FLOW SHOPS: F2 ||Cmax

1. FLOW SHOPS: F2||Cmax

2. n JOBS BANK OF m MACHINES (SERIES) Mm M1 M2 1 2 3 4 n FLOW SHOP SCHEDULING(n JOBS, m MACHINES) FLOW SHOPS: JOHNSON'S RULE

3. FLOW SHOPS PRODUCTION SYSTEMS FOR WHICH: A NUMBER OF OPERATIONS HAVE TO BE DONE ON EVERY JOB. THESE OPERATIONS HAVE TO BE DONE ON ALL JOBS IN THE SAME ORDER, i.e., THE JOBS HAVE TO FOLLOW THE SAME ROUTE. THE MACHINES ARE ASSUMED TO BE SET UP IN SERIES. COMMON ASSUMPTIONS: UNLIMITED STORAGE OR BUFFER CAPACITIES IN BETWEEN SUCCESIVE MACHINES (NO BLOCKING). A JOB HAS TO BE PROCCESSED AT EACH STAGE ON ONLY ONE OF THE MACHINES (NO PARALLEL MACHINES). FLOW SHOPS: JOHNSON'S RULE

4. PERMUTATION FLOW SHOPS FLOW SHOPS IN WHICH THE SAME SEQUENCE OR PERMUTATION OF JOBS IS MAINTAINED THROUGHOUT: THEY DO NOT ALLOW SEQUENCE CHANGES BETWEEN MACHINES. PRINCIPLE FOR Fm||Cmax: THERE ALWAYS EXISTS AN OPTIMAL SCHEDULE WITHOUT SEQUENCE CHANGES BETWEEN THE FIRST TWO MACHINES AND BETWEEN THE LAST TWO MACHINES. THERE ARE OPTIMAL SCHEDULES FOR F2||Cmax AND F3||Cmax THAT DO NOT REQUIRE SEQUENCE CHANGES BETWEEN MACHINES. FLOW SHOPS: JOHNSON'S RULE

5. JOHNSON’S F2||Cmax PROBLEM FLOW SHOP WITH TWO MACHINES IN SERIES WITH UNLIMITED STORAGE IN BETWEEN THE TWO MACHINES. THERE ARE n JOBS AND THE PROCESSING TIME OF JOB j ON MACHINE 1 IS p1j AND THE PROCESSING TIME ON MACHINE 2 IS p2j. THE RULE THAT MINIMIZES THE MAKESPAN IS COMMONLY REFERRED TO AS JOHNSON’S RULE. FLOW SHOPS: JOHNSON'S RULE

6. JOHNSON’S PRINCIPLE ANY SPT(1)-LPT(2) SCHEDULE IS OPTIMAL FOR Fm||Cmax. (THE SPT(1)-LPT(2) SCHEDULES ARE NOT THE ONLY SCHEDULES THAT ARE OPTIMAL. THE CLASS OF OPTIMAL SCHEDULES APPEARS TO BE HARD TO CHARACTERIZE AND DATA DEPENDENT). FLOW SHOPS: JOHNSON'S RULE

7. DESCRIPTION OF JOHNSON’S ALGORITHM • IDENTIFY THE JOB WITH THE SMALLEST PROCESSING TIME (ON EITHER MACHINE). • IF THE SMALLEST PROCESSING TIME INVOLVES: • MACHINE 1, SCHEDULE THE JOB AT THE BEGINNING OF THE SCHEDULE. • MACHINE 2, SCHEDULE THE JOB TOWARD THE END OF THE SCHEDULE. • IF THERE IS SOME UNSCHEDULED JOB, GO TO 1. OTHERWISE STOP. FLOW SHOPS: JOHNSON'S RULE

8. CONSIDER THE FOLLOWING INSTANCE OF THE JOHNSON’S (Fm||Cmax) PROBLEM: SEQUENCE: EXAMPLE FLOW SHOPS: JOHNSON'S RULE

9. SEQUENCE: 5 1 4 3 2 M1 M2 t EXAMPLE: SCHEDULE FLOW SHOPS: JOHNSON'S RULE

10. FOR JOHNSON’S PROBLEM: A BOUND ON THE MAKESPAN FLOW SHOPS: JOHNSON'S RULE

11. LET U = {1, 2,..., n} BE THE SET OF UNSCHEDULED JOBS. k =1, l = n, Ji = 0, i = 1, 2, ..., n. STEP 1: IDENTIFICATION OF SMALLEST PROCESSING TIME IF U = , GO TO STEP 4. LET IF i* = 1 GO TO STEP 2; OTHERWISE GO TO STEP 3. JOHNSON’S ALGORITHM FLOW SHOPS: JOHNSON'S RULE

12. STEP 2: SCHEDULING A JOB ON EARLIEST POSITION • SCHEDULE JOB j* IN THE EARLIEST AVAILABLE POSITION: Jk • = j*. • UPDATE k: k = k + 1. • REMOVE THE JOB FROM THE SCHEDULABLE SET, U = U – {j*}. • GO TO STEP 1. STEP 3: SCHEDULING A JOB ON LATEST POSITION • SCHEDULE JOB j* IN THE EARLIEST AVAILABLE POSITION: Jl • = j*. • UPDATE l: l = l - 1. • REMOVE THE JOB FROM THE SCHEDULABLE SET, U = U – {j*}. • GO TO STEP 1. JOHNSON’S ALGORITHM(CONTINUED) FLOW SHOPS: JOHNSON'S RULE

13. STEP 4: SEQUENCE OF JOBS THE SEQUENCE OF JOBS IS GIVEN BY Ji, WITH J1 THE FIRST JOB, AND SO FORTH. JOHNSON’S ALGORITHM(CONTINUED) FLOW SHOPS: JOHNSON'S RULE

14. Fm||Cmax Fm||Cmax IS A STRONGLY NP-HARD PROBLEM. AN EXTENSION OF JOHNSON’S ALGORITHM YIELDS AN OPTIMAL SOLUTION FOR THE F3||Cmax PROBLEM WHEN THE MIDDLE MACHINE IS DOMINATED BY EITHER THE FIRST OR THIRD MACHINE. FLOW SHOPS: JOHNSON'S RULE

15. A MACHINE IS DOMINATED WHEN ITS LARGEST PROCESSINGTIME IS NO LARGER THAN THE SMALLEST PROCESSING TIME ON ANOTHER MACHINE. FOR F3||Cmax PROBLEM: WHICH IMPLIES THAT MACHINE 2 (DOMINATED MACHINE) CAN NEVER CAUSE A DELAY IN THE SCHEDULE. MACHINE DOMINANCE: F3||Cmax FLOW SHOPS: JOHNSON'S RULE

16. FOR F3||Cmax, WHENEVER MACHINE 2 IS DOMINATED, i.e., OR SOLVING AN EQUIVALENT TWO-MACHINE PROBLEM WITH PROCESSING TIMES: p’1j = p1j + p2j AND p’2j = p2j + p3j GIVES THE OPTIMAL MAKESPAN SEQUENCE TO THE DOMINATED THREE-MACHINE PROBLEM. JOHNSON’S ALGORITHM FOR 3 MACHINES FLOW SHOPS: JOHNSON'S RULE

17. CONSIDER F3||Cmax WITH THE FOLLOWING JOBS: EXAMPLE: F3||Cmax FLOW SHOPS: JOHNSON'S RULE

18. SEQUENCE: EXAMPLE: PROCESSING TIMES, DUMMY MACHINES FLOW SHOPS: JOHNSON'S RULE

19. SEQUENCE: 1 4 5 2 3 M1 M2 M3 t EXAMPLE: SCHEDULE FLOW SHOPS: JOHNSON'S RULE