Slid

e 1

A T

our

of M

odel

ing

Tec

hniq

ues

John

Hoo

ker

Car

negi

e M

ello

n U

nive

rsity

Mar

ch 2

008

Slid

e 2

Ou

tlin

e

•M

ixed

inte

ger

linea

r(M

ILP

) m

odel

ing

•D

isju

nctiv

e m

odel

ing

•E

xam

ples

: fix

ed c

harg

e pr

oble

ms,

faci

lity

loca

tion,

lo

t siz

ing

with

set

up c

osts

.

•K

naps

ack

mod

elin

g

•E

xam

ples

: Fre

ight

pac

king

and

tran

sfer

•C

onst

rain

t pro

gram

min

g m

odel

s

•E

xam

ple:

Em

ploy

ee s

ched

ulin

g

•In

tegr

ated

Mod

els

•E

xam

ples

: Pro

duct

con

figur

atio

n, m

achi

ne s

ched

ulin

g

Slid

e 3

Mix

ed In

teg

er/L

inea

r M

od

elin

g

MIL

P M

odel

ing

Sys

tem

sM

ILP

Mod

els

Dis

junc

tive

Mod

elin

gK

naps

ack

Mod

elin

g

Slid

e 4

MIL

P M

od

elin

g S

yste

ms

•C

omm

erci

al m

odel

ing

syst

ems

•AM

PL

•G

AM

S

•AIM

MS

Slid

e 5

MIL

P M

od

elin

g S

yste

ms

•C

omm

erci

al m

odel

ing

syst

ems

with

ded

icat

ed s

olve

rs

•O

PL

Stu

dio

(run

s C

PLE

X)

•X

pres

s-B

CL

(run

s X

pres

s-M

P)

•X

pres

s-M

osel

(ru

ns X

pres

s-M

P)

•E

xcel

and

Qua

ttro

Pro

, Fro

ntlin

e S

yste

ms

(spr

eads

heet

bas

ed)

•LI

NG

O

•M

INO

PT

(al

so n

onlin

ear)

Slid

e 6

MIL

P M

od

elin

g S

yste

ms

•N

on-c

omm

erci

al m

odel

ing

syst

ems

•Z

IMP

L

•G

nu M

athp

rog

(GM

PL)

Slid

e 7

An

mix

ed in

teg

er li

nea

r p

rog

ram

min

g

(MIL

P)

mod

el h

as th

e fo

rmm

in ,0

inte

ger

cxdy

Ax

byb

xy

y

++

≥≥

MIL

P m

od

els

Slid

e 8

A p

rin

cip

led

ap

pro

ach

to

MIL

P m

od

elin

g

•M

ILP

mod

elin

g co

mbi

nes

two

dist

inct

kin

ds o

f mod

elin

g.

•M

odel

ing

of s

ubse

ts o

f con

tinuo

us s

pace

, usi

ng 0

-1 a

uxili

ary

varia

bles

.

•K

naps

ack

mod

elin

g, u

sing

gen

eral

inte

ger

varia

bles

.

•M

ILP

can

mod

el s

ubse

ts o

f con

tinuo

us s

pace

that

are

uni

ons

of

poly

hedr

a.

•…

that

is, r

epre

sent

ed b

y di

sjun

ctio

ns o

f lin

ear

syst

ems.

•S

o a

prin

cipl

ed a

ppro

ach

is to

ana

lyze

the

pro

blem

as

disj

unct

ions

in

tege

r

of li

near

+

kn

apsa

ck

sy

stem

s

i

nequ

aliti

es

Slid

e 9

Dis

jun

ctiv

e M

od

elin

g

Th

eore

m. A

sub

set o

f con

tinuo

us s

pace

can

be

repr

esen

ted

by a

n M

ILP

mod

el if

and

onl

y if

it is

the

unio

n of

fini

tely

man

y po

lyhe

dra

havi

ng t

he s

ame

rece

ssio

n co

ne.

Pol

yhed

ron

Rec

essi

on c

one

of p

olyh

edro

n

Uni

on o

f pol

yhed

ra w

ith th

e sa

me

rece

ssio

n co

ne

(in th

is c

ase,

the

orig

in)

Slid

e 10

Mo

del

ing

a u

nio

n o

f p

oly

hed

ra

Sta

rt w

ith a

dis

junc

tion

of li

near

sy

stem

s to

rep

rese

nt th

e un

ion

of p

olyh

edra

.

The

kth

pol

yhed

ron

is {

x | A

k x ≥

b}

()

min

kk

k

cx

Ax

b≥

∨

Intr

oduc

e a

0-1

varia

ble

yk

that

is

1 w

hen

xis

in p

olyh

edro

n k.

Dis

aggr

egat

e x

to c

reat

e an

xk

for

each

k.

{}

min

, al

l

1

0,1

kk

kk

kk

k

k

k

cx

Ax

by

k

y

xx

y

≥ =

= ∈

∑

∑

Slid

e 11

Tig

ht

Rel

axat

ion

s

•B

asic

fac

t:T

he c

ontin

uous

rel

axat

ion

of th

e di

sjun

ctiv

e M

ILP

m

odel

pro

vide

s a

con

vex

hu

ll re

laxa

tio

nof

the

disj

unct

ion.

•T

his

is th

e tig

htes

t pos

sibl

e lin

ear

mod

el fo

r th

e di

sjun

ctio

n.

Uni

on o

f pol

yhed

raC

onve

x hu

ll re

laxa

tion

(tig

htes

t lin

ear

rela

xatio

n)

Slid

e 12

Exa

mp

le: F

ixed

ch

arg

e fu

nct

ion

Min

imiz

e a

fixed

cha

rge

func

tion:

x 1

x 2

2

12

11

1min

0if

0

if 0

0x

xx

fcx

x

x

=

≥

+

>

≥

Slid

e 13

Fix

ed c

harg

e pr

oble

m

Min

imiz

e a

fixed

cha

rge

func

tion:

2

12

11

1min

0if

0

if 0

0x

xx

fcx

x

x

=

≥

+

>

≥

x 1

x 2

Fea

sibl

e se

t

Slid

e 14

Fix

ed c

harg

e pr

oble

m

Min

imiz

e a

fixed

cha

rge

func

tion:

2

12

11

1min

0if

0

if 0

0x

xx

fcx

x

x

=

≥

+

>

≥

x 1

x 2

Uni

on o

f tw

o po

lyhe

dra

P1,

P2

P1

Slid

e 15

Fix

ed c

harg

e pr

oble

m

Min

imiz

e a

fixed

cha

rge

func

tion:

2

12

11

1min

0if

0

if 0

0x

xx

fcx

x

x

=

≥

+

>

≥

x 1

x 2

Uni

on o

f tw

o po

lyhe

dra

P1,

P2

P1

P2

Slid

e 16

Fix

ed c

harg

e pr

oble

m

Min

imiz

e a

fixed

cha

rge

func

tion:

2

12

11

1min

0if

0

if 0

0x

xx

fcx

x

x

=

≥

+

>

≥

x 1

x 2

The

po

lyhe

dra

have

di

ffere

nt

rece

ssio

n co

nes.

P1

P1

rece

ssio

nco

ne

P2

P2

rece

ssio

nco

ne

Slid

e 17

Fix

ed c

harg

e pr

oble

m

Min

imiz

e a

fixed

cha

rge

func

tion:

Add

an

uppe

r bo

und

on x

1

2

12

11

1

min

0if

0

if

0

0

x

xx

fcx

x

xM

=

≤

≥

+

>

≤

x 1

x 2

The

po

lyhe

dra

have

the

sam

e re

cess

ion

cone

.P

1

P1

rece

ssio

nco

ne

P2

P2

rece

ssio

nco

neM

Slid

e 18

Fix

ed c

harg

e pr

oble

m

Sta

rt w

ith a

dis

junc

tion

of

linea

r sy

stem

s to

rep

rese

nt

the

unio

n of

pol

yhed

ra

2

11

22

1

min

00

0x

xx

M

xx

fcx

=≤

≤

∨

≥≥

+

x 1

x 2

P1

P2

M

Slid

e 19

Fix

ed c

harg

e pr

oble

m

Sta

rt w

ith a

dis

junc

tion

of

linea

r sy

stem

s to

rep

rese

nt

the

unio

n of

pol

yhed

ra

2

11

22

1

min

00

0x

xx

M

xx

fcx

=≤

≤

∨

≥≥

+

{}

2

12

11

21

22

21

22

12 1

21

21

11

22

2

min

00

0

1,

0,1

,k

x

xx

My

xcx

xfy

yy

y

xx

xx

xx

=≤

≤≥

−+

≥+

=∈

=+

=+

Intr

oduc

e a

0-1

varia

ble

yk

that

is 1

whe

n x

is in

po

lyhe

dron

k.

Dis

aggr

egat

e x

to c

reat

e an

xk

for

each

k.

Slid

e 20

{}

2

12

11

21

22

21

22

12 1

21

21

11

22

2

min

00

0

1,

0,1

,k

x

xx

My

xcx

xfy

yy

y

xx

xx

xx

=≤

≤≥

−+

≥+

=∈

=+

=+

To s

impl

ify, r

epla

ce

w

ith x

1 si

nce

2 1x1 1

0x

=

Slid

e 21

{}

2

21

22

22

12 1

22

2

1

1

2

min

0

0

1,

0,1

k

x

My

xc

xfy

yy

y

xx

x

x

x≤≤

≥−

+≥

+=

∈=

+

To s

impl

ify, r

epla

ce

w

ith x

1 si

nce

2 1x1 1

0x

=

Slid

e 22

{}

2

12

12

21

22

12 1

22

22

min

0

0

1,

0,1

k

x

xM

y

xcx

xfy

yy

y

xx

x

≤≤

≥−

+≥

+=

∈=

+

Rep

lace

with

x2

beca

use

play

s no

rol

e in

the

mod

el

2 2x 1 2x

Slid

e 23

{}

2

12

12

1

2

2

min

0

1,

0,1

k

x

xM

y

cxfy

x

yy

y

≤≤

−+

≥+

=∈

Rep

lace

with

x2

Bec

ause

pla

ys n

o ro

le in

the

mod

el

2 2x 1 2x

Slid

e 24

{}

2

12

12

2

12

min

0

1,

0,1

k

x

xM

y

cxx

fy

yy

y

≤≤

−+

≥+

=∈

Rep

lace

y2

with

y

Bec

ause

y2

play

s no

rol

e in

the

mod

el

Slid

e 25

{}

2

1

21

min

0

0,1

x xM

xcx

y yf

y

≤≤

≥+

∈Rep

lace

y2

with

y

Bec

ause

y2

play

s no

rol

e in

the

mod

el

{}

min

0

0,1

cxfy

xM

y

y

+≤

≤∈

or

“Big

M”

Slid

e 26

Exa

mp

le:

Un

cap

acit

ated

faci

lity

loca

tio

n

ij

f ic i

j

Fix

ed

cost

Tra

nspo

rt

cost

mpo

ssib

le

fact

ory

loca

tions

nm

arke

tsLo

cate

fact

orie

s to

ser

ve

mar

kets

so

as to

min

imiz

e to

tal f

ixed

cos

t and

tr

ansp

ort c

ost.

No

limit

on p

rodu

ctio

n ca

paci

ty o

f eac

h fa

ctor

y.

Slid

e 27

Unc

apac

itate

d fa

cilit

y lo

catio

n

ij

f ic i

j

Fix

ed

cost

Tra

nspo

rt

cost

nm

arke

tsD

isju

nctiv

e m

odel

:

min 0

1, a

ll 0,

all

, a

ll 0

1, a

ll

iij

iji

ij

ijij

ii

i

iji

zc

x

xj

xj

iz

fz

xj

+

≤≤

=

∨

≥=

=∑∑

∑

Fac

tory

at

loca

tion

iN

o fa

ctor

yat

loca

tion

i

Fra

ctio

n of

m

arke

t j’s

dem

and

satis

fied

from

lo

catio

n i

mpo

ssib

le

fact

ory

loca

tions

Slid

e 28

Unc

apac

itate

d fa

cilit

y lo

catio

n

MIL

P f

orm

ulat

ion:

Dis

junc

tive

mod

el:

{}

12

12

12

12

min

0,

all

,0,

all

,

, all

0, a

ll

,

,

0,1

1, a

ll

iij

iji

ij

iji

ij

ii

ii

ijij

iji

ii

i

iji

zc

x

xy

ij

xi

j

zfy

iz

i

xx

xz

zz

y

xj

+

≤≤

=≥

==

+=

+∈

=∑∑

∑min 0

1, a

ll 0,

all

, a

ll 0

1, a

ll

iij

iji

ij

ijij

ii

i

iji

zc

x

xj

xj

iz

fz

xj

+

≤≤

=

∨

≥=

=∑∑

∑

Slid

e 29

Unc

apac

itate

d fa

cilit

y lo

catio

n

Let

si

nce

1 ijij

xx

=2

0ijx

=

Let

si

nce

1 ii

zz

=2

0iz

=

MIL

P f

orm

ulat

ion:

{}

12

12

12

12

min

0,

all

,0,

all

,

, all

0, a

ll

,

,

0,1

1, a

ll

iij

iji

ij

iji

ij

ii

ii

ijij

iji

ii

i

iji

zc

x

xy

ij

xi

j

zfy

iz

i

xx

xz

zz

y

xj

+

≤≤

=≥

==

+=

+∈

=∑∑

∑

Slid

e 30

Unc

apac

itate

d fa

cilit

y lo

catio

n

Let

si

nce

{}

min

0,

all

,

, all

0,1 1,

all

iij

iji

ij

i

ii

i

ij

ij

i i

x

z

zc

x

yi

j

fyi

y

xj

+

≤≤

≥ ∈=∑

∑

∑

1 ijij

xx

=2

0ijx

=

Let

si

nce

1 ii

zz

=2

0iz

=

MIL

P f

orm

ulat

ion:

Slid

e 31

Unc

apac

itate

d fa

cilit

y lo

catio

n

Let

si

nce

{}

min

0,

all

,

, all

0,1 1,

all

iij

iji

ij

i

ii

i

ij

ij

i i

x

z

zc

x

yi

j

fyi

y

xj

+

≤≤

≥ ∈=∑

∑

∑

1 ijij

xx

=2

0ijx

=

Let

si

nce

1 ii

zz

=2

0iz

=

{}

min

0,

all

,

0,1 1,

all

ii

ijij

iij

iji

i

iji

cx

xy

i

f

j

y

x

y

j+

≤≤

∈=∑

∑

∑

or

MIL

P f

orm

ulat

ion:

Slid

e 32

Unc

apac

itate

d fa

cilit

y lo

catio

n

MIL

P f

orm

ulat

ion:

Beg

inne

r’s m

odel

:

{}

min

, a

ll

0,1 1,

all

ii

ijij

iij

iji

j i

iji

fyc

x

xny

i

y

xj+

≤

∈=∑

∑

∑ ∑

Bas

ed o

n ca

paci

tate

d lo

catio

n m

odel

.

It ha

s a

wea

ker

con

tin

uo

us

rela

xati

on

Thi

s be

ginn

er’s

mis

take

can

be

avoi

ded

by

star

ting

with

dis

junc

tive

form

ulat

ion.

Max

imum

out

put

from

loca

tion

i

{}

min

0,

all

,

0,1 1,

all

ii

ijij

iij

iji

i

iji

fyc

x

xy

ij

y

xj+

≤≤

∈=∑

∑

∑

Slid

e 33

Exa

mp

le:

Lo

t si

zin

g w

ith

set

up

co

sts

Det

erm

ine

lot s

ize

in e

ach

perio

d to

min

imiz

e to

tal

prod

uctio

n, in

vent

ory,

and

set

up c

osts

.

t =0

12

34

56

Dem

and

=D

0D

1D

2D

3D

4D

5D

6

Max

pr

oduc

tion

leve

l

Set

up c

ost i

ncur

red

Slid

e 34

00

00

0t

tt

tt

tt

tt

t

vf

vv

xC

xC

x

≥≥

≥

∨

∨

≤

≤≤

≤=

(1)

Sta

rt

prod

uctio

n(in

curs

set

up

cost

)

(2)

Con

tinue

pr

oduc

tion

(no

setu

p co

st)

(3)

Pro

duce

no

thin

g(n

o pr

oduc

tion

cost

)

Fix

ed-c

ost

varia

ble

Fix

ed

cost

Pro

duct

ion

leve

lP

rodu

ctio

n ca

paci

ty

Logi

cal c

ondi

tions

:

(2)

In p

erio

d t⇒

(1)

or (

2) in

per

iod

t−1

(1)

In p

erio

d t⇒

neith

er (

1) n

or (

2) in

per

iod

t−1

Slid

e 35

00

00

0t

tt

tt

tt

tt

t

vf

vv

xC

xC

x

≥≥

≥

∨

∨

≤

≤≤

≤=

(1)

Sta

rt

prod

uctio

n

(2)

Con

tinue

pr

oduc

tion

(3)

Pro

duce

no

thin

g

11

11

0t

tt

tt

t

vfy

xC

y

≥≤

≤

2 22

0

0t t

tt

v xC

y

≥≤

≤

3 3

0 0t tv x

≥ =

33

3

11

1

,,

{0,1

},

1,2,

3

kk

tt

tt

ttk

kk

k

tk

vv

xx

yy

yk

==

=

==

=

∈=

∑∑

∑

Con

vex

hull

MIL

P m

odel

of d

isju

nctio

n:

Slid

e 36

11

11

0t

tt

tt

t

vfy

xC

y

≥≤

≤

2 22

0

0t t

tt

v xC

y

≥≤

≤

3 3

0 0t tv x

≥ =

33

3

11

1

,,

{0,1

},

1,2,

3

kk

tt

tt

ttk

kk

k

tk

vv

xx

yy

yk

==

=

==

=

∈=

∑∑

∑

Con

vex

hull

MIL

P m

odel

of d

isju

nctio

n:

To s

impl

ify, d

efin

e

z t=

yt1

y t=

yt2

Slid

e 37

1

10

t

ttt

t

tvf

x

z

zC

≥≤

≤

2 2

0

0t

t tt

v xy

C

≥≤

≤

3 3

0 0t tv x

≥ =

33

11

,,

{0,1

},

1,

1

2,3

,kk

tt

tt

tt

tt

kk

vv

xx k

zy

zy

==

+

∈=

≤=

=∑

∑

Con

vex

hull

MIL

P m

odel

of d

isju

nctio

n:

To s

impl

ify, d

efin

e

z t=

yt1

y t=

yt2

= 1

for

star

tup

= 1

for

cont

inue

d pr

oduc

tion

Slid

e 38

1

10

tt

t

tt

t

vfz

xC

z

≥≤

≤

2 2

0

0t t

tt

v xC

y

≥≤

≤

3 3

0 0t tv x

≥ =

33

11

,,

1

,{0

,1},

1,

2,3

kk

tt

tt

tt

kk

tt

vv

xx

zy

zy

k=

=

==

+≤

∈=

∑∑

Con

vex

hull

MIL

P m

odel

of d

isju

nctio

n:

Sin

cese

t

30

tx= 1

2t

tt

xx

x=

+

Slid

e 39

1

0(

)t

tt

t

tt

tx

Cy

vz z

f

≤≤

+≥

20

tv≥

30

tv≥

3

1

,1

,{0

,1},

1,

2,3

kt

tt

tk

tt

vv

zy

zy

k=

=+

≤

∈=

∑

Con

vex

hull

MIL

P m

odel

of d

isju

nctio

n:

Sin

cese

t

30

tx= 1

21

12

xx

x=

+

Slid

e 40

1

0(

)t

tt

tt

tt

vfz

xC

zy

≥≤

≤+

20

tv≥

30

tv≥

3

1

,1

,{0

,1},

1,

2,3

kt

tt

tk

tt

vv

zy

zy

k=

=+

≤

∈=

∑

Con

vex

hull

MIL

P m

odel

of d

isju

nctio

n:

Sin

ce v

t occ

urs

posi

tivel

y in

the

obje

ctiv

e fu

nctio

n,

and

d

o no

t pla

y a

role

, let

2

3,

tt

vv

1t

tv

v=

Slid

e 41

0(

)t

tt

tt

tt

vfz

xC

zy

≥≤

≤+ 1

,{0

,1},

1,

2,3

tt

tt

zy

zy

k

+≤

∈=

Con

vex

hull

MIL

P m

odel

of d

isju

nctio

n:

Sin

ce v

t occ

urs

posi

tivel

y in

the

obje

ctiv

e fu

nctio

n,

and

d

o no

t pla

y a

role

, let

2

3,

tt

vv

1t

tv

v=

Slid

e 42

0(

)t

tt

tt

tt

vfz

xC

zy

≥≤

≤+

11

11

1

,{0

,1},

1,

2,3

1t

tt

t

t

t

t

t

tt

yz

y

zy

z

zz

y

y

k

−−

−−

+

≤∈ ≤

−≤=

+−

For

mul

ate

logi

cal c

ondi

tions

:

(2)

In p

erio

d t⇒

(1)

or (

2) in

per

iod

t−1

(1)

In p

erio

d t⇒

neith

er (

1) n

or (

2) in

per

iod

t−1

Slid

e 43

1

m

0(

)

in(

)n

tt

tt

tt

tt

t

tt

tt

vfz

x

px

hs

z

v

Cy

=

≥≤

≤+

++

∑

11

11

1

,{0

,1},

1,

2,3

1tt

tt

tt

t

tt

t

zy

zy

k

yz

y

zz

y−

−

−−

+≤

∈=

≤+

≤−

−

Add

obj

ectiv

e fu

nctio

n

Uni

t pro

duct

ion

cost

Uni

t hol

ding

cos

t

Slid

e 44

Kn

apsa

ck M

od

els

Inte

ger

varia

bles

can

als

o be

use

d to

exp

ress

cou

ntin

g id

eas.

Thi

s is

tota

lly d

iffer

ent f

rom

the

use

of 0

-1 v

aria

bles

to

expr

ess

unio

ns o

f pol

yhed

ra.

Slid

e 45

Exa

mp

le: F

reig

ht

Tran

sfer

•T

rans

port

42

tons

of f

reig

ht u

sing

8 tr

ucks

, whi

ch c

ome

in

4 si

zes…

Tru

ck

size

Num

ber

avai

labl

eC

apac

ity

(ton

s)

Cos

t pe

r tr

uck

13

790

23

560

33

450

43

340

Slid

e 46

Tru

ck

type

Num

ber

avai

labl

eC

apac

ity

(ton

s)

Cos

t pe

r tr

uck

13

790

23

560

33

450

43

340

12

34

12

34

12

34

min

90

6050

40

75

43

42

8

{0,1

,2,3

}i

xx

xx

xx

xx

xx

xx

x

++

++

++

≥+

++

≤∈

Num

ber

of tr

ucks

of t

ype

1

Kna

psac

k co

verin

g co

nstr

aint

Kna

psac

k pa

ckin

g co

nstr

aint

Slid

e 47

Exa

mp

le: F

reig

ht

Pac

kin

g a

nd

Tra

nsf

er

•T

rans

port

pac

kage

s us

ing

ntr

ucks

•E

ach

pack

age

jhas

siz

e a j

.

•E

ach

truc

k ih

as c

apac

ity Q

i.

Slid

e 48

Kna

psac

k co

mpo

nent

The

truc

ks s

elec

ted

mus

t hav

e en

ough

cap

acity

to

carr

y th

e lo

ad.

1n

ii

ji

j

Qy

a=

≥∑

∑

= 1

if tr

uck

iis

sele

cted

Slid

e 49

Dis

junc

tive

com

pone

nt (

with

em

bedd

ed k

naps

ack

cons

trai

nt)

0 0

01,

all

ii

ij

iji

jij

ijzc

za

xQ

x

xj

≥

≥

≤∨

=

≤≤

∑

Tru

ck i

sele

cted

Tru

ck i

not

sele

cted

= 1

if p

acka

ge j

is

load

ed o

n tr

uck

i

Cos

t of o

pera

ting

truc

k i

Cos

t var

iabl

e

Use

con

tinuo

us

rela

xatio

n be

caus

e w

e w

ant

a di

sjun

ctio

n of

lin

ear

syst

ems

Slid

e 50

0 0

01,

all

ii

ij

iji

jij

ijzc

za

xQ

x

xj

≥

≥

≤∨

=

≤≤

∑

Tru

ck i

sele

cted

Tru

ck i

not

sele

cted

0

ii

i

jij

ii

j

iji

zc

y

ax

Qy

xy

≥≤

≤≤

∑

Con

vex

hull

MIL

P

form

ulat

ion

Dis

junc

tive

com

pone

nt (

with

em

bedd

ed k

naps

ack

cons

trai

nt)

Slid

e 51

The

res

ultin

g m

odel

Dis

junc

tive

com

pone

nt

Logi

cal c

ondi

tion

(eac

h pa

ckag

e m

ust b

e sh

ippe

d)

Kna

psac

k co

mpo

nent

1

1 1min

, a

ll

0,

all

,

1 ,

all

,{0

,1}

n

ii

i

jij

ii

j

iji

n

iji

n

ii

ji

j

iji

cy

ax

Qy

i

xy

ij

xj

Qy

a

xy

=

= =

≤

≤≤ =

≥ ∈∑

∑

∑ ∑∑

Slid

e 52

The

res

ultin

g m

odel

1

1 1min

, a

ll

0,

all

,

1 ,

all

,{0

,1}

n

ii

i

jij

ii

j

iji

n

iji

n

ii

ji

j

iji

cy

ax

Qy

i

xy

ij

xj

Qy

a

xy

=

= =

≤

≤≤ =

≥ ∈∑

∑

∑ ∑∑

The

yiis

red

unda

nt b

ut m

akes

th

e co

ntin

uous

rela

xatio

n tig

hter

.

Thi

s is

a m

odel

ing

“tric

k,”

part

of

the

folk

lore

of m

odel

ing.

Slid

e 53

The

res

ultin

g m

odel

1

1 1min

, a

ll

0,

all

,

1 ,

all

,{0

,1}

n

ii

i

jij

ii

j

iji

n

iji

n

ii

ji

j

iji

cy

ax

Qy

i

xy

ij

xj

Qy

a

xy

=

= =

≤

≤≤ =

≥ ∈∑

∑

∑ ∑∑

The

yiis

red

unda

nt b

ut m

akes

th

e co

ntin

uous

rela

xatio

n tig

hter

.

Thi

s is

a m

odel

ing

“tric

k,”

part

of

the

folk

lore

of m

odel

ing.

Con

vent

iona

l mod

elin

g w

isdo

m

wou

ld n

ot u

se th

is c

onst

rain

t, be

caus

e it

is th

e su

m o

f the

firs

t co

nstr

aint

ove

r i.

But

it r

adic

ally

red

uces

sol

utio

n tim

e, b

ecau

se it

gen

erat

es

knap

sack

cut

s.

Thi

s ar

gues

for

a pr

inci

pled

ap

proa

ch to

mod

elin

g.

Slid

e 54

Co

nst

rain

t P

rog

ram

min

g M

od

els

CP

Mod

elin

g S

yste

ms

Glo

bal C

onst

rain

tsE

mpl

oyee

Sch

edul

ing

Slid

e 55CP

Mo

del

ing

Sys

tem

s

•C

omm

erci

al m

odel

ing

syst

ems

with

ded

icat

ed s

olve

rs

•O

PL

Stu

dio

(run

s IL

OG

Sol

ver,

ILO

G S

ched

uler

)

•C

HIP

(ru

ns C

HIP

sol

ver)

•M

osel

(ru

ns X

pres

s-K

alis

)

•M

ozar

t (us

es O

z la

ngua

ge)

•N

on-c

omm

erci

al m

odel

ing

syst

em w

ith d

edic

ated

sol

vers

•E

CLi

PS

e (r

uns

EC

LiP

Se

CP

sol

ver)

Slid

e 56Glo

bal

co

nst

rain

ts

•A

glo

bal

co

nst

rain

tre

pres

ents

a s

et o

f con

stra

ints

with

sp

ecia

l str

uctu

re.

•T

he s

truc

ture

is e

xplo

ited

by f

ilter

ing

alg

orith

ms

in th

e C

P

solv

er.

Slid

e 57

So

me

gen

eral

-pu

rpo

se g

lob

al c

on

stra

ints

Alld

iff

-R

equi

res

that

all

the

liste

d va

riabl

es t

ake

diffe

rent

va

lues

.

Am

on

g-

Bou

nds

the

num

ber

of li

sted

var

iabl

es t

hat t

ake

one

of

the

valu

es in

a li

st.

Car

din

alit

y-

Bou

nds

the

num

ber

of li

sted

var

iabl

es t

hat t

ake

each

of t

he v

alue

s in

a li

st.

Ele

men

t -

Req

uire

s th

at a

giv

en v

aria

ble

take

the

yth

valu

e in

a

list,

whe

re y

is a

n in

tege

r va

riabl

e.

Pat

h-

Req

uire

s th

at a

giv

en g

raph

con

tain

a p

ath

of a

t mos

t a

give

n le

ngth

.

Slid

e 58

So

me

glo

bal

co

nst

rain

ts f

or

sch

edu

ling

Dis

jun

ctiv

e -

Req

uire

s th

at n

o tw

o jo

bs o

verla

p in

tim

e.

Cu

mu

lati

ve-

Lim

its th

e re

sour

ces

cons

umed

by

jobs

run

ning

at

any

one

time.

In

part

icul

ar, i

t can

lim

it th

e nu

mbe

r of

jobs

ru

nnin

g at

any

one

tim

e.

Str

etch

-B

ound

s th

e le

ngth

of a

str

etch

of c

ontig

uous

per

iods

as

sign

ed th

e sa

me

job.

Seq

uen

ce –

A s

et o

f ove

rlapp

ing

amo

ng

con

stra

ints

.

Reg

ula

r–

Gen

eral

izes

str

etch

and

seq

uen

ce.

Dif

fn -

Req

uire

s th

at n

o tw

o bo

xes

in a

set

of m

ultid

imen

sion

al

boxe

s ov

erla

p.

Use

d fo

r sp

ace

or s

pace

-tim

e pa

ckin

g.

Slid

e 59Exa

mp

le:

Em

plo

yee

Sch

edu

ling

•S

ched

ule

four

nur

ses

in 8

-hou

r sh

ifts.

•A n

urse

wor

ks a

t mos

t one

shi

ft a

day,

at l

east

5 d

ays

a w

eek.

•S

ame

sche

dule

eve

ry w

eek.

•N

o sh

ift s

taffe

d by

mor

e th

an tw

o di

ffere

nt n

urse

s in

a w

eek.

•A n

urse

can

not w

ork

diffe

rent

shi

fts o

n tw

o co

nsec

utiv

e da

ys.

•A n

urse

who

wor

ks s

hift

2 or

3 m

ust d

o so

at l

east

two

days

in

a ro

w.

Slid

e 60

Two

way

s to

vie

w th

e p

rob

lem

Sun

Mon

Tue

Wed

Thu

Fri

Sat

Shi

ft 1

AB

AA

AA

A

Shi

ft 2

CC

CB

BB

B

Shi

ft 3

DD

DD

CC

D

Ass

ign

nurs

es to

shi

fts

Sun

Mon

Tue

Wed

Thu

Fri

Sat

Nur

se A

10

11

11

1

Nur

se B

01

02

22

2

Nur

se C

22

20

33

0

Nur

se D

33

33

00

3

Ass

ign

shift

s to

nur

ses

0 =

day

off

Slid

e 61U

se b

oth

form

ulat

ions

in th

e sa

me

mod

el!

Firs

t, as

sign

nur

ses

to s

hifts

.

Let w

sd=

nur

se a

ssig

ned

to s

hift

son

day

d

12

3al

ldiff

(,

,),

all

dd

dw

ww

dT

he v

aria

bles

w1d

, w2d

, w

3dta

ke d

iffer

ent v

alue

s

Tha

t is,

sch

edul

e 3

diffe

rent

nur

ses

on e

ach

day

Slid

e 62

()

12

3al

ldiff

(,

,),

all

card

inal

ity|(

,,

,),

(5,5

,5,5

),(6

,6,6

,6)

dd

dw

ww

wA

BC

d

D

Aoc

curs

at l

east

5 a

nd a

t mos

t 6

times

in th

e ar

ray

w, a

nd s

imila

rly

for

B, C

, D.

Tha

t is,

eac

h nu

rse

wor

ks a

t lea

st

5 an

d at

mos

t 6 d

ays

a w

eek

Use

bo

th fo

rmul

atio

ns in

the

sam

e m

odel

!

Firs

t, as

sign

nur

ses

to s

hifts

.

Let w

sd=

nur

se a

ssig

ned

to s

hift

son

day

d

Slid

e 63

()

()

()

12

3

,Sun

,Sat

alld

iff,

,,

all

card

inal

ity|(

,,

,),

(5,5

,5,5

),(6

,6,6

,6)

nval

ues

,...,

|1,2

, a

ll

dd

d

ss

ww

w

w

d

AB

CD

ww

s

The

var

iabl

es w

s,S

un, …

, ws,

Sat

take

at

leas

t 1 a

nd a

t mos

t 2 d

iffer

ent

valu

es.

Tha

t is,

at l

east

1 a

nd a

t mos

t 2

nurs

es w

ork

any

give

n sh

ift.

Use

bo

th fo

rmul

atio

ns in

the

sam

e m

odel

!

Firs

t, as

sign

nur

ses

to s

hifts

.

Let w

sd=

nur

se a

ssig

ned

to s

hift

son

day

d

Slid

e 64R

emai

ning

con

stra

ints

are

not

eas

ily e

xpre

ssed

in th

is

nota

tion.

So,

ass

ign

shift

s to

nur

ses.

Let y

id=

nur

se a

ssig

ned

to s

hift

son

day

d

()

12

3,

alld

iff,

all

,d

dd

yy

yd

Ass

ign

a di

ffere

nt n

urse

to e

ach

shift

on

each

day

.

Thi

s co

nstr

aint

is r

edun

dant

of

prev

ious

con

stra

ints

, but

re

dund

ant c

onst

rain

ts s

peed

so

lutio

n.

Slid

e 65

()

()

12

3

,Sun

,Sat

alld

iff,

all

stre

tch

,,

|(2,

3),

(2,2

),(6

,6),

, al

l

,,

dd

d

ii

y

Pi

y

yy

dy

…

Eve

ry s

tret

ch o

f 2’s

has

leng

th b

etw

een

2 an

d 6.

Eve

ry s

tret

ch o

f 3’s

has

leng

th b

etw

een

2 an

d 6.

So

a nu

rse

who

wor

ks s

hift

2 or

3 m

ust d

o so

at l

east

tw

o da

ys in

a r

ow.

Rem

aini

ng c

onst

rain

ts a

re n

ot e

asily

exp

ress

ed in

this

no

tatio

n.

So,

ass

ign

shift

s to

nur

ses.

Let y

id=

nur

se a

ssig

ned

to s

hift

son

day

d

Slid

e 66

()

()

12

3

,Sun

,Sat

alld

iff,

all

stre

tch

,,

|(2,

3),

(2,2

),(6

,6),

, al

l

,,

dd

d

ii

y

Pi

y

yy

dy

…

Her

e P

= {

(s,0

),(0

,s)

| s=

1,2

,3}

Whe

neve

r a

stre

tch

of a

’s im

med

iate

ly p

rece

des

a st

retc

h of

b’s

, (a

,b)

mus

t be

one

of th

e pa

irs in

P.

So

a nu

rse

cann

ot s

witc

h sh

ifts

with

out

taki

ng a

t lea

st o

ne d

ay o

ff.

Rem

aini

ng c

onst

rain

ts a

re n

ot e

asily

exp

ress

ed in

this

no

tatio

n.

So,

ass

ign

shift

s to

nur

ses.

Let y

id=

nur

se a

ssig

ned

to s

hift

son

day

d

Slid

e 67N

ow w

e m

ust c

onne

ct th

e w

sdva

riabl

es to

the

y id

varia

bles

.

Use

ch

ann

elin

g c

on

stra

ints

:

, a

ll ,

, a

ll ,

i dd s

d

wy

dy

ii

wd

ss

d

= =

Cha

nnel

ing

cons

trai

nts

incr

ease

pro

paga

tion

and

mak

e th

e pr

oble

m e

asie

r to

sol

ve.

Slid

e 68T

he c

ompl

ete

mod

el is

:

, a

ll ,

, a

ll ,

i dd s

d

wy

dy

ii

wd

ss

d

= =

()

()

()

12

3

,Sun

,Sat

alld

iff,

,,

all

card

inal

ity|(

,,

,),

(5,5

,5,5

),(6

,6,6

,6)

nval

ues

,...,

|1,2

, a

ll

dd

d

ss

ww

w

w

d

AB

CD

ww

s

()

()

12

3

,Sun

,Sat

alld

iff,

all

stre

tch

,,

|(2,

3),

(2,2

),(6

,6),

, al

l

,,

dd

d

ii

y

Pi

y

yy

dy

…

Slid

e 69

Inte

gra

ted

Mo

del

s

Mod

elin

g S

yste

ms

Pro

duct

Con

figur

atio

nM

achi

ne A

ssig

nmen

t and

Sch

edul

ing

Slid

e 70Inte

gra

ted

Mo

del

ing

Sys

tem

s

•C

omm

erci

al m

odel

ing

syst

ems

with

ded

icat

ed s

olve

rs

•O

PL

Stu

dio

(run

s C

PLE

X, I

LOG

Sol

ver/

Sch

edul

er)

•M

osel

(ru

ns X

pres

s-M

P, X

pres

s-K

alis

)

•N

on-c

omm

erci

al m

odel

ing

syst

ems

with

ded

icat

ed s

olve

rs

•E

CLi

PS

e (r

uns

EC

LiP

Se

CP

sol

ver,

Xpr

ess-

MP

)

•S

IMP

L (u

nder

dev

elop

men

t)

Slid

e 71

Thi

s ex

ampl

e co

mbi

nes

MIL

P m

od

elin

g w

ith v

aria

ble

ind

ices

, us

ed in

con

stra

int p

rogr

amm

ing.

•It

can

be s

olve

d by

com

bini

ng M

ILP

and

CP

tech

niqu

es.

Exa

mp

le:

Pro

du

ct C

on

fig

ura

tio

n

Slid

e 72

Mem

ory

Mem

ory

Mem

ory

Mem

ory

Mem

ory

Mem

ory

Pow

ersu

pply

Pow

ersu

pply

Pow

ersu

pply

Pow

ersu

pply

Dis

k dr

ive

Dis

k dr

ive

Dis

k dr

ive

Dis

k dr

ive

Dis

k dr

ive

Cho

ose

wha

t typ

e of

eac

h co

mpo

nent

, and

how

man

y

Per

sona

l com

pute

r

The

pro

blem

Slid

e 73

min

, al

l

, al

l

i

jj

j

ji

ijtik

jj

j

cv

vq

Aj

Lv

Uj

= ≤≤

∑ ∑

Am

ount

of a

ttrib

ute

jpr

oduc

ed

(< 0

if c

onsu

med

):

mem

ory,

hea

t, po

wer

, w

eigh

t, et

c.

Qua

ntity

of

com

pone

nt i

inst

alle

d

Inte

grat

ed m

odel

Am

ount

of a

ttrib

ute

jpr

oduc

ed b

y ty

pe t

iof

com

pone

nt i

Uni

t cos

t of p

rodu

cing

at

trib

ute

j

Slid

e 74

min

, al

l

, al

l

i

jj

j

ji

ijtik

jj

j

cv

vq

Aj

Lv

Uj

= ≤≤

∑ ∑

Inte

grat

ed m

odel

t i is

a v

aria

ble

in

dex

()

1

, al

l

elem

ent

,(,

,,

),,

all

,

ji

i

ii

iji

ijni

vz

j

tq

Aq

Az

ij

=∑

…

Thi

s is

ref

orm

ulat

ed

Slid

e 75

min

, al

l

, al

l

i

jj

j

ji

ijtik

jj

j

cv

vq

Aj

Lv

Uj

= ≤≤

∑ ∑

Inte

grat

ed m

odel

t i is

a v

aria

ble

in

dex

()

1

, al

l

elem

ent

,(),

, al

l ,

,,

i

ji

i

iij

iin

ij

vz

j

tz

qA

qA

ij

=∑

…

Thi

s is

ref

orm

ulat

ed

Set

zieq

ual t

o th

e t ith

item

in th

e re

dlis

t.

Slid

e 76

Mac

hin

e A

ssig

nm

ent

and

Sch

edu

ling

•Ass

ign

jobs

to m

achi

nes

and

sche

dule

the

mac

hine

s as

sign

ed

to e

ach

mac

hine

with

in t

ime

win

dow

s.

•T

he o

bjec

tive

is to

min

imiz

e m

akes

pan

.

•C

ombi

ne M

ILP

and

CP

mod

elin

g

Tim

e la

pse

betw

een

star

t of f

irst j

ob a

nd

end

of la

st jo

b.

Slid

e 77

Mac

hine

Sch

edul

ing

()

min

, al

l , al

l

disj

unct

ive

(),

()

, al

l

j

j

jx

j

jj

jx

j

jj

ijj

M

Ms

pj

rs

dp

j

sx

ip

xi

i

≥+

≤≤

−

==

Sta

rt ti

me

varia

ble

for

job

j

Mak

espa

n

The

mod

el is

Pro

cess

ing

time

of jo

b j

on m

achi

ne x

jM

achi

ne

assi

gned

to jo

b j

Slid

e 78

Mac

hine

Sch

edul

ing

()

min

, al

l , al

l

disj

unct

ive

(),

()

, al

l

j

j

jx

j

jj

jx

j

jj

ijj

M

Ms

pj

rs

dp

j

sx

ip

xi

i

≥+

≤≤

−

==

Rel

ease

tim

e fo

r jo

b j

Tim

e w

indo

ws

The

mod

el is

Dea

dlin

e fo

r jo

b j

Slid

e 79

Mac

hine

Sch

edul

ing

()

min

, al

l , al

l

disj

unct

ive

(),

()

, al

l

j

j

jx

j

jj

jx

j

jj

ijj

M

Ms

pj

rs

dp

j

sx

ip

xi

i

≥+

≤≤

−

==

Sta

rt ti

mes

of

jobs

ass

igne

d to

mac

hine

i

Dis

junc

tive

glob

al

cons

trai

nt r

equi

res

that

Jo

bs d

o no

t ove

rlap

The

mod

el is

Slid

e 80

Mac

hine

Sch

edul

ing

()

min

, al

l , al

l

disj

unct

ive

(),

()

, al

l

j

j

jx

j

jj

jx

j

jj

ijj

M

Ms

pj

rs

dp

j

sx

ip

xi

i

≥+

≤≤

−

==

The

pro

blem

can

be

solv

ed b

y lo

gic-

base

d B

ende

rs d

ecom

posi

tion.

Mas

ter

prob

lem

is

this

plu

s B

ende

rs

cuts

, sol

ved

as a

n M

ILP

Slid

e 81

Mac

hine

Sch

edul

ing

()

min

, al

l , al

l

disj

unct

ive

(),

()

, al

l

j

j

jx

j

jj

jx

j

jj

ijj

M

Ms

pj

rs

dp

j

sx

ip

xi

i

≥+

≤≤

−

==

The

pro

blem

can

be

solv

ed b

y lo

gic-

base

d B

ende

rs d

ecom

posi

tion.

Mas

ter

prob

lem

is

this

plu

s B

ende

rs

cuts

, sol

ved

as a

n M

ILP

Sub

prob

lem

is th

is, s

olve

d by

CP