Production Planning and Scheduling for Work Centers with Multiple Manufacturing Machines Considering Operating Mode

Transcription

1 05 Inrnaonal onfrnc on Managmn Scnc & ngnrng ( h ) Ocobr -, 05 Duba, Und rab mras roducon lannng and Schdulng for Wor nrs wh Mulpl Manufacurng Machns onsdrng Oprang Mod KONG J-l,YUN hun-hu,yng Fu-xng,JI Guo-zhu jng Unvrsy of oss and Tlcommuncaons, jng 0876,.R.hna School of conomcs and Managmn, hang Unvrsy, jng 0,.R.hna bsrac: In ordr o ma a prcs producon plannng and schdulng, opmz manufacurng rsourcs, and shorn h maxmum complon m of bach jobs for wor cnrs wh mulpl manufacurng machns, h producon plannng and schdulng for wor cnrs wh mulpl manufacurng machns consdrng oprang mod s rsarchd dply. novl mahmacal modl calld rocssng Tm Modls s sablshd, followd by a nw opmzaon modl for h numbr of manufacurng machns. numrc xampl s usd o s h fasbly of h procssng m modls and h opmzaon modl for h numbr of manufacurng machns n solvng h problms n hs bacground. Th rsuls show ha hs sudy can accuraly schdul manufacurng machns n ach wor cnr as wll as ma a prcs plannng of bach jobs undr h paralll-ransfr mod for wor cnrs wh mulpl manufacurng machns. Kywords: producon plannng and schdulng, wor cnr, procssng m modl, oprang mod, maxmum complon m Inroducon Th whol producon plannng managmn of a ypcal manufacurng nrprs could b classfd no hr yps: long-rm plannng managmn, mdum-rm plannng managmn and shor-rm plannng managmn. Th shop floor managmn and conrol s h cor par of shor-rm plannng managmn []. Th subsysm of shop floor managmn and conrol consss of hr man funcons: producon plannng, producon schdulng and producon monorng []. Th producon plannng and schdulng s h mphass and dffculy of h shop floor managmn and conrol [3]. Th producon plannng and schdulng s submd o h producon Suppord by h Naonal Naural Scnc Foundaon of hna (737007), h rojc Fundd by hna osdocoral Scnc Foundaon (05M57080), h Spcalzd Rsarch Fund for h Docoral rogram of Hghr ducaon (005) and h Mnsry of Naonal Scnc & Tchnology Suppor rogram of hna (0H3F07) worshop basd on wor cnr. Wor cnr, whch s consud by on or svral manufacurng machns, s h logcal ny of h manufacurng un o prform h spcfc manufacurng as []. Th producon plannng and schdulng s a rahr dffcul as n h cas of a wd rang of procssng procsss whn hr ar a lo of manufacurng machns n h producon worshop, spcally on wor cnr s composd of many manufacurng machns n h sam cagory [5]. In ordr o solv hs dffcul problm, h producon plannng and schdulng of wor cnr wh mulpl manufacurng machns s our nrs n hs papr. Th producon plannng and schdulng of wor cnr wh mulpl manufacurng machns s no only rlad o h procss roung of bach jobs, bu also rlad o oprang mods [6] whch ar usd o ransfr bach jobs bwn wo adjacn wor cnrs. Oprang mods could b classfd no hr nds: squnc-ransfr mod, paralll-ransfr mod, and paralll-squnc-ransfr mod. In ral manufacurng sysm, howvr, h squnc-ransfr mod, whch lads o h long maxmum complon m of bach jobs, s usually usd o ma a producon plannng and schdulng for wor cnrs wh mulpl manufacurng machns. Thrfor, hs papr focuss on h paralll-ransfr mod of bach jobs o rduc h maxmum complon m of bach jobs dramacally. Scholars sard o rsarch hs acadmc flds arlr. Th plannng and schdulng problm can b dvdd no hr lvls: sragc lvl, facory lvl, and wor cnr lvl. Morovr, h rm of wor cnr s an mporan concp n R sysm [7-]. Hnc, hr ar a lo of rsarch achvmns on h producon plannng and schdulng of wor cnr [-8]. Th rsarch rsuls n h aspcs of maxmum complon m of bach jobs ar as follows: llahvrd and ydl [] rsarchd a m-machn no-wa flowshop schdulng problm wh h objcv of mnmzng oal complon m. Huang and Ln [0] sudd a wo-sag hr-machn dffrnaon flow shop o fnd a schdul ha aand h mnmum maspan. Ruz-Torrs al [] rsarchd a nw unrlad paralll machn schdulng problm wh h objcv of maspan mnmzaon. Lu al [] proposd a /5/$ I - 3 -

2 dffrnaon flow-shop modl wh h objcv of mnmzng maspan. Zhu [3] nvsgad a wo sag schdulng problm wh h objcv of mnmzng maspan of bach jobs. Th abov rsarch achvmns dd no consdr h ffc of oprang mods upon h maxmum complon m of bach jobs. Thr ar also a plny of rsarch achvmns abou oprang mods. Kh [] provdd formulas of maspan for bach job undr dffrn oprang mods. L [5] proposd formulas of maspan for bach job undr dffrn oprang mods and dsgnd h marx forms usd o compung h maspan for h dffrn yps of bach jobs. Zhang al [6] prsnd hr formulas of maspan for manual procssng sysm undr hr oprang mods, ncludng h group-procssng mod, h producon-ln-paralll-ransfr mod and h producon-ln-paralll-squnc-ransfr mod. In oprang mods rsarch, Kong al [6-3] hav bn achvd a srs of rsarch achvmns, ncludng m modls, producon plannng and schdulng for bach jobs and qupmn rsourcs and so on. Th abov rsarch achvmns only consdrd wor cnr wh a sngl manufacurng machn, bu dd no a wor cnr wh mulpl manufacurng machns no accoun. nd s h man nnovaon of hs papr. Th rs of hs papr s organzd as follows. In scon, h problm formulaon s prsnd, ncludng problm dscrpon, assumpons, and noaons. procssng m modl s proposd n scon 3, followd by an opmzaon modl for h numbr of manufacurng machns n scon. In scon 5, a numrc xampl s sd and analyzd, ncludng h dals of h xampl, h prcs producon plannng of bach jobs and ohr nformaon, h accura schdulng of manufacurng machns n ach wor cnr, and h comparson abou h maxmum complon m of bach jobs bwn wo dffrn oprang mods. Ths papr nds wh som conclusons and a dscusson of lmaons and furhr rsarch drcons. roblm formulaon Th sudy focuss on h wor cnrs wh mulpl manufacurng machns and h paralll-ransfr mod of bach jobs. Th accura schdulng of manufacurng machns n ach wor cnr as wll as h prcs producon plannng for bach jobs could b oband usng h modls prsnd n hs papr.. roblm dscrpon Thr ar n jobs n h sam yp ha nd o b manufacurd n a s of wor cnrs W = { W =,,, m}. Th h wor cnr conans manufacurng machns and h numbr of manufacurng machns s usually dffrn among dffrn wor cnrs. ach job s o b manufacurd n h sam squnc undr h paralll-ransfr mod n all wor cnrs, from wor cnr W o W m. Th objcvs consdrd n hs papr ar o ma a prcs producon plannng for bach jobs on h bass of mnmzng h maxmum complon m and schdul manufacurng machns rasonably on h foundaon of mnmzng h numbr of manufacurng machns for ach wor cnr. Snc h paralll-ransfr mod of bach jobs n prvous rsarch was no dfnd undr hs bacground, hnc, an xac dfnon of h paralll-ransfr mod of bach jobs for wor cnrs wh mulpl manufacurng machns. Th paralll-ransfr mod of bach jobs rfrs o a nd of jobs ransfr mod bwn wo adjacn wor cnrs n whch on fnshd jobs on on cran manufacurng machn n on wor cnr mmdaly ransfr o h nx adjacn wor cnr o b manufacurd mly. Th purpos of hs mod s o mnmz h maxmum complon m of bach jobs by h way of h paralll opraon among wor cnrs.. ssumpons Som ssnal assumpons assocad wh hs problm ar as follows: ()Thr ar a fw manufacurng machns n on wor cnr, and h yps of manufacurng machns ar h sam n on cran wor cnr. ()ll jobs ar h sam yp and prpard a m zro. (3)ll wor cnrs run n good condon whou bradowns. ()ach job mus b manufacurd by only on manufacurng machn n on wor cnr. (5)ach job mus only b manufacurd on on manufacurng machn a any m and ach manufacurng machn could only manufacur on job a a m. (6)Onc h procss of manufacurng on job bgns, h manufacurng machn s no nrrupd unl h complon of hs job. (7)Th procssng m of ach wor cnr s nown and fxd. (8) Th procss roung of manufacurng bach jobs could b oband from h procss roung sh..3 Noaons Th noaons for hs problm ar dfnd as follow: n : oal numbr of bach jobs. m : oal numbr of wor cnrs usd o manufacurng bach jobs. : ndx for wor cnrs, {,,, m}. j : ndx for bach jobs. : ndx for manufacurng machns. : procssng m pr job n h h wor cnr. - -

3 (, j, ) : sarng m for manufacurng h j h job on h h manufacurng machn n h h wor cnr. : ndng m for manufacurng h j h (, j, ) job on h h manufacurng machn n h h wor cnr. : maxmum numbr of manufacurng machns n h h wor cnr. : oal numbr of manufacurng machns ha may b usd o prform h manufacurng as n h h wor cnr bfor dong a mor nns opmzaon,. T : maxmum complon m of bach jobs undr paralll-ransfr mod. : acual mnmum numbr of manufacurng machns ha could b usd o prform h manufacurng as n h h wor cnr afr opmzaon,. a : round up o h smalls ngr ha s no lss han a. c : h smalls ngr ha s no lss han half of. flag : daa for prformng h mard ffc. j : h j h manufacurng as on h h manufacurng machn n h h wor cnr bfor dong a mor nns opmzaon. n : oal numbr of manufacurng ass on h h manufacurng machn n h h wor cnr bfor dong a mor nns opmzaon. : sarng m of m pocs afr h (, j, ) xcuon of h j h manufacurng as on h h manufacurng machn n h h wor cnr bfor dong a mor nns opmzaon. : ndng m of m pocs afr h (, j, ) xcuon of h j h manufacurng as on h h manufacurng machn n h h wor cnr bfor dong a mor nns opmzaon. : duraon of m pocs afr h (, j, ) xcuon of h j h manufacurng as on h h manufacurng machn n h h wor cnr bfor dong a mor nns opmzaon. 3 rocssng m modl procssng procdur of h procssng m modl for wor cnr wh mulpl manufacurng machns could b dvdd no wo sags, ncludng h procssng procdurs for h procssng m modl n h frs wor cnr and h procssng m modl n h h ( m)wor cnr. Sp : Th procssng procdurs for h procssng m modl n h frs wor cnr Sp -: S =, j = and =, l (, j, ) = 0, and compar h valu of n wh n, hn s = n; hn go o Sp -. = ; f n Sp -: S (,, ) (,, ) j j : f <, hn s = +, and assss h valu of j : If < j < n, hn go o Sp -3; f j = n, hn go o Sp. Sp -3: ssss h valu of : f < <, hn go o Sp -; f Sp -: S (, j, ) (, j, ) =, hn go o Sp -5. = +, j = j+ and =, hn go bac o Sp -. Sp -5: S =, j j =, hn go bac o Sp -. (, j, ) (, j, ) = + and Sp : Th procssng procdurs for h procssng m modl n h h ( m ) wor cnr Sp -: ssss h valu of : f < m, hn go o Sp -; f = m, hn s T = (, j, ) and h procssng procdur of procssng m modl for wor cnr wh mulpl manufacurng machns s soppd. Sp -: S = +, j = and =, l (, j, ) (, j, ) : f =, and compar h valu of n wh n, s hn go o Sp -3. = ; f n Sp -3: S (,, ) (,, ) <, s j j = n; = +, and assss h valu of j. If < j < n, hn go o Sp -; f j = n, hn go bac o Sp. Sp -: ssss h valu of : f < <, hn go o Sp -5; f =, hn go o Sp -6. Sp -5: S = +, j = j+ and compar h valu of j wh : f j, hn s j = and go bac o Sp -3; f >, (, j, ) (, j, j ') hn go o Sp -7. Sp -6: S = and j j Sp -7. = +, hn go o (, j, ) Sp -7: ompar h valu of wh - 5 -

4 (, j, j ') : f (, j, ) (, j, j '), hn go o Sp j jj -8; f < (,, ) (,, '), hn go o Sp -. Sp -8: S = 0, hn go bac (, j, ) (, j, ) o Sp -3. Sp -: S (, j, ) = (, j, j '), hn go bac o Sp -3. Opmzaon modl for h numbr of manufacurng machns Th daa of could b oband usng h procssng m modl prsnd n scon 3. u s h maxmum numbr of manufacurng machns ha may b usd o prform h manufacurng as n h h wor cnr. On h foundaon of rmanng h maxmum complon m of bach jobs undr paralll-ransfr mod unchangd, h acual numbr of manufacurng machns ha could b usd o prform h manufacurng as n h h wor cnr can b opmzd. nw procssng procdur of opmzng h acual numbr of manufacurng machns namd opmzaon modl for h numbr of manufacurng machns s sablshd. Th procssng procdur of opmzaon modl for h numbr of manufacurng machns s dvdd no fv sps. Sp : S =, =, j =, flag = 0, =, and c =, hn go o Sp. Sp : ssss h valu of : f m m, hn go o Sp 3; f >, hn h procssng procdur of opmzaon modl for h numbr of manufacurng machns s soppd. Sp 3: ompar h valu of wh : f >, hn go o Sp 3-; f, hn go o Sp 3-. Sp 3-: Th numbr of manufacurng machns couldn b opmzd n h h procssng procss, hn s =, = +, =, j =, flag = 0, =, and c =, hn go bac o Sp. Sp 3-: ompar h valu of wh c : f c, hn go o Sp 3-3; f > c, hn go o Sp. j Sp 3-3: ompar h valu of < n, hn go o Sp 3-3-; f j j wh n : f = n, hn go o Sp 3-3-; f j Sp 3-3-: S > n, hn go o Sp =, (, j, ) (, j, ) =, and (, j, ) (, j +, ) =, hn go o Sp (, j, ) (, j, ) (, j, ) Sp 3-3-: S = T, and (, j, ) =, (, j, ) (, j, ) =, (, j, ) (, j, ) (, j, ) hn go o Sp Sp 3-3-3: S = + and j =, hn go bac o Sp 3-. Sp 3-3-: ompar h valu of wh : f (, j, ) j (, j, ) (, j, ), hn s flag = flag + ; f <, hn s flag flag j = ; hn s = + and go bac o Sp 3-3. Sp : ssss h valu of flag : f flag = 0, flag, hn go o Sp hn go bac o Sp 3-; f 0 5. Sp 5: S = and j =, hn go o Sp 5-. Sp 5-: ompar h valu of wh c : f c, hn go o Sp 5-; f > c, hn assss h valu of : f m, hn s = +, =, =, and c = j =, flag = 0,, hn go bac o Sp ; f = m, hn h procssng procdur of opmzaon modl for h numbr of manufacurng machns s soppd. Sp 5-: ssss h valu of : f (, j, + c) (, j, + c) s nohngnss, hn go bac o Sp 5-; ohrws compar h valu of (, j, + c) Sp 5-3; f : f (, j, ) (, j, c) (, j, ) (, j, c) wh (, j, ) +, hn go bac o +, hn h ( + c )h manufacurng machn n h procssng procss h canno b rplacd by h h manufacurng machn n h sam procssng procss, and s =, = +, j =, and go bac o Sp 5-. Sp 5-3: ompar h valu of wh (, j, + c) : f (, j, ) (, j, c) (, j, ), hn go bac o

5 Sp 5-; f <, hn h ( c + (, j, ) (, j, c) + )h manufacurng machn n h procssng procss h canno b rplacd by h h manufacurng machn n h sam procssng procss, and s =, = +, j =, and go bac o Sp 5-. Sp 5-: ompar h valu of j wh ( n ): f j ( n ), hn s j = j + and go bac o Sp 5-; f j > ( n ), hn h ( c + )h manufacurng machn n h procssng procss h can b rplacd by h h manufacurng machn n h sam procssng procss, and s =, = +, j =, and go bac o Sp numrc xampl and analyss 5. Dals of h xampl Thr ar jobs n h sam yp ha nd o b manufacurd n a s of wor cnrs W = { W, W, W3, W}. Th procssng m pr job and h maxmum numbr of manufacurng machns n h h wor cnr ar prsnd n Tab.. Tm uns of h procssng ms n all wor cnrs ar mnus. Tab. Th procssng m pr job and h maxmum numbr of manufacurng machns n all wor cnrs W W W 3 W /(mn) Th prcs producon plannng of bach jobs and ohr nformaon Usng h procssng m modl proposd n scon 3, w could ma h prcs producon plannng for bach jobs, and oban h maxmum complon m of bach jobs undr h paralll-ransfr mod and h oal numbr of manufacurng machns ha may b usd o prform h manufacurng as n h h wor cnr bfor dong a mor nns opmzaon. Th Gan har of bach jobs manufacurd n all wor cnrs wh mulpl manufacurng machns by h way of h paralll-ransfr mod whou h opmzaon for h numbr of manufacurng machns s shown n Fg.. Th nformaon for h oal numbr of manufacurng machns ha may b usd o prform h manufacurng as n h h wor cnr bfor dong a mor nns opmzaon and h maxmum complon m of bach jobs undr h paralll-ransfr mod ar lsd n Tab. n dal W W W W 8 Fg. Th Gan har of bach jobs undr h paralll-ransfr mod whou h opmzaon for h numbr of manufacurng machns - 7 -

6 Tab. Th nformaon for 3 and T T /(mn) 3 3 Th nformaon of h oal numbr of bach jobs manufacurd by ach manufacurng machn n all wor cnrs whou h opmzaon for h numbr of manufacurng machns ar oband from h abov nformaon. 3 N = n n n = [ 3 3] N = n n = [ 5 5] 3 N3 = n3 n3 n3 n 3 = [ 3 3 ] 3 N = n n n n = [ 3 3 ] Th rsuls of h prcs producon plannng of bach jobs ar as follows: T = whr T s h marx of sarng ms for manufacurng ach job n all wor cnrs T = whr T s h marx of ndng ms for manufacurng ach job n all wor cnrs. 5.3 Th accura schdulng of manufacurng machns n ach wor cnr Usng h opmzaon modl for h numbr of manufacurng machns prsnd n scon, h acual mnmum numbr of manufacurng machns ha could b usd o prform h manufacurng as n all wor cnrs could b gand accordng o h abov nformaon. Th acual mnmum numbr of manufacurng machns n ach wor cnr s shown n n Tab.3. Th accura schdulng schm of manufacurng machns n ach wor cnr could b oband basd on h abov nformaon and analyss. Tab.3 Th acual mnmum numbr of manufacurng machns n ach wor cnr 3 3 Th worng prod of h frs manufacurng machn n frs wor cnr s [0, 0]. Th worng prods of h scond and hrd manufacurng machn n frs wor cnr ar boh [0, 5]. Th worng prod of h frs manufacurng machn n scond wor cnr s [5, 0]. Th worng prods of h scond manufacurng machn n scond wor cnr ar [5, 8] and [, ], rspcvly. Th worng prod of h frs manufacurng machn n hrd wor cnr s [8, ]. Th worng prods of h scond manufacurng machn n hrd wor cnr ar [8, 5] and [6, 30], rspcvly. Th worng prods of h hrd and fourh manufacurng machn n hrd wor cnr ar [, 5] and [3, 7], rspcvly. Th worng prods of h frs manufacurng machn n fourh wor cnr ar [5, 7], [8, 0], [, ], [5, 7] and [, 3], rspcvly. Th worng prods of h scond manufacurng machn n fourh wor cnr ar [5, 7], [0, ], [3, 5], [7, ] and [30, 3], rspcvly. Th Gan har of h accura schdulng schm of manufacurng machns n ach wor cnr by h way of h paralll-ransfr mod s shown n Fg Machn Machn Machn 3 W Machn Machn W Machn Machn Machn 3 Machn W Machn W 6 8 Machn Fg. Th Gan har of h accura schdulng schm of manufacurng machns n ach wor cnr undr h paralll-ransfr mod - 8 -

7 5. Ulzaon of wor cnr For wor cnr wh mulpl manufacurng machns, a novl formula for calculang h ulzaon rao of wor cnr s dfnd as: whr U W acual usd W = avalabl 0% W usd avalabl and acual W W () ar h acual usd capacy and h avalabl capacy of h h wor cnr, rspcvly. whr avalabl W can b calculad as h followng: ( ) = T ST () avalabl W W W T and W ST W ar h ndng m and h sarng m for manufacurng bach jobs n h h wor cnr, rspcvly. acual usd can b calculad as follows: W acual usd W = n (3) Th ulzaon rao of ach wor cnr and ohr rlvan nformaon ar shown n Tab.. No. Tab. Th ulzaon rao of ach wor cnr and ohr rlvan nformaon ST W T W avalabl W acual usd W U W % % % % 5.5 omparson abou h maxmum complon m of bach jobs bwn wo dffrn oprang mods Th mos commonly usd oprang mod s h squnc-ransfr mod n ral manufacurng sysm du o h smplfcaon for mang a producon plannng. Th maxmum complon m of bach jobs undr h squnc-ransfr mod can b calculad by h followng formula: T S n = m = () Thus, h maxmum complon m of bach jobs undr h squnc-ransfr mod s 6 mnus. Howvr, h maxmum complon m of bach jobs undr h paralll-ransfr mod s jus 3 mnus. Th maxmum complon m of bach jobs undr h paralll-ransfr mod s qu small rlav o h on undr h squnc-ransfr mod, accouns for only 5.6% compard o h on undr h squnc-ransfr mod. Thrfor, h procssng m modl proposd n scon 3 could mprov h manufacurng sysm's rapd rspons ably o m h cusomrs urgn rqurmns. 6 oncluson Th wor cnr wh mulpl manufacurng machns has bn wdly usd n h manufacurng sysm. Succssful xprncs of hory and pracc a hom and abroad show ha h paralll-ransfr mod can shorn h maxmum complon m of bach jobs. asd on hs das, a novl procssng m modl s dsgnd n ordr o ma a prcs producon plannng for bach jobs undr h paralll-ransfr mod and oban h maxmum complon m of bach jobs. In ordr o opmz manufacurng rsourcs, a nw opmzaon modl for h numbr of manufacurng machns s proposd o mnmz h acual numbr of manufacurng machns usd o manufacurng bach jobs and schdul manufacurng machns n ach wor cnr accuraly. Fnally, a numrc xampl s usd o s h fasbly of h procssng m modl and h opmzaon modl for h numbr of manufacurng machns n solvng h problms of h producon plannng and schdulng for wor cnr wh mulpl manufacurng machns. Thr ar also som ssus ndd o furhr rsarch. I s ll-consdrd ha only h paralll-ransfr mod and h squnc-ransfr mod ar consdrd, h paralll-squnc-ransfr should also b an no accoun. Th xampl only provds h sam yp of bach jobs. If hr ar mor dffrn yps of bach jobs, h procssng m modl may b modfd sgnfcanly. ll of hs problms should b furhr xplord and rsarchd. Rfrncs []H F Zhu, J R Tan, T L. Rsarch on R shop job schdulng mhod basd on OT. hna Mchancal ngnrng, 005, 6(7): (n hns) []H F Zhu, J R Tan, T L, M L. closd loop dynamc job shop plannng. hna Mchancal ngnrng, 00, 3(6): (n hns) [3]L W Lu. roducon and opraon managmn. jng: Tsnghua Unvrsy rss, 8. (n hns) []Q F Wang, Yan, F Lu. Knowldg ngraon basd opraon mod for worshop manufacurng sysm. ompur Ingrad Manufacurng Sysms, 00, 5(): (n hns) [5]J L Kong, H Yuan, F X Yang, G Z Ja. Mul-objcv flow shop group schdulng consdrng sup m and handlng m sparabl. ompur Ingrad Manufacurng Sysms, hp:// hml. (n hns) [6] X Zhang, Y Y Guan, J. Song. Opmzaon of bachs procssng sysm consdrng mods for manual oprang larnng ra. Sysms ngnrng-thory & racc, 0, 30(): (n hns) [7] nussornnsarn, S Y Nof. -wor: Th challng of h nx gnraon R sysms. roducon lannng & onrol, 003, (8):

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