Inthem-machine flow shop problem, a set of jobs, each

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1 THE ASYMPTOTIC OPTIMALITY OF THE SPT RULE FOR THE FLOW SHOP MEAN COMPLETION TIME PROBLEM PHILIP KAMINSKY Industra Engneerng and Operatons Research, Unversty of Caforna, Bereey, Caforna 9470, DAVID SIMCHI-LEVI Massachusetts Insttute of Technoogy, Cambrdge, Massachusetts , Receved August 1997; revsons receved January 1999, October 1999; accepted February 000) In the fow shop mean competon tme probem, a set of jobs has to be processed on m-machnes. Every machne has to process each one of the jobs, and every job has the same routng through the machnes. The objectve s to determne a sequence of the jobs on the machnes so as to mnmze the sum of the competon tmes of a jobs on the fna machne. In ths paper, we prove the asymptotc optmaty of the Shortest Processng Tme agorthm for any contnuous, ndependent, and dentcay dstrbuted job processng tmes. Inthem-machne fow shop probem, a set of jobs, each consstng of m operatons, must be sequentay processed on m machnes. Each machne can hande at most one job at a tme, and a job can ony be processed on one machne at a tme. The jobs have to be processed on each of the machnes wthout preempton, and every machne serves the arrvng jobs n a frst-come, frst-served fashon. Gven the processng tmes of each of the jobs on each of the machnes, the Fow Shop Mean Competon Tme Probem nvoves determnng a sequence of the jobs on the machnes that mnmzes the average or, equvaenty, the sum, of the competon tmes of the jobs on the fna machne n the sequence. It s we nown see Garey et a. 1976) that ths probem s NP-hard, even n the twomachne case. Muchprevous researchon the Fow Shop Mean Competon Tme probem has focused on optma soutons to sma-sze probems, sometmes wthas many as 10 machnes and 50 jobs but most often wth ony machnes. Some of these approaches were adapted to fnd heurstc soutons. For exampe, Koher and Stegtz 1975) combned branchand bound technques wthoca search heurstcs to, approxmatey, sove -machne probems for up to 50 jobs. For arger probems, dspatchrues are typcay used to fnd reasonabe sequences. Bhasaran and Pnedo 199) suggest a varety of smpe and compound dspatchrues usefu for arger-probem nstances. Morton and Pentco 1993) compare bottenec dynamcs and OPT-e rues, whch adjust the schedue based on the perceved bottenecs, to dspatch rue-based heurstc schedung approaches. Smuaton experments demonstrated the reatve advantage of each of the approaches, athough the absoute performance of eachapproachwas not nown. Most wor nvovng the use of dspatch rues for the fow shop mode and, ndeed, for mutpe-machne modes n genera, s expermenta n nature. Recenty, Kamnsy and Smch-Lev 1998) used a dfferent approach to anayze ths mode. Utzng the toos of probabstc anayss, they characterzed the underyng structure of the asymptotc optma souton to the Fow Shop Competon Tme Probem weghted, n that case) as the number of jobs ncreases to nfnty and deveoped a smpe agorthm for the probem based on ths anayss. Ther agorthm s a modfcaton of the Shortest Processng Tme SPT) sequence, whch s defned as foows: Sequence the jobs n ncreasng order of ther sum of the processng tmes on a of the machnes. Indeed, they show that a specfc Shortest Processng Tme sequence of the jobs s optma for a speca dscretzed verson of the probem. Ths speca dscretzed verson has many dfferent SPT sequences, however, so the queston of the asymptotc optmaty of any Shortest Processng Tme agorthm for genera fow shop probems, and n partcuar for probems n whch processng tmes are contnuous random varabes, remaned open. The dffcuty n answerng ths queston s ustrated by the foowng exampe. Consder a two-machne schedung probem wththe same number, 3, of four types of jobs. The frst type, whch we ca the 1 3 type, has processng tme of 1 on Machne 1 and 3 on Machne ; the second type, the 3 1 type, has a processng tme of 3 on Machne 1 and 1 on Machne ; and the remanng two types, 1 1 and 3 3, have a processng tme of ether 1 or 3 on both machnes. Ceary, n an SPT sequence, the 1 1 jobs w be schedued frst, and the 3 3 jobs w be schedued ast. However, any sequence of 1 3 and 3 1 jobs s an SPT sequence. Interestngy, Subject cassfcatons: Producton/schedung: mutpe machne sequencng, Fow shop weghted competon tme probem, shortest processng tme dspatch rue. Anayss of agorthms: probabstc anayss. Area of revew: Optmzaton X/01/ $ eectronc ISSN 93 Operatons Research 001 INFORMS Vo. 49, No., March Apr 001, pp

2 94 / Kamnsy and Smch-Lev Fgure 1. Comparng SPT sequences. aternatng between 1 3 and 3 1 jobs reduces tota competon tme reatve to other SPT sequences, as ustrated n Fgure 1. Bothsequences n ths Fgure are SPT, but Sequence A has an objectve vaue of, whereas Sequence B has an objectve vaue of 5. Thus, the SPT sequence can ether be very good and n fact, asymptotcay optma, as n Kamnsy and Smch-Lev 1998) or, dependng on whch SPT sequence s used, far from optma. Prevous computatona resuts see Kamnsy and Smch-Lev 1998) ndcate that the SPT rue s n fact effectve for varous randomy generated fow shop probems, partcuary as the number of jobs gets to be arge. For exampe, for 5,000 jobs wthprocessng tmes generated from a unform dstrbuton, the SPT sequence has a cost whch, on average, s between 1% and 7% arger than that of a ower bound, dependng on the number of machnes n the fow shop. Specfcay, n the case of a three-machne fow shop probem, the average ncrease n cost s about 8% when the number of jobs s 500. Our objectve n ths paper s to characterze the condtons under whch any Shortest Processng Tme sequence s asymptotcay optma, as the number of jobs tends to nfnty, for the Fow Shop Mean Competon Tme Probem, and thus provde an anaytca, rather than an expermenta, understandng of the effectveness of ths partcuar dspatch rue. Indeed, the ey assumpton n the probabstc mode descrbed n the next secton s that processng tmes are generated from a contnuous dstrbuton. Ths mpes that, wth hgh probabty, a job processng tmes are dfferent, and thus, the mode avods the dffcuty of havng to seect the correct SPT that arses n the exampe shown above. To put our wor nto perspectve, we hghght other researchreatng to probabstc anayss of schedung probems. Much of ths wor has focused on parae machne probems, ncudng the wor of Coffman et a. 198), Louou 1984), and Fren and Rnnooy Kan 1987), who anayze the parae machne schedung probem when the objectve s to mnmze the maespan, and Spaccamea et a. 199) and Webster 1993), who anayze the parae machne weghted competon tme mode. The ony wor we are aware of reated to fow shop modes s the recent wor by Ramudhn et a. 1996), who anayze the twomachne fow shop maespan mode. Fnay, Ha 1997) surveys the deveopment of agorthms wth guaranteed worst-case bounds for varous reated schedung modes. 1. THE MODEL AND THE MAIN RESULT To formay present our mode, whch s smar to the mode presented n Kamnsy and Smch-Lev 1998), consder a set of n jobs that have to be processed on m machnes. Job = 1 n, has a processng tme t on Machne = 1 m. The processng tmes are drawn from an dentca and bounded dstrbuton wth nonzero densty, defned on the nterva 0 1. Eachjob must be processed wthout preempton on each of the machnes sequentay. That s, each job must be processed on Machne 1 through Machne m n that order. Jobs are avaabe for processng at tme zero, and wththe excepton of the frst machne a other machnes process the jobs n a frst-come, frst-served manner, a so-caed permutaton schedue. Aso, there s unmted ntermedate storage between successve machnes, and we are nterested n semactve schedues, or schedues n whch no operaton on any machne can be competed earer wthout aterng the processng sequence on any of the machnes see Pnedo 1995). For the objectve we are about to descrbe, there s aways an optma semactve schedue. The objectve s to determne a schedue, or sequence of jobs, such that the tota competon tmes of a the jobs on the fna machne s mnmzed. Note that athough the processng tmes of eachof the n jobs are drawn from a random dstrbuton as descrbed above, a of the processng tmes are avaabe before the schedue s determned. We ca ths probem Probem P, and use Z to denote ts optma objectve functon vaue. That s, Z s the mnmum possbe tota competon tme of a jobs n Probem P. Smary, gven a heurstc H for the Fow Shop Competon Tme Probem, we use Z H to denote the sum of the competon tme n the resutng schedue. Specfcay, Z SPT represents the sum of competon tmes of the jobs when they are sequenced from smaest to argest tota processng tmes, an SPT sequence. In ths paper we prove the foowng. Theorem 1. Let the processng tmes t 1t tm, = 1 n, be ndependent random varabes havng the same contnuous and bounded dstrbuton wth nonzero densty defned on 0, 1]. Then wth probabty one we have Z m n = m Z SPT n Theorem 1 thus mpes that the objectve vaue of the souton generated by the SPT sequence converges to the

3 optma objectve vaue as the number of jobs tends to nfnty. Unfortunatey, the rate of convergence remans an open queston. However, the computatona resuts cted above Kamnsy and Smch-Lev 1998) provde some nsght as to the rate at whch the objectve vaue of the SPT souton approaches the optma objectve vaue. For exampe, consder the three-machne fow shop, n whch processng tmes of eachjob are generated from a unform dstrbuton. For 500 jobs, the SPT sequence has a cost whch, on average, s about 8% hgher than that of a ower bound on the optma souton. Ths decreases to 5% for 1,000 jobs, 3% for,500 jobs, and 1% for 5,000 jobs. To prove Theorem 1, we start n bypresentng a smpfed dscrete mode. For ths smpfed mode, we prove a resut anaogous to our man resut. Ths proof heps to provde some of the ntuton for the proof of Theorem 1 n 3, whch uses, among other resuts, the smpfed dscrete resut. Certan Lemmas and Propertes to whch we refer throughout the anayss are ncuded n Appendx A. For the subsequent anayss, t s usefu to defne the concept of an assocated snge-machne mode to the fow shop mode we have defned. In partcuar, consder the foowng snge-machne mode, assocated wth Probem P defned above. Gven Job = 1 n, wthprocessng tmes t 1t tm on Machne 1 m, respectvey, et t = m =1 t. Consder a snge-machne schedung probem wth n tass eachhavng a processng tme t, = 1 n. As wth the orgna fow shop probem, the objectve of the snge-machne probem s to sequence the tass so as to mnmze the sum of ther competon tmes.. THE DISCRETE MODEL To prove our man theorem, Theorem 1, we begn by ntroducng a dscrete mode, frst ntroduced n Kamnsy and Smch-Lev 1998), wth a fnte number of dfferent possbe processng tmes and a carefuy defned reatonshp between certan subsets of the jobs. The foowng anayss of ths Cycc Dscrete Mode has two purposes. It provdes some ntuton as to why Theorem 1 s true, and t provdes an upper bound that s usefu n the proof of Theorem The Mode Consder an m-machne fow shop mode for whch the objectve s to mnmze the sum of the competon tmes. Eachjob has an assocated vector t 1 t t m, where t s the processng tme on the th machne. The tota processng tme of a job s the quantty m =1 t. We say that two jobs are dentca when the vectors representng each job are equa, eement wse, and we ca a set of dentca jobs, whch can a be represented by the same vector t 1 t t m ), a job type. Gven a job type represented by t 1 t t m ), we construct a number of new jobs types through a cycc shft of the processng tmes. That s, gven the job type represented by vector t 1 t t m ), new job types are created by shftng the processng tmes over one machne n a Kamnsy and Smch-Lev / 95 cycc manner. In that process we create the foowng job types, represented by the vectors t t 3 t m t 1 t 3 t 4 t m t 1 t t m t 1 t t m 1 Of course, when some of the processng tmes t = 1 m, are equa, some of the job types n the process may be dentca. If, on the other hand, the processng tmes t = 1 m, are a dfferent, the shfted cycc process w generate m 1 new nondentca job types. To smpfy the exposton, n ths mode we w restrct ourseves to job types for whch a of the processng tmes are dfferent, athough the resuts can be qute easy generazed to ncude job types for whch two or more processng tmes are the same. We defne a group type g to be a job type, whch we ca j g 1, and ts m 1 cycc shfted job types, jg jg 3 jg m, where job type j g s shfted eft one poston from jg 1 jg 3 s shfted eft two postons from j g 1, and so on. Thus, each group type g conssts of m job types, each of whch has the same tota processng tme, t g. In addton, when we refer to the next job type wthn a group, we are referrng to the job type that s shfted one addtona tme to the eft. That s, j g s the next job type after jg 1 jg 3 s the next job after j g, and so on, notng especay that jg 1 s the next job type after jm g. Fnay, we refer to a groups wththe same tota processng tme as a famy and defne t d to equa the tota processng tme of eachjob n Famy d. Now, consder a mode n whch there s a fnte number, G, of group types and, thus, a fnte number, f,of fames. Let n g be the number of jobs of type j g, for = 1 m, and g = 1 G. Thus, a job types n a group have the same number of jobs assgned to each one of them, so n = m G g=1 n g s the tota number of jobs, out of whch mn g are assocated wth group type g. Aso, et ñ, = 1 f be the number of jobs n Famy. Ceary, there s a strong reatonshp between groups and fames. That s, ñ = m t = t n. Let Z be the optma souton to ths m machne fow shop probem, where the objectve s to mnmze tota competon tmes of a jobs. In what foows, we refer to ths probem as the orgna Cycc Dscrete probem. In the probabstc anayss that foows, we consder a Cycc Dscrete mode n whch groups of m jobs are added to the mode by seectng a group type g wthprobabty p g, for g = 1 G, and then generatng m jobs, one for each job type wthn that group. That s, wth probabty one, we have p g = m n g / G =1 n for g = 1 G. Ths mpes that the probabty that a job beongs to Famy p, equas amost surey m ñ / f j=1 ñj for = 1 f. Fnay, gven an nstance of ths Cycc Dscrete mode, defne an assocated snge-machne mode as descrbed n Secton 1, wthoptma objectve vaue Z1... The Man Dscrete Resut Consder any SPT orderng of n jobs n the Cycc Dscrete probem descrbed above. Assocated wthsuchan orderng

4 96 / Kamnsy and Smch-Lev s a K n vaue, whch we determne as foows: Startng wth the frst job n the sequence, determne ts forward match by fndng n the sequence the frst job that s the next job type, as defned n Secton.1. Contnue through the remanng jobs, notng that to fnd the forward match of job j, fnd the frst job n the sequence foowng job j that has not been the forward match of a job precedng job j and s the next job type n the same group as job j. When a job has no forward match, we defne the ast job n ts famy n the SPT sequence to be ts forward match. In partcuar, the ast job n every famy s the forward match of tsef. Consequenty, wth the possbe excepton of the ast job, a job can ony be the forward match of a snge other job. However, each job must have a forward match, regardess of whether or not t s the forward match of a precedng job. Defne the dstance between two jobs as one pus the number of jobs between these two jobs. The K n vaue assocated wtha partcuar SPT orderng of n jobs s defned as the argest dstance between any job and ts forward match. Now, et Z SPT K be the tota competon tme of a jobs n an n-job SPT orderng wthts assocated K n vaue. We note that a partcuar set of jobs may have more than one assocated SPT orderng dependng on how tes are broen) and thus may have more than one possbe Z SPT K vaue. Fnay, reca our defnton of famy, whch mpes that n any SPT orderng, a jobs wthn the same famy are sequenced consecutvey. We prove the foowng. Theorem. For any SPT sequence whose assocated K n vaue satsfes amost surey K m n n = 0 we have wth probabty one Z m n = m Z SPT K n Z1 = m mn = for some constant >0. Ths theorem mpes that regardess of how tes are broen n a partcuar SPT sequence, as ong as the K n vaue of that sequence meets the condton requred above, the sequence s asymptotcay optma. Proof of Theorem. We prove the theorem by fndng a ower bound on the asymptotc vaue of Z /n and an upper bound on Z SPT K /n whch converge to the same vaue. Frst we appy Lemma A.1 to obtan the foowng ower bound, whch aso characterzes the vaue of n Theorem. Lemma 1. We have amost surey Proof. Consder the orgna Cycc Dscrete probem and an assocated snge machne schedung probem constructed as descrbed above. In the atter mode, we have ñ j jobs eachhavng a processng tme t j, for j = 1 f. The mnmum tota competon tme of n jobs on a snge machne s obtaned usng the SPT frst rue. Let ñ j + 1ñ j G j = t j j = 1 f The optma objectve vaue of the snge machne probem s ceary f f 1 Z 1 = G j + ñ ñ t ) j=1 =1 = Dvdng by mn, tang the mt as the number of jobs, n, tends to nfnty, and notng that wth probabty one, p j = m ñ j /n j = 1 f we get wthprobabty one, m Z 1 mn = 1 m [ f j=1 t j p j + f = 1 p =1 p t ] ) Ths, together wth Lemma A.1 competes the proof. We now construct an upper bound on Z SPT K and show that asymptotcay ths upper bound converges to the asymptotc ower bound from Equaton 1). For ths purpose, consder the orgna Cycc Dscrete probem. We schedue the jobs usng an arbtrary SPT orderng, and determne the K n vaue assocated wthths orderng. To smpfy exposton, we round up K n to the nearest mutpe of m, the number of machnes. We aso ndex the jobs from 1 to n, accordng to ther appearance n the sequence. To construct an upper bound we hod that part of the processng tme of the fna job departng from Machne m statonary and shft a the other jobs on Machne m as far to the rght as possbe, wth the sequence remanng the same. In other words, we eep the startng tme of job n on Machne m the same, and begnnng wth job n 1 and gong bacwards to job 1, we ncrease the startng tmes of eachjob on machne m ony as muchas possbe, wthout overappng jobs and whe mantanng the same order. Fgure provdes a smpe two-machne exampe of ths shftng procedure. Sequence A represents the orgna sequence, whereas Sequence B s the shfted sequence. Fgure. Comparng the orgna and shfted sequences. Z SPT K m n m m n [ f = 1 m Z j=1 Z 1 mn t j p j + f = 1 p =1 p t ] 1)

5 Let Z SHIFT be the tota competon tme of a jobs n the above shfted strategy. Ceary, Z SPT K Z SHIFT 3) Note that a of the de tme on machne m n ths new shfted sequence occurs before the frst job s processed on that machne. We defne the ength of the de tme on Machne m to be I m. To construct an upper bound on Z SHIFT, consder each famy, f j, j = 1 f, and dvde the famy nto s j 1 sets of exacty K n consecutve jobs and one addtona set that contans at most K n consecutve jobs. Number the sets consecutvey wthn a famy and et Sj, j = 1 f, = 1 s j be the thset n the jthfamy. We note the foowng mportant observatons: A sets, wth the possbe excepton of the ast set wthn each famy, contan K n jobs. An upper bound on the tota processng tme of any set on Machne m s K n, because a processng tmes are bounded by 1. Any job wthn Famy j and ts m 1) forward matches that s, a job, ts forward match, th e forward match job s forward match, and so on for a tota of m jobs) have a tota processng tme on Machne m of t j. Ths s true because ths coecton of jobs must beong to one group wthn Famy j. For every famy wth s j >m, consder the tme Machne m competes processng the ast job n the set Sj, = m s j, j = 1 f accordng to the shfted sequence. Ths tme can be dvded nto three components. The frst s the de tme of Machne m; the second s the tme t taes for Machne m to process a jobs pror to the frst job n Famy j; the thrd s the tota processng tme of competed jobs wthn the same famy, Famy j. The atter s no more than K n m t j + 1 m + m 1K n Ths s true because, for each = m s j, at east + 1 mk n jobs are part of coectons of m forward matched jobs, as descrbed n the prevous pont; each coecton has a tota processng tme of t j. On the other hand, at most m 1K n jobs are not part of any coecton and, therefore, the ony thng we can say s that each one of these m 1K n jobs has a processng tme no greater than 1. To fnd an upper bound on Z SHIFT, round the competon tme of eachjob wthn a set Sj, = 1 s j, j = 1 f on Machne m up to the competon tme of the entre set on Machne m. Thus, we get that j 1 [ ] f Z SHIFT ni m Kn + ñ j m t s + 1 m + m 1K n j= =1 + fm 1 K n + K n s f j [ ] Kn m t j + 1 m + m 1K n j=1 =m Kamnsy and Smch-Lev / 97 where the frst component n the above upper bound represents tota de tme, the second represents tota processng tme unt a specfc famy s processed, the thrd s an upper bound on the sum of competon tmes on Machne m of a jobs n Sj, = 1 m 1 and j = 1 f, and the ast component n the above upper bound represents the tota processng tmes on Machne m of a jobs n Sj, = m s j and j = 1 f. Hence, j 1[ ] f Z SHIFT ni m Kn + ñ j j= =1 m t s +m 1K n s f j [ ] +fm 1 K n +K Kn n j=1=m m t j+m 1K n ni m + K j 1 f n ñ m j t s +m 1K n n ff 1 j= =1 +fm 1 K n + K s f j n t m j +m 1K n n j=1 =1 ni m + 1 j 1 f ñ m j t ñ +K n +m 1K n n ff 1 j= =1 f +fm 1 K n + K s n j +1 t m j s j +m 1K n n Dvdng by n, tang the mt as the number of jobs, n, tends to nfnty, recang the assumpton that K m n n = 0 and notng that wth probabty one, j=1 p j = m s j K n /n = m ñ j /n j = 1 f we get that wth probabty one, Z SHIFT m n m I m n + 1 m [ f j=1 t j p j + f = 1 p =1 p t ] 4) Fnay, usng Lemma 1, Equaton 3), and Equaton 4) we get that wth probabty one m Z1 m mn Z Z SPT K m n n I m m n + m Z1 5) mn Thus, the dfference between the ower and upper bounds deveoped s a functon of I m, the de tme on Machne m obtaned n the shfted strategy. We characterze ths de tme beow. Lemma. For any SPT schedue n the orgna Cycc Dscrete fow shop probem descrbed above wth an assocated K n vaue, the tota de tme on Machne m satsfes I m = OK n. Proof. The proof proceeds by nducton on the number of machnes. We begn wth the two-machne case.

6 98 / Kamnsy and Smch-Lev Lemma A., combned wththe defnton of a sem-actve schedue, tes us that I = 0 t 1 1) } t = 3n = Gven, we anayze the functon = t 1 t 1) Because the dstance between any job and ts forward matchs no more than K n, the sequence of jobs 1 3 has no more than fk n jobs, eachof whch has the property that ts forward match s the ast job n ts famy. That s, every job, except for at most fk n jobs, has a forward match that s ts next job type, as defned n Secton.1. Ths, together wth the fact that the processng tme on a machne s no more than one, mpes that t 1 ) t 1 = can never be arger than fk n. Hence, accountng now for the de tme before the frst job begns processng, the de tme on the second machne s no arger than fk n + 1. Next, we assume that I m 1 = OK n, and we anayze I m. Lemma A. tes us that I m = 0 = 3n I m 1 + = t m 1 t m 1) )} By the nducton hypothess, we now that the I m 1 = OK n. Aso, by the same argument as above, = ) t m 1 t m 1 can never be arger than fk n. Hence, accountng for the de tme on Machne m before the frst job s processed, de tme on Machne m can never be arger than fk n + m 1 + OK n = OK n. To compete the proof of Theorem, we utze Equaton 5), Lemma, and the assumpton, K m n n = 0 3. PROOF OF THE MAIN THEOREM We prove Theorem 1 by constructng a number of dscretzed versons of Probem P. We begn by dscretzng the orgna probem and removng just enough jobs to obtan a Cycc Dscrete mode, descrbed n the prevous secton. These dscretzed modes aow us to deveop an expresson for an upper bound on the asymptotc objectve vaue of the SPT orderng assocated wth an nstance of Probem P. We show that under the condton stated n Theorem 1, ths upper bound on the SPT sequence converges to a ower bound on the optma vaue of Probem P deveoped n Lemma A Dscretzaton Frst, we tae the orgna contnuous probem, Probem P, and dscretze t so there are a fnte number of possbe job types. When dscretzng the probem, however, we need to ensure that gven an nstance of Probem P and an SPT sequence, ths sequence remans an SPT orderng n the dscretzed mode. That s, suppose t and t j represent the tota processng tmes of Jobs and j n Probem P, and t t j. Aso, suppose t d and t jd represent the tota processng tmes of the dscretzed versons of Jobs and j. We must ensure that t d t jd for a and j, suchthat t t j. To do ths, we round up each of the processng tmes, usng the foowng two-step process. In Step 1, we begn by subdvdng the 0 1 nterva nto s subntervas, eachof ength. WeuseA, = 1 s, to denote the thsubnterva, that s, A = 1. For every Job n Probem P, = 1 n, and Machne, = 1 m, suchthat t A for some, = 1 s, we round ts processng tme, t, up to the vaue, and ca ths new processng tme t. Let t be equa to the sum of the rounded processng tmes of Job. Ceary, ths step s not suffcent to ensure that an SPT orderng of the dscretzed job set s the same as the orgna sequence. In Step, we utze the foowng technque to ensure that the SPT orderng remans the same. For every Job = 1 n, n the orgna Probem P, et t be the tota processng tme of the job before roundng. Next, subdvde the nterva 0m] nto ms subntervas, eachof ength, and defne B to be the thnterva, = 1 ms. That s, B = 1. For every Job n Probem P, = 1 n, suchthat t B for some = 1 ms, et t =. Ceary, t t. To mantan the SPT sequence, every job that has the same assocated tme t as defned above must have the same tota processng tme n the dscretzed probem. Defne r = t t / and for eachjob n the rounded probem created n the frst step, add an addtona to the m 1 r argest unrounded processng tmes wthn that job. That s, gven a Job, order ts processng tme n Probem P on the m machnes from the argest to smaest. Now, for the m 1 r argest processng tmes of ths job n Probem P, ncrease the correspondng processng tmes n the rounded probem by exacty. Ths process s ustrated n Fgure 3 for m =. In ths fgure, each job s represented by a pont on the graph, where the x-axs represents processng tme on Machne 1, and the y axs represents processng tme on Machne. At the end of the two-step roundng process descrbed above, a of the ponts n the shaded area are rounded up to ponts on the dotted ne, as ustrated by the arrows. The remander of the processng tme pars are rounded n a smar fashon. Thus, for every Job, the tota processng tme of the rounded verson of that job s, t d = t + m 1 r = t + m 1

7 Fgure 3. The roundng strategy. Kamnsy and Smch-Lev / 99 In the new probem, Probem P CD, we assgn exacty We ca ths new rounded probem Probem P D, whose optma objectve vaue s ZD Gven an nstance of Probem P and an SPT sequence, construct Probem P D usng the roundng technque descrbed above. Ths roundng procedure mpes that the snge) orgna SPT sequence assocated wthprobem P s aso one of possby many SPT orderngs of Probem P D. We use ZD SPT to denote the objectve vaue of ths SPT sequence when apped to Probem P D. Ceary, Z Z SPT Z SPT D 6) Because n Probem P D, processng tmes tae ony dscrete vaues, we can construct an assocated Cycc Dscrete probem caed Probem P CD. As n the prevous secton, et a job type be represented by a vector t 1 t t m ). Observe that n Probem P D, every job type has a correspondng vector whose eements t satsfy t = for every, = 1 m, and for some, = 1, s + 1. In Probem P CD, we consder ony job types from Probem P D represented by vectors that have no two equa eements. We partton the set of a job types from Probem P D wththe above property nto groups g 1 g g G and n addton create the remanng job types necessary so that a of these groups are compete. That s, each group must ncude a of the job types that are obtaned by a cycc shft of each one of the others. Ceary, each such group conssts of exacty m job types, and a of the job types wthn a snge group correspond to the job types defned n Secton.1. Let n g be the number of jobs n Probem P D whose processng tmes are represented by the thjob type of group g, = 1 m, and = 1 G. Let G m ñ = n n g j=1 =1 that s, ñ s the number of jobs n Probem P D, eachof whch has at east two machnes on whch ts processng tmes are equa. n g = mn =1m n g jobs to eachone of the job types assocated wthgroup g. Eachjob n Probem P CD has a correspondng job n Probem P D. Let ZCD be the optma souton vaue of the resutng probem, et ZCD SPT be the objectve vaue of the resutng probem when jobs are sequenced n the same order as ther correspondng jobs n the SPT sequencng of jobs n Probem P D and observe that ths probem s a Cycc Dscrete mode, as defned n Secton. We note the foowng reatonshp between Probem P CD and Probem P D. For eachjob deeted from the SPT sequencng of Probem P D to obtan Probem P CD, th e competon tme of eachsubsequent job n the sequence decreases by no more than m1 +. Ths s true because the processng tme on each machne s bounded by one. Because a tota of G m ) ñ + n g n g =1 =1 jobs are deeted, the foowng reatonshp hods: [ ] G m Z SPT CD ZSPT D nm1 + ñ + n g n g 7) =1 =1 Dvdng Equaton 7) by n, tang the mt as n goes to nfnty, and usng Equaton 6) we obtan m Z m n Z SPT CD n + m n 1 [ nm1 + ] G m ñ + n g n g =1 =1 A smar argument to the one empoyed n Kamnsy and Smch-Lev 1998) can be used to show that the second term n the above upper bound s amost surey O, and thus, amost surey, m Z ZCD SPT m + O 8) n n Because Probem P CD s a Cycc Dscrete probem, we can utze Theorem to prove the foowng Lemma. Lemma 3. Consder Probem P CD, and ts SPT orderng. We have wth probabty one ZCD m = m Z CD SPT n n Of course, to appy Theorem, the K n vaue assocated wth the specfc SPT must have the property that amost surey K m n n = 0

8 300 / Kamnsy and Smch-Lev Indeed, n Appendx B we prove the foowng resut: Lemma 4. Consder an arbtrary sequence of jobs whose processng tmes are generated accordng to Theorem 1. Order the jobs accordng to the SPT schedue and construct Probem P CD and ts assocated SPT as descrbed above. The K n vaue assocated wth ths SPT satsfes K n = on amost surey. 3.. Competng the Proof To compete the proof we utze Lemma 3 and Equaton 6) and 8) to get m Z Z SPT m n n m Z CD n + O 9) To fnd an upper bound on ZCD, reca that every nstance of the fow shop mean competon tme has an assocated snge-machne mode, as defned n 1. Startng wth probem P CD, generate a snge-machne mode, Probem P 1CD wthoptma objectve vaue Z1CD, n exacty the same way that Probem P 1 was generated from Probem P. It foows from Theorem and Lemma 4 that wth probabty one, Z1CD m mn = m ZCD 10) n Next, we reate Z1CD to Z 1, the optma souton to Probem P 1, the snge-machne mode assocated wth Probem P. For ths purpose, note that each tas n Probem P 1CD has a correspondng tas n Probem P 1 athough the opposte s not true). Furthermore, the tota processng tme of eachtas n Probem P 1CD s no more than m arger than the tota processng tme of ts correspondng tas n Probem P 1. Consequenty, Z 1CD nn + 1 Z 1 + m and ths, together wth Equatons 9) and 10), shows that amost surey: m Z Z SPT m n n m Z 1CD mn + O Z 1 + O 11) mn On the other hand, Lemma A.1 tes us that Z m n Z 1 1) mn Thus, combnng Equatons 11) and 1) and choosng sma enough show that wth probabty one, Z m n = m Z SPT n Ths competes the proof of Theorem EXTENSIONS AND CONCLUDING REMARKS To ustrate the effectveness of the SPT rue, t s mportant to pont out that n Kamnsy and Smch-Lev 1998), we consder some ndustra data that ceary do not conform to a of the parameters of ths mode. We apped the SPT rue to two-, three-, and sx-machne probems wth169, 143, and 11 jobs, respectvey. These are sma nstances so t s not surprsng that SPT does not wor as we as for arger nstances. Indeed, for the two- and sxmachne nstances, SPT yeds soutons wth cost about 40% hgher than that of a ower bound. However, as we dscuss n more deta n the paper by Kamnsy and Smch- Lev, we suspect that at east some of ths gap s attrbutabe to the weaness of the ower bound. For the three-machne nstance, SPT performs better, yedng a souton that s about 5% arger than that of the ower bound. Fnay, we note that the anayss performed n ths paper can be carred over to a more genera verson of the Fow Shop Weghted Competon Tme Probem n whch one s aowed to process jobs on dfferent machnes n dfferent sequences, a nonpermutaton schedue. In addton, the toos of probabstc modeng have ony been apped n a mted way to schedung probems. In the future, we hope to extend the nds of approaches demonstrated n ths paper to more compex schedung modes. APPENDIX A. PRELIMINARY RESULTS In ths secton we present severa Lemmas and Propertes that we refer to throughout the paper. A.1. A Lower Bound Gven Probem P as defned n 1, we defne ts assocated snge-machne probem as defned n 1. We ca ths snge-machne schedung probem Probem P 1, wthoptma souton vaue Z1, the mnmum tota competon tme of a of the tass. Ths optma souton s acheved by sequencng the tass n Shortest Processng Tme frst order see, for exampe, Pnedo 1995). Probem P and Probem P 1 are reated through the foowng ower bound, whose proof s gven n Kamnsy and Smch-Lev 1998). Lemma A.1. Consder Probem P, the genera Fow Shop Mean Competon Tme Probem, and ts assocated sngemachne schedung probem, Probem P 1. For every nstance we have 1 m Z 1 Z A.. Tota Ide Tme Consder any semactve permutaton sequence of the jobs n the m machne fow shop probem and ndex the jobs accordng to ther appearance n that sequence. Our objectve s to characterze Ij a, a = m, j = n,the tota de tme ncurred on Machne a, between the tme the frst job starts on that machne and the tme Job j departs

9 from that machne. We show Lemma A.. For every jj we have I a j 0 = t a 1 )} = 3j We note that because there s no de tme on Machne 1,.e., Ij 1 = 0 for every j, I j = 0 = 3j = = t 1 t 1 } Proof. Defne j a, a = 1 m, j = n to be equa to the de tme on Machne a between the competon of Job j 1 on Machne a and the competon of Job j on Machne a. By defnton, j I a j = a = The proof proceeds by nducton on j. Ceary, I a = 0Ia 1 + t a 1 t a 1 Assume = 0 I a = 3 = t a 1 )} for a j. We dstngushbetween two cases. Case 1. Job j starts on Machne a mmedatey after fnshng on Machne a 1. Obvousy, f t a 1 +a 1 ta j, then I a = I a j = 0 = 3 j = = t a 1 t a 1 )} because t a 1 ta j. On the other hand, f ta 1 + a 1 >ta j, then, I a = = t a 1 To see why ths s true, note that Job 1 starts processng on Machne a mmedatey after t competes on Machne a 1 and that Job j + 1 starts processng on Machne a mmedatey after t competes on Machne a 1. Thus, the tota eapsed tme on Machne a between the start of processng of Job 1 and the start of processng of Job j +1 s exacty + = t a 1 Subtractng the tme devoted to processng yeds the de tme. Hence, we need to show that = = 3j t a 1 Kamnsy and Smch-Lev / 301 = t a 1 ) For ths purpose, we dentfy the atest Job j, whose processng tme on Machne a starts after ths machne has ncurred a deay. By the nducton assumpton and the fact that no addtona de tme s ncurred after Job starts processng unt Job j competes processng, I a = 0 = 0 = 3 = 3 j = = t a 1 )} t a 1 )} Smary, because the tota tme that eapses between the start of processng of Job on Machne a and the start of processng of Job j + 1 on Machne a s =+1 a 1 + t a 1 and t s cear that a 1 =+1 and, hence, I a = = I a >I a + t a 1 t a 1 >0 = + a 1 =+1 = 3j t a 1 + t a 1 t a 1 = t a 1 ) where the second equaty foows because the second term n the addton captures the de tme that occurs after Job competes processng on Machne a, and the fna nequaty foows from the nducton assumpton. Case. Job j has to wat n front of Machne a before ts processng starts. Let be the amount of tme Job j has to wat after fnshng on Machne a 1 and before beng processed on Machne a. We consder two cases dependng on the vaue + tj a.if + ta j ta 1 + a 1, then I a = = t a 1

10 30 / Kamnsy and Smch-Lev whch, foowng the same argument as n the second part of the prevous case, mpes that I a 0 = t a 1 )} = 3 = On the other hand, f + tj a >ta 1 + a 1, then I a = I a j Agan, we dentfy the atest Job < j whose processng tme on Machne a starts after ths machne has ncurred a deay and we use a smar approachto the prevous case. Note that n ths case, there s no addtona de tme on Machne a between the tme that Job starts processng and the tme that Job j + 1 competes processng. By the nducton assumpton and the choce of Job, I a = 0 = 0 = 3 = 3j = = t a 1 )} t a 1 )} On the other hand, by comparng eapsed tmes on Machne a and a 1 as before, a 1 =+1 and, hence, I a = I a j = 0 + t a 1 t a 1 <0 = 3j = by the nducton assumpton. Hence, I a = I a = = t a 1 >Ia 1 t a 1 )} = t a 1 where the frst equaty foows from the choce of, and the second equaty foows by comparng eapsed tme on both machnes, as n the prevous case. A.3. Usefu Inequates The foowng two propertes, gven here wthout proof, are used throughout the paper. Property A.1. Booe s Inequaty See, for exampe, Rohatg 1976). Consder event E, = 1 b, for some postve nteger b. We have b ) b Pr E 1 1 PrE =1 =1 Property A.. For any numbers a b, and c such that a 0, b 0, and 0 c 1, 1 1 a1 b1 c a + b + c APPENDIX B. PROOF OF LEMMA 4 To prove the Lemma, we fnd for every n arge enough a ower bound on the probabty that K n,themum dstance between any job and ts forward match n the SPT sequence consstng of n jobs, s no more than D, where D = on. In partcuar, we show that for D = on, PrK n D < n=b where B s an arbtrary constant. Hence, by the Bore- Cante Lemma, we have amost surey K n = on. Our strategy n cacuatng the probabty PrK n D s to consder three dfferent random varabes and then combne them to fnd our bound. The frst random varabe concerns X, the mum dstance between two consecutve jobs n the same group n the SPT sequence. Property A.3. For every x we have Pr X x 1 n1 c x 13) for some constant c. Proof. Gven Job n Group g j, et X g j be the dstance between Job and the frst job n Group g j that foows t n the SPT sequence. Gven a Job wthtota processng tme arger than that of Job, and whose famy s the one assocated wthgroup g j, et p g j be the condtona probabty that ths job s a member of Group g j. Because the job processng tmes are contnuous random varabes wthnonzero denstes, there exsts a constant c suchthat c p g PrX g j for a and j. Thus, we have that x 1 1 c x Usng Property A.1 we have Pr X x = PrX g j Hence, Pr X x 1 n1 c x n G gj x j 1 1 c x j=1 =1 To ntroduce the second random varabe, empoyed ony when m 3, we brea eachgroup nto subgroups, one for eachpar of job types wthn a group. EachSubgroup r conssts of a jobs wthn that group that are a job type and ts next job type, as defned n Secton.1. Ths mpes that f there are m dfferent machnes, there w be exacty m subgroups wthn a group. Aso, each job type, and therefore eachjob, w be n two subgroups; n one t w be the frst type, and n another t w be the second type. We number the jobs wthn a subgroup consecutvey and et Y g j r represent the number of jobs n Group g j that are sequenced between Job and Job +1 n Subgroup r. Note that we are countng ony jobs n Group g j ; for ths purpose, we gnore a jobs n other groups n the famy.

11 Observe that a job types wthn a group occur wth the same probabty. Hence, the condtona probabty that a partcuar job wthn a group s n a subgroup, equas /m. Defnng Y = jr Y g j r, and usng Property A.1 exacty as we dd above, we get Property A.4. Pr Y y 1 n1 /m y 14) For the thrd random varabe, note that gven a job, say, n Subgroup r and Group g j, t may be foowed by many jobs from the two job types n r unt ts forward match arrves. Let W g j r represent the number of jobs n Subgroup r that are n between Job and ts forward match, as defned n Secton. Let W = gj rw g j r.weshow Property A.5. Pr W w 1 4Gme w 8n 15) Proof. Let W g j r = W g j r. We start by cacuatng a ower bound on the probabty Pr W g j r w For ths purpose, consder the job types n the rthsubgroup of Group g j. We refer to one type as a pus type and the other type as a mnus type. Index a the jobs n ths subgroup accordng to ther appearance n the SPT sequence. Assocated wtheachsuchjob s a random varabe V. The random varabe V equas 1 when t beongs to the pus job type, and t s equa to 1 when t beongs to the mnus job type. Let S = =1 V. It s easy to see that W g j r = n g j S The random varabe S s we understood, see Theorem.7 n Coffman and Lueer 1991). They show that ) Pr S w e 8ng w j 1n gj and smary ) Pr mn S w e w 1n gj 8ng j Hence, because n n gj,wehave Pr W g j r w 1 4e w 8n Fnay, usng Property A.1 agan, we have Pr W w = Pr rg j W g j r w 1 rg j 1 PrW g j r w 1 4Gme w 8n 16) Kamnsy and Smch-Lev / 303 To fnsh the proof, we combne the three random varabes as foows. The above upper bounds deveoped n Equatons 13), 14), and 16) mpy that for any xy, and w suchthat D = xyw we have PrK n D Pr W wpr Y ypr X x and thus, 1 Gm4e w 8n 1 n1 /m y 1 n1 c x PrK n D 1 1 Gm4e w 8n 1 n1 /m y 1 n1 c x 17) Choosng x = C 1 n 1/10 y= C n 1/10 w= 16n n n for some Constants C 1 and C, notng that for these partcuar vaues, D = on, and utzng Property A., we get PrK n D 1 1 4Gm ) n + 1 ) 1 n1 /m C n 1/ ) 1 n1 c C 1n 1/10 ) 4Gm + ) n1 /m C n 1/10 n=b n + n1 c C 1n 1/10 ) Fnay, tang the sum over a n n B 4Gm ) PrK n D + n1 /m C n 1/10 n n=b + n=b ) n1 c C 1 n 1/10 n=b Because each of the terms on the rght hand sde s fnte, the proof s compete. ACKNOWLEDGMENTS Researchn ths study was supported n part by ONR Contracts N J-1649 and N , NSF Contracts DDM-9388 and DMI , and a grant from S&C Eectrc Corporaton. REFERENCES Bhasaran, K., M. Pnedo Dspatchng. G. Savendy, ed. Handboo of Industra Engneerng. Wey, New Yor, Coffman, E. G., G. N. Fredercson, G. S. Lueer Probabstc anayss of the LPT processor schedung heurstc. M. A. H. Dempster et a., eds. Determnstc and Stochastc Schedung. D. Rede Pubshng Company,

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