8/9/2019 Scheduling 7 and sequencing

1/25

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

Fl w Sh p Scheduling

8/9/2019 Scheduling 7 and sequencing

2/25

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,! (i,2! (i ,"!#..(i, m!

$ot processed b% machine k, &( i , k! = '

8/9/2019 Scheduling 7 and sequencing

3/25

Production Scheduling P.C. Chang, IEM, YZU."

Flow Shop Schedulingaker p.")

The processin* se+uence on each machine are all the same.

2.....

M

2 " -

2 " -

low shop

/ob shop

n0 1 flow shop permutation schedulen0.n0 ##.n0 1 /ob shop

m!0n(

k

!0n( mk constraint

( routin* problem!

" 2 -

or

8/9/2019 Scheduling 7 and sequencing

4/25

Production Scheduling P.C. Chang, IEM, YZU.-

Workflow in a flow shop

Machine

1

Machine

2

Machine

3

Machine

M1

Machine

M!.

Input

output

Machine

1

Machine

2

Machine

3

Machine

M1

Machine

M!.

Input

outputoutputoutputoutputoutput

Input Input Input Input

Type 1.

Type 2.

aker p."3

8/9/2019 Scheduling 7 and sequencing

5/25

Production Scheduling P.C. Chang, IEM, YZU.

Johnsons Rule

$ote

/ohnson4s rule can find an optimum with two machines

low shop problem for makespan problem.

aker p.-2

.filledarese+uenceinpositions

alluntilsteptoreturnandionconsideratfromjobassi*nedthe5emo6e"7tep ."stepto*o.se+uencein

positiona6ailablefirsttheinjobtheplace,2machinere+uiresminthe8f27tep

."stepto*o.se+uencein

positiona6ailablefirsttheinjobtheplace,machinere+uiresminthe8f27tep

ind7tep

tb

ta

}t,{tmin i2i1i

8/9/2019 Scheduling 7 and sequencing

6/25

Production Scheduling P.C. Chang, IEM, YZU.)

Ex.

j 2 " -

tj " ) 3

tj2 ) 2 2 )

7ta*e 9 Min tjk :ssi*nment &artial 7chedule

,2,",-, t" "=;< "

2 ,2,-, t22 2=;< " 2

" ,-, t =;2< " 2

- -, t2 =;-< " 2

- t -=;"< " - 2

8/9/2019 Scheduling 7 and sequencing

7/25

8/9/2019 Scheduling 7 and sequencing

8/25Production Scheduling P.C. Chang, IEM, YZU.>

The BB for !akespan "ro#le$

The Ignall-Schrage Algorithm (Baker p.149)1 : lower bound on the makespan associated with an%

completion of the correspondin* partial se+uence ? is

obtained b% considerin* the work remainin* on each

machine. To illustrate the procedure for m=".

or a *i6en partial se+uence ?, let

q1= the latest completion time on machine amon* jobs in ?.

q2= the latest completion time on machine 2 amon* jobs in ?.q3= the latest completion time on machine " amon* jobs in ?.

The amount of processin* %et re+uired of machine is @j

jt

8/9/2019 Scheduling 7 and sequencing

9/25Production Scheduling P.C. Chang, IEM, YZU.A

The %gnall&Schrage 'lgorith$

8n the most fa6orable situation, the last job! Encounters no dela% between the completion of one operation

and the start of its direct successor, and

2! Bas the minimal sum (tj2tj"! amon* jobs j belon*s to ?4

Bence one lower bound on the makespan is

: second lower bound on machine 2 is

: lower bound on machine " is

The lower bound proposed b% 8*nall and 7chra*e is

DttEmint+b "j2j@j@j

j +++=

DtEmint+b "j@j@j

2j22

++=

+= @j"j"" t+b

Db,b,bma=E "2=

8/9/2019 Scheduling 7 and sequencing

10/25Production Scheduling P.C. Chang, IEM, YZU.'

The %gnall&Schrage 'lgorith$

M1

M2

M3

tk1

tk2

tk3

.

.

.

.

.

.

q1

q2

q3 b1

M2

M3

tk2

tk3

.

.

.

.

q2

q3 b2

/ob in ?4

8/9/2019 Scheduling 7 and sequencing

11/25Production Scheduling P.C. Chang, IEM, YZU.

Ex. BB

j 2 " -

tj " 3 '

tj2 - A 2

tj" ' " 2

m=3or the first node ? =

"3!"3,","3ma=("32'3b

"2223b

"3)2>"b

boundlowerThe3ttt+

3tt+

"t+

"

2

"2"

22

== =+=

=++==++=

==++=

=+===

+

++=

22

"A

min)

8/9/2019 Scheduling 7 and sequencing

12/25Production Scheduling P.C. Chang, IEM, YZU.2

Ex. &artial7e+uence( + , +2 , +" ! (b,b2,b"!

( " , 3 , 3 ! ( "3 , " ,"3 !

"3

2 ( , 2 , 3 ! ( - , "A ,-2 !

-

" ( 3 , ) , 2A ! ( "3 , " ,-) !

-)

- ( ' , 22 , 2- ! ( "3 , - ,2 !

2

2 ( - , , 22 ! ( - , "> ,"3 !

-

" ( ' , A , "2 ! ( "3 , "- ,"A ! "A

- ( " , 2 , 23 ! ( "3 , -' ,- !

-

"2 ( 2 , 22 , "3 ! ( - , ") ,

"A !

-

1 2

1 2

1 2

' " -

3

3 22 -

22

"Amin!'3(-

DttEmint+b "j2j@j@jj

=

+

++++=

+++=

8/9/2019 Scheduling 7 and sequencing

13/25Production Scheduling P.C. Chang, IEM, YZU."

Ex. BB

&'

2 " -

B=

37

B=

45

B=

46

B=

52

12xx 13xx 14xx

B=

45

B=

45

B=

39

132x 134x

B=

45

B=

39

8/9/2019 Scheduling 7 and sequencing

14/25Production Scheduling P.C. Chang, IEM, YZU.-

Refined Bounds

+= @j

j22

8/9/2019 Scheduling 7 and sequencing

15/25Production Scheduling P.C. Chang, IEM, YZU.

Refined Bounds

&re6ious Machine1based bound

Modification2 /ob1based bound

[ ]

[ ]

Db,b,ma=E@

t,tminttma=@+b

t,tmintttma=+b

-

kj@j

"j2j"k2k@k

2

kj@j

"jj"k2kk@k

-

=

+++=

++++=

Modification2 (McMahon and urton!

8/9/2019 Scheduling 7 and sequencing

16/25

8/9/2019 Scheduling 7 and sequencing

17/25Production Scheduling P.C. Chang, IEM, YZU.3

(w.

a. ind the min makespan usin* the basic 8*nall17chra*eal*orithm. Count the nodes *enerated b% the branchin*

process.

b. ind the min makespan usin* the modified al*orithm.

j 2 " -

tj " 3 2) 2

tj2 " 2 A )

tj" 2 ) 3

Consider the followin* four1job three1machine problem

8/9/2019 Scheduling 7 and sequencing

18/25

Production Scheduling P.C. Chang, IEM, YZU.>

(euristic 'pproaches

Traditional H

The computational re+uirements will be se6ere for lar*e

problems

E6en for relati6el% small problems, there is no *uaranteethat the solution can be obtained +uickl%,

Beuristic :pproaches

can obtain solutions to lar*e problems with limited

computational effort.

Computational re+uirements are predictable for problem of

a *i6en siIe.

8/9/2019 Scheduling 7 and sequencing

19/25

Production Scheduling P.C. Chang, IEM, YZU.A

"al$er

&almer proposed the calculation of a slope inde, sj, foreach job.

,j2,j2m,jm,jm,jj t!m(t!"m(t!m(t!"m(t!m(s +++=

Then a permutation schedule is constructed usin* the

job orderin*

8/9/2019 Scheduling 7 and sequencing

20/25

Production Scheduling P.C. Chang, IEM, YZU.2'

)upta

Jupta thou*ht a transiti6e job orderin* in the form of followsthat would produce *ood schedules. Khere

Dtt,ttmin

es

"j2j2jj

jj

++

=

Khere

8/9/2019 Scheduling 7 and sequencing

21/25

Production Scheduling P.C. Chang, IEM, YZU.2

)upta

JeneraliIin* from this structure, Jupta proposed that for mG",the job inde to be calculated is

Dttmin

es

k,jjkmk

jj

+ +

=

Khere

8/9/2019 Scheduling 7 and sequencing

22/25

8/9/2019 Scheduling 7 and sequencing

23/25

Production Scheduling P.C. Chang, IEM, YZU.2"

*DS

8n other words, /ohnson4s rule is applied to the first and mth

operations and intermediate operations are i*nored. :t sta*e

2,m,jjm2j2jjj tt@tandtt@t +=+=

That is, /ohnson4s rule is applied to the sums of the first two

and last two operation processin* times. 8n *eneral at sta*e i,

===

+

=

i

k

km,j2j

i

k

jkj t@tandt@t

8/9/2019 Scheduling 7 and sequencing

24/25

Production Scheduling P.C. Chang, IEM, YZU.2-

Ex.

&almer

j 2 " -

tj ) - " A

tj2 > A )tj" 2 > )

( ) ( )

"32-"

22-)>

22

-"2

""

=

=====

==

M

sssss

tttmtmsjjjjj

Jupta

")M2-"

s

"

s

2

s2

s

'

s -"2

=

=====

CN7 "11-112 M="

8/9/2019 Scheduling 7 and sequencing

25/25

Production Scheduling PC Chang IEM YZU2

(W.

Oet

. 9se 8*nall17chra*e H McMahon1urtonto sol6e

2. 9se &almer, Jupta, CN7 to sol6e thisproblem.

j 2 " -

tj > 3 ) A

tj2 " 2 3

tj" ) 3 " '

D",E=

2 2

2 " - " ", , , , ,xxx xxxb b b b b of P P