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1 Ths s the Pre-Publshed Verson. Abstract In ths paper we consder the problem of schedulng obs wth equal processng tmes on a sngle batch processng machne so as to mnmze a prmary and a secondary crtera. We provde optmal polynomal algorthms for varous combnatons of the prmary and secondary crtera.

2 Bcrteron schedulng wth equal processng tmes on a batch processng machne L. L. Lu, C. T. Ng, T. C. E. Cheng Department of Logstcs, The Hong Kong Polytechnc Unversty, Hung Hom, Kowloon, Hong Kong Abstract In ths paper we consder the problem of schedulng obs wth equal processng tmes on a sngle batch processng machne so as to mnmze a prmary and a secondary crtera. We provde optmal polynomal algorthms for varous combnatons of the prmary and secondary crtera. Key words: Batch, Bcrteron schedulng, Equal processng tmes 1. Introducton In the last few years, there has been an ncreasng nterest n multcrteron schedulng problems because of ther applcaton potental. For example, decson makers may need to consder several crtera at the same tme such as customer satsfacton, on-tme delvery and work-n-process nventory. T kndt and Bllaut [18] provde some examples of practcal applcatons of multcrteron schedulng. As to bcrteron schedulng problems, two dfferent crtera are consdered together. Ths can be accomplshed n a number of ways. One approach s to mnmze the less mportant crteron, subect to the restrcton that the most mportant crteron s Correspondng author. Tel.: ; fax: E-mal address: Danel.Ng@net.polyu.edu.hk 1

3 optmzed. The two crtera are assumed to be prortzed as prmary and secondary wth the obectve of fndng the best schedule for the secondary crteron among all alternatve optmal schedules for the prmary crteron. The optmal soluton obtaned from ths approach s called a herarchcal schedule, and the problem s denoted by 1 γ / γ 1, where γ 1 s the prmary crteron and γ s the secondary crteron. The second approach s to generate all effcent (nondomnated, pareto-optmal) schedules for a problem whereby a schedule s sad to be effcent f there does not exst another schedule that s better than t on one crteron and no worse on the other. Ths problem s denoted by 1 γ 1, γ. The last approach s to use a weghtng functon to combne the two crtera. Here the decson maker assgn dfferent values of mportance to crtera γ 1 and γ. A schedulng problem wth two crtera γ 1, γ and a gven weghtng functon f s denoted by 1 f ( γ γ ) 1,. For schedulng problems, t makes sense to restrct f to the lnear f γ, γ = λ γ + λ γ, where 0 λ, λ 1 1 combnatons of varous regular crtera,.e., ( ) 1 1 1and λ + λ 1. 1 = In ths paper we study the problems of schedulng obs wth equal processng tmes on a sngle batch processng machne to mnmze a prmary and a secondary crtera, as well as to mnmze a lnear weghted crteron. A batchng machne can process several obs smultaneously. The processng tme of a batch s equal to the sum of the setup tme and the largest processng tme among all the obs n the batch. All the obs contaned n the same batch start and complete at the same tme. Once processng of a batch s ntated, t

4 can nether be nterrupted, nor can other obs be added to the batch. Ths model s motvated by the problem of schedulng burn-n operatons for very large-scale ntegrated crcut manufacturng [1]. A lot of work has been done on bcrteron schedulng problems and batch schedulng problems. In the next secton, we wll present a revew of prevous related studes. Secton descrbes the assumptons and notaton that wll be followed throughout the paper. In Secton 4 and Secton 5, we provde effcent solutons for sngle crteron schedulng problems and bcrteron schedulng problems wth varous combnatons of regular crtera. Results of ths paper are summarzed and future research drectons are suggested n Secton 6.. Prevous related work Most of the work reported about multcrteron schedulng has been concernng bcrteron schedulng problems on a sngle machne. Smth [17] may be the frst to study bcrteron schedulng problem on a sngle machne. He developed an algorthm for mnmzng the total completon tme, subect to the constrant that no ob s late. Chen and Bulfn [4] studed the problem of schedulng obs wth unt processng tmes on a sngle machne and developed algorthms for varous combnatons of crtera. Cheng [5] developed a soluton methodology for mnmzng the flow tme and mssed due dates. Surveys on algorthms and complexty results of bcrteron schedulng problems have been gven by Chen and Bulfn [], Lee and Varaktaraks [1], and Nagar et al. [14]. For parallel-machne bcrteron schedulng problems, Eck and Pnedo [6] consdered the

5 problem of searchng for the best schedule to the makespan crteron among the set of schedules that are optmal wth respect to the total completon tme. Sarn and Harharan [15] presented a heurstc to fnd the best schedule for the crteron of mnmzng the number of tardy obs under the constrant that the mum tardness s mnmzed for the two-machne case. Sarn and Prakash [16] consdered the problem of schedulng obs wth equal processng tmes on parallel dentcal machnes to mnmze a prmary and a secondary crtera, and provded optmal algorthms for some combnatons of the prmary and secondary crtera. Batch schedulng problems have ganed wde attenton of the schedulng research communty over the years. Ikura and Gmple [10], probably the frst researchers to study the problems of schedulng on a batch processng machne, provded a polynomal tme algorthm to determne whether a feasble schedule exsts for the case where release dates and due dates are agreeable and all the obs have the same processng tmes. Brucker et al. [] provded an extensve dscusson on mnmzng varous regular cost functons for the bounded model and the unbounded model wth all the obs havng the same release dates. Baptste [1] studed the problems of schedulng obs wth dfferent release tmes and equal processng tmes on a batch processng machne and showed that mnmzng the weghted number of tardy obs, the total weghted completon tme, the total tardness and the mal tardness are polynomally solvable.. Assumptons and notaton In ths paper we make assumptons about obs and the machne as follows. 4

6 There are n obs to be processed on a sngle machne, all of whch are avalable smultaneously. The dentcal processng tme s denoted by p, the weght of ob s denoted by w and the due date by d. The machne can process up to B obs at the same tme. Batchng refers to groupng the n obs nto batches, each of whch contans at most B obs. A batch s called full f t contans B obs, and a batch that s not full s called a partal batch. It s easy to see that there are at most n and at least n B batches, where r denotes the smallest nteger greater than or equal to r. The processng tme of a batch B, denoted by case under study, p p, s equal to the largest processng tme of the obs n the batch. For the = p. A schedule of a batch processng machne conssts of a batchng decson and a sequence of the batches. Gven a schedule, for each ob, we defne lateness, T { 0, } L C as ts completon tme, L = C d ts = ts tardness and U ts unt penalty, whch s defned as U = 1 f C > d and zero otherwse. The tradtonal crtera for machne schedulng nclude makespan, total completon tme, mum tardness, number of tardy obs and total tardness, denoted by C, C, T, U and T, respectvely. Snce the obs may not be equally mportant, addtve crtera may be used, whch are formed by assgnng a weght to each ob. Such crtera are denoted by w C, w U and w T, respectvely. 5

7 To be able to refer to the problems under study n a concse manner, we shall use the three-feld notaton of Graham et al. [7] to descrbe the batch processng schedulng problem, where the frst feld represents the machne envronment, the second feld represents problem characterstcs and the thrd feld denotes the crteron to be optmzed. For example, 1B, p = p w C / T represents the problem of schedulng obs wth equal processng tmes on a sngle batch processng machne to fnd the best schedule for the total weghted completon tme crteron, subect to mnmzng the mum tardness. 4. Sngle crteron schedulng problems on a sngle batch processng machne We frst develop a very useful characterzaton of a class of optmal schedules for mnmzng any regular sngle crteron and bcrteron schedulng problems. Lemma 1. For mnmzng any regular sngle crteron and bcrteron schedulng problems, there exsts an optmal schedule wth the frst k-1 batches contanng exactly B obs and the last batch contanng the remanng n-(k-1)b obs, where n B k =. Proof. Consder an optmal schedule σ = ( B, B,..., 1 B l,..., B r ) wth n B r and l < k, where batch B l s the frst batch that contans fewer than B obs. We move obs from other batches followng B l nto B l untl t s full. Snce all the obs have equal processng tmes, the completon tmes of the removed obs are decreased whle the completon tmes of the other obs do not change. Snce all the crtera we consder are regular, the 6

8 new schedule s optmal too. Repeatng ths procedure, we can obtan an optmal schedule wth the desred property. Lemma 1 shows that we may restrct our search for optmal solutons to schedules havng the property of Lemma 1. We start by dscussng the sngle crteron problems on a sngle batch processng machne wth all the obs havng equal processng tmes. The crtera we consder nclude w C, T, U, T, w U, w T, and do not nclude C and C snce any schedule wth the property of Lemma 1 s optmal for the C and C crtera Mnmzng the total weghted completon tme As for the problem of mnmzng the total weghted completon tme, we propose the followng algorthm, Full Batch Largest Weght (FBLW) frst. Algorthm FBLW Step 1. Arrange the obs n nonncreasng order of ther weghts. Step. Allocate the frst adacent unscheduled B obs as a batch and assgn them to the machne. Repeat ths step untl all the obs have been assgned. Note that except for the last one, all other batches contan exactly B obs. The tme complexty of ths algorthm s ( n n) O log. 7

9 Theorem 1. Algorthm FBLW generates an optmal schedule for the problem1 B, p = p w C. Proof. By contradcton. 4.. Mnmzng the number of tardy obs We modfy the Moore-Hodgson algorthm to mnmze the number of tardy obs and refer to t as the Modfed Moore-Hodgson (MMH) algorthm. Algorthm MMH Step 1. Arrange the obs n nondecreasng order of ther due dates. Ths sequence of obs s denoted by S and the complement of S s denoted by S. Intally, set S =φ. Step. Locate the frst tardy ob n S f we assgn the adacent B obs as a batch from the begnnng, and move t to S. Repeat ths step untl all the obs n S are early, and place ob set S after S. Step. Allocate the frst adacent unscheduled B obs as a batch and assgn them to the machne. Repeat ths step untl all obs have been assgned. Note that the tme complexty of algorthm MMH s ( n n) O log. Theorem. Algorthm MMH yelds an optmal schedule for the problem 1 B, p = p U. Proof. Let S be the sequence of obs scheduled early by algorthm MMH and S be the correspondng sequence of obs scheduled early of an optmal schedule. If necessary, we 8

10 can reorder the obs n Let ob l be the frst ob n S n EDD order and the number of early obs does not decrease. S that does not belong to S. When constructng S, ob l must be tardy and s elmnated from S. There must exst another ob h n S wth d d that h l does not belong to S, otherwse ob l wll not be tardy n S. Ths s mpossble accordng to the procedures of algorthm MMH. We delete ob h from S and ob l from S and repeat the approach. We know S s not worse than S and the result follows. 4.. Mnmzng the mum tardness and total tardness In ths secton, we provde an algorthm called Full Batch Earlest Due Date (FBEDD) to mnmze the mum tardness and total tardness. Algorthm FBEDD Step 1. Arrange the obs n nondecreasng order of ther due dates. Step. Allocate the frst adacent unscheduled B obs as a batch and assgn them to the machne. Repeat ths step untl all the obs have been assgned. Theorem. Algorthm FBEDD yelds an optmal schedule for the problems 1 B, p = p T and B, p = pt 1. Proof. Suppose there exsts an optmal schedule wth two batches P and Q, where P s processed before Q and there are two obs and such that P, Q and d > d. Let C 1 and C denote the completon tmes of batches P and Q, respectvely. We can 9

11 exchange and by movng to Q and to P. Snce all the obs have equal processng tmes, the completon tmes of the other obs wll not change after the exchange. Let T, T and T, T denote the tardness of obs and before and after the exchange, respectvely. Then T T { 0, C }, T { 0, C } = 1 d { 0, C } = d = d, 1 d, T = { 0, C }. For the mum tardness crteron, snce we have T T and T T, the mum tardness wll not ncrease after the exchange. For the total tardness crteron, snce d > d, 1 C C <, we have C1 d and C d both lyng wthn the nterval ( C1 d, C ). Moreover, ( C1 d ) ( C1 d ) = ( ) C d = ( ) C d = d d + d and x = { 0, x} + T ( T + T ) = ( ) + + ( ) + + C d C1 d - ( C1 d ) + ( C d ) T 0. s a convex functon, we have + ( ) Hence, the total tardness wll not ncrease after the exchange. The tme complexty of algorthm FBEDD s ( n n) O log Mnmzng the weghted number of tardy obs We adopt a smlar approach to treatng the problem of mnmzng the number of tardy obs to solve the problem of mnmzng the weghted number of tardy obs. The algorthm s referred to as the Weghted Modfed Moore-Hodgson (WMMH) algorthm. 10

12 Algorthm WMMH Step 1. Arrange the obs n nondecreasng order of ther due dates. Ths sequence of obs s denoted by S and the complement of S s denoted by S. Intally, set S =φ. Step. Locate the frst tardy ob l n S f we group the adacent B obs as a batch from the begnnng. Fnd ob h wth the largest possble due date among those n S up to and ncludng l that has the mnmum weght. Assgn ob h to S and repeat ths step untl all the obs n S are nontardy. Place ob set S after S. Step. Allocate the frst adacent unscheduled B obs as a batch and assgn them to the machne. Repeat ths step untl all the obs have been assgned. Theorem 4. Algorthm WMMH yelds an optmal schedule for the problem 1 B, p = p w U. Proof. Let S be the sequence of obs scheduled early by algorthm WMMH and S be the EDD order of obs scheduled early of an optmal schedule. Let ob l be the frst ob n that does not belong to S. When constructng S, ob l must be elmnated by some ob h (note that ob l may be elmnated by tself,.e., h=l). Let H be the set of obs n S between ob l and ob h (h ncluded) at the tme when ob l was elmnated. Due to the S procedures of algorthm WMMH, we have w > w for all H \ {} l. Thus, all the obs l n H must belong to S ; otherwse, replacng ob l by ob would yeld a better schedule than S. Hence, there must exst a ob e n S wth due date d d that does not belong e l to S. Otherwse, ob h n S wll be tardy, a contradcton. If w > w, we can move ob e forward and let ob l to be tardy. S wll be mproved, a contradcton. If e l w < w, at e l 11

13 the tme of ob h was tardy, ob e should be elmnated nstead of ob l. Hence, we have w e = w l. Delete ob e from S and ob l from S and repeat the procedure. We know schedule S s not worse than S and the result follows. The tme complexty of algorthm WMMH s ( n n) O log Mnmzng the total weghted tardness In order to solve the problem of mnmzng the total weghted tardness crteron, we transform ths schedulng problem nto an extended assgnment problem. The constructon s easy: n obs are assgned to k batches wth batch 1 to batch k-1 contanng B obs and batch k contanng the remanng n-(k-1)b obs and batch B completng at tme p. If ob s assgned to batch B, the cost or weghted tardness s c = w, programmng. { 0 p d }. Ths model can be formulated as the followng mathematcal Mnmze c x Subect to, { 1 n} x = 1,...,, x = B {,..., k 1} 1, ( k ) B k x = n 1 =, x { 0,1}, for all,. 1

14 The bnary varable x = 1 mples that ob s assgned to batch. Obvously, the optmal soluton of the extended assgnment problem corresponds to an optmal schedule for the problem 1 B, p p w T. Hung and Rom [9], and Kennngton and Wang [11] = proposed a polynomal tme algorthm, respectvely, whch can solve the extended assgnment problem n ( n ) O tme. We refer to ths algorthm for the extended assgnment problem as the Weghted Tardness to Assgnment Problem (WTAP). 5. Bcrteron schedulng problems on a sngle batch processng machne In ths secton we frst transform bcrteron schedulng problems, except those wth T, nto two extended assgnment problems, and for some problems, we develop even more effcent algorthms. In addton, the crtera we consder n ths secton do not nclude C and C as one crteron snce optmal schedules for the bcrteron schedulng problems 1 B, p = p g / f and 1 B, p = p f / g wth g {, C } and C f { w C, T,, 1 B, p = p f. Let 1 c and U w U, T, w T } are the same as problems c be the contrbutons to the prmary crteron and secondary crteron, respectvely, f ob s assgned to batch. The bcrteron schedulng problems can be formulated as the followng mathematcal programmng. Mnmze c x Subect to, 1

15 Mnmum c 1 x, { 1 n} x = 1,...,, x = B {,..., k 1} 1, ( k ) B k x = n 1 =, x { 0,1}, for all,. We extend Chen and Bulfn s approach [4] developed for the problem of schedulng unt processng tme obs on a sngle machne to solve our problem. The procedure to solve the above mathematcal programmng s as follows. Step 1. Solve the extended assgnment problem wth cost c 1. Let ( 1 k) v =,..., be ts optmal dual soluton. Step. Solve another extended assgnment problem wth cost u ( 1,..., n) = and c c = f c 1 = u otherwse + v. The optmal soluton of the second extended assgnment problem provdes an optmal schedule for the addtve bcrteron schedulng problem snce costs c are fnte only for optmal solutons to the frst assgnment problem. Hence, solvng the addtve bcrteron schedulng problems s equvalent to solvng two extended assgnment problems. Thus those addtve bcrteron schedulng problems can be solved n ( n ) O tme. 14

16 5.1. Problems wth total weghted completon tme as the prmary crteron To solve the problem 1B, p pt / w C, we frst apply algorthm FBLW to = generate an optmal schedule for the total weghted completon tme crteron. If there exst obs wth equal weght beng assgned to dfferent batches, apply algorthm FBEDD to those obs. Ths algorthm s referred to as the FBEDD/FBLW rule. Obvously, ths rule can yeld an optmal schedule for the problem 1B, p / n = pt w C ( n n) O log tme. Usng a smlar procedure, we can solve the other bcrteron schedulng problems wth total weghted completon tme as the prmary crteron. The problem 1 B, p p U / w C can be solved by the MMH/FBLW rule n ( n n) O log tme. = The problem 1 B, p p T / w C can be solved by the FBEDD/FBLW rule n ( n n) O log tme. = The problem 1 B, p p w U / w C can be solved by the WMMH/FBLW rule n ( n n) O log tme. = As to the problem 1 B, p p w T / w C, we frst apply algorthm FBLW. If = there are obs wth equal weght beng assgned to dfferent batches, apply algorthm WTAP to those obs. The tme complexty of ths approach s ( n ) O. 15

17 5.. Problems wth mum tardness as the prmary crteron For the problem 1B, p = p w C / T, we extend Heck and Robert s result [8] by assgnng several obs nstead of one ob each tme. Step 1. Apply algorthm FBEDD to obtan the optmal value T for the mum tardness crteron. Step. Locate n-(k-1)b obs wth the smallest weghts that satsfy kp- d T and assgn them to the last batch. Repeat ths step untl all the obs have been scheduled. Note that n the -th repetton, we select B obs wth the smallest weghts that satsfy ( k ) p d wth { 1,..., k 1} (+1)-th last batch. T and place them n the Ths algorthm yelds an optmal schedule for the problem 1B, p = p w C / T n ( n n) O log tme. For the problems = / wth γ { U, T, w U, w T } 1B, p p γ T, we frst set a deadlne for each ob and no ob can be completed after ts deadlne. The procedure for these problems s as follows. Step 1. Apply algorthm FBEDD to obtan the optmal value T for the mum tardness crteron. 16

18 Step. Defne the deadlne of ob as d + T. Apply the correspondng algorthm presented n Secton 4 to the problem 1B, p = p γ wth no ob exceedng ts deadlne. Obvously, ths algorthm yelds optmal schedules for the problems 1B, p = p γ T. The tme complexty of the problems 1B, p = p U / T, 1B, p = p T / T and 1 B, p p w U /T = s ( n n) O log, and 1B, p p w T / T = s ( n ) O. / 5.. Problems wth number of tardy obs as the prmary crteron For the bcrteron schedulng problems 1 B, p / and 1 B, p = p = p w C U w T / U, there does not exst a better algorthm than the ones for solvng the two extended assgnment problems. For the problems 1B, p = pt / U and 1 B, p = pt / U, we propose the followng algorthm EDD/MMH. Step 1. Apply algorthm MMH to generate an ntal schedule. Let S and S denote the sequences of the early obs and tardy obs, respectvely. Step. Pck the frst ob h from set S, f none exsts, stop. Otherwse, fnd the earlest batch B l n S such that the new completon tmes of obs wth the largest due date n each batch followng B l are no greater than ther due dates after ob h s 17

19 nserted nto batch B l. If no such batch exsts, set S = S {} h ; otherwse, nsert ob h nto batch B and set S = S {} h. Repeat ths step. l Theorem 5. Algorthm MMH/EDD yelds an optmal schedule for the problems 1B, p = pt / U and B, p pt / 1. = U Proof. The contrbuton of the early obs to the T and T crtera s zero. In order to mnmze the secondary crteron, accordng to Theorem, the due dates of the tardy obs should be as large as possble, and should be processed as early as possble wthout ncreasng the number of tardy obs. From the procedure of algorthm MMH, we know that t generates an optmal schedule wth the tardy obs havng the largest possble due dates. By the Step of the algorthm, the tardy obs are moved forward as much as possble n the EDD order. Ths s optmal to the T and T crtera accordng to Theorem Problems wth total tardness as the prmary crteron Snce algorthm FBEDD optmzes both the T and T crtera, the problem 1 B, p = p T / T can be solved by FBEDD n ( n n) O log tme. For the problems 1 B, p = p w C / T, 1 B, p = p U / T and 1 B, p = p w U / T, there does not exst a better algorthm than the ones for solvng the two assgnment problems. 18

20 5.5. Problems wth weghted number of tardy obs as the prmary crteron p For the bcrteron schedulng problems 1 B, p p w C / w U and 1 B, = = w T / w U, no better algorthm exsts than the ones for solvng the two assgnment problems. p For the problems 1B, p = pt / w U and B, p pt / 1, we = w U present an algorthm smlar to MMH/EDD. Ths algorthm s referred to WMMH/EDD. The dfference between these two algorthms s that WMMH/EDD apples algorthm WMMH nstead of MMH n the frst Step. Theorem 6. Algorthm WMMH/EDD yelds an optmal schedule for the problems 1B, p = pt / w U and B, p pt / 1. = w U Proof. From the proof of Theorem 4, we know that any two optmal schedules for the problem 1B, p = p w U have the same number of tardy obs and the tardy obs have the same weght. The contrbuton of the early obs to the T and T crtera s zero. Accordng to the procedure of algorthm WMMH, the tardy obs generated by algorthm WMMH have the largest possble due dates. By Step, the tardy obs n the EDD order are moved forward as much as possble wthout ncreasng the weghted number of tardy obs. Ths s optmal for the T and T crtera. 19

21 5.6. Problems wth total weghted tardness as the prmary crteron For the bcrteron schedulng problems U, w U } γ wth γ { w C, 1B, p = p / w T, no better algorthm exsts than the ones for solvng two assgnment problems. For the problem 1B, p pt / w T, we present the followng algorthm, whch = s referred to as the Bsecton Search Weghted Tardness Assgnment Problem (BSWTAP). Step 1. Apply algorthm FBEDD to obtan the optmal value LB for the problem 1B, p = pt. Apply algorthm WTAP to solve the problem 1 B, p = p w T wth S beng the optmal schedule, Y beng ts optmal value and UB beng ts mum tardness. Step. If UB LB < 1, replace S wth the new schedule and UB ts mum tardness, stop. Otherwse, solve the problem 1 B, p by WTAP algorthm = p w T wth ob havng the deadlne d ( UB + LB) w s greater than Y, set LB ( UB + LB) T set UB ( UB + LB) =, and repeat ths step. +. If the optmal value of = and repeat ths step; otherwse ( ) The tme complexty of ths algorthm s O n log( kp) where kp s the wdth of the search nterval of the bsecton search. The stoppng crteron s vald because we assume all the data to be nonnegatve ntegers. Obvously, the lower bound of ths nterval s zero 0

22 and the upper bound of ths nterval s less than the completon tme of the last batch. ( ) Thus, the complexty of ths algorthm s O n log( kp) A summary of the tme complexty obtaned for the bcrteron batch schedulng problems s gven n Table Weghted bcrteron schedulng problems The weghted bcrteron schedulng problems 1B, p = p λ 1γ 1 + λγ can be transformed nto the followng mathematcal programmng wth γ, γ { C, U, T, w } U, w T and λ,λ 1 beng nonnegatve ntegers. + 1 Mnmze ( λ c c ), 1 λ x { 1 n} x = 1,...,, {,..., k 1} x = B 1, ( k ) B k x k = n 1 =, x { 0,1}, for all,, w C, 1 where 1 c and c are the contrbutons to crtera γ 1 and γ, respectvely, f we assgn ob to batch. Ths s an extended assgnment problem and can be solved n ( n ) O tme. 1

23 6. Dscusson In ths paper we examned bcrteron schedulng problems wth equal processng tmes on a sngle batch processng machne. All the crtera often encountered n schedulng theory are consdered. We presented effcent polynomal tme algorthms for varous combnatons of crtera. Note that all the results for the sngle-machne case can be extended to the case of m parallel dentcal machnes. Three dfferent approaches from the sngle machne case need to be made clear. We let l = k m. Smlar to Lemma 1 for the sngle-machne case, there exsts a class of optmal schedules wth the frst ( l 1) m 1 k machnes processng l full batches, the k ( l 1)m th machne processng l 1 full batches and one possble partal batch comprsng n ( k 1)B obs and the other machnes processng l 1 full batches. The approach of assgnng batches to machnes s whenever a machne becomes free, the batch at the head of the lst s assgned to ths machne. The exchange of obs can be performed between the precedng and succeedng batches on the same machne and dfferent machnes. When a schedulng problem s transformed nto an extended assgnment problem, the cost of ob assgned to batch s c w m { 1 },...,k. { p d,0} = wth {,...,n} 1 and

24 Another mportant extenson would be to consder obs havng arbtrary processng tmes. Obvously, most of the bcrteron schedulng problems are strongly NP-complete even for the sngle-machne case. Future work on studyng the complexty of these problems, as well as desgnng effcent heurstc algorthms, s needed.

25 References [1] Baptste P. Batchng dentcal obs. Mathematcal Methods of Operaton Research 000;5: [] Brucker P, Gladky A, Hoogevreen H, Kovalyov MY, Potts CN, Tautenhahn T, Van de Velde S. Schedulng a batchng machne. Journal of Schedulng 1998;1: [] Chen CL, Bulfn RL. Complexty of sngle machne, mult-crtera schedulng problems. European Journal of Operatonal Research 199;70: [4] Chen CL, Bulfn RL. Schedulng unt processng tme obs on a sngle machne wth multple crtera. Computers and Operatons Research 1990;17:1-7. [5] Cheng TCE. Mnmzng the flow tme and mssed due dates n a sngle machne sequencng. Mathematcal and Computer Modelng 1990;1:71-7. [6] Eck BT, Pnedo M. On the mnmzaton of the makespan subect to flowtme optmalty. Operatons Research 199;41: [7] Graham RL, Lawler EL, Lenstra JK, Rnnooy Kan AHG. Optmzaton and approxmaton n determnstc sequencng and schedulng: a survey. Annals of Dscrete Mathematcs 1979;5:87-6. [8] Heck H, Roberts S. A note on the extenson of a result on schedulng wth a secondary crtera. Naval Research Logstcs Quarterly 197;19:40-5. [9] Hung MS, Rom WO. Solvng the assgnment problem by relaxaton. Operatons Research 1980;8: [10] Ikura Y, Gmple M. Effcent schedulng algorthms for a sngle batch processng machne. Operatons Research letters 1986;5:61-5. [11] Kennngton J, Wang Z. A shortest augmentng path algorthm for the sem-assgnment problem. Operatons Research 199;40: [1] Lee CY, Uzsoy R, Martn-Vega LA. Effcent algorthm for schedulng semconductor burnn operatons. Operaton Research 199;40: [1] Lee CY, Varaktaraks GL. Complexty of sngle machne herarchcal schedulng: A survey. Complexty n Numercal Optmzaton 199;19: [14] Nagar A, Haddock J, Heragu S. Multple and bcrtera schedulng: A lterature survey. 4

26 European Journal of Operatonal Research 1995;81: [15] Sarn SC, Harharan R. A two machne bcrtera schedulng problem. Internatonal Journal of Producton Economcs 000;65:15-9. [16] Sarn SC, Prakash D. Equal processng tme bcrtera schedulng on parallel machnes. Journal of Combnatoral Optmzaton 004;8:7-40. [17] Smth WE. Varous optmzers for sngle stage producton. Naval Research Logstcs Quarterly 1956;: [18] T kndt V, Bllaut JC. Multcrtera Schedulng: Theory, Models and Algorthms, Sprnger- Verlag: Berln, 00. 5

27 Table 1 The tme complexty of bcrteron schedulng problems Secondary Prmary C () n C ( n n) C C C w C T U T w U w T O log O ( n ) O nlog n O( nlog n) O( nlog n) O( nlog n) O ( n ) O nlog n O( nlog n) O( nlog n) O( nlog n) O ( n ) O nlog n O( nlog n) O( nlog n) O( nlog n) O ( n ) O nlog n O( nlog n) O( nlog n) O ( n ) O ( n ) O nlog n O ( n ) O( nlog n) O ( n ) O ( n ) O nlog n O ( n ) O( n log n) O( nlog n) O ( n ) np ( ) log O n O ( n ) O ( n ) O ( n ) O O( nlog n) O( nlog n) O( nlog n) O( nlog n) O( nlog n) ( n n) O log () n O O( nlog n) ( ) w O( nlog n) O( nlog n) O( nlog n) ( ) T O( nlog n) O( n log n) O( nlog n) ( ) U O( n log n) O( nlog n) ( n ) T O( nlog n) O( nlog n) ( n ) w U ( ) O nlog n O( n log n) ( n ) w T O ( n ) O ( n ) ( n ) O ( ) O ( ) O ( ) O O n B 6