production schedulingp.c. chang, iem, yzu. 1 how to schedule ?? how to find 1. an efficient...

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

How to schedule ??

How to find 1. an efficient Heuristic? 2. the optimal solution?

InventoryFFflowtimeSPT j ,

TardinessEDD

schedulelistmachineparallelLPT


Composite priority rule that is mixture of 3 basic priority rules:

• ATC ( apparent tardiness rule ) is comb. of:• 1. Weighted Shortest Processing Time First• 2. Earliest Due Date First• 3. Minimal slack

• ATCS ( ATC with setups )• 4. Shortest Setup Time First


Composite dispatching: Apparent Tardiness Cost (ATC)

• ATC combines MS rule and WSPT rule• k1=due date scaling par.• (look-ahead parameter)

k1 function of Due Date Range factor:

))0,max(

(1)( pk

tpd

j

jj

jj

epw

tI

5.0Rfor,R26k5.0Rfor,R5.4k

1

1

maxminmax C

)dd(R


Composite dispatching: Apparent Tardiness Cost with Setups (ATCS)

• ATCS combines MS rule, WSPT rule and SST rule:• k1=due date scaling par.• k2=setup time scaling pa

r. k1 and k2 functions of:

• Due Date tightness

• Due Date Range

• Setup Time Severity

maxminmax C

)dd(R

maxCd1

ps

)sk

s()pk

)0,tpdmax((

j

jj

2

lj

1

jj

eepw)t(I


HW.

• Please solve the following problem using ATC and ATCS Rules.

j pj dj1 5 92 3 103 7 124 6 185 2 5

Sij 1 2 3 4 51 0 5 7 2 32 5 0 4 6 53 7 4 0 3 44 2 6 3 0 25 3 5 4 2 0


Dispatching rules: multiple passes

• drawback of priority rules:may yield bad solutions

• SOLUTION: Use multiple passes

• Multi-pass priority rule based methods:• 1. Multi-priority rule procedures (repeat dispatc

hing procedure with different disp. rules)• 2. Sampling procedures (each job has a probabili

ty to be dispatched)


Weighted Problem

jjjjwj CwfwF jjj Cfr ,0

]1[P ]2[P ]4[P]3[P

)1(W )2(W )3(W )4(W

)(

)(

)(

]4[]3[]2[]1[]4[]4[

]3[]2[]1[]3[]3[

]2[]1[]2[]2[

]1[]1[]1[

ppppwf

pppwf

ppwf

pwf

]4[]4[]3[]4[]2[]4[]1[]4[

]3[]3[]2[]3[]1[]3[

]2[]2[]1[]2[

]1[]1[

pwpwpwpw

pwpwpw

pwpw

pw

... ... ... ... ...

j

n

1j

n

ji]j[]i[j p)w(f

n

j

n

j

n

jjjj pwpwpw

1 2 3]3[][]2[][]1[][


WSPT – Weighted SPT

j

j

wp

WSPT

j

j

WP

WP

WP

.....2

2

1

1

,SPTonBasedwp

j

j Sort from small to large.j

jw

p


Example

wj F/0r/1/4

SPT rule:jFwjF

WSPT rule:

j pj wj pj/wj

1 3 1 32 2 2 13 7 3 2.3334 5 2 2.5

2 5 10 17 = 342*2 5*1 10*2 17*3 = 80

2 9 14 17 = 422*2 9*3 14*2 17*1 = 76

jFwjF

412 　 3　

4 12 　 3　


Dynamic WSPT

jj corfrn /0/1/

Problem: jj cr-cf,0 jjjrwhen

Problem toConverted jc

wjjj corcrn /0/1/

makespanC max

j

jjcwMinMinZ

Min Total Completion Time


Heuristic HP [Hariri and Potts]

• HP procedure Step1:

Step2:

Step3:

Step4:

Step5:

.

},{,,}{\

,,

,

,

.,,,,,

3steptogo

rMintMaxtotherwisestopUIf

iUUtwptt1kk

QjwpMinwithijobFind

UjtrjQptt

rMinwithijobFind0k0tS

n21UN

jj

iii

j

j

j

j

i

jj

EWSPT


Ex.j rj pj wj1 4 1 0.42 9 8 0.63 1 9 0.34 3 10 0.75 2 12 0.8

WSPTthen

QUU

},,,{}Uj,taj{Q.

CC

ptt

},{Max}t,r{Maxt,j,rMin.

.t},,,,{US},,,,{N:HP

j

j

j

jjUj

54212

10

1091

101311

05432154321

3

3 1 10


Is HP an optimum? Why?

jj

jj

jj

p)(r

prmin

prmin

HP

1

2

…


Heuristic PPC

Step1:Step2:

Step3:

Step4:

Step5:

.steptogo

}rMin,t{Maxt,otherwisestop,UIf}i{\UU

tw,ptt,kk

Qj),Y,X(MinwithijobFind

timeidle:xw

pxMinY,

wp

MinX

Uj,trjQ

ptt

wpr

Minor)pr(MinorrMinwithijobFind

.k,t,Sn,,,UN

jj

iii

jj

jj

jj

j

j

j

i

j

jj

jjjjjj

3

1

0021


HW.Use to solve the problem

jjj CW/0r/1/5

j rj pj wj

1 4 1 0.42 9 8 0.63 1 9 0.34 3 10 0.75 2 12 0.8

PPCHeuristic.

p)(r.

prmin.

jj

jj

3

12

21


For Another Tardiness Problems…


I. Smith Rule

n

jjkki

n

jji

pdthatsuchkjobsallamongpp

pd

position. last the assigned be can i Job

1

1

).2(

).1(

Baker p.26


EX. j pj dj

1 4 162 7 163 1 84 6 215 3 9

SPTusejobsother

palllastjobppbut

},,,{j,pdd).(

.positionlastthetoassignedisjob

},,,,{jpd).(

j

jj

jj

82

532115162

4

5432121211

21

21

4

42 8 15 21

153

SPT


II. Hodgson’s Algorithm

.2

.,max.3][,

.,.2,.1

steptogoEintimescompletiontherevise

LtojobthisremovekipwithjobtheFindStep

kjobdelayfirstthefindotherwisestoplateareEinjobsnoIfStep

LEDDwithEinjobsallPlaceStep

i

Baker p.27

Sule p.37

TNMinimizeTo [Minimize the number of tardy jobs.]


EX.1j pj dj

1 1 22 5 73 3 84 9 135 7 11Stage 1

Step1. Initialize E={1-2-3-5-4} L=ψStep2. Job 3 is the 1st late jobStep3. Job 2 is removed from E. E={1-3-5-4} L={2}

Stage 2Step1. Job 4 is the 1st late jobStep2. Job 4 is removed from E. E={1-3-5} L={2-4}

Stage 3Step1. No jobs in the E are late. An optimal sequence is

1-3-5-2-4 (NT=2)


EX.2j pj dj

1 10 152 15 253 8 274 12 325 22 40

Step1. Select the 1st job, Ttemp=T+10=10 < d1 , so S={1}, T= Ttemp=10

Step2. Examine job 2. Ttemp=10+15=25 d≦ 2, so S={1,2}T= Ttemp=25

Step3. For job 3. Ttemp=25+8=33 > d3, find job 2 with Max P in S ∵d2>d3 ,so remove it, T=25-15=10, Ttemp=10+8=18 < d3, so S={1,3}, T=Ttemp=18

Step4. For job 4 Ttemp=18+12=30 < d4, so S={1,3,4}T= Ttemp=30

Step5. For job 5 Ttemp=52 > d5, find job 4 with Max P in S, but d4<d5, so job 5 is not selected.

Step6. The Max number of jobs can be processed on time is three. And the sequence is {1,3,4}


III.Wilkerson-IrwinBaker p.30

iC jC

N : Set of all jobsS : Scheduled set Q : Unscheduled setS Q = N∪

ji dd

iBi Ptd

ji

S

Q

Bt

avaiablemachinetB

T/n/m

n

TT

n

ii

1


Test two exception :

ij,ttdandttd,dd iBjiBiji

or when

jBjiBiji ttdandttd,dd

Use SPT rule


S

Q

: the index of the last job on the schedule list: the index of the pivot job

: the index of the first job on the unscheduled list

F


Test 0: Place all the jobs on the unscheduled list in EDD order

Test 1: },d,dmax{}t,tmax{F or if tt

then job and and repeat test 1,other wise test2

Test 2: }d,dmax{}t,tmax{F and tt unscheduled list , and , and proceed to test 3

Test 3: }d,dmax{}t,tmax{tF or if tt and go to test 1&2 otherwise test 4

Test 4: }d,dmax{}t,tmax{tF and ttjump , remove

?thanbettertest

?thanbettertest


EX.j Pj=tj

dj

1 5 62 7 83 6 94 4 10Stage 1- }d,dmax{}t,tmax{F

8,6max7,5max0

)1( jobArrange

test 1:

Stage 2- test 1: 9,8max6,7max5 test 1 fail )( thanbetternotis

and,max,max 68575 5t7t test 2:test 2 success )( thanbetteris

3

21

4…

remove 2,3


EX.Stage 3- )3( joblistscheduledtoArrange

test 3:

65and9,6max6,5max55

Stage 3’- test 1: 10,8max4,7max11 fail

test 2: 67and9,8max6,7max11 success 2,4

test 3: 10,9max4,6max611 fail

test 4: 46and10,9max4,6max611 success

stage Scheduled list Unscheduled list Decision result1 Empty 0 - 1 2 2-3-42 1 5 1 2 3 3-43 1-3 11 3 2 4 4 jump3‘ 1 5 1 - 4 2-34 1-4 9 4 2 3 3

F 1

43

3

Final sequence is 1-4-3-2

Job 1 enter Scheduled list

2 vs. 4

2 vs. 3

3 vs. 4

4 Enter

2 leave


HW.

• Using Wilkerson & Irvine to solve n/1/

j tj dj1 2 102 7 143 5 184 6 205 4 23

T

production schedulingp.c. chang, iem, yzu. 1 how to schedule ?? how to find 1. an efficient...

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