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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
Production Scheduling P.C. Chang, IEM, YZU.2
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
Production Scheduling P.C. Chang, IEM, YZU.3
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
Production Scheduling P.C. Chang, IEM, YZU.4
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
Production Scheduling P.C. Chang, IEM, YZU.5
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
Production Scheduling P.C. Chang, IEM, YZU.6
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)
Production Scheduling P.C. Chang, IEM, YZU.7
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[][
Production Scheduling P.C. Chang, IEM, YZU.8
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
Production Scheduling P.C. Chang, IEM, YZU.9
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
Production Scheduling P.C. Chang, IEM, YZU.10
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
Production Scheduling P.C. Chang, IEM, YZU.11
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
Production Scheduling P.C. Chang, IEM, YZU.12
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
Production Scheduling P.C. Chang, IEM, YZU.13
Is HP an optimum? Why?
jj
jj
jj
p)(r
prmin
prmin
HP
1
2
…
Production Scheduling P.C. Chang, IEM, YZU.14
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
Production Scheduling P.C. Chang, IEM, YZU.15
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
Production Scheduling P.C. Chang, IEM, YZU.16
For Another Tardiness Problems…
Production Scheduling P.C. Chang, IEM, YZU.17
I. Smith Rule
n
jjkki
n
jji
pdthatsuchkjobsallamongpp
pd
position. last the assigned be can i Job
1
1
).2(
).1(
Baker p.26
Production Scheduling P.C. Chang, IEM, YZU.18
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
Production Scheduling P.C. Chang, IEM, YZU.19
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.]
Production Scheduling P.C. Chang, IEM, YZU.20
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)
Production Scheduling P.C. Chang, IEM, YZU.21
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}
Production Scheduling P.C. Chang, IEM, YZU.22
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
Production Scheduling P.C. Chang, IEM, YZU.23
Test two exception :
ij,ttdandttd,dd iBjiBiji
or when
jBjiBiji ttdandttd,dd
Use SPT rule
Production Scheduling P.C. Chang, IEM, YZU.24
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
Production Scheduling P.C. Chang, IEM, YZU.25
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
Production Scheduling P.C. Chang, IEM, YZU.26
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
Production Scheduling P.C. Chang, IEM, YZU.27
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
Production Scheduling P.C. Chang, IEM, YZU.28
HW.
• Using Wilkerson & Irvine to solve n/1/
j tj dj1 2 102 7 143 5 184 6 205 4 23
T