Integrating Production Scheduling and Maintenance: Practical Implications

Transcription

1 Proceedgs of the 2012 Iteratoal Coferece o Idustral Egeerg ad Operatos Maagemet Istabul, Turkey, uly 3 6, 2012 Itegratg Producto Schedulg ad Mateace: Practcal Implcatos Lath A. Hadd ad Umar M. Al-Turk Departmet of Systems Egeerg Kg Fahd Uversty of Petroleum & Merals Dhahra 31261, Saud Araba M. Abdur Rahm Faculty of Busess Admstrato Uversty of New Bruswck Fredercto, NB E3B 5A3, Caada Abstract Ths work dscusses the tegrato of the schedulg decsos regardg producto ad mateace operatos for mmzg total cost of producto cludg product holdg cost ad mateace costs. Delays producto occur wheever the mache goes through prevetve mateace or correctve mateace actvtes whch effect creases the total producto cost. Prevetve mateace may be doe betwee job processg wth the purpose of reducg the chace of uplaed mache falures whle processg a job. I ths paper we dscuss the case volvg a sgle mache processg a umber of jobs dfferg processg tme requremets. I practce, decsos regardg job schedulg are usually dealt wth dsjotly from mateace related decsos ad models that optmze such decsos are scarce the lterature. I ths paper we preset a modfed verso of the tegrated model developed by Hadd, et al. (2011) ad demostrate ts utlzato uder varous codtos to emphasze the practcal mplcatos of the model. Key words Producto schedulg; mateace schedulg; prevetve mateace; deteroratg process; tegrated models; urelable mache. 1. Itroducto I ths paper we cosder a producto system cosstg of a sgle mache or a sgle producto le that produces a certa type of product batches (jobs) of dfferet szes ad processg requremets. A fxed umber of dfferet jobs ready for processg at the begg of the plag horzo, are to be scheduled a way that mmzes total delays ad hece total process vetory (holdg) costs. The mache may be terrupted by breakdows that causes uplaed delays producto ad hece, mssg deadles ad losg customer goodwll. To reduce such uplaed terruptos, plaed prevetve mateace (PM) operatos s chose to be the strategy for reducg delays that are ot accouted for. Such mateace operatos are assumed to retur the mache back to ts orgal codto (as good as ew) ad has to be coducted betwee job processg,.e., wthout terruptg processg a job. However, PM does ot elmate breakdows completely ad radom mache breakdow may stll occur. I such case, mmal repar s appled to retur the mache back to ts codto just before the start of processg the curret job. All stoppages, plaed or uplaed, adds to the total cost of producto. The objectve s to costruct, at the begg of the plag horzo, a schedule for processg jobs ad prevetve mateace operatos smultaeously such that the total expected cost s mmzed. Oce the schedule s set ad producto starts, o chages are made o that schedule. The case cosdered above s qute commo ad exsts almost all types of producto systems. The lterature cludes several practcal cases establshg the lk betwee producto schedulg ad mateace plag such as Aksoy ad Ozturk (2010) ad Karamatsouks ad Kyrakds (2010). However, the producto schedulg ad PM are, practce, hadled dsjotly ad most cases by dfferet ettes wth the orgazato. The 336

2 producto departmet, for example, costructs a producto schedule, wthout takg mache terruptos to cosderato. The schedule s forwarded to the mateace departmet for schedulg PMs wthout alterg the producto schedule. Alteratvely, the mateace departmet sets a schedule for PM operatos ad leaves t to the producto departmet to schedule ther producto wth the tme slots whch the mache s avalable. Such dsjot schedulg results a suboptmal schedule terms of ts total cost ad also coflcts betwee two fuctoal departmets wth the orgazato. Itegratg the two schedulg decsos to a sgle global schedule resolves coflcts ad mmzes the total cost. The resultg plas of a specfc fucto may dsrupt the other fucto plas. For example, the mateace fucto assgs scheduled shut-dow tervals. These tervals wll be commucated to the producto ut. The suggested mateace tervals may maxmze the mache avalablty, but they wll affect producto plas. Smlarly, producto schedulers may have the tedecy to utlze maches to ther full capacty to meet demad. Uder ths codto, productvty may crease, but mache avalablty wll decrease, due to havg more breakdows. I the lterature, Models developed for optmzg decsos related to producto schedulg ad mateace plag smultaeously s scarce. Cassady ad Kutaoglu (2003, 2005) were amog the frst to cosder such models. Kuo ad Chag (2007) cosdered the problem uder cumulatve damage falure process. Yula et al. (2008) cosdered multple performace crtera ther objectve fucto the tegrated problem. I ther revew, Hadd, et al. (2012) developed a classfcato scheme for problems related to producto schedulg ad mateace plag ad revewed the lterature related to both. Hadd, et al. (2011) formulated a cost model that smultaeously cosder, mateace, mmal repar ad holdg costs for several producto jobs wth the objectve to mmze the expected total costs. The mateace s assumed to be perfect ad restores the mache to a as good as ew codto. Hece, after each mateace acto a ew cycle wll commece wthout beg affected by the mache codto before mateace. The expected cycle legth cludes: average holdg tme, average mateace tme, ad average mmal repar tme. Smlarly, total expected cycle cost cludes: expected mateace cost, expected mmal repar cost ad expected holdg cost. The focus of ths work s to show the practcal mplcato of the tegrated model. Ths s doe by cosderg two specal cases. The orgazato of the rest of ths paper s as follows. Secto 2 presets model otatos. Secto 3 defes, detal, the tegrated problem ad t s assumptos. Secto 4 presets ad solves a example for the tegrated problem. Secto 5 shows practcal mplcatos for the model ad fally Secto 6 dscusses the coclusos. 2. Notatos P j t j a [0] a [j] t r c [j] P [j] N(τ) m(τ) c p C m h Number of jobs to be scheduled Processg tme of job j Tme eeded to perform PM o the mache Age of the mache at the begg of the plag horzo Age of the mache after the th job the sequece Tme requred to repar the mache Completo tme of the th job the sequece Processg tme of the th job the sequece Number of falures τ tme uts of cotuous operato Expected umber of falures τ tme of cotuous operato Cost of coductg PM Mmal repar cost ob order sze Holdg cost for each of producto ut 337

3 3. Problem Formulato Cosder job orders to be processed o a sgle mache that s subject to a Prevetve Mateace (PM) requremet. Each job order wll represet processg oe batch of sze. Raw materals for all job orders are assumed to be released to the shop floor at tme 0. These job orders are avalable at tme zero wth o precedece costrats. Moreover, the mache s avalable cotuously alog the tme horzo uless a mache breakdow occurs durg job processg. If so, a mmum repar that restores the mache to ts codto before breakdow s coducted. Also, the job that was terrupted by the breakdow should resume the remag porto of the job, after mache repar. Each breakdow wll delay completo tme of successve jobs by the eeded tme to repar t r (assumed to be costat). To reduce the chace for mache breakdow, a PM actvty ca be performed before startg ay job. If so, t wll restore mache codto to as good as ew. Ths PM actvty wll delay successve jobs by the tme of PM t p (assumed to be costat). The mache may or may ot fal, causg the completo tme for each job to be stochastc. The PM decsos affect the stochastc process goverg mache falure. Hece, chage the expected value of job completo tme E(c ). Each mateace acto costs a fxed PM cost c p. Smlarly, each break dow wll cost a fxed mmal repar cost c m. It s assumed that all jobs are released to the shop floor ready to be processed at the start of the schedule. Hece, gve that s the average work--process vetory, each job order watg o the shop floor wll cur a holdg cost of h per ut tme. The problem would be to detfy a set of PM decsos, as well as a set of job sequecg decsos a way that reduces the total weghted expected completo tmes. Pror to the job startg, a decso has to be made whether to perform PM or ot. If so, PM wll take a costat tme that wll delay cosequet jobs by tp. If ot, the job wll start at the completo tme of the prevous job. However, the sgle mache ca fal durg job processg. The umber of falures, durg job processg, s strogly affected by the mache age.e., whe the mache ages t has a hgher probablty to fal. The followg assumptos are also cosdered to complete the descrpto of the problem. obs caot be preempted obs terrupted by falure are resumed after repar Mache has creasg hazard rate Upo falure, mmum repar s coducted wthout chage ts age status PM restores the mache to a `as good as ew' codto PM ad Repar tmes are determstc ad kow advace Mache breakdows (falures) may occur radomly ad depedet from each other 4. Problem formulato Let N(τ) be the umber of mache falures τtme uts of cotuous mache operato. Gve that τ p s the tme uts of mache operato over job, the E[N(τ p )] s the expected umber of mache falures durg ts processg of job. The decso of whether to perform a PM or ot before the start of processg a job, ca be modeled by a bary varable y. Let 1 f PM s performed before job y = 0 Otherwse = 12,,, (1) The Gatt chart Fgure 1 resembles the tme spa for mache actvtes whle processg jobs a gve sequece j 1, j 2,, j -1, j, wth hatched area represetg PM decsos ad uplaed breakdows may or may ot occur durg processg jobs. For a gve sequece of jobs the problem reduces to fdg the optmum values of y that mmzes the expected total processg tme. 338

4 1 2 c 1 c2 c c 1 2 c 1 c 2 c Fgure 1 Gatt charts for job processg wth ad wthout mateace for a gve sequece c Let a [] be the mache age ad uder the assumpto that PM restores the mache to a `as good as ew' codto,.e., mache age after PM wll reduce to 0, the a [] = (1-y ) a [-1] + P [] =1,2,, (2) Suppose the mache used to process the jobs s subject to falure, ad the tme to falure for the mache, s govered by a Webull probablty dstrbuto, havg shapg parameter β greater tha 1. Whe the mache fals, t s assumed that the mache s mmally repared.e., the mache s restored to a operatg codto, but mache age s ot altered. Ths mples that, upo mache falure, the mache operator does just eough mateace to resume mache fucto. The assumpto that PM restores the mache to `as good as ew' mples that PM s a more comprehesve acto tha repar that may clude the replacemet of may key parts the mache. Also, the operato ad mateace of the mache (betwee two successve PM's) ca be modeled as a reewal process. Due to the mmal repar, the occurrece of falures durg each `cycle' of the reewal process ca be modeled usg a o homogeeous Posso process. Gve that the job I starts at age (1-y ) a [-1] ad eds at age a [] the a[ ] a[ ] β β 1 E [ N ( τ P )] = ( ) [ ] z t dt = t dt (1 y ) a [ 1] (1 y ) a β [ 1] η (3) = ma ( ) m((1 y) a ) [ ] [ 1] β β [ ] (1 ) [ 1] a y a = η η Where z(t) correspods to the hazard fucto. Fgures 2, presets the effect of PM o the age of the mache ad o the completo tme of job. Mache Age a [0] 0 a [2] a a [ ] [ a 1] [ ] 1 2 c 1 c 2 c Fgure 2. The effect of PM o mache age ad job completo tme. c 339

5 A geeral formula for E[c [] ] ca be foud as follows { τ } [ k ] Ec [ ] = t y + P + ten [ ( )] = 1, 2,, [ ] p k [ k ] r P { p k k r [ k ] k [ k 1] } = t y + P + t ma ( ) m((1 y ) a ) = 1, 2,, (4) The objectve s to provde jobs sequece, as well as PM schedules to mmze the expected total cost that cludes: expected holdg cost (HC), expected mmal repar cost (MRC), ad PM cost (PMC). Fgure 3 shows the vetory level for job order. The average holdg tme for a job deped o t s startg ad completo tmes. The average holdg amout per ut tme s equal to the sum of areas I ad II ( Fgure 3) over E[c [] ],.e. I + II [ ] = Ec ( ) [ ] Batch sze [ ] 0 I II 1 2 c 1 c 2 c Fgure 3 holdg amout of job c [ ] = E( c [ P[ ) + t[ ma ( r [ ) m((1 y ) a[ )]] k ] ] 1] 2 [ ] [ ] E( c ) [ ] [ P + t[ ma ( ) m((1 y) a )]] [ P + t[ ma ( ) m((1 y) a )]] [ ] r [ ] [ 1] [ ] r [ ] [ 1] = 1 1 [ ] = [ ] 2 E( c ) [ ] 2 ( ty + P + t( ma ( ) m((1 y ) a )) p k [ k ] r [ k ] k [ k 1] (5) The holdg cost s usually estmated by holdg cost ut h, whch s expressed moetary ut per work pece ut per tme ut. Let E(c [] ) be the expected tme to complete all jobs. The, the expected holdg cost wll be, HC = h E ( c ) [ ] [ ] = 1 Gve that c m s the expected mmal repar cost per falure. The expected mmal repar cost (MRC) s, MRC = c [ m( a ) m((1 y ) a )] m [ ] [ 1] = 1 Gve that c s the cost for each PM wth a total cost of, PMC = c p y p Itegratg all costs results the followg model. (6) (7) 340

6 M he ( c ) + c [ m ( a ) m ((1 y ) a )] + c y Subject to [ ] [ ] m [ ] [ 1] p = 1 = 1 = 1 a = (1 y ) a + P = 1, 2,, (8) [ ] [ 1] [ ] { } Ec ( ) = t y + P + t ma ( ) m((1 y ) a ) (9) [ ] p k [ k ] r [ k ] k [ k 1] [ P + t[ ma ( ) m((1 y) a )]] = = (10) r [ ] [ ] [ 1] 1 1, 2,, [ ] [ ] 2 ( ty + P + t( ma ( ) m((1 y ) a )) [ ] j j [ ] j j j= 1 j= 1 k = 1 p k [ k ] r [ k ] k [ k 1] P = ( Px ), = ( x ) = 1, 2,, j j j= 1 = 1 a [0] = a x, y = 0 or 1; = 1, 2,, ; j = 1, 2,, j x = 1, x = 1 = 1, 2,, ; j = 1, 2,, (11) (12) The producto schedulg s modeled by the decso varable x j where, th 1 f the job performed s job j x j = 0 Otherwse Costrat (8) keeps track of the age of the mache as t processes jobs ad as t goes through mateace operatos. Costrats (9) ad (10) defe the objectve fucto. Costrat (11) defes the ordered set of jobs wth respect to ther processg tmes ad completo tmes. The model has two logcal sets costrat (12), states that job caot seze two postos at the same tme ad the other states that oe posto caot hold more tha oe job. The developed model s llustrated by a example the ext secto. 5. Solvg the Itegrated Problem The problem formulato ca be solved through oe of the mathematcal programmg laguages. I the followg example GAMS laguage was used to put the model ad the BARON solver was used to reach the optmal soluto. The BARON solver s a computatoal system desged for solvg o-covex NLP optmzato problems to global optmalty. Whe β>1, t may be practcal to perform prevetve mateace o the mache order to reduce the creasg rsk of mache falure. We wll llustrate the formulato ad ts soluto the same example preseted Hadd et al. (2011). Cosder processg 3 job orders cosstg of 1 = 2 = 3 = 500 work pece. ob order 1 eeds 6 mutes for each work pece. ob order 2 eeds 3 mutes for each work pece. ob order 3 eeds 2 mutes for each work pece. The mache age s a [0] =88 hours. The prevetve mateace tme s t p = 5 hours. The mache falure rate follows a Webull dstrbuto wth the followg parameters β=2, η=100. Upo falure, a mmal repar s coducted wth a repar tme t r =15 hours. For the Webull dstrbuto: t1 β ( β 1) t1 m () t = t dt = 0 β η η h = 1.50$ / workpece / hour, c c m p = $1500 ad = $500 β 341

7 ob Order order sze Processg tme per Processg tme per (work pece) work pece (mute) job order (hour) The Formulato wll be: M he ( c[3] ) [ ] + cm[ m ( a[ ]) m ((1 y ) a[ 1] )] + c p y = 1 = 1 = 1 Subject to P[1] = 53x x x13 P[2] = 53x x x 23 P[3] = 53x x x 33 [1] = 500x x x13 [2] = 500x x x 23 [3] = 500x x x 33 a[0] = 88 a[1] = P[1] + (1 y1)88 a[2] = P[2] + (1 y 2 ) P[1] + (1 y1)88 a[3] = P[3] + (1 y 3 ) P[2] + (1 y 2 ) P[1] + (1 y1)88 3 { p k k r k k } Ec ( ) = t y + P + t ma ( ) m((1 y) a ) [3] [ ] [ ] [ 1] [ P + t [ m( P + (1 y )88) m((1 y )88)]] [1] r [1] 1 1 = 1 2( t y + P + t ( m( P + (1 y )88) m((1 y )88))) [1] [1] = 1 [2] [2] 2 = 1 [ 3] [3] 3 p 1 [1] r [1] 1 1 [ P + t [ m( a ) m((1 y ) a )]] [2] r [2] 2 [1] 2 ( t y + P + t ( m( a ) m((1 y ) a )) p k [ k ] r [ k ] k [ k 1] [ P + t [ m( a ) m((1 y ) a )]] [3] r [3] 3 [2] 2 ( t y + P + t ( m( a ) m((1 y ) a )) p k [ k ] r [ k ] k [ k 1] bary varables x, x, x, x, x, x, x, x, x, y, y, y Soluto: x 13 =x 22 =x 31 =1 ad x 11 =x 12 =x 21 =x 23 =x 32 =x 33 =0 ad y 1 =y 3 =1 ad y 2 =0 wth a total expected cost equal to $178,030. Hece, to get the mmum expected cost, producto should start wth {ob 3, PM, ob 2 the ob 1}. 342

8 6. Practcal Implcatos of the Model I ths secto we wll dscuss two extreme cases of the model. Case I: Cosder the case where the holdg cost s so small to be eglgble. The problem ca be approxmated wth mmzg the total expected weghted completo tmes whch was preseted Cassady ad Kutaoglu (2005). The followg example was show ther work. Cosder processg 3 jobs of processg tmes 8, 48 ad 41. Each job has a dfferet weght w 1 =2, w 2 =10, w 3 =10. Processg starts whe the mache has a age of a [0] =88 tme ut ad prevetve mateace tme t p =5 ad correctve mateace (mmal retur) tme t r =15 uts of tme. The mache falure rate follows a Webull dstrbuto wth followg parameters β=2, η=100. Upo falure a mmal repar s coducted. They used full eumerato to fd the optmal soluto. The model results the schedule, x 13 =x 21 =x 32 =1 s the optmal sequece ad y 1 =y 3 =1 ad total cost of Case II: We cosder as the case where mateace cost parameters c m s much hgher tha other cost parameters c p ad h. It worth to meto that β s greater tha 1, otherwse PM wll have o effect. I ths case y 1 =y 2 = =y =1, to mmze repar cost to the mmum. The problem wll reduce to a typcal schedulg problem of fdg the mmum average completo tme wth WSPT (Wated Shortest Processg Tme) as a optmal schedule. 7. Coclusos ad Future Research Itegrated models are expected to provde better savgs over dsjot models. Research tegrated modelg stll has great potetal to further cotrbute to more effcet utlzato of resources justfed by the expected savgs provded by tegrated modelg. Ths work preseted a revsed verso of the prevous work of Hadd et al. (2011) ad demostrated ts utlzatos through two practcal specal cases. A mathematcal model was developed that gves a optmal soluto for both producto schedulg ad prevetve mateace polcy wth respect to mmzg product holdg cost ad mateace costs smultaeously. Both prevetve mateace ad correctve mateace are assumed as a exteso to ths work, ths assumpto ca be geeralzed where repar ad prevetve mateace tmes are radom varables followg a geeral dstrbuto. The model ca be studed uder the effect of dfferet repar ad prevetve mateace polces. Ths effect ca be vestgated future work. Ackowledgmet The authors ackowledge the support of Kg Fahd Uversty of Petroleum ad Merals ad Uversty of New Bruswck, Fredercto, Caada. Refereces Aksoy, A. ad Ozturk, N., Smulated aealg approach schedulg of vrtual cellular maufacturg the automotve dustry, Iteratoal oural of Vehcle Desg, vol. 52, os. 1-4, pp , Cassady, C. ad Kutaoglu, E., Mmzg job tardess usg tegrated prevetve mateace plag ad producto schedulg, IIE Trasactos, vol. 35, o. 2, pp , Cassady, C. ad Kutaoglu, E., Itegratg prevetve mateace plag ad producto schedulg for a sgle mache, IEEE Trasactos o Relablty, vol. 54, o. 2, pp , Hadd, L.A., Al-Turk, U.M. ad Rahm, M.A. (2011) A tegrated cost model for producto schedulg ad perfect mateace, It.. Mathematcs Operatoal Research, Vol. 3, No. 4, pp Hadd, L.A., Al-Turk, U.M. ad Rahm, M.A., Itegrated Models Producto Plag ad Schedulg, Mateace ad ualty: A Revew, Iteratoal oural of Idustral ad Systems Egeerg, vol. 10, o.1, pp , Karamatsouks, C. ad Kyrakds, E., Optmal mateace of two stochastcally deteroratg maches wth a termedate buffer, Europea oural of Operatoal Research, vol. 207, o. 1, pp , Kuo, Y. ad Chag, Z-A., Itegrated producto schedulg ad prevetve mateace plag for a sgle mache uder a cumulatve damage falure process, Naval Research Logstcs, vol. 54, o. 6, pp , Yula,., Zuhua,. ad Weru, H., Mult-objectve tegrated optmzato research o prevetve mateace plag ad producto schedulg for a sgle mache, Iteratoal oural Advaced Maufacturg Techology, vol. 39, os. 9-10, pp ,