FLOW SHOPS: F2||Cmax

FLOW SHOPS: JOHNSON'S RULE2 FLOW SHOP SCHEDULING (n JOBS, m MACHINES) n JOBS BANK OF m MACHINES (SERIES) n M1 M2Mm

FLOW SHOPS: JOHNSON'S RULE3 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 RULE4 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 RULE5 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 RULE6 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 RULE7 DESCRIPTION OF JOHNSON’S ALGORITHM 1.IDENTIFY THE JOB WITH THE SMALLEST PROCESSING TIME (ON EITHER MACHINE). 2.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. 3.IF THERE IS SOME UNSCHEDULED JOB, GO TO 1. OTHERWISE STOP.

FLOW SHOPS: JOHNSON'S RULE8 EXAMPLE CONSIDER THE FOLLOWING INSTANCE OF THE JOHNSON’S (Fm||Cmax) PROBLEM: SEQUENCE:

FLOW SHOPS: JOHNSON'S RULE9 EXAMPLE: SCHEDULE SEQUENCE: t M1 M2

FLOW SHOPS: JOHNSON'S RULE10 A BOUND ON THE MAKESPAN FOR JOHNSON’S PROBLEM:

FLOW SHOPS: JOHNSON'S RULE11 JOHNSON’S ALGORITHM 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.

FLOW SHOPS: JOHNSON'S RULE12 JOHNSON’S ALGORITHM (CONTINUED) 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.

FLOW SHOPS: JOHNSON'S RULE13 JOHNSON’S ALGORITHM (CONTINUED) STEP 4: SEQUENCE OF JOBS THE SEQUENCE OF JOBS IS GIVEN BY Ji, WITH J1 THE FIRST JOB, AND SO FORTH.

FLOW SHOPS: JOHNSON'S RULE14 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 RULE15 MACHINE DOMINANCE: F3||Cmax A MACHINE IS DOMINATED WHEN ITS LARGEST PROCESSING TIME 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.

FLOW SHOPS: JOHNSON'S RULE16 JOHNSON’S ALGORITHM FOR 3 MACHINES FOR F3||Cmax, WHENEVER MACHINE 2 IS DOMINATED, i.e., OR SOLVING AN EQUIVALENT TWO-MACHINE PROBLEM WITH PROCESSING TIMES: p’ 1j = p 1j + p 2j AND p’ 2j = p 2j + p 3j GIVES THE OPTIMAL MAKESPAN SEQUENCE TO THE DOMINATED THREE-MACHINE PROBLEM.

FLOW SHOPS: JOHNSON'S RULE17 EXAMPLE: F3||Cmax CONSIDER F3||  Cmax WITH THE FOLLOWING JOBS:

FLOW SHOPS: JOHNSON'S RULE18 EXAMPLE: PROCESSING TIMES, DUMMY MACHINES SEQUENCE:

FLOW SHOPS: JOHNSON'S RULE19 EXAMPLE: SCHEDULE SEQUENCE: t M1 M2 M3