N job m machine.pdf OR job Sequencing problems
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The document discusses solving scheduling problems involving multiple jobs and machines, outlining a method for optimizing job sequences on two machines. It details the steps of the Johnson's algorithm to minimize elapsed time and minimize idle time across machines. Several examples illustrate the application of the algorithm to compute optimal sequences and total elapsed times.
N job m machine.pdf OR job Sequencing problems
- 1. @stat acficing Sequencing oblemns ype TProblems with n Jobs hsough 2 machune Type I Psoblems with Jobs thro ugh 3 mac hine gpe: soblems oith Jobs thwough R machume Jobs thsough K machine Tpe Y Problems with 2 Jobs 2 Machine A A A A3 An Machine B , B Bs B, Fin cipal Assumbtrions (d No machune Com psocess mose tham one 0pexaton at a tme. must be omed tl Combleton tye Jobs sBavled ges only ome T (i Each operation bnce ) Jhese mac hin e each between machenes is megi ble. ( The tume eq, wed to ron her he
- 2. Pooblemswith n Jobs Thaough k Machine @stant sacticing Con veet the bxo blem into tuoo mac hime bso ble solve us vmg Johmsom's algoHthm Steb 4: fund Min. Mi amd Min Mk ond Max. o each Ma.M3 Mk- (Mks last machine) amd then Step 2 : Check whethex Min M 2 Max. M 03 (j:2,3. K-1) Min Mx 2 Max M tep 3: It the n eg,ualities in steb 2 ne mot saihied , the method ails okherwise 30 t0 mext step. Gteb 4:1n addihom to step 2 ib M,+Ms+.. MR- C hee C is bositve constant or all i:1,2.. n. Then detesmine the optvmal sequumce ot m Jobs, wheu two machines Miand Mk by sUng the optimal sequnce olgoruthm ate Step 5 I the condikom M,+My + Mk-1 # C ne dekime too machuine G and H Such that i =M + M2 + Mk- H M2 +Mgtt M Detetmine the obtvm al sequmce us Ung two maC hune ulgoruthm or Jo hmsom's agotuthm.
- 3. @stort fsacticing 30 blems wth m Jobs thzough K machene Q: Four Jobs oeto be brocessed on each o the 5 machines Fundl the total minimum elabsed tme it no bass ing oh Jobs is pexmited Als0 und the dle time ok each mac hune Sinte, Bi+C+DL =10 (Vf B, C+ D2 = 15 Machines 3 B+C+Di #C we dekhe tuwo machines q and H Such thad G=AtB+C +D H B CtD +E D 5 Solutio: Sinee he pro blem is to be sequnced onive machines , ' we Convertthe o blem tuoo- machine o blem Min (A) -5 Min E) = 6 Max (8,c, D) = (6, 5, ¢) Jebs 1 nto 2 20 16 2S 4 Optimal Sequene Min (E) Max (G,S.6) 66 (T) 3 2 4
- 4. kind he mim todal elapsed tme and Idle time o machuine A,B 2C Machine A Machine B MachiineC Machuine D Machune 6 Job IN OUT IN OUT IN OUT IN OUT IN OUT 12 12 14 17 26 27 27 33 35 35| 45 12 12 6 16 1 21 35 18 24 24 29 28 2 12 18 18 45 26 26 29 29 32 33 5 Total ela bseol tme 5Iunis Tdle time A s1- 26) ums =25 nis o mac hine me or machine B (51 29)+tt2t2 22tIl 33k 1dle Tdle tme mac hune C (51 - 32)+12 2t3+1 = 14 18 =3 7w machie D - (S1 -35)+14+ t 1 3 uit machneE (sl-s1)+ 7t1 18 unis Ldle Jle tme
- 5. Example-2 BCD Jobs A Machinee Ma Job My 24 22 2 M 24 21 I 92 My2 1S 15 6 22 14 21 32 Min (M) = 16 Min (M) =4 Max (Ma, M,) = ( ?) M (M4) 2 Max(AMs) (1,) CT 14 (M Me 3t6 =1y (M,H)6+?ciy M+Ms) 1+5 =1y