FLOW SHOPS: F2||Cmax

FLOW SHOPS: JOHNSON'S RULE 2

FLOW SHOP SCHEDULING(n JOBS, m MACHINES)

n JOBS BANK OF m MACHINES (SERIES)

12

3

4 n

M1 M2 Mm

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

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 RULE 8

EXAMPLE

CONSIDER THE FOLLOWING INSTANCE OF THE JOHNSON’S (Fm||Cmax) PROBLEM:

JOB 1 2 3 4 5 p1j 4 4 10 6 2 p2j 5 1 4 10 3

SEQUENCE:

FLOW SHOPS: JOHNSON'S RULE 9

EXAMPLE: SCHEDULE

SEQUENCE:5 1 4 3 2

JOB 1 2 3 4 5 p1j 4 4 10 6 2 p2j 5 1 4 10 3

t

M1

M2

FLOW SHOPS: JOHNSON'S RULE 10

A BOUND ON THE MAKESPAN

FOR JOHNSON’S PROBLEM:

n

1jj2j1

n,..,1j

n

1jj1j2

n,..,1jmax ppmin,ppminmax)OPT(C

FLOW SHOPS: JOHNSON'S RULE 11

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

j2n,..,1j

,j1n,..,1j

*j*i pminpminminp

IF i* = 1 GO TO STEP 2; OTHERWISE GO TO STEP 3.

FLOW SHOPS: JOHNSON'S RULE 12

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 RULE 13

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 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

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:

j3j1j

j2 pmin,pminmaxp

WHICH IMPLIES THAT MACHINE 2 (DOMINATED MACHINE) CAN NEVER CAUSE A DELAY IN THE SCHEDULE.

FLOW SHOPS: JOHNSON'S RULE 16

JOHNSON’S ALGORITHM FOR 3 MACHINES

FOR F3||Cmax, WHENEVER MACHINE 2 IS DOMINATED, i.e.,

OR }p{max}p{min j2j

j1j

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.

}p{max}p{min j2j

j3j

FLOW SHOPS: JOHNSON'S RULE 17

EXAMPLE: F3||Cmax

CONSIDER F3||Cmax WITH THE FOLLOWING JOBS:

JOB 1 2 3 4 5 p1j 4 9 8 6 5 p2j 5 6 2 3 4 p3j 8 10 6 7 11

}p{min j1j

}p{max j2j

}p{min j3j

FLOW SHOPS: JOHNSON'S RULE 18

EXAMPLE: PROCESSING TIMES, DUMMY MACHINES

JOB 1 2 3 4 5 p1j 4 9 8 6 5 p2j 5 6 2 3 4 p3j 8 10 6 7 11 p'1j p'2j

SEQUENCE:

FLOW SHOPS: JOHNSON'S RULE 19

EXAMPLE: SCHEDULE

SEQUENCE:1 4 5 2 3

JOB 1 2 3 4 5 p1j 4 9 8 6 5 p2j 5 6 2 3 4 p3j 8 10 6 7 11

t

M1

M2

M3