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FLOW SHOPS: F2 ||Cmax. n JOBS BANK OF m MACHINES (SERIES). Mm. M1. M2. 1. 2. 3. 4. n. FLOW SHOP SCHEDULING (n JOBS, m MACHINES). FLOW SHOPS. PRODUCTION SYSTEMS FOR WHICH: A NUMBER OF OPERATIONS HAVE TO BE DONE ON EVERY JOB.

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FLOW SHOPS: F2 ||Cmax

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Flow shops f2 cmax l.jpg

FLOW SHOPS: F2||Cmax


Flow shop scheduling n jobs m machines l.jpg

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


Flow shops l.jpg

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


Permutation flow shops l.jpg

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


Johnson s f2 cmax problem l.jpg

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


Johnson s principle l.jpg

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


Description of johnson s algorithm l.jpg

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


Example l.jpg

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

SEQUENCE:

EXAMPLE

FLOW SHOPS: JOHNSON'S RULE


Example schedule l.jpg

SEQUENCE:

5 1 4 3 2

M1

M2

t

EXAMPLE: SCHEDULE

FLOW SHOPS: JOHNSON'S RULE


A bound on the makespan l.jpg

FOR JOHNSON’S PROBLEM:

A BOUND ON THE MAKESPAN

FLOW SHOPS: JOHNSON'S RULE


Johnson s algorithm l.jpg

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


Johnson s algorithm continued l.jpg

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


Johnson s algorithm continued13 l.jpg

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


F m cmax l.jpg

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


Machine dominance f3 cmax l.jpg

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


Johnson s algorithm for 3 machines l.jpg

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


Example f3 cmax l.jpg

CONSIDER F3||Cmax WITH THE FOLLOWING JOBS:

EXAMPLE: F3||Cmax

FLOW SHOPS: JOHNSON'S RULE


Example processing times dummy machines l.jpg

SEQUENCE:

EXAMPLE: PROCESSING TIMES, DUMMY MACHINES

FLOW SHOPS: JOHNSON'S RULE


Example schedule19 l.jpg

SEQUENCE:

1 4 5 2 3

M1

M2

M3

t

EXAMPLE: SCHEDULE

FLOW SHOPS: JOHNSON'S RULE


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