A Branch and Bound Method for Sum of Completion Permutation Flow Shop

Transcription

1 UNLV Theses, Dssetatons, Pofessonal Paes, and Castones A Banch and Bound ethod fo Sum of Comleton Pemutaton Flow Sho Swana Kodmala Unvesty of Nevada, Las Vegas, swanakodmala@gmal.com Follow ths and addtonal woks at: htts://dgtalscholash.unlv.edu/thesesdssetatons Pat of the Comute Scences Commons Reostoy Ctaton Kodmala, Swana, "A Banch and Bound ethod fo Sum of Comleton Pemutaton Flow Sho" (04). UNLV Theses, Dssetatons, Pofessonal Paes, and Castones. 0. htts://dgtalscholash.unlv.edu/thesesdssetatons/0 Ths Thess s bought to you fo fee and oen access by Dgtal Scholash@UNLV. It has been acceted fo ncluson n UNLV Theses, Dssetatons, Pofessonal Paes, and Castones by an authozed admnstato of Dgtal Scholash@UNLV. Fo moe nfomaton, lease contact dgtalscholash@unlv.edu.

2 A BRANCH AND BOUND ETHOD FOR SU OF COPLETION PERUTATION FLOW SHOP By Swana Kodmala Bachelo of Technology, Infomaton Technology Jawahalal Nehu Technologcal Unvesty, Inda 0 A thess submtted n atal fulfllment of the equements fo the aste of Scence n Comute Scence School of Comute Scence Howad R. Hughes College of Engneeng The Gaduate College Unvesty of Nevada, Las Vegas ay 04

3 Swana Kodmala, 04 All Rghts Reseved

4 THE GRADUATE COLLEGE We ecommend the thess eaed unde ou suevson by Swana Kodmala enttled A Banch and Bound ethod fo Sum of Comleton Pemutaton Flow Sho s aoved n atal fulfllment of the equements fo the degee of aste of Scence n Comute Scence School of Comute Scence Wolfgang Ben, Ph.D., Commttee Cha Ajoy K. Datta, Ph.D., Commttee embe Laxm Gewal, Ph.D., Commttee embe Zhyong Wang, Ph.D., Gaduate College Reesentatve Kathyn Hausbeck Kogan, Ph.D., Intem Dean of the Gaduate College ay 04

5 ABSTRACT A Banch and Bound ethod fo Sum of Comleton Pemutaton Flow Sho By Swana Kodmala D. Wolfgang Ben, Examnaton Commttee Cha Pofesso of Comute Scence Unvesty of Nevada, Las Vegas We esent a new banch and bound algothm fo solvng thee machne emutaton flow sho oblem whee the otmzaton cteon s the mnmzaton of sum of comleton tmes of all the jobs. The emutaton flow sho oblem (F C ) belongs to the class of NP-had oblems; fndng the otmal soluton s thus exected to be hghly comutatonal. Fo each soluton ou scheme gves an aoxmaton ato and fnds nea otmal solutons. Comutatonal esults fo u to 0 jobs ae gven fo machne flow sho oblem when the objectve s mnmzng the sum of comleton tmes. The thess also dscusses a numbe of elated but ease flow sho oblems whee olynomal otmzaton algothms exst..

6 ACKNOWLEDGEENTS I wsh to exess my dee gattude to my advso and cha, D. Wolfgang Ben, fo hs excellent gudance, suevson and temendous suot. Hs gudance heled me all the tme and hs nvaluable suot dung my TA wok heled me to comlete my thess. I would lke to secally thank D. Laxm P. Gewal, D. Ajoy K. Datta and D. Zhyong Wang fo the wllngness to seve on my commttee. I am gateful to D. Ajoy K Datta fo hs moal suot and encouagement. I am ndebted to school of comute scence, Unvesty of Nevada Las Vegas fo ovdng the fnancal suot. Fnally I would lke to thank my aents, both elde sstes and my dea fends fo the endless suot and motvaton. They wee always wllng to hel me and encouage me wth the best wshes. v

7 TABLE OF CONTENTS ABSTRACT... ACKNOWLEDGEENTS... v TABLE OF CONTENTS... v LIST OF TABLES... v LIST OF FIGURES... x CHAPTER INTRODUCTION TO SCHEDULING.... Schedulng..... Notatons.... Classes of Schedulng..... achne Envonment (α)..... Job Chaactestcs (β) Otmalty Cteon (γ) Dsjunctve Gah odel... 6 CHAPTER FLOW SHOP SCHEDULING Flow Sho Poblem Pemutaton Flow Sho F C max and F C Fm C max.... Johnson s Algothm fo F C max Poblem....4 Johnson s Algothm fo F C Poblem... 7 CHAPTER BRANCH AND BOUND ALGORITH FOR PERUTATION FLOW SHOP Poblem Statement Usng Banch and Bound Algothm Lowe Bound Calculaton Calculaton of LB..... Calculaton of LB..... Calculaton of LB... v

8 .4 Paametes of the Algothm Algothm Illustaton Banch and Bound Fm-em C (m )... 5 CHAPTER 4 RESULTS Assumtons Paametes that Detemne Pefomance of the algothm Results fo Vaous, and Values Random, and Values Lage Values of, and Inceasng, and Deceasng Values , Inceases and then Deceasng Effcency of the Algothm CHAPTER 5 CONCLUSION AND FUTURE WORK APPENDIX BIBLIOGRAPHY... 5 VITA v

9 LIST OF TABLES. Notatons fo achne Envonment (α) Examle to Illustate Flow Sho Poblem Examle to Illustate Pemutaton Flow Sho Poblem Examle to show same job ode fo Fm C does not hold fo m Examle to Demonstate Johnson s F C max Poblem....5 Executon Stes fo Johnson s F C max oblem....6 Examle To Show Johnson s Algothm s bad fo F C Poblem Calculaton of LB.... Calculaton of LB.... Calculaton of LB....4 Basc Notatons fo Banch and Bound Algothm Examle to Demonstate Banch and Bound Calculatng Intal Feasble Schedule Calculatng Intal Ue Bound Calculaton of Comleton Tmes fo Patal Sequence [ * * *] Calculaton of LB fo Patal sequence [ * * *] Calculaton of LB fo Patal sequence [ * * *] Calculaton of LB fo Patal sequence [ * * *] Calculaton of Comleton Tmes fo Patal Sequence [ * * *] Calculaton of LB fo Patal sequence [ * * *] Calculaton of LB fo Patal sequence [ * * *] Calculaton of LB fo Patal sequence [ * * *] Calculaton of Comleton Tmes fo Patal Sequence [ * *] Calculaton of LB fo Patal sequence [ * *]....8 Calculaton of LB fo Patal sequence [ * *]....9 Calculaton of LB fo Patal sequence [ * *]....0 Calculaton of Comleton Tmes fo Patal Sequence [ 4 *].... Calculaton of LB fo Patal sequence [ 4 *].... Calculaton of LB fo Patal sequence [ 4 *].... Calculaton of LB fo Patal sequence [ 4 *]....4 Calculaton of C fo Schedule [ 4 ] Executon Stes fo F C Random, and values fo n u to C Results fo Random Values of, and Results of Banch and Bound Algothm fo n u to v

10 4.4 Lage Values of, and C Results fo Lage Values of, and Inceasng, and Deceasng Values C Results fo Inceasng, and Deceasng Values , Inceasng and then Deceasng C Results fo, Inceasng and then Deceasng v

11 LIST OF FIGURES. achne Oented Gantt Chat.... Job Oented Gantt Chat.... Gantt Chat fo Flow Sho Poblem Gantt Chat fo Pemutaton Flow Sho Poblem Schedule Reesentng Same Job Ode fo Fst k Jobs Schedule wth same job ode on last -machnes....5 Schedule wth dffeent job ode on last -machnes....6 Dsjunctve Gah fo F C max....7 Otmal Schedule fo Johnson s F C max Poblem....8 Dsjunctve Gah fo Otmal Soluton fo F C max Case (a) f j s scheduled mmedately afte Case (b) f j s scheduled mmedately afte Case(c) f j s scheduled mmedately afte Schedule fo F C based on Johnsons Algothm Gantt Chat fo Otmal Schedule F C Geneal Banch and Bound Seach Tee Enumeaton tee fo 4 jobs- machne usng banch and bound algothm Sceenshot of the Outut fo the,, Values Gven n Table Banchng Tee Showng the Hghest Level Nodes... 7 x

12 CHAPTER INTRODUCTION TO SCHEDULING. Schedulng A schedulng oblem can be descbed as follows. Gven m dentcal machnes j (j=,, m) and n jobs J (=,, n) wth ocessng tmes. A schedule s an otmal allocaton of jobs to machnes ove tme. The schedulng estctons ae a job cannot be ocessed by moe than one machne at a tme and a machne can ocess at most one job at a tme. Gantt chats ae used to gahcally eesent a schedule. Thee ae two tyes of Gantt chats, namely machne oented Gantt chats and job-oented Gantt chats. In machne oented Gantt chats X-axs eesents the tme and Y-axs eesents the machnes. In job oented Gantt chats X- axs eesents the tme and Y-axs eesents the jobs. Fgue. and Fgue. eesent the machne oented and job oented Gantt chat esectvely fo machne and 4 jobs oblem J 4 J J J J 4 J J J Fgue. achne Oented Gantt Chat J J J J 4 Fgue. Job Oented Gantt Chat.. Notatons Accodng to Pete Bucke [], the followng notatons ae used to descbe a basc schedulng oblem. J eesents the set of n jobs whee = {, n. j eesents the set of m machnes whee

13 j = {, m. Each job J has k numbe of oeatons and ae denoted as O, O,, O k. Assocated wth each oeaton s a ocessng tme denoted by j. Comleton tme of oeaton of job on machne j s denoted as c j. Comleton tme of job J s the tme taken by the job to comlete all ts oeatons and s denoted by C. In addton each job has a weght w, deadlne d and elease tme. A schedule s sad to be feasble f no two oeatons of a job ae ocessed at the same tme and a machne can ocess at most one job at a tme. A schedule s sad to be otmal f t mnmzes the otmalty ctea.. Classes of Schedulng Schedulng oblems ae defned by a thee feld notaton α β γ [] whee α descbes machne envonment β descbes job chaactestcs and γ secfes otmalty ctea.. achne Envonment (α) The machne envonment s descbed by the stng α = α α whee α {o, P, Q, R, PP, QP, G, J, O, F, X and α secfes numbe of machnes. Case : If α {o, P, Q, R, PP, QP each job J conssts of a sngle oeaton. α o Sngle machne o eesents the emty symbol. When α = o, α = α and hee only sngle machne s avalable fo ocessng the jobs. α P Identcal aallel machnes Thee ae m aallel machnes wth dentcal seeds avalable fo ocessng the jobs. The ocessng tme j of job J on machne j s, j =. α Q Unfom aallel machnes Fo ocessng the jobs thee ae m aallel machnes avalable wth each machne havng an ndvdual ocessng seed s j. The ocessng tme j of job J on machne j s, j = / s j. α R Unelated aallel machnes Fo ocessng the jobs thee ae m aallel machnes avalable wth each machne havng an ndvdual ocessng seed s j. The ocessng tme j of job J on machne j s, j = / s j.

14 α PP o QP If α = PP o QP then they ae mult-uose machnes wth dentcal seeds and unfom seeds esectvely. Case : If α {G, J, O, F, X then each job J s assocated wth a set of oeatons {O, O,,O k and each oeaton must be ocessed on a dedcated machne. α G Geneal sho In geneal sho thee s ecedence elaton between the oeatons. α J Job sho Job sho s a secal case of geneal sho. In job sho each job has a edetemned oute and the ecedence elaton between the oeatons s of the fom O O O k. Thus fo the job sho oblem, fo each machne j we need to fnd a job ode. α F Flow sho In flow sho each job J conssts of m oeatons O,O, O m and the j th oeaton of job has to be ocessed on machne j fo j tme unts. The ecedence elaton between the oeatons s, a job can stat ocessng on machne j, only afte comletng ts oeaton on machne (j-). Hee all the jobs follow the same machne ode m. Thus fo the flow sho oblem we need to fnd the job ode fo each machne. If all the machnes follow the same job ode then s called emutaton flow sho. Fo emutaton flow sho we use the notaton F em. α O Oen sho In oen sho each job J conssts of m oeatons O, O,, O m and the j th oeaton of job has to be ocessed on machne j fo j tme unts. Thee ae no ecedence elatons between the oeatons. Thus n case of oen sho we need to fnd both the job as well as machne odes. α X xed sho xed job s the combnaton of job sho and oen sho.

15 Symbol Descton Sngle machne P Identcal aallel machne Q Unfom aallel machne R Unelated aallel machne PP ult-uose machne wth dentcal seeds QP ult-uose machne wth unfom seeds G Geneal sho oblem J Job sho O Oen sho F Flow sho X xed sho Table. Notatons fo achne Envonment (α).. Job Chaactestcs (β) Job chaactestcs ae secfed by the set β {β, β, β, β 4, β 5, β 6 []. β mtn Peemton Peemton means that the ocessng of the jobs can be nteuted and can be esumed late even on othe machne. If β = mtn then eemton s allowed, othewse eemtons ae not allowed. β ec Pecedence constants Job J j cannot stat ocessng untl the job J has comleted. Ths constant on jobs s secfed usng ecedence constants. Pecedence constants ae gven by gah G = (V, A) whee each vetex coesonds to a job and each ac eesents a ecedence constant. Chans, ntee, outee, s-gah gves estcted ecedence constant between the jobs. We set β = chans f each node has atmost one edecesso and one successo. We set β = ntee f each node has atmost one successo and β = outee f each node has atmost one edecesso. Accodng to Pete Bucke [], A gah G = (V, A) s called a sees aallel gah f t conssts of a sngle vetex o f t s fomed by the aallel combnaton of two gahs G = (V, A) and G = (V, A) such that G = (V V, A A) o by the sees combnaton of two gahs 4

16 G = (V, A) and G = (V, A) such that G = (V V, A A T S). Hee T s set of snks n gah G and S s set of souces n gah G. We set β = s-gah f the gven gah s a sees aallel gah. β Release dates Release dates secfes the tme when the fst oeaton of the job J s avalable fo ocessng. If each job s assocated wth a elease tme then t s secfed by β = []. β 4 j Pocessng tmes If thee ae estctons on the ocessng tmes of the jobs then we eesent t usng β 4. If β 4 = j = then the ocessng tmes of all the jobs s. If β 4 = j = then the ocessng tmes of all the jobs s equal to. β 5 d Deadlnes Deadlne s the tme by whch the job J has to comlete ts executon. If the jobs ae subjected to deadlne constant then t s secfed by β 5 = d. Β 6 {s-batch, -batch Batch ocessng In batch oblems the jobs ae goued togethe and ae scheduled. Thee ae two tyes of batches namely s-batch and -batch. The comleton tme of jobs n the batch s equal to fnshng tme of the batch. In s-batch the fnshng tme of the batch s the sum of ocessng tmes of all the jobs n the batch and n -batch, the fnshng tme of the batch s maxmum of ocessng tmes of jobs... Otmalty Ctea (γ) Accodng to Pete Bucke [], the thd feld efes to otmalty ctea. A schedule s sad to be otmal f t mnmzes the objectve functon. c j denotes the comleton tme of oeaton of job on machne j. C denotes comleton tme of job J. Comleton tme of a job s the tme at whch the job comletes ts ocessng and exts the system. The commonly used objectves ae to mnmze the makesan o the sum of the comleton tmes of the jobs. akesan (Cmax) akesan s the maxmum of the comleton tmes of all the jobs. It s eesented as C max. C max = max {C, =,,,n 5

17 Sum of Comleton Tme ( C ) Sum of comleton tme s the summaton of the comleton tmes of all the jobs. Lateness (L ) n C = C Lateness s the dffeence between the comleton tme of a job and ts due date. It s used to detemne whethe a job s comleted befoe o afte ts due date. If lateness s ostve mles a job s comleted afte the due date and s called tadness. If lateness s negatve, t s ealness and mles that the job s cometed befoe the due date []. L = C - d Tadness (T ) Tadness occus f the job J s comleted afte ts deadlne. It s gven as, T = max {0, C - d Ealness (E ) Ealness occus f the job J s comleted befoe ts deadlne. It s gven as, E = max {0, d - C Unt Penalty (U ) If a job J s comleted afte the deadlne, then a enalty of one unt s mosed on the job. 0 C d U = { othewse Absolute Devaton (D ) D = C d Squaed Devaton (S ) S = (C d ). Dsjunctve Gah odel Dsjunctve gah dects all the feasble solutons of the sho oblems. The feasble soluton set always contans the otmal soluton. Theefoe dsjunctve gah model can be used to fnd the otmal soluton. Accodng to Pete Bucke [], fo a dsjunctve gah G (V, C, D) 6

18 V Set of vetces V s the set of vetces contanng the oeatons of all jobs. In addton to these vetces, t also contans a souce (0) and a snk (*) vetex. Weght of souce and snk ae zeo whle the weghts of all the othe nodes ae thee coesondng ocessng tmes. C Set of conjunctve acs C s the conjunctve ac set eesentng the ecedence constant between the oeatons. Addtonally conjunctve acs ae dawn between souce and all oeatons wthout a edecesso and between snk and all oeatons wthout a successo. D Set of dsjunctve acs Dsjunctve acs ae dawn between a of oeatons belongng to the same job whch ae not connected by conjunctve acs and between a of oeatons whch ae to be ocessed on the same machne and whch ae not connected by conjunctve acs. 7

19 CHAPTER FLOW SHOP SCHEDULING. Flow Sho Poblem A flow sho oblem s defned as follows. Thee ae n jobs J (J, J, J,.,J n ) and m machnes j (,,,. m ). Each job J conssts of m oeatons O,O, O m and the j th oeaton of job has to be ocessed on machne j fo j tme unts. The ecedence elaton between the oeatons s, a job can stat ocessng on machne j, only afte comletng ts oeaton on machne (j-) [5]. No two oeatons of a job ae ocessed at the same tme and a machne can ocess at most one job at a tme. In flow sho all the jobs follow the same machne ode m but the job ode fo each machne dffes. The common objectves ae to mnmze the makesan o the sum of the comleton tmes of the jobs. Thus fo the flow sho oblem, fo each machne j we need to fnd a job ode. In case of n-job m-machne flow sho oblem thee exsts (n!) m schedules and fndng an otmal schedule n that case s lkely had. Theefoe we estct ou attenton to emutaton schedules. Examle fo Flow Sho Poblem Job J J 4 J 5 Table. Examle to Illustate Flow Sho Poblem J J J J J J J J J Fgue. Gantt Chat fo Flow Sho oblem 8

20 . Pemutaton Flow Sho Pemutaton flow sho s a secal case of flow sho oblem wth an addtonal constant that the job sequence s same on all the machnes. Wth ths constant the numbe of sequences educes to (n!). Examle fo Pemutaton Flow Sho Schedulng Poblem Job J 5 J 6 Table. Examle to Illustate Pemutaton Flow Sho Poblem J J J J Fgue. Gantt Chat fo Pemutaton Flow Sho Poblem.. F C max and F C Accodng to Pete Bucke [], Fo the F C max and F C oblem thee exsts an otmal schedule n whch both the machnes ocess the jobs n the same ode. Poof: Assume an otmal schedule wth the same ode fo fst k jobs on both machnes and k<n. Let be the k th job, and let j be the job mmedately afte job on machne. Then we have the otmal schedule as follows: e h j j Fgue. Schedule Reesentng Same Job Ode fo Fst k Jobs 9

21 If we eschedule job j to the oston mmedately afte job on machne and move all jobs scheduled between job and job j by j tme unts to the ght, (we can do ths wthout nceasng the comleton tme of any job on machne ) we get anothe otmal schedule[]. We can contnue ths awse swtchng of jobs on the machne untl the job ode of machne matches wth machne [6]. Thus fo the F C max and F C oblem thee exsts an otmal schedule n whch both the machnes ocess the jobs n the same ode... Fm C max Accodng to Lemma 6.8 [], Fo oblem Fm C max otmal schedule exsts wth the followng oetes: () The job sequence on the fst two machnes s the same. () The job sequence on the last two machnes s the same. Fo two o thee machnes, the otmal soluton of the flow sho oblem s not bette than that of the coesondng emutaton flow sho. Ths s not the case f thee ae moe than thee machnes. Poof: The oof of () s smla to F C max. In case of (), fom [], f the job ode dffes on last two machnes; eschedule the jobs on machne m so that t matches wth the ode on machne m-. We contnue ths awse swtchng of jobs on the machne m untl the job ode of machne m and machne m- s dentcal. Theefoe when m the numbe of sequences educes fom (n!) m to (n!). But when m the above oety s not tue fo the sum of comleton tme of all jobs, Fm C Examle: Job J 4 J 4 Table. Examle to show same job ode fo Fm C does not hold fo m 0

22 J J J J J J Fo same job ode J -J C = C +C = =9 Fgue.4 Schedule wth same job ode on last -machnes J J J J J J Fo dffeent job ode C = C +C = 7 + =8 Fgue.5 Schedule wth dffeent job ode on last -machnes Fom Fgue.4 and Fgue.5 we see that, f we follow a dffeent job ode on the machne, we get anothe schedule whee C = C + C = 7 + =8. Theefoe the above examle makes the ont that, when m oety () does not hold fo the total comleton tme, Fm C.. Johnson s Algothm fo F C max Poblem Accodng to Pete Bucke [], Johnson s algothm fnds the otmal schedule fo F C max oblem. The algothm uses the same job ode on both the machnes. It constucts two lsts L and R, whee lst L contans jobs such that < and lst R contans jobs such that >. The otmal schedule s constucted by concatenatng T = L and R [6]. Fom the lst of unscheduled jobs dentfy the job wth the smallest ocessng tme. If the job wth smallest ocessng tme nvolves machne, then concatenate the job at the end of the lst L. If the job wth the smallest ocessng tme nvolves machne concatenate the job at the begnnng of the lst R. Then delete the job fom the lst. Ths ocess contnues on untl all jobs have been scheduled. Fnal schedule s obtaned by combnng the lsts L and R.

23 Algothm. Johnson s Algothm. Let S = {, n be the lst of unscheduled jobs. Let L, R denote two othe lsts. Fnd the job wth mnmum ocessng tme.e j. If j =, concatenate job at the end of lst L 4. Else concatenate job at the begnnng of lst R 5. Remove the job fom the lst S. 6. If thee s an unscheduled job GO TO ste 7. Else concatenate L and R Examle fo F C max : To exlan Johnson s algothm the followng 5 jobs and machnes oblem has been used. Job J 4 5 J 6 J 9 J 4 8 J Table.4 Examle to Demonstate Johnson s F C max Poblem Let S = {, n, L = {, R = { n j achne j Lst L Lst R Set of job s S j= { { {,,4,5 P j= { { {,4,5 P 4 j= { {4, {,5 P j= {, {4, {5 P 5 j= {,,5 {4, { Table.5 Executon Stes fo Johnson s F C max oblem

24 Dsjunctve Gah fo F C max O O O O 0 O O * O 4 O 4 O 5 O 5 Fgue.6 Dsjunctve Gah fo F C max Theefoe usng Johnson s algothm the otmal sequence s, T = {J, J, J 5, J 4, J J J J 5 J 4 J J J J 5 J 4 J T=J, J, J 5, J 4, J C max =8 Fgue.7 Otmal Schedule fo Johnson s F C max Poblem

25 Dsjunctve Gah of Otmal Soluton fo F C max O O O O 0 O O * O 4 O 4 O 5 O 5 Fgue.8 Dsjunctve Gah fo Otmal Soluton fo F C max Lemma.. Accodng to Lemma 6.9 [], to solve F C max oblem Johnson oosed a ule called Johnson s ule. If T s the lst constucted by the algothm then, fo any two jobs J and J j f mn {a,b j < mn{a j,b then job J s scheduled eale than job J j n the lst T. Poof: Case : If a s mn, a < mn{a j,b then a < b mles Job J belongs to lst L. If job J j s added to lst R we ae done. Othewse f Job J j goes nto L, t aeas afte J because a < a j. Case : If b j s mn, b j <mn{a j,b then b j < b mles Job J j belongs to lst R. If job J s added to lst L we ae done. Othewse f Job J goes nto R, t aeas befoe J j because b > b j. 4

26 Lemma.. Accodng to Lemma 6.0 [], Consde a schedule n whch job j s scheduled mmedately afte job, then mn{ j, mn{, j mles that and j can be swaed wthout nceasng the C max value. Poof: If j s scheduled mmedately afte, then we have thee ossble cases as shown n fgue. Let w j be the length of the tme eod fom the stat of job to the fnshng tme of job j. Then, Case : Fgue.9 Case (a) f j s scheduled mmedately afte Fo case, w j = max{ + j + j Case : Fgue.0 Case (b) f j s scheduled mmedately afte Fo case, w j = max{ + + j 5

27 Case : w j = max{x + + j Fgue. Case(c) f j s scheduled mmedately afte Fo case, w j = max{x + + j Fom case, case and case, the ossble w j s, w j = max{ + j + j, + + j, x + + j = max { j + + max {, j, x + + j Smlaly w j = max { + j + max { j,, x + j +, f s scheduled mmedately afte j Accodng to Lemma 6.0 [], we see that mn{ j, mn{, j can be wtten as, max{-,- j max{- j,- Addng,, j, j to both sdes of the above nequalty we get, + j + max{-,- j + + j j + + max{- j,- + + j = max{, j + + j max{ j, + + j = w j w j As, w j w j mles that we can swa and j wthout nceasng C max value. Theoem.. Accodng to Theoem 6. [], the lst L: L(), L(),L(n) constucted by the Johnson s algothm fo F C max oblem s otmal Poof: To ove the above theoem we use the Lemma.. and Lemma.. Assume that the lst L constucted by the Johnson s algothm was not otmal. Let us consde then, an otmal soluton S such that, S matches wth L as much as ossble n the followng way []: 6

28 L(v) = S(v) fo v =,,,,(s-). Let L(s) = and S(s) = j. In S, s not an mmedate successo of j. Let the job k be schedule between job j and job. Thus we have, L: L(), L(),,L(s-),, k, j and S: S(), S(),,S(s-), j, k, Hee all we have to show s that we can swa k and wthout nceasng C max value of S. We need to contnue swang untl S matches wth L, then we can say that the lst L constucted by Johnson s algothm s otmal. Snce k s not befoe n L, usng the Lemma.. we say that, mn{ k, mn{, k Now alyng the lemma.. to S, we can swa k and wthout nceasng the C max value. We contnue swang n S untl, S matches wth L. Thus the lst L constucted by Johnson s algothm s otmal..4 Johnson s Algothm fo F C Poblem Johnson s algothm gves abtaly bad soluton fo F C oblem. Fom [5], fo examle let us consde a two machne flow sho oblem wth n jobs. The value ϵ s consdeed vey small and value k s vey lage. Job J ϵ ϵ J ϵ ϵ J ϵ ϵ J n ϵ/ k Table.6 Examle to Show Johnson s Algothm s bad fo F C Poblem Johnson s algothm schedules the n th job fst, followed by jobs J, J, Jn. 7

29 n J J J J 4 J 5 J 6 k J J J n- n- Fgue. Schedule fo F C based on Johnsons Algothm C = C +C +C +.C n- +C n = ( ϵ/+k)+( ϵ/+k+ϵ)+( ϵ/+k+ϵ)+ +( ϵ/+k+(n-)ϵ) = n(ϵ/)+nk+(n(n-)ϵ)/ = nk+ϵ/(n+n(n-)) Theefoe the soluton constucted by ths algothm s abtaly bad as n gows. The otmal soluton fo F C oblem would schedule the n th job last J J J J 4 J 5 J J J J 4 n- n n- K Fgue. Gantt Chat fo Otmal Schedule F C C =C +C +C +.C n- +C n = (ϵ+ϵ)+(ϵ+ϵ+ϵ)+ nϵ+ (nϵ+ ϵ/+k) = (n(n+)-)( ϵ/)+k 8

30 CHAPTER BRANCH AND BOUND ALGORITH FOR PERUTATION FLOW SHOP Fom ths chate we consde only emutaton flow sho Fm-em. Ths chate s oganzed as follows: In the next secton we defne the oblem statement. In secton. we esent the banch and bound algothm; then n secton. we deve the thee ossble lowe bounds. In secton.4 we ntoduce the notatons used fo banch and bound algothm. In secton.5 we llustate the banch and bound algothm wth an examle. In secton.6 we genealze the banch and bound aoach when m.. Poblem Statement Gven a thee machne emutaton flow sho schedulng oblem F-em, and the objectve s to fnd a emutaton schedule that mnmzes the sum of the comleton tme C of all the jobs. The thee machne flow sho oblem F s defned as follows: Thee ae n Jobs J (J, J, J J n ) and machnes,,. Each job must be ocessed on the thee machnes, fst on machne, then on and then on. The ocessng tmes of job on machne j s denoted as j. The comleton tme of job on machne j s denoted as c j. The comleton tme of job J ; C, s the tme when ts last oeaton has comleted on the last machne C = C. The oblem F-em C belongs to the class of NP-had and thus fndng the otmal soluton s lkely had. We constuct a new banch and bound algothm fo solvng t. Banch and bound ntellgently enumeates emutatons of the schedule. Ths algothm s obvously an exonental algothm, but t efoms much bette n actce than the comlete enumeaton.. Usng Banch and Bound Algothm Gven a Poblem P and all feasble solutons of the oblem P ae defned by the set S, whch s called the soluton sace fo that oblem. The oblem P s dvded nto sub oblems S such that S S []. These sub oblems ae agan dvded nto smalle sub oblems. Thus banchng s a ecusve ocess and ente soluton sace s oganzed as a tee. The basc comonents needed fo banch and bound algothm ae: 9

31 Banchng Stategy: Banchng stategy dvdes the soluton sace S n to smalle and smalle sub oblems S ( =,, ) such that S =. Lowe Boundng: Then we aly an algothm to calculate the lowe bound fo each sub oblem geneated n the banchng tee. Punng Stategy: If the lowe bound of the sub oblem s geate than o equal to ue bound, then ths sub oblem cannot yeld a bette soluton and we sto banchng fom the coesondng node and all othe nodes that emege fom t n the banchng tee. The ncle of banch and bound algothm s to make an mlct seach though all feasble solutons. Banch and bound tee stats wth an ntal oot node whee no jobs have been scheduled. Then we ty to banch n ths tee by tyng to fx each of the jobs as the fst job n the sequence. The ossble banches ae n snce thee ae n jobs. Each of these n nodes emanate n to (n-) banches as thee ae (n-) ossble jobs that can occuy the second lace n the sequence. Thus ths s a ecusve ocess. In banch and bound algothm, each node eesents a atal schedule whee k jobs ae scheduled n fxed ode. Banchng fom a node conssts of takng each of the unallocated jobs n tun and lacng t next to the atal schedule. Each of these new atal schedules s then eesented by a new node. The lowe bound values fo each node ae then calculated. Fgue. Geneal Banch and Bound Seach Tee 0

32 . Lowe Bound Calculaton: Each node n the seach tee contans a set of jobs k that ae aleady scheduled and set of jobs that need to be scheduled. Let J (J, J, J,,J n ) eesents the set of jobs. Suose we ae at a node at whch the jobs n the set {, k ae aleady scheduled n that ode; =. Let U {+, +,,n eesents the set of unscheduled jobs. Sum of the comleton tmes fo ths schedule can be dvded nto, C C S (.) Comutng the second sum s vey dffcult, theefoe we estmate ts lowe bound based on the followng assumtons:.. Calculaton of LB :. Evey job stats ocessng on machne wthout any delay tme. That s, afte the fst job fnshes ts ocessng on machne, the followng job stats mmedately wthout any watng tme. LB = n k k k k k n ] ) ( [ Consde the jobs +, + n comletes ts ocessng wthout any delay on machne.,,, +, +, k, n,, +, +, k, n,, +, +, k, n, Fgue. Calculaton of LB LB = C + + C + + C C n C C... k k k k C

33 n n n k n C Theefoe, LB = n k k k k k n ] ) ( [ Fo LB schedule the jobs n U n nceasng ode of values... Calculaton of LB :. Evey job stats ocessng on machne wthout any delay tme. That s, afte the fst job fnshes ts ocessng on machne, the followng job stats mmedately wthout any watng tme. The exesson mn, max{ C s a lowe bound on the stat of fst job on machne. LB = n k k k k n C ] ) ( mn, [max{ Consde the jobs +, + n comletes ts ocessng wthout any delay on machne.,,, +, +, k, n,,, +, +, k, n,, +, +, k, n, Fgue. Calculaton of LB LB = C + + C + + C C n mn, max{ C C mn, max{ C C... mn, max{ k k k C C

34 mn, max{ k n k n C C Theefoe, LB = n k k k k n C ] ) ( mn, [max{ Fo LB schedule the jobs n U n nceasng ode of values... Calculaton of LB :. Evey job stats ocessng on machne wthout any watng tme. That s, afte the fst job fnshes ts ocessng on machne, the followng job stats mmedately wthout any watng tme. The exesson mn mn, max{, max{ C C s a lowe bound on the stat of fst job on machne. ] ) ( mn mn, max{, max{ [ k n k k n C C LB Consde the jobs +, + n comletes ts ocessng wthout any delay on machne.,,, +, +, k, n,,, +, +, k, n,,, +, +, k, n, Fgue. Calculaton of LB LB = C + + C + + C C n mn mn, max{, max{ C C C mn mn, max{, max{ C C C... mn mn, max{, max{ k k C C C mn mn, max{, max{ n k n C C C

35 n Theefoe, LB k k [ max{ C, max{ C, mn mn ( n k ) ] Fo LB schedule the jobs n U n nceasng ode of values. Theefoe the lowe bound s max (LB, LB, LB ) Fom (.), we obtan, S = S C C C max (LB, LB, LB ) s the cost of the schedule..4 Paametes of the Algothm Inut The nut to the algothm s gven n a fle, whee the fst aamete ndcates the numbe of jobs n, second aamete ndcates the numbe of machnes m. Fom the thd aamete the ocessng tmes of jobs follows. The numbe n ow and column j s the ocessng tme of job on machne j. Notatons Followng notatons ae used to mlement the algothm N Job_a = {J, J, J J n {, U {+, +, n j 0 c j Notatons Numbe of jobs Numbe of machnes Set of jobs Odeed set of scheduled jobs Set of unscheduled jobs Pocessng tme of job on machne j Comleton tme of oeaton of job of on machne j C S Comleton tme of job J Sum of the comleton tme of the schedule LB Lowe bound based on machne LB Lowe Bound based on machne LB Lowe bound based on machne Table.4 Basc Notatons fo Banch and Bound Algothm 4

36 Algothm. Banch and Bound. Intalze Job_a = {J, J, J J n. Calculate ntal uebound = sum of comleton tmes of ntal feasble schedule cb_ode = ntal feasble schedule. sotng_jobs() 4. geneate_node(fxed_joba, level) a. IF level = n (.e. leaf) then cuent soluton = comleton tme of the schedule. If cuent soluton < ue bound, udate ue bound b. ELSE. CALCULATE the lowebound. IF lowebound >= uebound THEN une the node ELSE CALL geneate_node(fxed_joba, level+) END IF END IF 5. Sto.5 The Algothm Illustaton To evaluate the banch and bound algothm the followng 5 jobs and machnes oblem has been used. Job Table.5 Examle to Demonstate Banch and Bound 5

37 In the above table, each ow eesents the job and each column eesents the machne j. The ocessng tme of an oeaton of job on machne j s mentoned n each cell and s denoted as j. Ou objectve s to obtan a emutaton schedule that mnmzes the sum of comleton tmes of all the jobs. Ste : Fnd ntal feasble schedule by aangng the jobs n the nceasng ode of the sum of ocessng tmes. The ntal feasble schedule s [,, 4, ] Job Sum of j 4++ = 6 ++ = = = 9 Table.6 Calculatng Intal Feasble Schedule Ste : Calculate ntal ue bound whch s sum of comleton tmes of ntal feasble schedule. n Fo ode [,, 4, ] ue bound (UB) = C = = 55 Job c c c Table.6 Calculatng Intal Ue Bound Ste : We now comute the lowe bound fo each node n the tee. In tee each node eesents a atal sequence S k whee jobs n the fst k ostons ae fxed. C (k), C (k), C (k), be the comleton tmes on machne, machne, machne esectvely fo the atal sequence. Calculatng Lowe Bound fo Patal Sequence [ * * *] Set of scheduled jobs = { and = = Set of unscheduled jobs U = {, 4,. 6

38 Cost of the schedule S = C max (LB, LB, LB ) Job C C C Table.8 Calculaton of Comleton Tmes fo Patal Sequence [ * * *] Fo LB fom the lst of unscheduled jobs U, schedule the jobs n the nceasng ode of the ocessng tmes on machne. Sequence of jobs wth nceasng values s [, 4, ]. Let k, k =,,,n be the ndex of these jobs. ( n k ( n k ) k k k ) k k k k= 4. = 6 = = 5 k= 4. 4 = 0 4 = 4 = 8 k=4 4. = 6 P = 5 = 6 Table.9 Calculaton of LB fo Patal sequence [ * * *] n LB = [ ( n k ) = = 49. k k k k ] Fo LB fom the lst of unscheduled jobs U, schedule the jobs n the nceasng ode of the ocessng tmes on machne. Sequence of jobs wth nceasng values s [4,, ]. The exesson, max{ C, mn = max {5, 4 + = max {5, 4 + = 6 ( n k ) k max{ C k, mn ( ) n k k k k=. 4 = P 4 = k=. = 6 P = 4 k=4. = 5 = Table.0 Calculaton of LB fo Patal sequence [ * * *] 7

39 n k k k ] LB = [max{ C, mn ( n k ) = +4+ = 8 Fo LB fom the lst of unscheduled jobs U, schedule the jobs n the nceasng ode of ocessng tmes on machne. Sequence of jobs wth nceasng values s [,, 4]. The exesson, max{ C, max{ C, mn mn = max {6, max {5, = max {6, max {5, = max {6, 6+ = 7 max{ C, max{ C, mn mn ( n k ( n k ) k ) k k=. = 0 k=. = 4 k=4. 4 = 0 Table. Calculaton of LB fo Patal sequence [ * * *] LB n [ k k max{ C, max{ C, mn mn ( n k ) ] = 0++0 = LB( * * * ) = C max (LB, LB, LB ) = C + max( 49, 8, ) = =55 Snce lowe bound of atal sequence ( * * *) = 55 ue bound, une the node [***] and all the banches that emege fom t. Calculatng Lowe Bound fo Patal Sequence [ * * *] Smlaly we calculate the lowe bound fo atal sequence ( * * *) Set of scheduled jobs = { and = = Set of unscheduled jobs U = {, 4,. Job C C C 5 7 Table. Calculaton of Comleton Tmes fo Patal sequence [ * * *] 8

40 Fo LB fom the unscheduled jobs U, schedule the jobs n the nceasng ode of the ocessng tmes on machne. Sequence of jobs wth nceasng values s [, 4, ]. ( n k ( n k ) k k k ) k k k k=. = P = = 6 k=. 4 = 0 4 = 4 = 6 k=4. = 6 P = 5 = 4 Table. Calculaton of LB fo Patal sequence [ * * *] n LB = [ ( n k ) = = 46. k k k k ] Fo LB fom the lst of unscheduled jobs U, schedule the jobs n the nceasng ode of the ocessng tmes on machne. Sequence of jobs wth nceasng values s [, 4, ]. The exesson, max{ C, mn = max {5, + = max {5, + 4 =6 max{ C, mn ( n k ( n k ) k k ) k k k=. = P 4 = 0 k=. 4 = P = k=4. = 5 = Table.4 Calculaton of LB fo Patal sequence [ * * *] n k k k ] LB = [max{ C, mn ( n k ) = 0++ = Fo LB fom the lst of unscheduled jobs U, schedule the jobs n the nceasng ode of ocessng tmes on machne. Sequence of jobs wth nceasng values s [,, 4]. The exesson, max{ C, max{ C, mn mn = max {7, max {5, + + = max {7, max {5, = max {7, 6+ = 7 9

41 ( n k ) max{ C k, max{ C, mn mn ( ) n k k k=. = 0 k=. = 9 k=4. 4 = 0 Table.5 Calculaton of LB fo Patal sequence [ * * *] LB n [ k k max{ C, max{ C, mn mn ( n k ) ] = = 9 LB( * * * ) = C max (LB, LB, LB ) = C + max( 46,, 9) = =5 Snce lowe bound of atal sequence ( * * *) = 5< ue bound, we banch to lowe level nodes fom atal sequence ( * * *). Banchng fom a node conssts of takng each of the unallocated jobs n tun and lacng t next to the atal schedule. Each of these new atal schedules s then eesented by a new node. Calculatng Lowe Bound fo Patal Sequence [ * *] Lowe bound fo the atal sequence [ * *] s calculated as follows: Set of scheduled jobs = {, and = = Set of unscheduled jobs U = {4,. Job C C C Table.6 Calculaton of Comleton Tmes fo Patal sequence [ * *] Fo LB fom the lst of unscheduled jobs U, schedule the jobs n the nceasng ode of the ocessng tmes on machne. Sequence of jobs wth nceasng values s [4, ]. 0

42 ( n k ) k k k ( n k ) k k k k= 6. 4 = 0 4 = 4 = 0 k=4 6. = 6 P = 5 = 8 Table.7 Calculaton of LB fo Patal sequence [ * *] n LB = [ ( n k ) = 0+8 = 8 k k k k ] Fo LB fom the lst of unscheduled jobs U, schedule the jobs n the nceasng ode of the ocessng tmes on machne. Sequence of jobs wth nceasng values s [4, ]. The exesson, max{, mn = max {7, = max {5, = C max{ C, mn ( n k ( n k ) k k ) k k k=. 4 = P 4 = 6 k=4. = 5 = 7 Table.8 Calculaton of LB fo Patal sequence [ * *] n k k k ] LB = [max{ C, mn ( n k ) = 6+7 = Fo LB fom the lst of unscheduled jobs U, schedule the jobs n the nceasng ode of ocessng tmes on machne. Sequence of jobs wth nceasng values s [, 4]. The exesson, max{ C, max{ C, mn mn = max {8, max {7, = max {8, max {7, = max {8, = ( n k ) max{ C k, max{ C, mn mn ( ) n k k k=. = 4 k=4. 4 = 5 Table.9 Calculaton of LB fo Patal sequence [ * *]

43 LB n [ k k LB( * * ) = max{ C, max{ C, mn mn ( n k ) ] = 4+5 = 9 C max (LB, LB, LB ) = C + C + max( 8,, 9) = =5 Snce lowe bound of atal sequence ( * *) = 5< ue bound, we banch to lowe level nodes fom atal sequence ( * *). Banchng fom a node conssts of takng each of the unallocated jobs n tun and lacng t next to the atal schedule. Each of these new atal schedules s then eesented by a new node. Calculatng Lowe Bound fo Patal Sequence [ 4 *] Lowe bound fo the atal sequence [ 4 *] s calculated as follows: Set of scheduled jobs = {,, 4 and = = Set of unscheduled jobs U = {. Job C C C Table.0 Calculaton of Comleton Tmes fo Patal sequence [ 4 *] Fo LB fom the lst of unscheduled the jobs aange jobs n the nceasng ode of ocessng tmes on machne. Sequence of jobs wth nceasng values s []. ( n k ( n k ) k k k ) k k k k=4. = 6 P = 5 = Table. Calculaton of LB fo Patal sequence [ 4 *] n LB = [ ( n k ) = k k k k ]

44 Fo LB fom the lst of unscheduled jobs U, schedule the jobs n the nceasng ode of the ocessng tmes on machne. Sequence of jobs wth nceasng values s {. The exesson, max{, mn = max {, + = max {, + 6 = 7 C ( n k ) k max{ C k, mn ( ) n k k k k=4. = 5 = Table. Calculaton of LB fo Patal sequence [ 4 *] n k k k ] LB = [max{ C, mn ( n k ) = Fo LB fom the lst of unscheduled jobs U, schedule the jobs n the nceasng ode of ocessng tmes on machne. Sequence of jobs wth nceasng values s {. The exesson, max{ C, max{ C, mn mn = max {5, max {, + + = max {5, max {, = max {5, = max{ C, max{ C, mn mn ( n k ( n k ) k ) k k=4. = Table. Calculaton of LB fo Patal sequence [ 4 *] LB n [ k k LB( 4 * ) = max{ C, max{ C, mn mn ( n k ) ] = C max (LB, LB, LB ) = C + C + C 4 + max(,, ) = = 5 Snce lowe bound of atal sequence ( 4 *) = 5< ue bound, we banch to lowe level nodes fom atal sequence ( 4 *). Hee when we banch to the lowe level, we fnd that the node wth atal sequence ( 4 ) s a leaf node, so we calculate the comleton tme of the schedule.

45 Job C C C Table.4 Calculaton of C fo Schedule [ 4 ] Comleton tme fo the schedule [ 4 ] =5 < ue bound. Theefoe, now we udate the ue bound and the cuent best ode. We now exloe othe nodes n the seach tee wth the udated ue bound. Smlaly the Lowe bound fo the atal sequence [ *] = 54, [ 4 * *] = 54, [ * *] = 56, [ * * *] = 60 and [4 * * *] = 57. Snce all lowe bounds ue bound (5), we une all these nodes. Fgue. Enumeaton tee fo 4 jobs- machne usng banch and bound algothm 4

46 Patal sequence Lowe bound Cuent best UB and ode Oeaton [ * * * ] 55 55, [,, 4, ] Cut node [ * * * ] 5 55, [,, 4, ] Banch fom node [ * * ] 5 55, [,, 4, ] Banch fom node [ 4 * ] 5 5, [,, 4, ] Leaf node, calculate C [ * ] 54 5, [,, 4, ] Cut node [ 4* * ] 54 5, [,, 4, ] Cut node [ * * ] 54 5, [,, 4, ] Cut node [ * * * ] 60 5, [,, 4, ] Cut node [4 * * * ] 57 5, [,, 4, ] Cut node Table.5 Executon Stes fo F C Fgue. Sceenshot of the Outut fo the,, Values Gven n Table.5.6 Banch and Bound Fm-em C (m ) Banch and bound aoach fo F-em C can be genealzed to m machnes. In ths case at each node we detemne m machne based lowe bounds and the oveall lowe bound s the maxmum of the m-lowe bounds [8]. 5

47 Thus the lowe bound at node T s, LB [T] = C C = C max (LB, LB, LB LB m ) Geneally to calculate the lowe bound on a machne x, we assume the ossblty that the ocessng on machne x s contnuous fo the unassgned job set {U. LB x = ealest tme that the fst job n the set U can stat on machne x + sum of comleton tmes of jobs n set U on machne x (hee schedule the jobs n the nceasng ode of ocessng tmes on machne x. Let +, +,,n eesent the sequence of jobs wth nceasng x values ) + sum of ocessng tmes of jobs n set U on emanng (m-x) machnes. Let be the last job n the odeed set of scheduled jobs, then x n, x, y j k, ) y x jy k LB x = max{ C, max { C mn ( n k k, x n m k jx k, j 6

48 CHAPTER 4 RESULTS Ths chate gves a detaled vew of the esults obtaned by alyng Banch and Bound algothm fo emutaton F-em C schedulng oblem. 4. Assumtons Followng assumtons ae made whle mlementng the algothm:. The algothm ntalzes the banch and bound tee wth an ntal feasble schedule and an ntal ue bound. Intal feasble schedule s obtaned by aangng the jobs n the nceasng ode of the sum of ocessng tmes.. Intal Ue bound can be obtaned by calculatng the sum of comleton tmes of ntal feasble schedule. 4. Paametes that Detemne Pefomance of the Algothm Intal Ue Bound Intal Ue bound s obtaned by calculatng the sum of comleton tmes of ntal feasble schedule. Global Lowe bound The mnmum lowe bound on the hghest level nodes coesonds to global lowe bound. We stat the banch and bound tee wth ntal ue bound. Then we ty to banch n ths tee by tyng to fx each of the jobs as the fst job n the sequence. Fom the data gven n Table.5 and Fgue., thee ae fou jobs, so ossble banches ae fou. Fgue 4. Banchng Tee Showng the Hghest Level Nodes 7

49 Then we calculate the lowe bound fo these nodes. The mnmum lowe bound on these hghest level nodes coesonds to the global lowe bound because all the coesondng banches emegng fom these nodes have lowe bound geate than o equal to t. Fo the above oblem, global lowe bound = 5. Cuent Best Soluton Cost of the schedule obtaned by alyng banch and bound algothm eesents the cuent best soluton. Pefomance Rato % ncease ove the otmal soluton s used to analyze the efomance of banch and bound algothm. Banch and bound s one of the heustc to detemne nea otmal soluton, but t does not guaantee to ovde an otmal soluton. Theefoe we use efomance ato n ode to detemne the ecentage of devaton of cuent best soluton obtaned fom otmal. Executon Tme Executon tme s the tme taken by the banch and bound ogam to detemne the cuent best soluton. Fom the above esults we see that, banch and bound efoms much bette n actce than the comlete enumeaton. Numbe of Elmnated Sequences If the lowe bound of the sub oblem s geate than o equal to ue bound, then ths sub oblem cannot yeld a bette soluton and we sto banchng fom the coesondng node n the banchng tee. Thus we une all the banches emegng fom that node. If thee ae n jobs and f a node at k th level s uned, then we elmnate (n-k)! sequences fom ocessng. 4. Results fo Vaous, and Values Followng esults ae obtaned by alyng banch and bound algothm. Comutatonal esults fo u to 0 jobs ae gven fo machne emutaton flow sho oblem when the objectve s mnmzng the sum of comleton tmes. 8

50 4.. Random, and Values Fo andomly chosen, and values gven n the Table 4. the esults obtaned by executng banch and bound algothm fo the objectve functon C ae esented n Table 4. and Table 4.. Job Table 4. Random, and values fo n u to 0 9

51 Fo the fst n jobs Intal UB Global LB Cuent Best soluton (Cuent soluton/global LB)*00 Executon tme(sec) n= n= n= n= n= n= n= n= n= n= n= Table 4. C Results fo Random Values of, and Fo the No of No of elmnated fst n ocessed Cuent best ode sequences jobs sequences n= [, 0, 8,, 9, 4,, 7, 5, 6] n= [, 0, 8, 4, 9,,, 7,, 5, 6] n= [,, 0, 9, 8, 4,,, 7,, 5, 6] n= [,, 0, 9, 8, 4,,,, 7,, 5, 6] n= [, 4, 0, 8,, 9,, 4,,, 7,, 5, 6] n= [, 5, 0, 8, 4,,, 9,,, 4,, 7, 5, 6] n= n= n= n= n= [, 6, 0, 9, 8, 5,,,, 4,, 4,, 7, 5, 6] [, 6, 0, 9, 8, 5,,,, 4,, 7, 4,, 7, 5, 6] [, 6, 0, 9, 8, 5,,,, 4,, 7, 4,, 7, 5, 8, 6] [0,, 6, 0, 9, 8, 5, 4,, 7, 9,,,, 7, 4, 5,, 8, 6] [, 6, 0, 9, 8, 5, 4,, 7, 9,,,, 7, 0, 4,, 5, 8, 6] Table 4. Results of Banch and Bound Algothm fo n u to 0 40

52 4.. Lage Values of, and Fo the lage values of, and gven n the Table 4.4 the esults obtaned by executng banch and bound algothm fo the objectve functon C ae esented n Table 4.5. Job Table 4.4 Lage Values of, and 4

53 Fo the fst n jobs Intal UB Global LB Cuent soluton (Cuent soluton/global LB)*00 Executon tme(sec) No of ocessed sequences n= n= n= n= n= n= n= n= n= n= n= Table 4.5 C Results fo Lage Values of, and Job Table 4.6 Inceasng, and Deceasng Values 4

54 4.. Inceasng, and Deceasng Values Fo a atcula case whee the values of, nceases and deceases as the job ndex nceases ae gven n the Table 4.6 and the esults obtaned by executng banch and bound algothm fo the objectve functon C ae esented n Table 4.7. Fo the fst n jobs Intal UB Global LB Cuent soluton (Cuent soluton/global LB)*00 Executon tme(sec) No of ocessed sequences n = n = n = n = n= n= n= n= n= n= Table 4.7 C Results fo Inceasng, and Deceasng Values 4..4, Inceases and then Deceasng The values of, nceases as the job ndex nceases and deceases afte > (n+) / and the values of deceases as the job ndex nceases and nceases afte > (n+) / ae gven n the Table 4.8. The esults obtaned by executng banch and bound algothm fo the objectve functon C ae esented n Table

55 Job Table 4.8, Inceasng and then Deceasng Fo the fst n jobs Intal UB Global LB Cuent soluton (Cuent soluton/global LB)*00 Executon tme(sec) No of ocessed sequences n= n= n= n= n= n= Table 4.9 C Results fo, Inceasng and then Deceasng 44

56 4.4 Effcency of the Algothm In ode to valdate actcally the effcency of banch and bound algothm, the esults of the algothm ae comaed wth the esults obtaned by geneatng all the n! emutatons sequences, when numbe of jobs (n ). Fom the esults obtaned, we see that banch and bound efoms much bette n actce than the comlete enumeaton. Fom the exements, we notce that nstead of seachng ente soluton sace banch and bound algothm uned many nodes and ths consdeably educed the comutatonal tme. Banch and bound algothm s used to detemne nea otmal soluton, but t does not guaantee to ovde an otmal soluton. Theefoe we use efomance ato n ode to detemne the ecentage of devaton of banch and bound soluton fom otmal. Fom the esults obtaned we see a dffeence of aoxmately.% between the banch and bound soluton and otmal soluton. 45

57 CHAPTER 5 CONCLUSION AND FUTURE WORK In ths thess, we have esented, evaluated and mlemented the banch and bound algothm to mnmze the sum of comleton tmes fo thee machne emutaton flow sho oblem. We esented the lowe bounds, ue bounds and efomance ato fo the vaous oblems. In geneal, ou esults consstently gve solutons wth a ato of bette than.% of otmal. We ndeed obseved that a sgnfcant numbe of sub oblems can be elmnated fom futhe consdeaton, f the ntal ue bound s tght. Theefoe we can use some good heustcs to obtan bette ntal ue bound. As n gows, the banch and bound algothm s obvously exonental n tme but efoms much bette n actce than the comlete enumeaton. The futue wok would be to move lowe bounds fo mnmzng the sum of comleton tmes of n jobs ove m machnes. 46

58 ackage sumc; APPENDIX mot java.o.*; mot java.utl.*; ublc class BanchAndBound { statc ublc nt njobs, nmachnes; statc ublc nt[][] ; statc nt[][] c; statc long cb_ub = ; statc long global_lb = ; statc Stng cb_ode; a<intege, Intege> soted_mac; a<intege, Intege> soted_mac; a<intege, Intege> soted_mac; statc nt[] job_a; statc long ocessd_node = 0; statc long count = 0; // ethod to ead data fom nut fle ublc statc vod eaddata(stng flename) { Scanne sc = null; ty { sc = new Scanne(new FleReade(flename)); catch (Exceton e) { System.out.ntln("could not fnd the fle "); njobs = sc.nextint(); nmachnes = sc.nextint(); = new nt[njobs + ][nmachnes + ]; System.out.ntln("The ocessng tmes ae:"); fo (nt j = ; j <= njobs; j++) { fo (nt m = ; m <= nmachnes; m++) { [j][m] = sc.nextint(); System.out.nt([j][m] + " "); System.out.ntln(); System.out.ntln("Numbe of Jobs ae:" + njobs); System.out.ntln("Numbe of achnes ae:" + nmachnes); sc.close(); // ethod to calculate comleton tme of the gven jobs and sumc fo the // gven schedule ublc nt calcom(nt[] a) { c = new nt[njobs + ][nmachnes + ]; nt sumc = 0; fo (nt j = ; j <= a.length - ; j++) { fo (nt m = ; m <= nmachnes; m++) { 47