Minimizing Makespan and Total Completion Time Criteria on a Single Machine with Release Dates

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1 Joural of Emergg Treds Egeerg ad Appled Sceces (JETEAS) (): Joural Scholarlk of Emergg Research Treds Isttute Egeerg Jourals, ad 200 Appled Sceces (JETEAS) (): jeteas.scholarlkresearch.org Mmzg Makespa ad Total Completo Tme Crtera o a Sgle Mache wth Release Dates E. O. Oyetuj ad 2 A. E. Oluleye Departmet of Computer Scece Uversty for Developmet Studes, Ghaa 2 Departmet of Idustral ad Producto Egeerg Uversty of Ibada, Ngera Correspodg Author: E. O. Oyetuj Abstract Ths paper cosders the schedulg problem of smultaeously mmzg the makespa ad total completo tme crtera o a sgle mache wth release dates. Sce the problem has bee characterzed as P-Hard, approxmato algorthms are desred for solvg the problem. A ew mplemetato of a exstg algorthm (GAlg) was adopted for ths schedulg problem. Ths s approprately amed algorthm. The algorthm was compared wth the Best Beta (BESTB) algorthm selected from the lterature. The two crtera were aggregated together to a lear composte objectve fucto (LCOF). Fve varats of the LCOF were utlzed. Performace evaluatos were based o both effectveess ad effcecy of the algorthms. Both ad BESTB algorthms were tested o a set of 900 radomly geerated sgle mache schedulg problems. Expermetal results show that the algorthm outperformed the BESTB algorthm uder all the fve dfferet LCOFs ad the cosdered problem szes. Keywords: crtera, algorthm, sgle mache, dmesoless, makespa, total completo tme I TRODUCTIO O the realzato of the fact that the total cost of a schedule s deed ot a fucto of oe objectve (crtero) but rather a fucto of two or more objectves (Frech, 982), research efforts have bee drected towards explorato of both bcrtera ad mult-crtera schedulg problems (Nagar et al., 995; Chakrabart et al., 996; Ste ad We, 997; Ehrgott ad Gradbleux, 2000; Hoogevee, 2005; Molar, 2005; T_kdt ad Bllaut, 2006; Petrovc et al., 2007). May of these studes dealt wth provg the NP-Hard ature of bcrtera ad mult-crtera schedulg problems as well as some exstece theorems (Chakrabart et al., 996; Ste ad We, 997; Aslam et al., 999; Rasala et al., 999). Oly a few have proposed some soluto methods usg dyamc programmg (DP), brach ad bouds (BB), evolutoary algorthms (EA) ad heurstc methods (Hoogevee ad Va de Velde, 995; Oyetuj ad Oluleye, 2008a; Oyetuj ad Oluleye (200a, 200b)). Despte all these research efforts, today several bcrtera/multcrtera schedulg problems are stll ope or ot well solved. Amogst others, the bcrtera schedulg problem of smultaeously mmzg the makespa ad total completo tme o a sgle mache wth release dates s oe of such problems. Hece, ths problem s explored ths paper. The paper s orgazed as follows: Secto covers troducto, whle lterature revew s 00 covered secto 2. Problem defto ad soluto methods are covered sectos 3 ad 4 respectvely. Data aalyss, dscusso of results ad coclusos are respectvely covered sectos 5, 6 ad 7. LITERATURE REVIEW Explorato of bcrtera ad multcrtera schedulg problems has attracted the atteto of may researchers (Ste ad We, 997; Aslam et al., 999; Rasala et al., 999; Ehrgott ad Gradbleux, 2000; Hoogevee, 2005). Aggarwal ad MeCarl (974) developed ad evaluated a cost-based composte schedulg rule. The rule takes care of four performace measures (objectves) amely: Iprocess vetory, faclty utlzato, lateess ad mea setup tme. Va Wassehove ad Gelders (980) explored the bcrtera schedulg problem of mmzg holdg cost ad maxmum tardess. They characterzed the set of effcet pots ad gave a pseudo-polyomal algorthm to eumerate all the effcet pots. Che et al. (993) studed the complexty of may bcrtera schedulg problems. The crtera cosdered are maxmal tardess, flowtme, umber of tardy jobs, tardess, ad the weghted couterparts of the last three measures. Complexty results for secodary crtero, bcrtera ad weghted crtera approaches for all combatos of measures were preseted. They also showed that of all the

2 Joural of Emergg Treds Egeerg ad Appled Sceces (JETEAS) (): problems examed oly sx rema ope. The bcrtera sgle-mache schedulg problem of mmzg total completo tme ad maxmum cost (f max ) was studed by Hoogevee ad Velde (995). The maxmum cost s defed as max j f j (C j ), where each f j deotes a arbtrary regular cost fucto for job J j. They proved that the problem s smultaeously solvable polyomal tme ad proposed two algorthms (called Algorthm I ad Algorthm II). Hoogevee (996a) explored the schedulg problem of mmzg maxmum promptess ad maxmum lateess o a sgle mache. He proposed O( 2 log ) algorthms for the varat whch dle tme s ot allowed ad for the specal case whch the objectve fucto s lear. Also, Hoogevee (996b) explored the sgle mache schedulg problem of mmzg a fucto of two or three maxmum cost crtera o a sgle mache that s cotuously avalable from tme zero oward ad that ca hadle o more tha oe job at a tme. He preseted a polyomal algorthm for the problem ad showed that these ca be used f precedece costrats exst betwee the jobs or f all pealty fuctos are o-creasg the job completo tmes. I 997, Ste ad We worked o the exstece of schedules that are ear-optmal for both makespa ad total weghted completo tme. They provded a proof to show that for ay stace of a very geeral class of schedulg problems, there exsts a schedule of makespa at most twce that of the optmal possble ad of total weghted completo tme at most twce that of the optmal. Three brach ad boud approaches (forward, backward, ad double-sded approaches) to solve bcrtera two-mache permutato flowshop problems wth the am of mmzg weghted combato of average flowtme ad makespa were developed by Svrkaya et al. (998). I 999, Say ad Karabat studed the bcrtera schedulg problem of mmzg makespa ad sum of completo tmes smultaeously a 2-mache flowshop evromet. It was show that the problem s NP-Hard. A Brach ad boud procedure to eumerate all of the effcet solutos was proposed. Improved bouds o the exstece of schedules that smultaeously optmze makespa ad average completo tme crtera was gve by Aslam et al. (999). Ther coceptual dea was based o the fact that a average completo tme schedule ca be vewed as a cotuous probablty desty fucto. They showed that, for ay schedulg problem, there exsts a (2,.582)-schedule, a (.695, 2)-schedule ad a (.806,.806)- schedule. Rasala et al. (999) studed exstece theorems, lower bouds ad algorthms for schedulg to meet two objectves. They studed the followg combato of objectves: makespa ad average weghted flow tme, maxmum flow tme ad average weghted completo 0 tme, makespa ad average completo tme, maxmum flow tme ad average flow tme. They showed that there exsts a ear-optmal schedule for both combatos of objectves (performace measures). A approxmato algorthm (BEST-β) was proposed for the combato of makespa ad total completo tme. Ths was a adaptato of the Best-Alpha (BEST-α ) approxmato algorthm of Chekur et al. (997). BEST-β s a ( β e β, e + β ) approxmato algorthm for the r (C max, C tot ) problem. Hoogevee (2005) explored a umber of dfferet combatos of the crtera. The crtera were aggregated together to a sgle scalar fucto called composte objectve fucto. He cosdered two types of the composte objectve fuctos (lear ad geeral composte objectve fuctos). Oyetuj ad Oluleye (2008a) proposed three heurstcs (HR4, HR5 ad HR6) for the b crtera problem of smultaeously mmzg the total completo tme (Ctot) ad umber of tardy jobs (NT) o sgle mache wth release dates. Also, the b crtera problem of mmzg the total completo tme (Ctot) ad umber of tardy jobs (NT) o sgle mache wth release dates was modeled as herarchcal mmzato problems by Oyetuj ad Oluleye (2008b). Two types of herarchcal mmzato models (the case of the total completo tme crtero beg more mportat tha the umber of tardy jobs crtero ad the case of the umber of tardy jobs crtero beg more mportat tha the total completo tme crtero) were explored. Oyetuj ad Oluleye (2009) proposed a methodology for evaluatg the performaces of soluto methods to bcrtera schedulg problems. A ormalzato scheme (whch coverts the values of oe crtero to the other) was also put forward. They also proposed a methodology for the determato of the maxmum ad mmum possble values of the two crtera. A ew heurstc (HR7) was proposed by Oyetuj (200a) for the b crtera problem of smultaeously mmzg the total completo tme ad umber of tardy jobs o sgle mache wth release dates. Hs expermetal results show that the HR7 outperformed a earler heurstc (HR6) for the same b crtera problem terms of both effectveess (for problems volvg 3 to 500 jobs) ad effcecy (for problems volvg less tha 30 jobs). Recetly, the techque proposed earler o for assessg the performace of soluto methods to b crtera problem by Oyetuj ad Oluleye (2009) was exteded to mxed mult-objectves schedulg problem by Oyetuj (200b). The composte objectve fucto was desged to hadle stuatos whch some of the

3 Joural of Emergg Treds Egeerg ad Appled Sceces (JETEAS) (): crtera were to be mmzed whle at the same tme some were to be maxmzed. Oyetuj ad Oluleye (200a) proposed two ew heurstcs (HR9 ad HR0) for solvg the b crtera schedulg problem of smultaeously mmzg the total completo tme ad umber of tardy jobs wth release dates o a sgle mache. HR9 ad HR0 were compared wth the HR7 heurstc proposed by Oyetuj (200a). Based o ther expermetal results, the HR7 heurstc was recommeded for the b-crtera problem volvg less tha 30 jobs whle the HR0 heurstc was recommeded for the b-crtera problems volvg 30 or more jobs. Very recetly, Oyetuj ad Oluleye (200b) proposed a geeralzed algorthm (called GAlg) for solvg multcrtera schedulg problems. They clamed that the GAlg algorthm ca be appled to wde classes of bcrtera/multcrtera schedulg problems ad t was tested o a very famlar b-crtera schedulg problem of smultaeously mmzg total completo tme ad umber of tardy jobs o a sgle mache wth release dates. Ther expermetal results show that the GAlg algorthm outperformed the selected soluto methods (HR7 ad HR0) whe the total completo tme crtero s much more mportat tha the umber of tardy jobs. To the best of our kowledge, perhaps oly Rasala et al. (999) has proposed ad approxmato algorthm for the b-crtera schedulg problem of mmzg the makespa ad total completo tme o a sgle mache wth release dates (.e. the r ( C, C ) problem). max tot The Problem Gve the followg for a sgle mache problem:. a set of jobs; J, J 2, J. processg tme (postve teger) of each job; P. release or ready tme (postve teger) of each job; r The tme at whch the processg of each jobs completes s C. The total completo tme (C tot ) s defed as: C tot = C + C C = = Whle the makespa (C max ) s defed as: C max = max( C, C 2 C,..., C ) = max( C ) The am s to mmze both the makespa ad total completo tme crtera o a sgle mache wth release tme. We assume that pre-empto s ot allowed ad that the problem s statc ad determstc.e. umber of jobs, ther processg tmes, ad ready tmes are all kow ad fxed. Usg the otatos of Graham et al. (979), the problem s represeted as r ( C max, C tot ) or r (max( C ), = C ) Soluto Methods Sce the b-crtera schedulg problem beg explored s NP-Hard, approxmato algorthm s desred order to solve the problem wth a acceptable (polyomal) tme. Oyetuj ad Oluleye (200a) proposed a geeralzed algorthm called GAlg. The algorthm was appled to the b-crtera schedulg problem of mmzg the total completo tme ad umber of tardy jobs o a sgle mache wth release dates. They cocluded that the algorthm ca be appled to wde rages/classes of b-crtera/multcrtera schedulg problems. Therefore, a ew mplemetato of the GAlg algorthm has bee adopted for the bcrtera problem beg explored ths study. Also selected for evaluato s the algorthm (BESTB) proposed by Rasala et al. (999) for the bcrtera schedulg problem of mmzg the makespa ad total completo tme o a sgle mache wth release dates. The two algorthms are dscussed below. The ew Implemetato of GAlg Algorthm The GAlg algorthm of Oyetuj ad Oluleye (200a) requred, as put, the schedules from the dvdual crtero that makes the bcrtera/multcrtera problem. A uque feature of the GAlg algorthm s ts self adaptve ature. For example, the GAlg algorthm was appled to r ( C, U ) problem by = costructg two schedules for = problem r C = (whch was solved usg the AEO algorthm of Oyetuj ad Oluleye (2007) ) ad r ( ) problem U = (whch was solved usg the EOO algorthm of Oyetuj ad Oluleye (2008c) ). I the case of the bcrtera problem at had (.e. = r (max( C ), C ) problem), we costruct two schedules by solvg the r max( C ) problem (whch ca be solved usg the NAL algorthm of Oyetuj ad Oluleye (200c) ) ad the r C = (whch ca be solved usg the AEO algorthm of Oyetuj ad Oluleye (2007) ). Oce the put schedules 02

4 Joural of Emergg Treds Egeerg ad Appled Sceces (JETEAS) (): have bee obtaed as descrbed above, the rest of the GAlg algorthm (Steps 2 6) should be carred out teratvely. Please, refer to Oyetuj ad Oluleye (200b) for detals of the GAlg algorthm. Ths ew mplemetato s approprately amed NGAlg algorthm. The BEST Beta (BESTB) Algorthm The Best Beta (BESTB) algorthm was proposed by Rasala et al. (999) for the bcrtera schedulg problem of mmzg the makespa ad total completo tme o a sgle mache wth release dates (.e. = r (max( C ), C ) problem ). The BESTB algorthm s a refemet of the Best alpha (BESTA) algorthm whch was proposed by Chekur et al. (997) for the sgle crtero problem of mmzg total completo tme o a sgle mache wth release dates (.e. problem). The BESTA algorthm frst r C = costructs a pre-emptve schedule usg the shortest remag processg tme (SRPT) rule, the costructs o-preemptve schedules by varyg the value of alpha (α = [0,]) proceedg to select the best out of the opreemptve schedules. For each value of beta (β ) wth the rage 0<β, the BESTB algorthm tres all <α β values of alpha (α ) wth the rage 0 ad the choose the best. Rasala et al. (999) proved that the BESTB algorthm s a ( β e β, e + β = ) approxmato algorthm for the r (max( C ), C ) problem. Therefore, the BESTB algorthm was selected for evaluato purposes. DATA A ALYSIS The two algorthms (NGAlg ad BESTB) were tested o a set of 900 radomly geerated sgle mache problems. Eghtee (8) problem szes (ragg from 0 to 500 jobs) were utlzed ad 50 staces per problem sze were solved. The processg tmes of the jobs were radomly geerated (usg radom umber geerator Mcrosoft vsual basc 6.0) wth values ragg betwee ad 00 clusve. Two classes of ready tmes were utlzed (0-24 ad 0-49). Ths meas that the ready tmes of the jobs were radomly geerated wth values ragg betwee 0 ad 24 (for early arrval) ad 0 ad 49 (for late arrval). Fve dfferet values of ormalzed lear composte objectve fucto were utlzed. These are: a case of the makespa crtero beg extremely more mportat tha the total completo tme crtero (.e. K= 0.0* C * max( C ) ), a case of the = makespa crtero beg more mportat tha the total completo tme crtero (.e. K 2= 0.05 * C * max( C ) ), a case of the = makespa crtero beg as mportat as the total completo tme crtero (.e. K3= 0.5 * C * max( C ) ), a case of the total = completo tme crtero beg more mportat tha the makespa crtero (.e. K 4= 0.95 * C * max( C ) ) ad a case of the = total completo tme crtero beg extremely more mportat tha the makespa crtero (.e. K5= 0.99 * C + 0.0* max( C ) ). = The algorthms were coded Mcrosoft vsual basc 6.0 ad the dfferet values of the ormalzed lear composte objectve fucto computed for each algorthm ad for each problem. The data was exported to Statstcal Aalyss System (SAS verso 9.2) for detaled aalyss. The hardware used for the expermet s a.73 GHz T2080 Itel CPU wth 024 MB of ma memory. I order to dsplay some useful statstcs (mea, meda, mmum ad maxmum) o the values of the ormalzed lear composte objectve fuctos, the box plot aalyss was carred out usg the box plot procedure SAS. Also, test of meas (t-test) was carred out usg the GLM procedure SAS. Ths s to eable us to determe whether or ot the dffereces observed the mea values of the ormalzed lear composte objectve fucto obtaed by varous soluto methods are statstcally sgfcat. The results obtaed are preseted ad dscussed the ext secto. RESULTS A D DISCUSSIO S The values of the composte objectve fuctos (K, K2, K3, K4 ad K5) obtaed usg each soluto methods for the 50 problem staces solved uder each problem sze are represeted o the box plots. The box plot has bee used to provde a better vsualzato of the dstrbuto of the value of each composte objectve fucto over the samples (50). The box plot cossts of a box wth thck horzotal le sde the box ad thck vertcal 03

5 Joural of Emergg Treds Egeerg ad Appled Sceces (JETEAS) (): les outsde the box at both the lower ad upper eds of the box. The lower ad upper edpots of the vertcal les dcates the mmum (0 th percetle) ad maxmum (00 th percetles) values (the spread) of the composte objectve fucto whle the lower ad upper eds of the box dcates the frst quartle (25 th percetles) ad thrd quartle (75 th percetles) respectvely. The horzotal thck le sde the box dcates the meda (50 th percetles) whle the symbol marker (+) dcate the mea of the dstrbuto. Whe the makespa crtero s extremely more mportat tha the total completo tme crtero (.e. composte objectve fucto ; K), the NGAlg algorthm outperformed (wth respect to effectveess.e. based o the values of K) the BESTB algorthm for all the problem szes cosdered (0 to 500 jobs) ad release dates ragg from 0 to 24 (Fgs. & 2). Smlar results (ot show) were obtaed uder composte objectve fuctos K2, K3 & K4. However, whe the total completo tme crtero s extremely more mportat tha the makespa crtero (.e. composte objectve fucto 5; K5), the BESTB algorthm outperformed (wth respect to effectveess.e. based o the values of K5) the NGAlg algorthm for 0 50 problems (Fg. 3). Whe the umber of jobs exceeds 50 ( > 50), the NGAlg algorthm outperformed the BESTB algorthm (Fg. 4). I order to determe the superorty of the NGAlg algorthm over the BESTB algorthm, the test of meas (t-test) was carred ad the results obtaed for the composte objectve fucto (K) s show Table. The mea value of the composte objectve fucto obtaed by NGAlg s sgfcatly dfferet (dcatg better performace) from that of BESTB algorthm at 5% level uder problem loadg ad whe release dates rages betwee 0 ad 24 (Table ). Smlar results (ot show) were obtaed uder composte objectve fuctos K2, K3 & K4. However, uder the composte objectve fucto 5 (K5), the mea value of K5 obtaed by NGAlg s ot sgfcatly dfferet (dcatg compettve performace) from that of BESTB algorthm at the same level, problem loadg, ad release dates values (Table 2). I order to determe the effcecy of the algorthms, the executo tme (secs) take to obta soluto to each problem stace was measured ad are show Fgs. 5 & 6. The mea value of executo tme take by the NGAlg algorthm was smaller tha that of BESTB algorthm for all the problem szes (0 to 500 jobs) cosdered (Fgs 5 & 6). It s observed that the BESTB algorthm exhbted expoetal tme-complexty fucto whe the umber of jobs exceeds 50 (Fg. 6). To determe the superorty of the NGAlg algorthm over the BESTB algorthm wth respect to effcecy, the test of meas was also carred ad the results obtaed are 04 show Tables 3 & 4. The mea value of executo tme take by NGAlg s ot sgfcatly dfferet (dcatg compettve performace wth respect to speed) from that of BESTB algorthm at 5% level for 0 jobs problem (Table 3). However, whe the umber of jobs exceeds 0, the mea value of executo tme take by NGAlg s sgfcatly dfferet (dcatg that NGAlg s faster) from that of BESTB algorthm at 5% level uder problem loadg ad whe release dates rages betwee 0 ad 24 (Table 4). C o m p o s t e O b j e c t v e F u c t o Problem Szes 0x 5x 20x 25x 30x 35x 40x 45x 50x Soluto Methods Fg. Box plot of composte objectve fucto for 0 50 problem szes ad release dates ragg from 0 24 C o m p o s te O b je c tv e F u c to Problem Szes 00x 50x 200x 250x 300x 350x 400x 450x 500x Soluto Methods Fg. 2 Box plot of composte objectve fucto for problem szes ad release dates ragg from 0 24

6 Joural of Emergg Treds Egeerg ad Appled Sceces (JETEAS) (): C o m p o s t e O b j e c t v e F u c t o Problem Szes 0x 5x 20x 25x 30x 35x 40x 45x 50x Table. Test of meas (probablty values) of ormalzed composte objectve fucto for problems Soluto Methods Soluto Methods BESTB BESTB - <0.000* <0.000* - ote * dcate sgfcat result at 5% level;sample sze = 50 - dcate ot ecessary 0.20 Soluto Methods Table 2 Test of meas (probablty values) of ormalzed composte objectve fucto 5 for problems Fg. 3 Box plot of composte objectve fucto 5 for 0 50 problem szes ad release dates ragg from 0 24 C o m p o s t e O b je c t v e F u c t o Problem Szes 00x 50x 200x 250x 300x 350x 400x 450x 500x Soluto Methods Soluto Methods BESTB BESTB - >0.x >0.x - Note x dcate o sgfcat result at 5% level; Sample sze = 50 - dcate ot ecessary 200 Problem Szes 0x 5x 20x 25x 30x 35x 40x 45x 50x Fg. 4 BestB Box NGAlgplot BestB NGAlg of composte BestB NGAlg BestB NGAlg objectve BestB NGAlg fucto BestB NGAlg BestB5 NGAlg for BestB 00 NGAlg BestB NGAlg 500 problem szes ad release dates ragg from 0 24 Soluto Methods T t m e Soluto Methods Fg. 5 Box plot of executo tme (secs) for 0 50 problem szes ad release dates ragg from 0 24

7 Joural of Emergg Treds Egeerg ad Appled Sceces (JETEAS) (): T t m e Problem Szes 00x 50x 200x 250x 300x 350x 400x 450x 500x Soluto Methods Fg. 6 Box plot of executo tme (secs) for problem szes ad release dates ragg from 0 24 Table 3 Test of meas (probablty values) of executo tme for 0 job problems Soluto Methods Soluto Methods BESTB BESTB - >0.5x >0.5x - ote x dcate o sgfcat result at 5% level; Sample sze = 50 - dcate ot ecessary Table 4 Test of meas (probablty values) of executo tme for problems Soluto Methods Soluto Methods BESTB BESTB - <0.000* <0.000* - CO CLUSIO Ths paper explored the bcrtera problem of mmzg the makespa ad total completo tme o a sgle mache wth release dates. Earler o Oyetuj ad Oluleye (200b) proposed a algorthm (called GAlg) whch they sad could be used to solve wde classes of b/multcrtera schedulg problems. The GAlg algorthm was appled to the bcrtera problem of mmzg the total completo tme ad umber of tardy jobs o a sgle mache wth release dates. The performace of the GAlg algorthm over the HR7 ad HR0 was mpressve. I vew of ths mpressve performace, a modfed verso of the GAlg algorthm has bee mplemeted for the bcrtera problem of mmzg the makespa ad total completo tme o a sgle mache wth release dates. The ew mplemetato s called NGAlg algorthm ad compared wth the Best Beta (BESTB) algorthm of Rasala et al. (999). Aga, fve dfferet composte objectve fuctos (K=a case of the makespa crtero beg extremely more mportat tha the total completo tme crtero, K2=a case of the makespa crtero beg more mportat tha the total completo tme crtero, K3=a case of the makespa crtero beg as mportat as the total completo tme crtero, K4=a case of the total completo tme crtero beg more mportat tha the makespa crtero ad K5=a case of the total completo tme crtero beg extremely more mportat tha the makespa crtero) were utlzed. Expermetal results showed that, based o effectveess ad effcecy, the NGAlg algorthm outperformed the BESTB algorthm uder the fve dfferet composte objectve fuctos. Therefore, the NGAlg algorthm s recommeded for the bcrtera schedulg problem of mmzg the makespa ad total completo tme crtero o a sgle mache wth release dates. ACK OWLEDGEME T The authors would lke to ackowledge the Assocato of Afrca Uverstes (AAU) for provdg grat for ths study uder ts 2009/200 Staff Exchage Programme. Note * dcate sgfcat result at 5% level; Sample sze = 50 - dcate ot ecessary 06

8 Joural of Emergg Treds Egeerg ad Appled Sceces (JETEAS) (): REFERE CES Aggarwal S. C. ad MeCarl Bruce A The Developmet ad Evaluato of a Cost-Based Composte Schedulg Rule, Naval Research Logstcs, 2(): Aslam J., Rasala A., Ste C. ad Youg N Improved Bcrtera Exstece Theorems for Schedulg, I proceedgs of the 0 th aual ACM-SIAM Symposum o Dscrete Algorthms, Chakrabart S., Phllps C.A., Schulz A.S., Shmoys D.B., Ste C. ad We J Improved schedulg algorthms for msum crtera. I F. Meyer auf der Hede ad B. Moe, edtors, Automata, Laguages ad Programmg, Lecture Notes Computer Scece 099, , Berl, 996. Sprger. Proceedgs of the 23rd Iteratoal Colloquum (ICALP'96). Chekur C., Motwa R., Nataraja B. ad Ste C Approxmato techques for Average Completo Tme Schedulg, I Proceedgs of the 8 th ACM-SIAM Symposum o Dscrete Algorthms, Che Chue-Lug ad Bulf R. L Complexty of sgle mache, mult-crtera schedulg problems, Europea Joural of Operatoal Research, 70(): Ehrgott M. ad Gradbleux X A Aotated Bblography of Multobjectve Combatoral Optmzato, Report Wrtschaftsmathematk, No. 62, Fachberech Mathematk - Uverstat Kaserslauter. Frech S Sequecg ad Schedulg. Ells Horwood Lmted. Hoogevee J. A. 996a. Mmzg maxmum promptess ad maxmum lateess o a sgle Mache, Mathematcs of Operatos Research, 2:00-4. Hoogevee J.A. 996b. Sgle-mache schedulg to mmze a fucto of two or three maxmum cost crtera, Joural of Algorthms, 2(2): Hoogevee J.A Multcrtera Schedulg, Europea Joural of Operatoal research, 67(3): Hoogevee J.A. ad va de Velde S.L Mmzg total completo tme ad maxmum cost smultaeously s solvable polyomal tme, Operatos Research Letters, 7: Hurkes C.A.J. ad Coster M.J O the makespa of a schedule mmzg total 07 completo tme for urelated parallel maches, Upublshed mauscrpt. Nagar A., Haddock J. ad Heragu S Multple ad bcrtera schedulg: A lterature survey. Europea Joural of Operatoal Research, 8(): Molar B Mult-crtera schedulg of order pckg processes wth smulato optmzato. Perodca Polytechca SER. TRANSP. ENG., 33( 2): Oyetuj E.O. 200a). Trucato ad Composto of Schedules: A Good Strategy for Solvg Bcrtera Schedulg Problems, Amerca Joural of Scetfc ad Idustral Research ( press). Oyetuj E.O. 200b. Assessg soluto methods to mxed mult-objectves schedulg problems, Iteratoal Joural of Idustral ad Systems Egeerg ( press). Oyetuj E.O. ad Oluleye A.E Heurstcs for mmzg total completo tme o sgle mache wth release tme. Advaced Materals Research, 8-9: Oyetuj E.O. ad Oluleye A.E. 2008a. Heurstcs for Mmzg Total Completo Tme ad Number of Tardy Jobs Smultaeously o Sgle Mache wth Release Tme, Research Joural of Appled Sceces, 3(2): Oyetuj E.O. ad Oluleye A.E. 2008b. Herarchcal Mmzato of Total Completo tme ad Number of Tardy Jobs Crtera. Asa Joural of Iformato Techology, 7(4): Oyetuj E.O. ad Oluleye A.E. 2008c. Heurstcs for mmzg umber of tardy jobs o Sgle mache wth release tme, South Afrca Joural of Idustral Egeerg, Vol. 9(2): Oyetuj E.O. ad Oluleye A.E Evaluatg Soluto Methods to Bcrtera Schedulg Problems, Advaced Materals Research, 62-64: Oyetuj E.O. ad Oluleye A.E. 200a. New Heurstcs for Mmzg Total Completo Tme ad Number of Tardy Jobs Crtera o Sgle Mache wth Release Tme, South Afrca Joural of Idustral Egeerg ( press). Oyetuj E.O. ad Oluleye A.E. 200b. A Geeralzed Algorthm for Solvg Multcrtera Schedulg Problems, I Proceedgs of 3 rd Iteratoal Coferece

9 Joural of Emergg Treds Egeerg ad Appled Sceces (JETEAS) (): o Egeerg Research ad Developmet: Advaces Egeerg, Scece & Techology held at the Uversty of Be, Be, Ngera; 7 th 9 th September, 200: Oyetuj E.O. ad Oluleye A.E. 200c. Mmzg makespa o a sgle mache wth release dates, A paper preseted at the 5 th Iter-Faculty lectures held at the Uversty for Developmet Studes, Tamale, Ghaa; 2 d 4 th September, 200. Petrovc D., Alejadra D., ad Saa P Decso support tool for mult-objectve job shop schedulg problems wth lgustcally quatfed decso fuctos. Decso Support Systems, 43(4): Rasala Aprl, Ste Clff, Torg Erc ad Uthasombut Patchrawat, 999. Exstece Theorems, Lower Bouds ad Algorthms for Schedulg to Meet Two Objectves, Techcal Report, Dartmouth College, Computer Scece Techcal Report, PCS-TR Say S. ad Karabat S A Bcrtera Approach to the 2-mache flowshop Schedulg problem, Europea Joural of Operatoal Research, 3: Svrkaya Fuda ad Ulusoy Guduz, (998), A bcrtera two-mache permutato flowshop problem, Europea Joural of Operatoal Research, 07(2), Ste C. ad We J O the Exstece of Schedules that are Near-Optmal for both Makespa ad Total Weghted Completo Tme, Operatos Research Letters, 2(3): T kdt V. ad Bllaut J. C Multcrtera Schedulg: Theory, Models ad Algorthms, 2 d Edto. Sprger, Berl. Va Wassehove L.N. ad Gelders F Solvg a bcrtero schedulg problem. Europea Joural of Operatos Research, 4: