Power Point Presentation on Flow Shop Scheduling

This is a ppt on flow shop scheduling

1. Production Scheduling P.C. Chang, IEM, YZU.1
Flow Shop Scheduling

2. Production Scheduling P.C. Chang, IEM, YZU.2
Definitions
•Contains mdifferent machines.
•Each job consists moperators in different machine.
•The flow of work is unidirectional.
•Machines in a flow shop = 1,2,…….,m
•The operations of job i, (i,1) (i,2) (i ,3)…..(i, m)
•Not processed by machine k, P( i , k) = 0

3. Production Scheduling P.C. Chang, IEM, YZU.3
Flow Shop Scheduling
Baker p.136
The processing sequence on each machine are all the same.
1
2
.....
M
23154
23154
Flow shop
Job shop
n!-flow shop permutation schedule
n!.n! …….n!-Job shopm
)!n( k
)!n(
m
k : constraint
(∵routing problem)
13245
or

4. Production Scheduling P.C. Chang, IEM, YZU.4
Workflow in a flow shop
Machine
1
Machine
2
Machine
3
Machine
M-1
Machine
M
….
Input
output
Machine
1
Machine
2
Machine
3
Machine
M-1
Machine
M
….
Input
outputoutputoutputoutputoutput
Input Input Input Input
Type 1.
Type 2.
Baker p.137

5. Production Scheduling P.C. Chang, IEM, YZU.5
Johnson’s Rule
Note:
Johnson’s rule can find an optimum with two machines
Flow shop problem for makespan problem.
Baker p.142.filledaresequenceinpositions
alluntil1steptoreturnandionconsideratfromjobassignedtheRemove:3Step
.3steptogo.sequencein
positionavailablefirsttheinjobtheplace,2machinerequiresmintheIf:2Step
.3steptogo.sequencein
positionavailablefirsttheinjobtheplace,1machinerequiresmintheIf:2Step
Find:1Step
tb
ta
}t,{tmin
i2i1i

6. Production Scheduling P.C. Chang, IEM, YZU.6
Ex.
j1 2 3 45
tj13 5 1 67
tj26 2 2 65
Stage U Min t
jkAssignment Partial Schedule
1 1,2,3,4,5t
31 3=[1] 3 x x x x
2 1,2,4,5 t
22 2=[5] 3 x x x 2
3 1,4,5 t
11 1=[2] 3 1 x x 2
4 4,5 t
52 5=[4] 3 1 x 5 2
5 4 t
11 4=[3] 3 1 4 5 2

7. Production Scheduling P.C. Chang, IEM, YZU.7
Ex.
31 4 5 2
3 1 4 5
24
M1
M2
The makespan is 24
2

8. Production Scheduling P.C. Chang, IEM, YZU.8
The B&B for Makespan Problem
The Ignall-Schrage Algorithm (Baker p.149)
-A lower bound on the makespan associated with any
completion of the corresponding partial sequence σ is
obtained by considering the work remaining on each
machine. To illustrate the procedure for m=3.
For a given partial sequence σ, let
q1= the latest completion time on machine 1 among jobs in σ.
q2= the latest completion time on machine 2 among jobs in σ.
q3= the latest completion time on machine 3 among jobs in σ.
The amount of processing yet required of machine 1 is 
'j
1jt

9. Production Scheduling P.C. Chang, IEM, YZU.9
The Ignall-Schrage Algorithm
In the most favorable situation, the last job
1)Encounters no delay between the completion of one operation
and the start of its direct successor, and
2)Has the minimal sum (tj2+tj3) among jobs j belongs to σ’
Hence one lower bound on the makespan is
A second lower bound on machine 2 is
A lower bound on machine 3 is
The lower bound proposed by Ignall and Schrage is}tt{mintqb
3j2j
'j'j
1j11 
 }t{mintqb
3j
'j'j
2j22

 
'j
3j33 tqb }b,b,bmax{B
321

10. Production Scheduling P.C. Chang, IEM, YZU.10
The Ignall-Schrage Algorithm
M1
M2
M3
tk1
tk2
tk3
……..
……..
……..
q1
q2
q3 b1
M2
M3
tk2
tk3
……..
……..
q2
q3 b2
Job in σ’

11. Production Scheduling P.C. Chang, IEM, YZU.11
Ex. B&B
j1234
tj1311710
tj241912
tj3105132
m=3
For the first node: σ =137)37,31,37max(B
372017b
312227b
376283b
boundlowerThe
17tttq
7ttq
3tq
3
2
1
1312113
12112
111







 













212
139
51
min6

12. Production Scheduling P.C. Chang, IEM, YZU.12
Ex.
Partial Sequence( q1 , q2 , q3 )(b1,b2,b3)B
1xxx ( 3 , 7 , 17 )( 37 , 31 , 37 )37
2xxx ( 11 , 12 , 17 )( 45 , 39 , 42 )45
3xxx ( 7 , 16 , 29 )( 37 , 35 , 46 )46
4xxx ( 10 , 22 , 24 )( 37 , 41 , 52 )52
12xx ( 14 , 15 , 22 )( 45 , 38 , 37 )45
13xx ( 10 , 19 , 32 )( 37 , 34 , 39 )39
14xx ( 13 , 25 , 27 )( 37 , 40 , 45 )45
132x ( 21 , 22 , 37 )( 45 , 36 , 39 )45
134x ( 20 , 32 , 34 )( 37 , 38 , 39 )39
1 2
1 2
1 2
0 3 14
7 15
172245
212
139
min)107(14
}tt{mintqb
3j2j
'j'j
1j11













13. Production Scheduling P.C. Chang, IEM, YZU.13
Ex. B&B
P0
1xxx 2xxx 3xxx 4xxx
B=
37
B=4
5
B=4
6
B=5
2
12xx 13xx 14xx
B=4
5
B=4
5
B=3
9
132x 134x
B=4
5
B=3
9

14. Production Scheduling P.C. Chang, IEM, YZU.14
Refined Bounds










'j
1j122 ]tmin[q,qmax'q
Modification1:










 'j
2j1j1
'j
2j233 ]ttmin[q,]tmin[q,qmax'q
The use of q2 in the calculation of b2 ignores the possibility that
the starting time of job j on the machine 2 may be constrained
by commitments on machine1. Hence:
consider idle time

15. Production Scheduling P.C. Chang, IEM, YZU.15
Refined Bounds
Previous : Machine-based bound
Modification2 : Job-based bound 
 
}b,b,Bmax{'B
t,tminttmax'qb
t,tmintttmaxqb
54
kj
'j
3j2j3k2k
'k
25
kj
'j
3j1j3k2k1k
'k
14



























Modification2: (McMahon and Burton)

16. Production Scheduling P.C. Chang, IEM, YZU.16
Refined Bounds
Obviously, B’>=B, This means that the combination
of machine-based and job-based bounds represented
by B’ will lead to a more efficient search of the
branching tree in the sense that fewer nodes will be
created.

17. Production Scheduling P.C. Chang, IEM, YZU.17
Hw.
a.Find the min makespan using the basic Ignall-Schrage
algorithm. Count the nodes generated by the branching
process.
b.Find the min makespan using the modified algorithm.
j1234
tj1137262
tj231296
tj3121671
Consider the following four-job three-machine problem

18. Production Scheduling P.C. Chang, IEM, YZU.18
Heuristic Approaches
Traditional B&B:
•The computational requirements will be severe for large
problems
•Even for relatively small problems, there is no guarantee
that the solution can be obtained quickly,
Heuristic Approaches
•can obtain solutions to large problems with limited
computational effort.
•Computational requirements are predictable for problem of
a given size.

19. Production Scheduling P.C. Chang, IEM, YZU.19
Palmer
Palmer proposed the calculation of a slope index, sj, for
each job.1,j2,j2m,j1m,jm,jj t)1m(t)3m(t)5m(t)3m(t)1m(s 
 
Then a permutation schedule is constructed using the
job ordering]n[]2[]1[ sss  

20. Production Scheduling P.C. Chang, IEM, YZU.20
Gupta
Gupta thought a transitive job ordering in the form of follows
that would produce good schedules. Where}tt,ttmin{
e
s
3j2j2j1j
j
j


Where





3j1j
3j1j
j
ttif1
ttif1
e

21. Production Scheduling P.C. Chang, IEM, YZU.21
Gupta
Generalizing from this structure, Gupta proposed that for m>3,
the job index to be calculated is}tt{min
e
s
1k,jjk
1mk1
j
j




Where





jm1j
jm1j
j
ttif1
ttif1
e

22. Production Scheduling P.C. Chang, IEM, YZU.22
CDS
Its strength lies in two properties:
1.It use Johnson’s rule in a heuristic fashion
2.It generally creates several schedules from which a “best”
schedule can be chosen.
The CDS algorithm corresponds to a multistage use if
Johnson’s rule applied to a new problem, derived from the
original, with processing times and . At stage 1, 1j't 2j't jm2j1j1j t'tandt't 

23. Production Scheduling P.C. Chang, IEM, YZU.23
CDS
In other words, Johnson’s rule is applied to the first and mth
operations and intermediate operations are ignored. At stage 2,1m,jjm2j2j1j1j tt'tandtt't

That is, Johnson’s rule is applied to the sums of the first two
and last two operation processing times. In general at stage i,



i
1k
1km,j2j
i
1k
jk1j
t'tandt't

24. Production Scheduling P.C. Chang, IEM, YZU.24
Ex.
Palmer:
j12345
tj164395
tj281956
tj321586
3712453
22468
2211
54321
1313



M
sssss
tttmtms
jjjjj
Gupta:36M21435
11
1
s
13
1
s
12
1
s
2
1
s
10
1
s
54321


CDS: 3-5-4-1-2 M=35

25. Production Scheduling P.C. Chang, IEM, YZU.25
HW.
Let
1.Use Ignall-Schrage & McMahon-Burton
to solve
2.Use Palmer, Gupta, CDS to solve this
problem.
j12345
tj1811769
tj2325711
tj36571310}3,1{ 22
1 2 3 4 5 13 31
, , , , ,
xxx xxx
b b b b b of P P