Modeling: Parameters

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


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


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


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


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)


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


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


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


Large Scale Problem (man-made)

available solution space

unavailable solution space

Upper Bound

Lower Bound

approach

approach

Optimum

(Heuristic)

(Release Constraints)


Performance Measure

1. Max Completion Time (Makspan)

Cmax = Max Ci = C6

2. Minimize Inventory

fi : Reduce Inv.

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)


Compute flow time

4321 fffffi 38131285

4１２　３　 0 5 8 12 13

5 5 5 53 3 3

4 41

45332411

5 3 4 1

1234 4321 ppppfi

]1[ inpf ii


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


Scheduling Problem Representation

4 / 1 / (n / m / o )

# job# machine

objective function

f

max

max

L

T

T

f

.....


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


Sol.

1. Using FCFS (First come first serve)

4１２　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


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


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


Bi-criterion

maxT

if

Frontier

EDD

SPT

1

'

21

2211

max2

1

OOO

TO

fO i


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


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


Dynamic Problem

0

4321

43214321321211

4321

43214321

4321

222

111

1234

,,,

)(

)()(

.

ppppf

ppppCpppCppCpCbut

CCCC

aaaaCCCC

fffff

aCf

aCf

problemstaticforaCf iii


Dynamic Problem

)()()()(

)()()()(1234

)()(

)()()(

)()(

):(.

41312111

413121114321

432143211

321121111

43214321

aaaaaaaafstatic

aaaaaaaapppp

aaaappppa

pppappapa

aaaaCCCCf

timeidlewithjobfirstonlycasespecialproblemdynamicfor

),0max(

),0max(

112112

122

PaaPaC

CaC

casegeneralfor


Sol.

4１２　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


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


Sol.

4 １２　３　 0 2 6 7 10 15

3. Using FCFS then SPT (ESPT)Use SPT to arrange jobs (available jobs)

3-4-2-1

24

12315

5510

347

426

1

2

4

3

if

f

f

f

f Static (SPT)

After arrange job 3, the dynamic problem will become a Static one. Then use SPT.

if/1/4


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


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


Sol.

4１２　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


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


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


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


JIT problem

4

１

２

　３　

Slackness Rule:

Find di-pi (Job j have to start before this time)

)min(/1/4 ii TE

d1

d2 d4d3

２２１

try

or

a1

a2 a4a3


Ex.

4 １２　３　 0 5 9 10 17 26

2-4-3-1

i ai pi di di-pi1 3 5 9 42 5 3 4 13 2 4 7 34 4 1 3 2

)min(/1/4 ii TE


HW.

1. 5 / 1 / 2. 5 / 1 / 3. 5 / 1 /f T Tf

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!

Date post:	04-Feb-2016
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Modeling: Parameters

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