Post on 02-Jan-2016
Production Scheduling P.C. Chang, IEM, YZU.1
Modeling: Parameters
• Typical scheduling parameters:
• Number of resources (m machines, operators)• Configuration and layout• Resource capabilities• Number of jobs (n)• Job processing times (pij)• Job release and due dates (resp. rij and dij )• Job weight (wij ) or priority• Setup times
Production Scheduling P.C. Chang, IEM, YZU.2
Modeling: Objective function
• Objectives and performance measures:
• Throughput, makespan (Cmax, weighted sum)• Due date related objectives (Lmax, Tmax, ΣwjTj)• Work-in-process (WIP), lead time (response time), finishe
d inventory• Total setup time• Penalties due to lateness (ΣwjLj)• Idle time• Yield
• Multiple objectives may be used with weights
Production Scheduling P.C. Chang, IEM, YZU.3
Modeling: Constraints
• Precedence constraints (linear vs. network)• Routing constraints• Material handling constraints• (Sequence dependent) Setup times• Transport times• Preemption• Machine eligibility• Tooling/resource constraints• Personnel (capability) scheduling constraints• Storage/waiting constraints• Resource capacity constraints
Production Scheduling P.C. Chang, IEM, YZU.4
Machine configurations:
• Single-machine vs. parallel-machine
• Flow shop vs. job shop
Processing characteristics:• Sequence dependent setup times and costs
– length of setup depends on jobs
– sijk: setup time for processing job j after k on machine i
– costs: waste of material, labor
• Preemptions
– interrupt the processing of one job to process another with a higher priority
Production Scheduling P.C. Chang, IEM, YZU.5
Generic notation of scheduling problem
• Machine Job Objective• characteristics characteristics function
• for example:• Pm | rj, prmp | ΣwjCj (parallel machines)• 1 | sjk | Cmax (sequence dependent• setup / traveling salesma
n)• Q2 | prec | ΣwjTj (2 machines w. different speed,
precedence rel., weighted tardiness)
Production Scheduling P.C. Chang, IEM, YZU.6
Scheduling models
• Deterministic models– input matches realization
• vs.
• Stochastic models– distributions of processing times, release and du
e dates, etc. known in advance– outcome/realization of distribution known at co
mpletion
Production Scheduling P.C. Chang, IEM, YZU.7
Symbol
: Job number
: Machine number
: Arrival time
: Processing time of job : Completion time of job
: due date
T : Tardiness
E : Earliness
ia
ip
ic
ij
ii
id
Production Scheduling P.C. Chang, IEM, YZU.8
Static V.S. Dynamic
Static
Assume all the jobs are ready at the beginning which means ai=0
Dynamic
Each job with a different arrival time. Which ai≠0
Production Scheduling P.C. Chang, IEM, YZU.9
Large Scale Problem (man-made)
available solution space
unavailable solution space
Upper Bound
Lower Bound
approach
approach
Optimum
(Heuristic)
(Release Constraints)
Production Scheduling P.C. Chang, IEM, YZU.10
Performance Measure
1. Completion Time
Cmax = Max Ci = C6
指工件集合 s 中，最晚之完工時間，即指 Cmax. (Makespan)
2. Minimize Inventory
fi : 庫存降低fi = Ci – ai ( Static Problem : ai=0)
3. Satisfy Due Date
Tardiness = Max(Ci-di , 0 )
Earliness = Max(di-Ci , 0 )
JIT = Ci-di
4. Bi-criteria Multi-Objective
(flow time = waiting time + process time)
Production Scheduling P.C. Chang, IEM, YZU.11
Compute flow time
4321 fffffi 38131285
4１ ２ ３ 0 5 8 12 13
5 5 5 53 3 3
4 41
45332411
5 3 4 1
1234 4321 ppppfi :ip
]1[ inpf ii
第 i 順位之 Pi
Production Scheduling P.C. Chang, IEM, YZU.12
Gantt Chart
６４５１２ ３
d3 c3 d1 c2 d2 c1d4 c5 c4 d5 d6
c6
＝
tardiness
c4 > d4
jobs are
ready
flow time
c2 – a2
Production Scheduling P.C. Chang, IEM, YZU.13
Scheduling Problem Representation
4 / 1 / (n / m / o )
# job# machine
objective function
f
max
max
L
T
T
f
.....
Production Scheduling P.C. Chang, IEM, YZU.14
Example:
A factory has receive 4 different orders as follows
i pi di
1 5 9
2 3 4
3 4 7
4 1 3
Please assign the production sequence of the 4 jobs to satisfy:
1. Due Date2. Min Inventory
Production Scheduling P.C. Chang, IEM, YZU.15
Sol.
1. Using FCFS (First come first serve)
4１ ２ ３ 0 5 8 12 13
38131285
13
12
8
5
4444
3333
2222
1111
if
acfc
acfc
acfc
acfc
19
100,313
50,712
40,48
00,40,
4
3
2
111
iT
MaxT
MaxT
MaxT
MaxdcMaxT
1-2-3-4
Production Scheduling P.C. Chang, IEM, YZU.16
Sol.
2. Using EDD (Earliest Due Date)
4 １２ ３ 0 1 4 8 13
2613841
13
8
4
1
44
33
22
11
if
fc
fc
fc
fc
5
40,913
10,78
00,44
00,31
4
3
2
1
iT
MaxT
MaxT
MaxT
MaxT
4-2-3-1
Production Scheduling P.C. Chang, IEM, YZU.17
Sol.
3. Using SPT (Shortest Processing Time)
The same with EDD Optimum
4 １２ ３ 0 1 4 8 13
4-2-3-1
EDD – Due Date – Tmax SPT – Inventory - Flow time
Production Scheduling P.C. Chang, IEM, YZU.18
Bi-criterion
maxT
if
Frontier
EDD
SPT
1
'
21
2211
max2
1
OOO
TO
fO i
Production Scheduling P.C. Chang, IEM, YZU.19
HW.
5 / 1 /
i pi di
1 3 13
2 2 8
3 5 9
4 4 7
5 6 10
ii fT 1
Draw the Frontier when 9.0~1.0
Production Scheduling P.C. Chang, IEM, YZU.20
Dynamic Problem Example:
4 / 1 /
i ai pi di
1 3 5 9
2 5 3 4
3 2 4 7
4 4 1 3
if
Production Scheduling P.C. Chang, IEM, YZU.21
Sol.
4１ ２ ３ 0 3 8 11 15 16
1. Using Job index 1-2-3-4
Ck > ai , C1≧ a2 - no idle timeElse, ifai > Ck, a2 > C1 - idle
36
12416
13215
6511
538
4
3
2
1
if
f
f
f
f
if/1/4
5 5-2 5+1 5-1 =18 3 3 3 = 9 4 4 = 8 1 = 1 36
or
3-5 3-2 3-4
Production Scheduling P.C. Chang, IEM, YZU.22
Sol.
4 １２ ３ 0 4 5 8 12 17
2. Using SPT. EDD 4-2-3-1
28
14317
10212
358
145
1
3
2
4
if
f
f
f
f
if/1/4
Production Scheduling P.C. Chang, IEM, YZU.23
Sol.
4 １２ ３ 0 2 6 7 10 15
3. Using FCFS then SPT (ESPT)從 Available jobs找 SPT
3-4-2-1
24
12315
5510
347
426
1
2
4
3
if
f
f
f
f Static (SPT)
排了工件 3 之後， Dynamic 問題變為 Static ，所以 SPT 每個工件輪流排第一來比較
if/1/4
Production Scheduling P.C. Chang, IEM, YZU.24
Rule ESPT
.2
1,
min.4
,.3
/.2
)(min.1
1 ,,3,2,1
1
togo
jjPCC
Pforkjobfind
iTT
UiCaforifind
stopUif
kSSkUU
Pawithkjobfind
jTSnU
kjj
kTk
ji
kkUk
Production Scheduling P.C. Chang, IEM, YZU.25
Ex:ESPT
1. find Min
2.
3. for min 531 124 PPPPi
24
12315
5510
347
426
1
2
4
3
if
f
f
f
f
23 aai
ikkk aallCpaC 6423
Production Scheduling P.C. Chang, IEM, YZU.26
Sol.
4１ ２ ３ 0 3 8 11 15 16
1. Using Job index 1-2-3-4
28
130,316
80,715
70,411
00,98
4
3
2
1
iT
MaxT
MaxT
MaxT
MaxT
iT/1/4
Production Scheduling P.C. Chang, IEM, YZU.27
Sol.
4 １２ ３ 0 4 5 8 12 17
2. Using SPT
3. Using EEDD (next slide)
4-2-3-1
19
80,917
50,712
40,48
20,35
1
3
2
4
iT
MaxT
MaxT
MaxT
MaxT
iT/1/4
Production Scheduling P.C. Chang, IEM, YZU.28
Rule EEDD
.2
1,
min.4
,.3
/.2
min.1
1 ,,3,2,1
1
togo
jjPCC
dforkjobfind
iTT
UiCaforifind
stopUif
kSSkUU
PaCawithkjobfind
jTSnU
kjj
kTk
ji
kkjkUk
Production Scheduling P.C. Chang, IEM, YZU.29
Ex:EEDD
1. find Min
2.
3. for min
let4. Return 3
23 aai
ikkk aallCpaC 6423
16
6)0,915(
6)0,410(
4)0,37(
0)0,76(
1
2
4
3
iT
MaxT
MaxT
MaxT
MaxT
6max T
34 ddi
7164 CPCC iki
4CCCC kik
1551091
103742
121
242
PCCd
PCCd
Production Scheduling P.C. Chang, IEM, YZU.30
HW.
1. 5 / 1 / 2. 5 / 1 /f T
i ai pi di
1 2 6 15
2 7 2 13
3 5 8 25
4 1 5 30
5 9 3 28
Find an optimal solution!