CAREER: Scheduling of Large-Scale Systems: Probabilistic Analysis and Practical Algorithms for Due Date Quotation

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1 NSF grat umber: DMI NSF Program Name: Operatos Research. CAREER: Schedulg of Large-Scale Systems: Probablstc Aalyss ad Practcal Algorthms for Due Date Quotato Phlp Kamsky Uversty of Calfora, Berkeley Our Kaya Uversty of Calfora, Berkeley Abstract: We preset a ovel sgle faclty due-date quotato model ad algorthm, ad characterze the effectveess of ths algorthm employg the tools of probablstc aalyss. Ths model s teded to be a buldg block for the models that we wll use to aalyze due date quotato make-to-order supply chas, ad we preset a summary of these models. 1. Itroducto: The vast majorty of recet supply cha research focuses o make-to-stock systems, ad performace measures bult aroud servce ad vetory levels. O the other had, a creasg umber of supply chas are better characterzed as make-to-order systems. Ths s partcularly true as more ad more supply chas move to a mass customzato-based approach to satsfyg customers (see Smch-Lev, Kamsky, ad Smch-Lev [31]). Mass customzato mples that at least the fal detals of project maufacturg must occur after specfc orders have bee receved, ad must thus be completed quckly ad effcetly. Clearly, make-to-order supply chas face may uque ssues. Oe of the most sgfcat s that sce customers place orders ad wat for products to arrve, due dates or delvery dates must typcally be quoted whe a product s ordered. Recetly, researchers have troduced a varety of models a attempt to uderstad effectve due date quotato. I some models, the assumpto s made that orders wll be placed depedet of the legth of the quoted lead tme (see [1] for a survey), where the quoted lead tme s typcally defed to be the tme spa from the arrval of a job utl ts quoted start-processg tme (or quoted due date). The objectve ths case s frequetly to mmze average quoted lead tme subject to some costrat o the umber or amout of tardy jobs. The majorty of papers based o these models have bee smulato based. For stace, Elo ad Chowdhury [2], Weeks [3], Myazak [4], Baker ad Bertrad [5], ad Bertrad [6] cosder varous due date assgmet ad sequecg polces, ad geeral demostrate that polces whch use estmates of shop cogesto ad job cotet formato lead to better shop performace tha polces based solely o job cotet. Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

2 Some aalytcal results do exst for ted versos of these models. Prmarly, these cosst of determstc, commo due date models, where a sgle due date must be assged for all jobs, ad statc models, where all jobs are avalable at tme 0. For these smplfed models, a varety of polyomal algorthms have bee developed (see Brucker[7], Kahlbacher[8], Pawalkar et al.[9], Hall ad Poser [10], Sedma et al.[11], ad Chad ad Chhajed [12]); however, these results do t exted a obvous way to more complex models. I other models, ot all potetal jobs are processed. I some of these models, a maxmum lead tme s assocated wth each customer. The frm decdes whether or ot to accept a partcular offer, ad f the offer s accepted, a perfectly relable lead tme (whch must be less tha the customers maxmum lead tme) s quoted. I other words, a accepted job has to start processg wth ts quoted lead tme. The objectve typcally volves a reveue fucto whch decreases wth creasg quoted lead tmes. Ths s kow as the Lead Tme Quotato (LTQ) problem (see Keskocak, Rav, ad Tayur [13] ad the refereces there). Aga, effectve algorthms exst for specal cases of ths problem, but ot for the most geeral cases. Some researchers approach LTQ models wth a queug theoretc framework. For example, We [14] cosders a multclass M/G/1 queug system uder the objectve of mmzg the weghted average lead tme subject to the costrats of the maxmum fracto of tardy jobs ad the maxmum average tardess. Spearma ad Zhag [17] further characterze heurstc performace for these types of systems. I order to capture the mpact of quoted lead tmes o demad, some models assume that gve a quoted lead tme, the customer decdes whether or ot to place a order. The probablty that a customer places a order decreases wth creasg lead tme. Dueyas ad Hopp [16] cosder such a system usg a queug model, ad provde effectve heurstcs uder varous problem characterstcs. I ther model, there s a sgle class of customers; the et reveue per customer s costat, customers have the same prefereces for lead tmes, ad the arrval ad processg tmes follow the same dstrbuto. Dueyas [15] exteds ther results to multple customer classes, wth dfferet et reveues ad lead tme prefereces. Kamsky ad Hochbaum [30] survey due date quotato models more detal. I Kamsky ad Lee [29], we troduce a ew due date quotato model (DDQP) whch captures some of the key elemets of the models ad approaches above. I ths model, jobs or orders arrve to a sgle server, represetg a maufacturg orgazato, over tme. All jobs are accepted, ad due dates must be quoted mmedately upo job arrval. The objectve s to mmze average quoted lead tme (or quoted due date), ad all due dates must be met. Based o ths 100% relable due date quotato model, they develop a o-le due date quotato algorthm (DDQ) wth several varatos, characterze the asymptotc performace of ths algorthm, ad the aalyze asymptotc probablstc bouds o ts performace. I Kamsky ad Lee [32], we exted the DDQP model to a more complex evromet, the flow shop. I that paper, we developed a flow shop o-le due date quotato algorthm, aalyze asymptotc bouds o ts performace uder some probablstc assumptos, ad the preseted computatoal results whch demostrate the effectveess of the algorthm. I our curret work, we are explorg due date Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

3 quotato supply chas. I partcular, we wll focus o a smple 2 party supply cha, whch a maufacturer works to satsfy customer orders. Customers arrve at the maufacturer over tme, ad the maufacturer produces to order. I order to complete producto, the maufacturer eeds to receve a compoet from a suppler. Each order takes a dfferet amout of tme to process at the maufacturer, ad at the suppler. The maufacturer s objectve s to determe a schedule ad quote due dates order to mmze a fucto of due date quoted, ad lateess. Our prmary objectve s develop effectve approaches for schedulg both the cetralzed ad decetralzed versos of ths model, so that we ca vestgate the relatve advatage of a cetralzed system uder varous codtos. To do ths, we wll focus o three models. I the frst model, the cetralzed model, both facltes are cotrolled by the same aget, who quotes a due date to the arrvg customer, ad the schedules jobs both facltes order to acheve the ed objectve. I the secod model, the smple decetralzed model, the maufacturer makes certa assumptos about the suppler order to estmate a due date, ad the suppler follows a smple schedulg prorty rule. I ths model, each faclty s workg to acheve ts ow goals. I the thrd model, the decetralzed model wth addtoal formato exchage, both maufacturer ad suppler quote due dates, the suppler to the maufacturer, ad the maufacturer to the ed customer based o the due date quoted to the maufacturer from the suppler. We wll model the stuato the cetralzed model as a flowshop. As poted out Kamsky ad Smch-Lev [21] ad Pedo [22], most of the flow shop related research has focused o mmzg the makespa sce the other objectves such as mmzg total completo tme s very dffcult to aalyze. However, we aalyze a due-date based flowshop model, ad our aalyss of the flowshop depeds o ovel aalyss of a sgle mache model. I ths model, jobs arrve at a sgle server over tme. Due dates must be quoted to arrvg customers, ad jobs must be sequeced o the server to mmze a fucto of quoted due dates, ad lateess. Our aalyss of both cetralzed models wll also rely o our aalyss of the same model. I ths paper, we preset our aalyss of ths sgle mache model, ad the troduce our supply cha models. Of course, these are very dffcult problems. Ideed, most versos of models related to those descrbed above are NP hard. Thus, we focus o developg effectve heurstcs for determstc versos of these models. Usg the tools of probablstc aalyss, as well as computatoal testg, we characterze the performace of these heurstcs uder varous codtos. I partcular, we determe codtos uder whch these heurstcs provde asymptotcally optmal solutos for these models. To put our work to perspectve, we hghlght other research relatg to probablstc aalyss of schedulg problems. To the best of our kowledge, oe of ths work has focused o due date quotato models, ad the vast majorty has focused o relatvely smple objectves, ad the aalyss of relatvely smple algorthms. Much of ths work has focused o parallel mache problems, cludg the work of Coffma, Frederckso ad Lueker [23], Loulou [24] ad Frek ad Rooy Ka [25], who aalyze the parallel mache schedulg problem whe the objectve s to mmze the makespa, ad Spaccamela et al.[26] ad Webster [27], who aalyze the parallel mache weghted completo tme model. Ramudh et al.[28] who aalyze the 2-mache flow shop makespa model. Kamsky ad Smch- Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

4 Lev(1999) [21], Kamsky ad Smch-Lev [18] ad Xa, Shathkumar, ad Gly [20] aalyze the flow shop average completo tme problem. Fally, Kamsky [33] cosders the flow shop delvery tme problem, whch s closely related to the flow shop maxmum lateess problem, a model whch volves due dates, although they are gve rather tha determed by the model. Also, we wll focus o o-le algorthms for these models. I ths cotext, o-le schedulg algorthms sequece jobs at ay tme usg oly formato pertag to jobs whch have bee released by that tme. Ths models may real world problems, where job formato s ot kow utl a job arrves, ad formato about future arrvals s ot kow utl these jobs arrve. I cotrast, off-le algorthms may use formato about jobs whch wll be released the future. It s clear that order to fd the optmal soluto to ths model, t may be ecessary to accout for future arrvals whe assgg due dates, ad t may be ecessary to sert dle tme to the schedule whle watg for certa jobs to arrve, both of whch suggest that off-le algorthms, whch ca utlze formato about future arrvals to assg due dates ad to decde whether or ot to sert dle tme the schedule, wll perform better tha o-le algorthms whch ca t. I the ext secto, we troduce the sgle faclty model, ad relevat heurstcs ad results. I secto 3, we troduce our supply cha models, whch we ted to aalyze the ear future. 2 The Sgle Mache Model ad Ma Results 2.1 The Model: We cosder a sgle-mache ole producto system. Customers arrve at the system over tme, ad place a order (equvaletly, jobs are released over tme). The processg tme of the order s kow whe the customer arrves, ad the producer quotes a due date at the tme of the arrval, wth the goal of mmzg costs assocated wth the tme utl the due date, ad costs assocated wth completg the processg of the job after ts quoted due date. I ths model, a set of jobs eeds to be processed o-preemptvely o a sgle mache. Each job has a assocated type l, l = 1, 2,...k, ad each type has a assocated processg tme p l. Each job, = 1, 2,..., also has a assocated release tme r. Also, the operator of the system quotes a due date for each job d. I partcular, we focus o a system whch due dates are quoted wthout ay kowledge of future arrvals - a ole system. However, formato about the curret state of the system ad prevous arrvals ca be used. As metoed above, we use the tools of probablstc aalyss, as well as computatoal testg, to characterze the performace of these heurstcs uder varous codtos. I ths type of aalyss, we cosder a sequece of radomly geerated determstc staces of the problem, ad characterze the objectve values resultg from applyg a heurstc to these staces as the sze of the staces (the umber of jobs) grows to fty. For ths probablstc aalyss, we geerate problem staces as follows. Each job has depedet probablty P l of beg job type l, where k l=1 P l = 1 ad job type l has kow processg tme p l. Arrval tmes are determed by geeratg ter-arrval tmes draw from detcal depedet dstrbutos bouded above by some costat, wth expected value ET. Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

5 The orders eed to be processed o a sgle mache wthout preempto ad o job ca beg processg before ts arrval tme. Let C be the completo tme of the order. For each job, the producer curs a cost of c d = cd d, where where c d s the ut cost of the quoted due date ad a lateess cost of c L = c L [C d ] + for order f the order completes processg after ts assged due date, where c L s the ut lateess cost. We assume that c L > c d ; otherwse, settg all due dates equal to 0 wll be optmal. The objectve of ths problem s thus to determe a sequece of jobs ad a set of due dates such that the total cost= =1 (cd + cl ) s mmzed. 2.2 The Heurstc: Ay algorthm for ths problem has to do two thgs: quote due dates for each job, ad sequece jobs for processg. The heurstc we propose, -SL, s based o sequecg the jobs accordg to the Shortest Processg Tme Avalable () rule. Uder the heurstc, each tme a job completes processg, the shortest avalable job whch has yet ot bee processed s selected for processg. Note that ths approach to sequecg does ot take quoted due date to accout, ad s thus easly mplemeted. Istead, the due date quotato rule takes the sequecg rule to accout. To quote due dates, we mata a ordered lst of jobs that have bee released ad are watg to be processed. I ths lst, jobs are sequeced creasg order of processg tme, so that the shortest job s at the head of the lst. Sce we are sequecg jobs, whe a job completes processg, the frst job o the lst s processed, ad each job moves up oe posto the lst. Whe a job arrves at the system at ts release tme r wth processg tme p ad the system s empty, t mmedately begs processg ad a due date equal to ts release tme plus ts processg tme s quoted: d = r + p However, f the system s ot empty whe a job arrves, t s serted to the approprate place the watg lst. Let r be the release tme of job, rt be the remag tme of the job process at the tme of arrval, pos[] be the posto of job the watg lst ad lst[] be the dex of the th job the watg lst. The, a due date s quoted for ths job as follows: pos[] d = r + rt + (p lst[j] ) + slack (0.1) where slack s some addtoal tme added to the due date order to accout for future arrvals before ths job s processed wth processg tmes less tha ths job these are the jobs that wll be processed ahead of ths job, ad cause a delay ts completo tme. The remader of ths subsecto focuses o determg a approprate value for slack. I the ext secto, we aalytcally demostrate the effectveess of the -SL approach. Defe the mwat tme whe a job arrves to be the remag tme of the job process plus the total processg tmes of all of the jobs to be processed ahead of ths arrval, so that pos[] 1 mwat = rt + (p lst[j] ). Now, the slack for job s calculated as follows, assumg a problem stace of sze. Let pr l be the probablty that a arrvg job has processg tme less tha p l. For job types Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

6 l ad l j, assume wthout loss of geeralty that p l p lj f ad oly f l l j. So, for a job type l, ad l 1 pr l = Pr{p < p l } = pr 1 = 0. P j l = 2, 3...k Also, let EP l the expected processg tme of a job gve that t s less tha p l, l 1 EP l = E[p p < p l ] = P j p j. ad let EA(t) = t ET be the expected umber of arrvals durg a tme terval of legth t. Sce ths s a job stace, t may be that all of the jobs have arrved before job s processed. So, we eed to cosder two cases accordg to the value of the total expected arrvals before job s processed. Let A 1 be the expected umber of all arrvals durg the mwat tme for job of type l. So, A 1 = EA(mwat )The multplyg t wth pr l gves us the expected umber of ew arrvals that wll be placed frot of job. These arrvals wll add a addtoal expected watg tme of s 1 = A 1 pr l EP l. Of course, durg ths tme, some ew addtoal jobs wll also arrve; let A 2 = EA(s 1 ) represet the umber of these addtoal jobs ad s 2 = A 2 pr l EP l wll be the expected durato of the jobs amog A 2 that wll be placed frot of job. More addtoal jobs may arrve whle these jobs are processed. Thus, we eed to determe the total umber of jobs that are expected to arrve assumg that we have a fte source of arrvg jobs, stead of oly jobs, ad compare t wth the remag umber of jobs,. Applyg the same logc as above, we estmate the expected umber of total arrvals : TA = (A j) where A j = EA(s j 1 ), s j = A j pr l EP l for j = 1, 2,...k ad s 0 = mwat. Startg from A 1 ad s 1 ad wrtg the equatos dyamcally, we see that: ad A j = mwat (pr l ) j 1 (EP l ) j 1 ET j TA = If pr l EP l ET mwat (pr l ) j 1 (EP l ) j 1 ET j. 1 the ths sum grows to fty, whch mples that all of the remag jobs are expected to arrve to the system before job goes to process. So, ths case, stead set TA =. If pr lep l ET TA = < 1, the mwat pr j 1 l EP j 1 l ET j = mwat ET pr l EP l Of course, eve ths case t s possble that mwat ET pr l EP l mght be greater tha the umber of remag jobs,, aga mplyg that the total expected umber of arrvals TA =. The, we quote the slack accordg to the followg formula: slack = TA pr l EP l Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

7 We summarze the due date quotato rule for job of type l as follows: If pr l EP l ET 1 else slack = ( ) pr l EP l slack = m{ mwat pr l EP l, ( )pr l EP l } ET pr l EP l (0.2) ad gve ths slack, we quote the followg due date: d = r + mwat + p + slack. 2.3 Aalyss ad Results For sets of radomly geerated problem staces as descrbed precedg sectos, let Z SL represet the objectve fucto value obtaed by applyg the -SL rule for a, ad let Z be the optmal objectve fucto value for that stace. I ths secto, we prove Theorem 1. Cosder a seres of radomly geerated problem staces meetg the requremets descrbed above. Almost surely, Z SL Z Z = 0 I other words, -SL s asymptotcally optmal for ths problem. To prove ths theorem, we utlze a seres of preary results ad observatos. Frst, we cosder a feasble schedule for a stace of ths problem, ad defe a cha to be a seres of jobs processed cosecutvely wthout dle tme betwee them, but wth dle tme before ad after the seres begs processg. Now cosder the followg: Observato 1. Cosder ay two o-delay schedules o a sgle mache. Number the chas cosecutvely both of the schedules.there wll be the same umber of chas both sequeces ad the k th cha ether sequece wll cota exactly the same jobs ad wll start ad ed at exactly the same tme o the mache. We cosder two cases, depedg o the relatoshp betwee the expected processg ad expected terarrval tmes, EP ad ET. Case 1: EP < ET For ths case, Gazmur [19] proved the followg lemma: Lemma 1. Cosder a sgle mache problem. Let terarrval tmes T 1, T 2,..., T 1 (T = r +1 r ) be..d. radom varables, bouded above by some costat, where T s a geerc radom varable represetg a terarrval tme; the processg tmes p 1, p 2,..., p be..d radom varables, bouded above by some costat, where p s a geerc radom varable represetg a processg tme; the processg tmes ad terarrval tmes be depedet of each other, ad let EP < ET. The f M s a radom varable represetg the umber of jobs a cha, E(M) ad E(M 2 ) are bouded by costats that are depedet of. Now, let l k be the cha dex, ad let N() ad M k be the umber of chas a job stace ad the umber of jobs cha k, respectvely. I the followg dscusso, we cosder a sgle cha, ad dex 1,2...M k cha k. For each cha, let Z SL l k ad Zl k be that porto of the objectve fucto from jobs the cha. The Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

8 Z SL l k Z l k = M k (c d d +c L (C d ) + ) Zl k =1 (0.3) or summg over all chas, ad revertg to orgal dexg: Z SL Z = (c d d +c L (C d ) + ) Z =1 (0.4) We ow boud both the lateess ad the due date portos of ths quatty. Observe that for each cha k, a job ca t be late by more tha the legth of that cha sce the release tme of ay job a cha s at least as bg as the start tme of the cha, we set a due date greater tha the release tme of job, ad the job completes by the ed of the cha. Thus, ad M k C d M k p max c L (C d ) + c L M k (M k p max ) =1 Summg ths quatty over all chas, we get: c L (C d ) + =1 N() k=1 c L M 2 k p max Next, we focus o the maxmum dfferece betwee the optmal objectve value ad the due date cost related porto of the heurstc objectve fucto value. Clearly, a job s earlest possble due date the optmal schedule s equal to that job s release date, plus ts processg tme. Also, recall that due dates are set accordg to the formula d = r + mwat + p + slack, so that a upper boud o the dfferece betwee the due date cost the heurstc soluto, ad the total cost the optmal soluto, s: M k (c d d ) Zl k c d (slack + mwat ) =1 M k =1 (0.5) Now, recall the quatty mwat, the mmum tme that job wll wat before processg uder algorthm -SL f there s o serted slack tme. Clearly, mwat s also smaller tha the legth of ts cha because whe job arrves, mwat ca be o loger tha the processg tme of all watg jobs, whch s o greater tha the legth of the cha. Thus, mwat M k p max (0.6) Summg over all the jobs the cha, we get: M k mwat Mk 2 p max (0.7) =1 Also, sce sce EP < ET ad pr 1, from (0.2), slack = m{ mwat pr l EP l ET pr l EP l, ( )pr l EP l } slack mwat pr l EP l ET pr l EP l mwat EP ET EP or, f we defe U = the EP ET EP, ad apply (0.6), slack M k p max U (0.8) Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

9 Summg over all the jobs the cha, we get M k slack Mk 2 p max U (0.9) =1 Substtutg (0.7) ad (0.9) to (0.5), we get: M k (c d d ) Zl k c d (Mk 2 p maxu + Mk 2 p max) =1 (c d + Uc d )M 2 k p max (0.10) Summg over the N() chas, ad substtutg to (0.4), N() Z SL Z = (c d + Uc d + c L )Mk 2 p max =1 Dvdg by 2 ad multplyg ad dvdg by N(), we get: Z SL 2 Z (c d + c d U + c L )p max N() N() k=1 2 M2 k N() Also, ote that the followg hold almost surely. N() = E(M) N() k=1 (M2 k ) = E(M 2 ) N() Takg the t as the umber of jobs goes to fty, ad substtutg: Z SL 2 Z (c d + c d U + c L )p max E(M 2 ) E(M) = 0 (0.11) Case 2: EP ET For ths case, we cosder the asymptotc dfferece betwee the heurstc ad optmal solutos, ad beg ths part of the proof by addg ad subtractg the value =1 cd C, where C deotes the completo tme of job wth the sequece. Also, C deotes the completo tme of job wth the optmal sequece. Z SL Z =1 cd d + c L [C SL d ] + c d C =1 cd d c d =1 C 2 + c d =1 C =1 + cl [C SL d ] + =1 cd C 2 (0.12) It s kow (see Kamsky ad Smch-Lev (2001)) that s asymptotcally optmal for mmzg total completo tme of a sgle mache problem uder the codtos cosdered ths paper. Thus, we ca wrte the followg lemma. Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

10 Lemma 2. For a seres of radomly geerated problem staces meetg the requremets descrbed above, almost surely, c d =1 C =1 cd C 0 Now, recall that the algorthm -SL sequeces jobs order, let S be the sum of completo tmes of all jobs sequeced accordg to a job stace, ad let S SL be the total cost of all due dates, =1 cd = =1 cd d. To prove our fal result, we employ the followg Lemma: Lemma 3. Cosder a seres of radomly geerated problem staces meetg the requremets descrbed above. Almost surely, cd S SL S S = =1 d C 0 Proof. For ay job, let Ãj be the umber of jobs of type j that arrved after r but before job of type l started processg. By, these jobs wll be processed before job. It s easy to see that the sequece, for a job of type l 2, l 1 C = r + mwat + p l + Ã j p j ad for a job of type 1 C = r + mwat + p 1. Now we dvde the job types to two sets. For a job problem stace, all jobs of type l such that ET EP l pr l 0 are set B1, ad all jobs of type l such that ET EP l pr l > 0 are B2. For all jobs B1, the quoted due date wll be d = r + mwat + p l + ( ) pr l EP l. Smlarly, for all jobs B2, the quoted due date wll be: d = r + mwat + p l +m{( )pr l EP l, mwat pr l EP l ET EP l pr l } Summg over all jobs ad takg the t as the umber of jobs goes to fty, a.s. we get: = S SL c d 2 = 2 S c d =1 d C 2 B j d C 2 (0.13) Now, we cosder the B1 ad B 2 jobs separately. Let S l deote the set of type l jobs B1, ad thus, for the B 1 jobs, c d B d C 1 c d B1 [( )pr lep l l 1 Ãj p j] c d k l=1 So, for all type l jobs, [( )pr l EP l l 1 Ãj p j] 2 d C [( )pr l EP l l 1 Ãj p j] Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

11 S l EP l pr l (pr l EP l l 1 Ãj p j) 2 By employg ths lemma, we ca wrte equato 0.14 as: = P l EP l pr l S l (pr l EP l + ÃEP l ) 2 (0.14) by usg the facts that, S l l 1 Ãj p j à = P l almost surely ad à = EP l a.s. where à = l 1 Ãj We also use the followg lemma to prove our result. Lemma 4. For the values descrbed above, almost surely, the followg holds. S l pr l + à 2 = pr l P l Proof. Sce pr l s the expected umber of jobs wth processg tme less tha p that arrved before job ad à s the umber of jobs wth processg tme less tha p that arrved after r whle job s watg the queue to be processed, ( pr l + Ã) deotes the umber of jobs wth processg tme less tha p that arrved before job goes to servce. Sce ET EP l pr l 0 ET pr l EP l, t s expected that all jobs wll arrve to the system before job of type l s processed ad pr l of them s expected to be located before job, that s S l pr l + à S l pr l 2 = = pr l P l S l pr l d C 2 S l (pr l EP l + ÃEP l ) 2 = P l EP l pr l = P l EP l pr l P l EP l pr l = 0 (0.15) The, f we add all types of jobs ad multply wth c d, we get: c d B d C 1 c d k l=1 d C 0 (0.16) Now, for the jobs B2 where ET EP l pr l > 0 for a job type l, the quoted slack wll be m{( ) pr l EP l, mwat pr l EP l ET EP l pr l }. Let F deote the tme of the last job arrval to the system, such that o more jobs wll arrve after that pot. The we ca dvde the jobs B2 to four groups accordg to the quoted due dates ad actual completo tmes of the jobs. For a job problem stace, 1. D1 deotes the set of jobs where the quoted slack for a job of type l s slack = ( ) pr l EP l ad the actual completo tme for that job satsfes C F. 2. D2 s the set of jobs where the quoted slack for a job of type l s slack = mwat pr l EP l ET EP l pr l ad C F. 3. D3 s the set of jobs where the quoted slack for a job of type l s slack = mwat pr l EP l ET EP l pr l ad C F. Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

12 4. D4 deotes the set of jobs where the quoted slack for a job of type l s slack = ( ) pr l EP l ad C F. The we ca wrte: c d B d C 2 c d 4 D j d C 2 (0.17) For each of these cases, we ca wrte the followg: 1. For the jobs D 1, let S l deote the set of type l jobs D 1. The, D 1 d C k l=1 d C 2 (0.18) à = ( ) pr l à + pr l = pr l S l ( pr l + Ã) = S l pr l Substtutg to equato 0.19, we get: d C S l pr l EP l Summg over all job types: D 1 d C S l pr l EP l = 0 (0.20) k l=1 d C 0 (0.21) 2. For the jobs D 2, let S l deote the set of type l jobs D 2. The, For each job type l, we ca wrte, d C (( )pr l EP l l 1 Ãj p j) S l pr l EP l S l (pr l + Ã)EP l 2 (0.19) Sce C F for all jobs D 1, almost surely D 2 d C k l=1 d C 2 (0.22) The, for each job type l, we have d C S l ( mwat pr l EP l ET EP l pr l 2 l 1 Ãj p j) Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

13 = = = S l mwat pr l EP l (ET EP l pr l ) 2 (ET EP l pr l ) l 1 Ãj p j (ET EP l pr l ) 2 (mwat + l 1 Ãj p j) (ET EP l pr l ) 2 ET l 1 pr l EP l Ãj p j (ET EP l pr l ) 2 (mwat + l 1 Ãj p j) (ET EP l pr l ) 2 S l ETEP l à pr l EP l (ET EP l pr l ) 2 (0.23) Sce C = r + mwat + p + l 1 Ãj p j, l 1 C r p = mwat + à j p j (0.24) where C r p s the tme terval betwee the release tme ad the servce start tme of job. Also, sce C F ad à s the umber of jobs that arrved durg ths tme wth à processg tme less tha p, pr l deotes the expectato of total umber of jobs that arrved durg that tme ad ET à pr l deotes the expected legth of that tme terval. So, we ca wrte almost surely that: S l C r p 2 = S l ETà pr l 2 (0.25) The, f we combe equatos 0.24 ad 0.25 ad substtute to equato 0.23, we get: d C S l ETà pr l (ET EP l pr l ) 2 Summg over all job types: D 2 d C S l ETà pr l (ET EP l pr l ) 0 k l=1 d C 0 (0.27) 3. For the jobs D 3, let S l deote the set of type l jobs D 3. The, D 3 d C k l=1 d C 2 (0.28) For each job type l, we ca wrte, d C S l ( mwat pr l EP l ET EP l pr l Sce slack = mwat pr l EP l ET EP l pr l, l 1 Ãj p j) 2 (0.29) (0.26) Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

14 mwat pr l < ( )pr l ET EP l pr l mwat + ( )pr l EP l < ( )ET r + mwat + ( )pr l EP l r + ( )ET (0.30) Sce C F, all of the jobs wll have arrved before job completes processg. So, Ã we ca wrte, almost surely, = ( ) pr l. The, the completo tme of a job of type l s a.s.: C = r + mwat + p + l 1 Ãj p j r + mwat + ( )pr l EP l (0.31) Also, observe the followg about the value of F wth respect to the release tme of job : F = r + ( )ET (0.32) Substtutg equato 0.31 ad 0.32 to equato 0.30, we get: C F However, we also kow that, for job D 3, C F. Thus, we have the followg lemma: = Lemma 5. The umber of jobs set S l, that satsfes the above relatos, almost surely satsfes the followg: S l = 0 Proof. Sce F s a sgle tme pot, f C C F ad F holds at the same tme, the C should satsfy the followg relato C = F + K where K s a costat depedet of. So, the umber of such jobs, ca ot deped o whch S gves us the result, l = 0. Also, we kow that for ay job of type l, mwat p max ad l 1 Ãj p j p max. The we ca wrte the followg equaltes: ad S j l 1 Aj p j 2 mwat pr l EP l S j ET EP l pr l 2 S j p max pr l EP l (ET EP l pr l ) 0 (0.33) S j p max 0 (0.34) Combg these results wth equato 0.29: d C S l ( mwat pr l EP l ET EP l pr l 2 l 1 Ãj p j) = 0 (0.35) Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

15 Summg over all job types: D 3 d C k l=1 d C 0 (0.36) 4. For the jobs D 4, let S l deote the set of type l jobs D 4. The, D 4 d C k l=1 d C 2 (0.37) For each job type l, we ca wrte, d C (( )pr l EP l l 1 Ãj p j) 2 (0.38) Sce slack = ( )pr l EP l, we have that mwat pr l ET EP l pr l > ( )pr l mwat > ( )(ET pr l EP l ) (0.39) However, sce ET EP l pr l > 0, we ca say that almost surely mwat < K for some costat K <, that s, the queue legth frot of job wll almost surely be fte. The, we ca wrte the followg lemma about the umber of jobs D 4. Lemma 6. The umber of jobs set S l that satsfes the above relatos, almost surely satsfes the followg: S l = 0 Proof. Sce there exsts K < s.t. mwat < K ad also mwat > ( )(ET pr l EP l ), ( ) wll be fte, the umber of such jobs s fte. Also, we kow that for ay job of type l, l 1 Ãj p j p max. The we ca wrte the followg equatos ad j 1 Ãj p j 2 S l ( )pr l EP l 2 S l pr l EP l 0 (0.40) S l p max 0 (0.41) So, substtutg to our orgal equato 0.38: d C (( )pr l EP l l 1 Ãj p j) 0 Summg over all job types: D 4 d C k l=1 d C 0 (0.42) Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

16 Substtutg all the results 0.21, 0.27, 0.36 ad 0.42 to equato 0.17 ad combg t wth equato 0.16, we get S SL c d 2 = = S S B j d C 2 c d B1 d C 2 + c d 4 D j d C 0 (0.43) To complete ths part, we eed to cosder the tardess porto of equato Whe we cosder oly the tardy jobs, we ca wrte the followg Lemma: Lemma 7. For a seres of radomly geerated problem staces meetg the requremets descrbed above, almost surely =1 cl [C SL d ] + 0 Proof. The proof follows the proof of Lemma 3. For Lemma 3, we have already prove that =1 d C 0 Smlarly, we ca follow smlar steps but sum exclusvely over tardy jobs, rather tha over all jobs. The oe ca show that L j c L (C SL d ) 0 where L j s the set of late jobs. Fally, by combg Lemmas 2, 3 ad 7, we prove Theorem 1. Z SL Z 0 (0.44) 3. The Supply Cha Models Utlzg our aalyss of the sgle stage model dscussed the prevous secto, subsequet work we wll cosder two partes, a suppler ad a maufacturer, workg to satsfy customer orders. Customer order, = 1, 2, come to the maufacturer at arrval tme(release tme) r ad the maufacturer quotes each customer a due date d at the arrval of the order. The release tmes are ot kow advace ad they are draw from a certa probablstc dstrbuto. To complete the processg ad to delver the product to the customer, the maufacturer requres materals from the suppler, ad orders these materals from the suppler at the tme of customer order arrval. The processg tme of a order at the suppler s deoted by p 1. The, a flow shop settg, as soo as a order s completed at the suppler, t s moved to the maufacturer, where t requres p 2 tme uts to complete processg. The processg tmes p 1 ad p 2 are kow at the arrval of the order. The orders are processed both at the suppler ad the maufacturer o a sgle mache wthout preempto, ad o job ca beg processg before ts arrval tme. The materals are set to the maufacturer as soo as they are completed at the suppler ad the orders are delvered to the customers as soo as they complete processg. The objectve of ths research wll be to determe a smple ad asymptotcally optmal ole schedule ad due date quotato heurstc for the maufacturer ad the suppler to mmze the Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

17 total cost fucto =1 (cd d + c L L ) where L = (C d ) + s the tardess of job ad c d ad c L are the ut due date ad tardess costs where c l > c d. Ths research wll utlze the results developed the prevous sectos of ths paper. Our goal s to develop effectve approaches for schedulg ad due date quotato for both the cetralzed ad decetralzed versos of ths model, so that we ca vestgate the relatve advatage of a cetralzed system uder varous codtos. To do ths, we ted to focus o three models. I the Cetralzed Model, we model the system as a two faclty flow shop model. It s assumed that the maufacturer ad the suppler works as a sgle etty ad they are both cotrolled by the same aget. The decsos about the schedulg of the jobs at both facltes ad due date settg for the customer are made by ths aget. The objectve ths model s to quote the due dates so that the cost fucto =1 (cd d + c L L ) s mmzed for the whole system. It s assumed that the cetralzed case, the mea of the dstrbuto of order terarrval tmes ad the mea processg tmes at both the maufacturer ad the suppler are kow by the aget. However, the actual terarrval ad processg tmes are ot kow utl the arrval of that order to the system. I the Smple Decetralzed Model, we ted to cosder the possblty that the maufacturer ad the suppler work depedetly, ad that they both try to mmze ther ow costs. Whe the customer comes to the maufacturer ad places the order, the maufacturer mmedately quotes a due date, but the maufacturer has o kowledge of the supplers operato. He does t kow the process tme of jobs at the suppler, or the schedule of the suppler. We assume that the maufacturer oly kows the mea terarrval tme of orders ad mea processg tme of the jobs at the suppler ad at hs ow faclty. Also, he kows the umber of jobs at the suppler, sce ths quatty s equal to the umber of orders arrved to the maufacturer mus the umber of orders the suppler fshed ad set to the maufacturer. The maufacturer therefore has to quote the due dates to the customers wthout kowg the schedule at the suppler ad thus wthout kowg whe the raw materals for that order wll arrve to hm from the suppler. I ths decetralzed case, we wll cosder the problem from the maufacturer sde, sce he quotes due dates to the customer, ad we wll attempt to fd a effectve schedulg rule ad a due date quotato heurstc for the maufacturer to mmze the total cost. I the Decetralzed Model wth Addtoal Iformato Exchage, we wll cosder aother verso of the decetralzed case whch the maufacturer has more formato about the suppler, ad thus ca quote more effectve due dates. I ths case, we wll assume that the suppler also sets a due date for the completo tme of jobs at hs ste. Whe a customer order arrves, the maufacturer asks whe the suppler wll delver the raw materals to hm ad the maufacturer uses ths formato to set due dates for hs cus- Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

18 tomers. However, the maufacturer does t kow what schedule the suppler uses, ad thus the maufacturer ca oly use the due date quoted by the suppler as addtoal formato. For each of these models, we wll focus o developg effectve heurstcs for schedulg ad due date quotato for these models, ad aalytcally characterzg the effectveess of these heurstcs. I partcular, buldg o the results descrbed secto, we wll use the tools of probablstc aalyss, as well as computatoal testg, to characterze performace of these heurstcs uder varous codtos. Fally, we wll aalytcally ad computatoally compare the varous models, to assess the beeft of cooperatve due date quotato maketo-order supply chas. Refereces [1] Cheg, T.C.E. ad M.C. Gupta (1989), Survey of Schedulg Research Ivolvg Due Date Determato Decsos. Europea Joural of Operatoal Research 38, pp [2] Elo, S. ad I.G. Chowdhury (1976), Due Dates Job Shop Schedulg. Iteratoal Joural of Producto Research 14, pp [3] Weeks, J.K. (1979), A Smulato Study of Predctable Due-Dates. Maagemet Scece 25, pp [4] Myazak, S. (1981), Combed Schedulg System for Reducg Job Tardess a Job Shop. Iteratoal Joural of Producto Research 19, pp [5] K.R. Baker ad J.W.M. Bertrad (1981), A Comparso of Due-Date Selecto Rules. AIIE Trasactos 13, pp [6] Bertrad, J.W.M. (1983), The Effect of Workload Depedet Due-Dates o Job Shop Performace. Maagemet Scece 29, pp [7] Brucker, P. (1998), Schedulg Algorthms. Sprger, New York. [8] Kahlbacher, H. (1992), Term-ud Ablaufplaug - e aalytscher Zugag, Ph.D. thess, Uversty of Kaserslauter, cted [7] [9] Pawalkar, S.S., M.L. Smth, ad A. Sedma (1982), Commo Due Date Assgmet to Mmze Total Pealty for the Oe Mache Schedulg Problem. Operatos Research 30, pp [10] Hall, N.G. ad M.E. Poser (1991), Earless-Tardess Schedulg Problems, I: Weghted Devato of Completo Tmes about a Commo Due Date. Operatos Research 39, pp [11] Sedma, A., S.S. Pawalker, ad M.L. Smth (1981), Optmal Assgmet of Due Dates For a Sgle Processor Schedlug Problem. Iteratoal Joural of Producto Research 19, pp [12] Chad, S. ad D. Chhajed (1992), A Sgle Mache Model for Determato of Optmal Due Date ad Sequece. Operatos Research 40, pp Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

19 [13] Keskocak, P., R. Rav, ad S. Tayur (1997), Algorthms for Relable Lead Tme Quotato. GSIA Workg Paper, Carege Mello Uversty, Pttsburgh, PA. [14] We, L. M. (1991) Due-date Settg ad Prorty Sequecg a Multclass M/G/1 Queue. Maagemet Scece 37, pp [15] Dueyas, I. (1995), Sgle Faclty Due Date Settg wth Multple Customer Classes. Maagemet Scece 41, pp [16] Dueyas, I ad W.J. Hopp (1995), Quotg Customer Lead Tmes. Maagemet Scece 41, pp [17] Spearma, M., ad R.Q. Zhag (1999), Optmal Lead Tme Polces. Maagemet Scece 45, pp [18] Kamsky, P. ad D. Smch-Lev (2001), Probablstc Aalyss of a Ole Algorthm for the Sgle Mache Completo Tme Problem wth Release Dates. Operatos Research Letters 29 pp [19] Gazmur (1985), Probablstc Aalyss of a Mache Schedulg Problem. Mathematcs of Operatos Research 10 pp [20] Xa, C., G. Shathkumar, ad P. Gly (2000), O The Asymptotc Optmalty of The SPT Rule for The Flow Shop Average Completo Tme Problem. Operatos Research 48, pp [21] Kamsky, P. ad D. Smch-Lev (1998), Probablstc Aalyss ad Practcal Algorthms for the Flow Shop Weghted Completo Tme Problem. Operatos Research, 46, pp [22] Pedo, M. (1995), Schedulg: Theory, Algorthms ad Systems, Pretce Hall, Ic. Eglewood Clffs, New Jersey. [23] Coffma, E.G., G. N. Frederckso ad G. S. Lueker (1982), Probablstc Aalyss of the LPT Processor Schedulg Heurstc. Determstc ad Stochastc Schedulg, D. Redel Publshg Compay, M. A. H. Dempster et al. (eds.), pp [24] Loulou, R. (1984), Tght Bouds ad Probablstc Aalyss of Two Heurstcs for Parallel Processor Schedulg. Mathematcs of Operatos Research 9, pp [25] Frek, J.B.G. ad A. H. G. Rooy Ka (1987), The Asymptotc Optmalty of the LPT rule. Mathematcs of Operatos Research 12, pp [26] Spaccamela, A.M., W.S. Rhee, L. Stouge ad S. va de Geer (1992), Probablstc Aalyss of the Mmum Weghted Flowtme Schedulg Problem. Operatos Research Letters 11, pp [27] Webster, S. (1993), Bouds ad asymptotc results for the uform parallel processor weghted flow tme problem. Operatos Research Letters, 41, pp Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa

20 [28] Ramudh, A, J.J. Barthold, J. Calv, J.H. Vade Vate, ad G. Wess (1996) A Probablstc Aalyss of 2-Mache Flowshops. Operatos Research 44, pp [29] Lee, Z.H. ad P. Kamsky (2003) Ole Algorthms for Due Date Quotato wth Lateess Pealtes. Proceedgs of the 2003 NSF Desg, Servce ad Maufacturg Gratees ad Research Coferece, pp [30] Kamsky, P ad D. Hochbaum (2004) Due Date Quotato Models ad Algorthms. To be publshed as a chapter the forthcomg book Hadbook o Schedulg Algorthms, Methods ad Models, Joseph Y. Leug (ed.), Chapma Hall/CRC. [31] Smch-Lev, D., P. Kamsky ad E. Smch-Lev(2004)Maagg The Supply Cha: The Deftve Gude for the Busess Professoal, McGraw-Hll Trade: New York. [32] Kamsky, P. ad Z-H Lee (2004) Aalyss of O-le Algorthms for Due Date Quotato. Submtted for publcato. [33] Kamsky, P. (2003) The Effectveess of the Logest Delvery Tme Rule for the Flow Shop Delvery Tme Problem. Naval Research Logstcs 50(3) pp Proceedgs of 2005 NSF DMII Gratees Coferece, Scottsdale, Arzoa