Constrained Renewable Resource Allocation in Fuzzy Metagraphs via Min- Slack

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1 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

2 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

3 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

4 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

5 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

6 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 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

7 Constrnd Rnwl Rsourc llocton 13 LS LF d ( mx(0, mn( lf mx(0, mn( lf mx(0, mn( lf mx(0, mn( lf , d, 4,mn( lf,mn( lf, mn( lf 3 3 1, d 4 d d Exmpl consdr th fuzzy mtgrph shown n fgur 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

8 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 (

9 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

10 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, 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(, Chn, C T, Hung, S F, (007 pplyng fuzzy mthod for msurng crtclty n proct ntwork Informton Scnc, 177(1, Kucht, D, (001 Us of fuzzy numrs n proct rsk (crtclty ssssmnt Intrntonl Journl of Proct Mngmnt, 19(5, 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, 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, Lootsm, F, (1989 Thory nd Mthodology stochstc nd fuzzy PERT Europn Journl of Oprtonl Rsrch, 143(, McChon, C S, L, E S, (1988 Proct ntwork nlyss wth fuzzy ctvty tms Computr nd Mthmtcs wth pplctons, 15(10, 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, su,, lnnng, R W, (1998 Th nlyss of ssumptons n Modl ss Usng Mtgrphs Mngmnt Scnc, 44(7,

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