tm ovr th xpctd duraton tm - Cacuat th duraton tm of th crtca path for ach projct: 3 u r x γ. 4 d (dag( u γ) yr) C 4 d (dag( u ) y ) C 2 2 γ r 2 - Dfn th progrss tm at th montorng stag wthout consdrng th ffct of th nput tm (tm usd): z ( Buγ ) z [ 3ε 9 ε ε ε ε] z ( B uγ ) z [ εεε3ε7 εε] Obtan th rato of th progrss tm ovr th duraton tm of th crtca path (tm usd %): 3 9 ε ε ε ε ε ε d r t z d2 r 2 εε ε ε ε ε t z. 2 2 h rato of buffr consumpton vrsus th corrspondng progrss tm s pottd on a fvr chart as shown n Fgur 5. h pottd ponts xprss th statuss of th comptd procsss aong th crtca paths. h chart mps th foowng: h prformanc of projct s xprssd by procsss () and (3). In partcuar procss (3) shows th atst status whch bongs to th rd zon. It mans that projct w possby b at and managrs shoud ntat th xtra acton to avod furthr buffr consumpton. h prformanc of projct 2 s xprssd by procsss (4) and (6). Procss (6) shows th atst status whch bongs to th grn zon. It mans that projct 2 s gong w and nd not tak an acton. Snc a capacty buffr was nsrtd btwn projcts and 2 at th transton of procss (3) (6) though procss (3) taks a ongr tm to compt whch ads projct to b at rsk projct 2 s st a rght. hus t sgnfs that managrs nd to tak th actons whch shoud focus on projct ony. Rd zon Yow zon Grn zon Fgur 5. Fvr chart for th mut-projct systm n Fgur CONCUSION W hav proposd an MP rprsntaton for dscrbng a buffr managmnt n th CCPM mthod focusng on mut-projct systms. By ntroducng an nspctd vctor whch rprsnts th actua compton tm of th comptd procsss at th montorng stag th managrs can survy th prformanc of a projcts frqunty. Morovr th proposd mthod can anayz th compx systm to stmat th rats of th consumd projct buffrs and th apsd progrss tms rady for xprssng th currnt status of a projcts and thr ntracton whch support managrs to asy montor ach projct ffctvy managng th ntr systm. Snc w hav dvopd a dtrmnstc mod for nsrtng and controng th tm buffrs for practca cass a stochastc mod whch taks nto account th

9 Buffr Managmnt Mthod for Mutp Projcts n th CCPM-MP Rprsntaton Vo No 4 Dcmbr 22 pp KII 45 anayss of th xtrna uncrtants w b an mportant futur work. ACKNOWDGMNS Yoshnor ak has bn supportd n part by a rsarch grant from JSPS Grants-n-Ad for Scntfc Rsarch No Hroyuk Goto has bn supportd n part by a rsarch grant from JSPS Grants-n-Ad for Scntfc Rsarch No Hrotaka akahash has usd th facts of arthquak Rsarch Insttut (RI) h Unvrsty of okyo. H has aso bn supportd n part by th JSPS Grant-n-Ad for Scntfc Rsarch No RFRNCS Cohn I. Mandbaum A. and Shtub A. (24) Mutprojct Schdung and Contro: A procss-basd comparatv study of th Crtca Chan Mthodoogy and Som Atrnatvs Projct Managmnt Journa 35(2) Godratt. M. (99) hory of Constrants North Rvr Corp. Grat Barngton MA. Goto H. (2) Modng mthods basd on dscrt agbrac systms. In: Got A. (d.) Dscrt vnt Smuatons Scyo Rjka Croata Hdrgott B. (26) Max-Pus nar Stochastc Systms and Prturbaton Anayss Sprngr Nw York NY. Hdrgott B. Jan Osdr G. and van dr Woud J. W. (26) Max Pus at Work: Modng and Anayss of Synchronzd Systms Prncton Unvrsty Prss Prncton NJ. Hrron W. us R. and Dmumstr. (22) Crtca chan projct schdung: do not ovrsmpfy Projct Managmnt Journa Kasahara M. akahash H. and Goto H. (29) On a buffr managmnt pocy for CCPM-MP rprsntaton Intrnatona Journa of Computatona Scnc 3(6) ach. P. (25) Crtca Chan Projct Managmnt Artch Hous Boston MA. akahash H. Goto H. and Kasahara M. (29) oward th appcaton of a crtca-chan-projctmanagmnt-basd framwork on max-pus nar systms Industra ngnrng and Managmnt Systms 8(3) ruc N.. N. Goto H. akahash H. and Yoshda S. (2) nhancd framwork for dtrmnng tm buffrs n th MP-CCPM rprsntaton Intrnatona Journa of Informaton Scnc and Computr Mathmatcs 4() -8. ruc N.. N. Goto H. akahash H. Yoshda S. and ak Y. (22) Crtca chan projct managmnt basd on a max-pus nar rprsntaton for dtrmnng tm buffrs n mutp projcts Journa of Advancd Mchanca Dsgn Systms and Manufacturng 6(5) uk O. I. Rom W. O. and ksogu S. D. (26) An nvstgaton of buffr szng tchnqus n crtca chan schdung uropan Journa of Opratona Rsarch 72(2) Yoshda S. akahash H. and Goto H. (2) Modfd max-pus nar rprsntaton for nsrtng tm buffrs Procdngs of th I Intrnatona Confrnc on Industra ngnrng and ngnrng Managmnt Macau Chna Yoshda S. akahash H. and Goto H. (2) Rsouton of tm and workr confcts for a sng projct n a max-pus nar rprsntaton Industra ngnrng and Managmnt Systms (4)