Intrntonl Journl of ppld Oprtonl Rsrch Vol 1, No 1, pp 7-17, Summr 011 Journl hompg: wwworlur Constrnd Rnwl Rsourc llocton n Fuzzy Mtgrphs v Mn- Slck S S Hshmn* Rcvd: Jnury 31, 011 ; ccptd: My 1, 011 strct Ths ppr dscusss tht th fuzzy mtgrphs cn usd s tool for schdulng nd control of fuzzy procts Oftn, vll rsourcs for xcutng procts my lmtd It s ssumd th rsourcs rqurd to ccomph ch ctvty of proct (mtgrph dgs s rnwl On of th common mthods for schdulng procts s usng th mn-slck So, frst, th forwrd nd ckwrd computtons for dtrmnstc mtgrphs r dscrd Thn, ssumng tht th complton tm of ctvts r postv trpzodl fuzzy numrs, th forwrd nd ckwrd computtons r gnrlzd for fuzzy mtgrphs Consquntly, complton tm of proct, rt nd ltst tm of strt nd nd of ctvts nd flotng tm of thm r otnd s fuzzy numrs o, crtcl ctvts nd crtcl pths r dfnd Thn, for schdulng, y usng th slctd rnkng mthod, th ctvts of proct (mtgrph dgs sd on mn-slck scndng wr ordrd Through som numrcl xmp, clculton stps nd th rsults r llustrtd Kywords Mtgrph, Fuzzy, Constrnd Rnwl Rsourc, Proct Schdulng 1 Introducton In mny rl world procts, th durton of ctvts s non-dtrmnstc nondtrmnstc chrctr my stochstc or fuzzy Rcnt works dfn th fuzzy chrctrs for th proct ntworks cus th fuzzy mod r closr to rlty nd smplr to us [1-7] On th othr hnd, complton of proct on tm hs sgnfcnt ffcts on ts cost, rvnu nd usfulnss Thrfor, th mn octv of proct mngrs s to vod ny dly Howvr, du to th lmtton, to chv ths gol, optml proct schdulng s vtl In rcnt yrs, mny rsrchrs usd th mtgrphs for nlyss of systms On of ths pplctons s proct plnnng nd control usng th mtgrph Mtgrph nd som of ts pplctons r dscrd n [8] Mtgrph nd ts spcfctons hv n dscrd n [9-11] Mtgrphs my hv othr pplctons pplctons of mtgrph n dcson support systms r shown n [1, 13] Mtgrphs r usd n workflow mngmnt [14-16] Hrrchcl modlng y mtgrphs s don n [17] Modl mngmnt usng th Ptr nt nd mtgrphs hs n llustrtd n [18] Rfrnc [19] dscrs th modl ntgrton usng mtgrphs y usng th mtgrphs, ntrprs modlng cn xcutd [0] Implct ntgrty constrnts usng mtgrphs hs n dscrd n [1] Trpzodl fuzzy numr s proposd for stmton of ctvty tm of proct [, ] proct cn * Corrspondng uthor ( E-ml: s_s_hshmn@yhoocom (S S Hshmn S S Hshmn Dprtmnt of Industrl Engnrng, rdl rnch, Islmc zd Unvrsty, rdl, Irn
8 S S Hshmn/ IJOR Vol 1, No 1, 7-17, Summr 011 (Srl #1 shown s mtgrph [16] To th st of our knowldg, thr s not ny ppr ddrssng th fuzzy mtgrph schdulng So, ths ppr dscusss th constrnd rnwl rsourc llocton n fuzzy mtgrphs Th m s th complton of proct wth lowst dly suct to rsourc lmtton Fuzzy mn-slck s crtron s dvlopd n ordr to rch th m Th proposd lgorthm s llustrtd through n xmpl Ths ppr hs th followng structur Scton ntroducs th mtgrphs Scton 3 dscrs th knds of pths n mtgrph Scton 4 dtrmns th crtcl pth mthod n dtrmnstc mtgrphs Scton 5 gnrlzs th crtcl pth mthod for fuzzy mtgrphs y usng th slctd mthod nd dfnton of nw s for comprson of fuzzy trpzodl numrs, Scton 6 schdu th mtgrph dgs sd on mnmum fuzzy slck tm whn th vll rsourc s rnwl nd constrnd Scton 7 s dvotd to conclusons nd rcommndtons Mtgrph Dfnton 1 mtgrph s dntfd wth F ( X, E, D X { x, 1,,, I} s clld th gnrtng st x s clld th lmnt of X E {, 1,,, n} s th st of dgs Ech dg s ordrd pr s ( V, W V X s clld th nvrtx of nd W X s clld th outvrtx of such tht, V W It s supposd tht th durton of dg s known nd rprsntd y d such tht d D Fgur 1 shows mtgrph In ths mtgrph X { x1, x, x3, x4, x5, x6, x7}, E { 1,, 3, 4, 5}, 1 ({ x1},{ x3}, =({x 3, x 4 }, {x 7 }, 3 ({ x1, x},{ x7}, 4 ({ x},{ x6}, 5 ({ x5, x6},{ x7} Fg 1 mtgrph
Constrnd Rnwl Rsourc llocton 9 Dfnton If Dfnton 3 If k V thn V { k } s th conput of lmnt k W thn W } s th cooutput of lmnt k k { k 3 Pths 31 Smpl Pth n lmnt x X s connctd to lmnt x X f th squnc of dgs ( k, k 1,,, K xsts such tht, x V 1, x W K nd k 1,,, K 1 W k V k 1 Ths squnc of dgs s clld smpl pth from x to x x s clld sourc nd x s clld trgt K s clld th lngth of smpl pth In fgur 1 45 s smpl pth from x to x 7 3 Mtpth mtpth s th st of dgs tht s shown wth M (, C s th sourc st nd C s th trgt st s th nvrtx lmnts tht r not o outvrtx lmnts C s th outvrtx lmnts tht r not o nvrtx lmnts Mtgrph n fgur 1 s mtpth wth x, x, x, } nd C x } { 1 4 x5 { 7 4 Crtcl pth mthod for dtrmnstc mtgrph Suppos tht th durton of dg dtrmnstc nd known It s shown y d Thn, y usng th followng lgorthm [16], complton tm of mtpth nd ts crtcl pth nd othr spcfctons such s th rt nd ltst tms of strtng nd fnshng nd crtcl dgs cn dtrmnd Stg 1 Forwrd computtons For ch lmnt x, st Q 0nd mrk x Lt Q 0 for ll othr lmnts Lt E M (, C Whl E for ch dg n E such tht ll lmnts n th nvrtx of r mrk st ES mx{ Q } xv Such tht ES s th rt strt tm of dg Thn, for ch xk W st Qk mx{ Qk,( ES d }, E E { } nd mrk t Rpt th ov oprtons whl E Thn, T (rt complton tm of mtpth s otnd l St T T mx{ Q }, x X
10 S S Hshmn/ IJOR Vol 1, No 1, 7-17, Summr 011 (Srl #1 Stg ckwrd computtons l l For ch lmnt x C, st L T nd mrk x Lt L T for ll othr lmnts Lt E0 M (, C Whl E 0 for ch dg n E0 such tht ll lmnts n th outvrtx of r mrk st LF mn{ L } xw LF s th ltst fnsh tm of dg Such tht Thn, for ch xk V st Lk mn{ Lk,( LF d }, E0 E0 { } nd mrk t Rpt th ov oprtons whl E 0 So, ordrd pr Q, L cn otnd for ch lmnt of mtgrph ( Exmpl 1 Consdr th mtgrph of fgur dg d for ch dg s wrttn undr th Fg Mtgrph of xmpl 1 y usng th lgorthm, rsults r otnd s fgur 3 Fg 3 Rsults for xmpl 1
Constrnd Rnwl Rsourc llocton 11 Dfnton 4 Elmnt x s crtcl f nd only f Q L In xmpl 1 lmnts x1, x, x6, x9, x10 r crtcl Dfnton 5 V s th crtcl nvrtx f nd only f mx( Q mn( L So, n xmpl 1, nvrtx of dgs 1, 4, 5 r crtcl Not tht, mx( Q mn( L xv xv Thorm 1 n nvrtx V s crtcl f t contns ny crtcl lmnts Proof: ssum tht V contns two lmnts Elmnt s crtcl nd lmnt s noncrtcl So, Q L nd mx( Q mn( L or mx( Q, Q mn( L, L or xv mx( Q, Q mn( Q, L xv Cs 1 Q Q L In ths cs, mx{ Q} Q nd mn{ L} Q Thrfor, Q Q nd consquntly, Q Q nd nvrtx V s crtcl Cs Q Q L In ths cs, mx{ Q} Q nd mn{ L} Q So, nvrtx V s crtcl Cs 3 Q L Q In ths cs, mx{ Q} Q nd mn{ L} L nd Q L Consquntly, Q L So, nvrtx V s crtcl Dfnton 6 Th slck tm of dg s dfnd s Slck( mn( L mx( Q d Dfnton 7 Edg s crtcl f nd only f Slck ( 0 Thorm If th nvrtx of dg contns crtcl lmnt wth complton tm T nd th outvrtx of ths dg contns crtcl lmnt wth complton tm T such tht T T d, thn s th crtcl dg Proof: Snc, r crtcl lmnts, thn xw xv xw xv mn ( L T, xv x V xw xv mx( Q T So, mn( L mx( Q T T d nd fnlly, Slck ( d d 0 Consquntly, Slck ( 0 5 Crtcl pth mthod n fuzzy mtgrph If th complton tm of dgs s not dtrmnstc, t my stochstc or fuzzy Hr, t s supposd tht th complton tm of dg s postv trpzodl fuzzy numr Som of rsrchrs support ths lf [, 3, 4, 5, 6, 7, 1] Rfrncs [, 3] ntroduc th oprtons on fuzzy numrs Som of thm r s follows If ( 1, 1, c1, d1 nd (,, c, d r two rtrry trpzodl fuzzy numrs thn
1 S S Hshmn/ IJOR Vol 1, No 1, 7-17, Summr 011 (Srl #1 ( 1, 1, c1 c, d1 d ( 1 d, 1 c, c1, d1 x(, (mx(,,mx( 1,,mx( c1, c,mx( d1, n(, (mn(,,mn(,,mn( c, c,mn( d, m 1 d m 1 1 1 1 d Stg 1 Forwrd computtons For ch lmnt x, st Q (0,0,0,0 nd mrk x Lt Q (0,0,0,0 for ll othr lmnts Lt E M (, C Whl E for ch dg n E such tht ll lmnts n th nvrtx of r mrk st ES mx{ Q } xv Such tht E S s th rt strt tm of dg Thn, for ch xk W st Qk mx{ Qk,( ES d }, E E { } nd mrk t Rpt th ov oprtons whl E Thn, T (th rt complton tm of mtpth s otnd l St T T mx{ Q }, x X Stg ckwrd computtons l For ch lmnt x C, st L T nd mrk l x Lt L T for ll othr lmnts Lt E0 M (, C Whl E 0 for ch dg n E0 such tht ll lmnts n th outvrtx of r mrk st LF mn{ L } xw Such tht L F s th ltst fnsh tm of dg Thn, for ch xk V st Lk mn{ Lk,( LF d }, E0 E0 { } nd mrk t Rpt th ov oprtons whl E 0 So, ordrd pr ( Q, L cn otnd for ch lmnt of mtgrph In ckwrd computtons, w must comput th sutrcton of two postv trpzodl fuzzy numrs Ths sutrcton must postv trpzodl fuzzy numr ut sd on sutrcton dfnton n ths scton, th sutrcton of two postv trpzodl fuzzy numrs my non-postv nd ths rsult n ckwrd computtons s not fsl So, w pply th othr sutl sutrcton oprtor tht supposd n [4] Ths sutrcton oprtor s dfnd s follows: If LF ( lf 1, lf, lf 3, lf 4, LS ( 1,, 3, 4 nd d ( d, d, d, d thn 1 3 4
Constrnd Rnwl Rsourc llocton 13 LS 4 3 1 LF d ( mx(0, mn( lf mx(0, mn( lf mx(0, mn( lf mx(0, mn( lf 1 4 4 3, d, 4,mn( lf,mn( lf, mn( lf 3 3 1, d 4 d d Exmpl consdr th fuzzy mtgrph shown n fgur 4 1 3 Fg 4 Th mtgrph of xmpl ftr th forwrd nd ckwrd computtons, rsults r shown n fgur 5 Fg 5 Rsult of computtons of crtcl pth mthod n xmpl
14 S S Hshmn/ IJOR Vol 1, No 1, 7-17, Summr 011 (Srl #1 Now, w cn gnrlz th prvous dfntons nd thorms for fuzzy mtgrphs Dfnton 8 If Q L thn lmnt x s crtcl Dfnton 9 If mx( Q mn( L thn V s th crtcl nvrtx xv xv Thorm 3 n nvrtx V s crtcl f t contns ny crtcl lmnts Thorm 4 If th nvrtx of dg contns crtcl lmnt wth complton tm T nd th outvrtx of ths dg contns crtcl lmnt wth complton tm T such tht T T d thn s th crtcl dg Consdr tht th nvrs of thorms 3, 4 my ncorrct Dfnton 10 Th slck tm of dg s dfnd s Slck( mn( L mx( Q d Dfnton 11 Edg s crtcl f nd only f Slck ( (0,0,0,0 In xmpl, complton tm of mtgrph s (10, 15, 155, 170 nd dgs, 4, 5 r crtcl So, Slck ( Slck( 4 Slck( 5 0,0,0,0 Edgs 1,3 r non-crtcl nd slck tms of thm r Slck ( 1 (3,15,5,7, Slck ( 3 (3,15,5,7 x W x V 6 Constrnd rnwl rsourc llocton n fuzzy mtgrphs Suppos tht vll rsourc for xcutng th mtgrph rnwl nd constrnd Thrfor, th mn octv of proct mngrs s to vod ny dly So, frst, crtcl dgs must xcutd o, othr dgs must n scndng ordr sd on slck tm of dgs Howvr, w nd sutl rnkng mthod for ordrng mthod tht dscrd n [5] s slctd for chvng ths purpos In th slctd mthod, t s supposd tht,, r ny two fuzzy numrs wth rtrry contnuous mmrshp functons ( x, x, ( x, x o, suppos tht nd s gvn numr G ( nd G ( for fuzzy numrs, r dfnd s whr U U G ( G ( mx( x x mx( x x U U L U U L, L, L ( x dx ( x dx ( x dx ( x dx mn( x x mn( x x Dfnton 1 f nd only f G ( G (
Constrnd Rnwl Rsourc llocton 15 Th ov dfnton cn xtndd for mor thn two fuzzy numrs Suppos tht th slck tms of non-crtcl dgs r shown wth Slck( ( s 1, s, s 3, s 4 In ths ppr, th followng vlu s proposd for Notc tht th slctd fuzzy rnkng mthod cn ordr th slck tms sd on s k NC 4 k1 NC Whr NC s th st of non-crtcl dgs nd NC s th numr of NC mmrs 1 4 Exmpl 3 Consdr th fuzzy mtgrph shown n fgur 6 Fg 6 Mtgrph of xmpl 6 y usng th proposd mthod dscrd n scton 5, fuzzy slck tm of dgs r otnd Thn, fuzzy slck tms r ordrd sd on slctd mthod nd proposd n 77 scton 6 n ths xmpl s Rsults r summrzd n tl 1 6
16 S S Hshmn/ IJOR Vol 1, No 1, 7-17, Summr 011 (Srl #1 Tl 1 Fuzzy slck tms n xmpl 3 Ordrd fuzzy slck tm of (0,0,0,0 (0,0,0,0 5 (0,0,0,0 8 (0,1,16,5 1 (0,1,16,5 3 (0,1,16,5 7 (8,15,5,7 4 (7,7,33,39 6 (7,7,33,39 9 7 Conclusons nd rcommndtons In ths ppr, t s shown tht th uncrtn procts cn dfnd s mtgrphs wth fuzzy dg tms Ths ppr y gnrlzng th crtcl pth mthod for fuzzy mtgrphs nd computng th fuzyy slck tms nd rnkng thm sd on nw crtr gvs th prml s for schdulng fuzzy mtgrphs whn th vll rsourc s rnwl nd constrnd In futur studs, comnton of fuzzy mn-slck wth othr known crtr such s mnltst fnsh tm n fuzzy nvronmnt or usng th othr rnkng mthod my usful Rfrncs 1 Chns, S, Zlnsk, P, (001 Crtcl pth nlyss n th ntwork wth fuzzy ctvty tms Fuzzy Sts nd Systms, 1(, 195-04 Chn, C T, Hung, S F, (007 pplyng fuzzy mthod for msurng crtclty n proct ntwork Informton Scnc, 177(1, 448-458 3 Kucht, D, (001 Us of fuzzy numrs n proct rsk (crtclty ssssmnt Intrntonl Journl of Proct Mngmnt, 19(5, 305-310 4 Ln, F T, Yo, J S, (003 Fuzzy crtcl pth mthod sd on sgnd-dstnc rnkng nd sttstcl confdnc ntrvl stmts Journl of supr computng, 4(3, 305-35 5 Lu, S Y, Lu, S C, Ln, J W, (004 Modl formulton nd dvlopmnt of fuzzy GERT ntworks Journl of th Chns Insttut of Industrl Engnrs, 1, 156-166 6 Lootsm, F, (1989 Thory nd Mthodology stochstc nd fuzzy PERT Europn Journl of Oprtonl Rsrch, 143(, 174-183 7 McChon, C S, L, E S, (1988 Proct ntwork nlyss wth fuzzy ctvty tms Computr nd Mthmtcs wth pplctons, 15(10, 89-838 8 su,, lnnng, R W, (007 Mtgrph nd Thr pplcton Sprngr US, Intgrtd Srs n Informton Systms 9 su,, lnnng, R W, (1994 Cyc n Mtgrphs 7 th Hw Intrntonl Confrnc on Systms Scnc 10 su,, lnnng, R W, (1995 Mtgrphs, Omg, 3(1, 13-5 11 su,, lnnng, R W, (1998 Th nlyss of ssumptons n Modl ss Usng Mtgrphs Mngmnt Scnc, 44(7, 98-995
Constrnd Rnwl Rsourc llocton 17 1 su,, lnnng, R W, (1994 Mtgrphs: Tool for modlng Dcson support systms Mngmnt Scnc, 40(1, 1579-1600 13 su,, lnnng, R W, (1996 Mtgrph- sd DSS nlyss Worknch 9th Hw Intrntonl Confrnc on Systm Scnc 14 su,, lnnng, R W, (1997 Mtgrph Trnsformton nd Workflow Mngmnt Procdngs of th Thrtth nnul Hw Intrntonl Confrnc on Systm Scnc 15 su,, lnnng, R W, (1999 Mtgrphs n workflow Support Systms Dcson Support Systms, 5(3, 199-08 16 su,, lnnng, R W, (001 Workflow nlyss Usng ttrutd Mtgrphs Procdngs of th 34 th Hw Intrntonl Confrnc on Systm Scnc 17 su,, lnnng, R W, Shtu,, (1997 Mtgrphs n Hrrchcl Modlng Mngmnt Scnc, 43(5, 63-639 18 su,, lnnng, R W, (199 Mtgrphs nd Prt Nts n Modl Mngmnt Procdngs of th Scond nnul workshop on Informton tchnologs nd systms, 64-73 19 su,, lnnng, R W, (1994 Modl ntgrton usng Mtgrphs Informton Systms Rsrch, 5(3, 195-18 0 su,, lnnng, R W, (199 Entrprs Modlng usng Mtgrphs Dcson support systms, Exprncs nd Expcttons, 183-19 1 su,, lnnng, R W, (1995 Dscovrng Implct Intgrty Constrnts n Rul ss usng Mthgrphs 8 th Hw Intrntonl Confrnc on Systm Scnc Chn, S J, Hwng, C L, (199 Fuzzy multpl ttrut dcson mkng: mthods nd pplctons Lctur nots n conomcs nd mthmtcl systms, Sprngr-Vrlg, rln, Grmny 3 Zmrmnn, H J, (1996 Fuzzy st thory nd ts pplcton, 3rd Edton, Kluwr cdmc, Puhrs, oston 4 Soltn,, H, R, (007 proct schdulng mthod sd on fuzzy thory Journl of Industrl nd Systm Engnrng, 1(1, 70-80 5 Modrrs, M, Sd-Nzhd, S, (001 Rnkng fuzzy numrs y prfrnc rto Fuzzy Sts nd Systms, 118(3, 49-436