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GE
7 S
ep 0
7 S
lide
1Com
bini
ng O
ptim
izat
ion
and
Con
stra
int P
rogr
amm
ing
John
Hoo
ker
Car
negi
e M
ello
n U
nive
rsity
GE
Res
earc
h C
ente
r7
Sep
tem
ber
2007
GE
7 S
ep 0
7 S
lide
2
Opt
imiz
atio
n an
d C
onst
rain
t Pro
gram
min
g
•O
ptim
izat
ion:
focu
s on
mat
hem
atic
al p
rogr
amm
ing.
–50
+ y
ears
old
•C
onst
rain
t pro
gram
min
g–
20 y
ears
old
–D
evel
oped
in c
ompu
ter
scie
nce/
AI c
omm
unity
–B
ette
r kn
own
in E
urop
e/A
sia
than
US
A
GE
7 S
ep 0
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lide
3
Mat
h pr
ogra
mm
ing
& C
onst
rain
t pro
gram
min
g
•M
athe
mat
ical
pro
gram
min
g m
etho
ds r
ely
heav
ily o
n nu
mer
ical
cal
cula
tion
.
–Li
near
pro
gram
min
g (L
P)
–M
ixed
inte
ger/
linea
r pr
ogra
mm
ing
(MIL
P)
–N
onlin
ear
prog
ram
min
g (N
LP)
•C
onst
rain
t pro
gram
min
g re
lies
heav
ily o
n co
nstr
aint
pr
opag
atio
n–
A fo
rm o
f log
ical
infe
renc
e
GE
7 S
ep 0
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lide
4
•C
onta
iner
por
t sch
edul
ing
(Hon
g K
ong
and
Sin
gapo
re)
•C
ircui
t des
ign
(Sie
men
s)
•R
eal-t
ime
cont
rol
(Sie
men
s, X
erox
)
CP
: Ear
ly c
omm
erci
al s
ucce
sses
GE
7 S
ep 0
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lide
5
•E
mpl
oyee
sch
edul
ing
•S
hift
plan
ning
•Ass
embl
y lin
e sm
ooth
ing
and
bala
ncin
g
•C
ellu
lar
freq
uenc
y as
sign
men
t
•M
aint
enan
ce p
lann
ing
•Airl
ine
crew
ros
terin
g an
d sc
hedu
ling
•Airp
ort g
ate
allo
catio
n an
d st
and
plan
ning
CP
: App
licat
ions
GE
7 S
ep 0
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lide
6
•P
rodu
ctio
n sc
hedu
ling
chem
ical
sav
iatio
noi
l ref
inin
gst
eel
lum
ber
phot
ogra
phic
pla
tes
tires
•T
rans
port
sch
edul
ing
(foo
d,
nucl
ear
fuel
)
•W
areh
ouse
man
agem
ent
•C
ours
e tim
etab
ling
CP
: App
licat
ions
GE
7 S
ep 0
7 S
lide
7
•In
mat
hem
atic
al p
rogr
amm
ing
, equ
atio
ns
(con
stra
ints
) de
scrib
e th
e pr
oble
m b
ut d
on’t
tell
how
to s
olve
it.
•In
con
stra
int p
rogr
amm
ing
, eac
h co
nstr
aint
in
voke
s a
proc
edur
e th
at s
cree
ns o
ut
unac
cept
able
sol
utio
ns.
Mat
h pr
ogra
mm
ing
& C
onst
rain
t pro
gram
min
g
GE
7 S
ep 0
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lide
8
•M
athe
mat
ical
pro
gram
min
g
•Li
near
, non
linea
r an
d m
ixed
inte
ger
prog
ram
min
g, g
loba
l op
timiz
atio
n
•C
PLE
X, O
SL,
Xpr
ess-
MP,
Exc
el s
olve
r, M
INT
O, B
AR
ON
•C
onst
rain
t pro
gram
min
g
•C
onst
rain
t pro
paga
tion,
dom
ain
redu
ctio
n, in
terv
al a
rithm
etic
•IL
OG
Sch
edul
er,
OP
L S
tudi
o, C
HIP
, Moz
art,
EC
LiP
Se,
New
ton
Mat
h pr
ogra
mm
ing
& C
onst
rain
t pro
gram
min
g
GE
7 S
ep 0
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lide
9
Why
uni
fy m
ath
prog
ram
min
g an
d co
nstr
aint
pr
ogra
mm
ing?
•O
ne-s
top
shop
ping
.–
One
sol
ver
does
it a
ll.
GE
7 S
ep 0
7 S
lide
10
Why
uni
fy m
ath
prog
ram
min
g an
d co
nstr
aint
pr
ogra
mm
ing?
•O
ne-s
top
shop
ping
.–
One
sol
ver
does
it a
ll.
•R
iche
r m
odel
ing
fram
ewor
k.–
Nat
ural
mod
els,
less
deb
uggi
ng
& d
evel
opm
ent t
ime.
GE
7 S
ep 0
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lide
11
Why
uni
fy m
ath
prog
ram
min
g an
d co
nstr
aint
pr
ogra
mm
ing?
•O
ne-s
top
shop
ping
.–
One
sol
ver
does
it a
ll.
•R
iche
r m
odel
ing
fram
ewor
k.–
Nat
ural
mod
els,
less
deb
uggi
ng
& d
evel
opm
ent t
ime.
•C
ompu
tatio
nal s
peed
up.
–A
sel
ectio
n of
res
ults
…
GE
7 S
ep 0
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lide
12
Com
puta
tiona
l Adv
anta
ge o
f In
tegr
atin
g M
P a
nd C
P
Usi
ng C
P +
rel
axat
ion
from
MIL
P
Pro
blem
Spe
edup
Foc
acci
, Lod
i, M
ilano
(19
99)
Less
on
timet
ablin
g2
to 5
0 tim
es fa
ster
th
an C
P
Ref
alo
(199
9)P
iece
wis
e lin
ear
cost
s2
to 2
00 ti
mes
fa
ster
than
MIL
P
Hoo
ker
& O
sorio
(1
999)
Flo
w s
hop
sche
dulin
g, e
tc.
4 to
150
tim
es
fast
er th
an M
ILP.
Tho
rste
inss
on &
O
ttoss
on (
2001
)P
rodu
ct
conf
igur
atio
n30
to 4
0 tim
es
fast
er th
an C
P,
MIL
P
GE
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ep 0
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lide
13
Com
puta
tiona
l Adv
anta
ge o
f In
tegr
atin
g M
P a
nd C
P
Usi
ng C
P +
rel
axat
ion
from
MIL
P
Pro
blem
Spe
edup
Sel
lman
n &
F
ahle
(20
01)
Aut
omat
ic
reco
rdin
g1
to 1
0 tim
es fa
ster
th
an C
P, M
ILP
Van
Hoe
ve
(200
1)S
tabl
e se
t pr
oble
mB
ette
r th
an C
P in
le
ss ti
me
Bol
lapr
agad
a,
Gha
ttas
&
Hoo
ker
(200
1)
Str
uctu
ral d
esig
n (n
onlin
ear)
Up
to 6
00 ti
mes
fa
ster
than
MIL
P.2
prob
lem
s: <
6 m
in
vs >
20 h
rs fo
r M
ILP
Bec
k &
Ref
alo
(200
3)S
ched
ulin
g w
ith
earli
ness
&
tard
ines
s co
sts
Sol
ved
67 o
f 90,
CP
so
lved
onl
y 12
GE
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ep 0
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lide
14
Com
puta
tiona
l Adv
anta
ge o
f In
tegr
atin
g M
P a
nd C
P
Usi
ng C
P-b
ased
Bra
nch
and
Pric
e
Pro
blem
Spe
edup
Yun
es,
Mou
ra &
de
Sou
za (
1999
)U
rban
tran
sit
crew
sch
edul
ing
Opt
imal
sch
edul
e fo
r 21
0 tr
ips,
vs.
12
0 fo
r tr
aditi
onal
br
anch
and
pric
e
Eas
ton,
N
emha
user
&
Tric
k (2
002)
Tra
velin
g to
urna
men
t sc
hedu
ling
Firs
t to
solv
e 8-
team
inst
ance
GE
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ep 0
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15
Com
puta
tiona
l Adv
anta
ge o
f In
tegr
atin
g M
P a
nd C
P
Usi
ng C
P/M
ILP
Ben
ders
met
hods
Pro
blem
Spe
edup
Jain
&
Gro
ssm
ann
(200
1)
Min
-cos
t pla
nnin
g &
sch
edui
ng20
to 1
000
times
fa
ster
than
CP,
M
ILP
Tho
rste
inss
on
(200
1)M
in-c
ost p
lann
ing
& s
ched
ulin
g10
tim
es fa
ster
th
an J
ain
&
Gro
ssm
ann
Tim
pe (
2002
)P
olyp
ropy
lene
ba
tch
sche
dulin
g at
BA
SF
Sol
ved
prev
ious
ly
inso
lubl
e pr
oble
m
in 1
0 m
in
GE
7 S
ep 0
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16
Com
puta
tiona
l Adv
anta
ge o
f In
tegr
atin
g M
P a
nd C
P
Usi
ng C
P/M
ILP
Ben
ders
met
hods
Pro
blem
Spe
edup
Ben
oist
, Gau
din,
R
otte
mbo
urg
(200
2)
Cal
l cen
ter
sche
dulin
gS
olve
d tw
ice
as
man
y in
stan
ces
as tr
aditi
onal
B
ende
rs
Hoo
ker
(200
4)M
in-c
ost,
min
-mak
espa
n pl
anni
ng &
cum
ulat
ive
sche
dulin
g
100-
1000
tim
es
fast
er th
an C
P,
MIL
P
Hoo
ker
(200
5)M
in ta
rdin
ess
plan
ning
& c
umul
ativ
e sc
hedu
ling
10-1
000
times
fa
ster
than
CP,
M
ILP
GE
7 S
ep 0
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lide
17
Mod
elin
g is
key
•C
P m
odel
s ex
plai
n by
rev
ealin
g pr
oble
m s
truc
ture
.–
Thr
ough
glo
bal c
onst
rain
ts.
•T
his
can
be e
xten
ded
to a
uni
fied
fram
ewor
k.–
Glo
bal c
onst
rain
ts b
ecom
e m
etac
onst
rain
ts.
–E
ach
met
acon
stra
int “
know
s” h
ow to
com
bine
MP
and
CP
to
expl
oit i
ts s
truc
ture
.
GE
7 S
ep 0
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lide
18
The
bas
ic a
lgor
ithm
•S
earc
h: E
num
erat
e pr
oble
m r
estr
ictio
ns–
Tre
e se
arch
(br
anch
ing)
–C
onst
rain
t-ba
sed
(nog
ood-
base
d) s
earc
h
GE
7 S
ep 0
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lide
19
The
bas
ic a
lgor
ithm
•S
earc
h: E
num
erat
e pr
oble
m r
estr
ictio
ns–
Tre
e se
arch
(br
anch
ing)
–C
onst
rain
t-ba
sed
(nog
ood-
base
d) s
earc
h
•In
fer:
D
educ
e co
nstr
aint
s fr
om c
urre
nt r
estr
ictio
n
GE
7 S
ep 0
7 S
lide
20
The
bas
ic a
lgor
ithm
•S
earc
h: E
num
erat
e pr
oble
m r
estr
ictio
ns–
Tre
e se
arch
(br
anch
ing)
–C
onst
rain
t-ba
sed
(nog
ood-
base
d) s
earc
h
•In
fer:
D
educ
e co
nstr
aint
s fr
om c
urre
nt r
estr
ictio
n•
Rel
ax:
Sol
ve r
elax
atio
n of
cur
rent
res
tric
tion
GE
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ep 0
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lide
21
Uni
fyin
g fr
amew
ork
•E
xist
ing
met
hods
ar
e sp
ecia
l cas
es o
f thi
s fr
amew
ork.
•In
tegr
ated
met
hods
ar
e al
so s
peci
al c
ases
.–
Sel
ect a
n ov
eral
l sea
rch
sche
me.
–S
elec
t inf
eren
ce
met
hods
as
need
ed f
rom
CP,
OR
.
–S
elec
t rel
axat
ion
met
hods
as
need
ed.
GE
7 S
ep 0
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lide
22
Som
e ex
istin
g m
etho
ds –
Bra
nchi
ng
•C
onst
rain
t sol
vers
(C
P)
–S
earc
h: B
ranc
hing
on
dom
ains
–In
fere
nce:
C
onst
rain
t pro
paga
tion,
filt
erin
g
–R
elax
atio
n:
Dom
ain
stor
e
•M
ixed
inte
ger
prog
ram
min
g (O
R)
–S
earc
h: B
ranc
h an
d bo
und
–In
fere
nce:
C
uttin
g pl
anes
–R
elax
atio
n:
Line
ar p
rogr
amm
ing
GE
7 S
ep 0
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lide
23
Som
e ex
istin
g m
etho
ds –
Con
stra
int-
base
d se
arch
•S
AT
sol
vers
(C
P)
–S
earc
h: B
ranc
hing
on
varia
bles
–In
fere
nce:
U
nit c
laus
e ru
le, c
laus
e le
arni
ng (
nogo
ods)
–R
elax
atio
n:
Con
flict
cla
uses
•B
ende
rs d
ecom
posi
tion
(OR
)–
Sea
rch:
Enu
mer
atio
n of
sub
prob
lem
s
–In
fere
nce:
B
ende
rs c
uts
(nog
oods
)
–R
elax
atio
n:
Mas
ter
prob
lem
GE
7 S
ep 0
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lide
24
•C
oope
ratin
g so
lver
s:
•IL
OG
’s O
PL
Stu
dio
•E
CLi
PS
e
•In
tegr
atio
n w
ith lo
w-le
vel m
odel
ing:
•D
ash
Opt
imiz
atio
n’s
Mos
el.
•In
tegr
atio
n w
ith h
igh-
leve
l mod
elin
g:
•S
IMP
L (C
MU
).
Sof
twar
e fo
r in
tegr
ated
met
hods
GE
7 S
ep 0
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lide
25
Out
line
Exa
mpl
e pr
oble
ms
to il
lust
rate
inte
grat
ed a
ppro
ach
•S
impl
er m
odel
ing
–Lo
t siz
ing
and
sche
dulin
g
•B
ranc
hing
sea
rch
–F
reig
ht tr
ansf
er
–P
rodu
ct c
onfig
urat
ion
–C
ontin
uous
glo
bal o
ptim
izat
ion
–A
irlin
e cr
ew s
ched
ulin
g
•C
onst
rain
t-ba
sed
sear
ch–
Mac
hine
sch
edul
ing
–S
ucce
ss s
torie
s fr
om B
AS
F, B
arbo
t, P
euge
ot-C
itroë
n
GE
7 S
ep 0
7 S
lide
26
Exa
mpl
e: L
ot s
izin
g an
d sc
hedu
ling
Sim
plifi
ed m
odel
ing
GE
7 S
ep 0
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lide
27
Day
:1
2
3
4
5
6
7
8
AB
A
Pro
duct
•A
t mos
t one
pro
duct
man
ufac
ture
d on
eac
h da
y.
•D
eman
ds fo
r ea
ch p
rodu
ct o
n ea
ch d
ay.
•M
inim
ize
setu
p +
hol
ding
cos
t.
Lot s
izin
g an
d sc
hedu
ling
GE
7 S
ep 0
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lide
28
0 ,
}1,0{ ,
,
al
l ,1
, al
l ,
, al
l
,
, al
l
,
, al
l
,1,
all
,1
, al
l
,
, al
l
,
, al
l
,
s.t.
min
1,
1,
1,
1,
1,,
≥∈
=≤≤≥
−+
≥−
≤≤−
≥+
=+
+
∑∑∑
−−
−
−
−
≠
itit
ijt
ititi
it
itit
jtij
t
ti
ijt
jtt
iij
t
ti
it
itit
ti
itit
itit
itt
iit
ij
ijt
ijit
it sx
zy
ty
ti
Cy
xt
iy
ti
yt
iy
yt
iy
zt
iy
zt
iy
yz
ti
sd
xs
qs
h
δ
δδδ
δ
Inte
ger
prog
ram
min
gm
odel
(Wol
sey)
Man
y va
riabl
es
GE
7 S
ep 0
7 S
lide
29()
()
ti
xi
y
ti
sC
x
ti
sd
xs
sh
q
itt
itit
itit
itt
iti
iti
yy
tt
, al
l ,
0
, al
l ,0
,0
, al
l ,
s.t.
min
1,
1
=→
≠≥
≤≤
+=
+
+
−
∑∑
−
Min
imiz
e ho
ldin
g an
d se
tup
cost
s
Inve
ntor
y ba
lanc
e
Inte
grat
ed m
odel
Pro
duct
ion
capa
city
GE
7 S
ep 0
7 S
lide
30Exa
mpl
e: F
reig
ht T
rans
fer
Bra
nch-
and-
boun
d se
arch
w
ith in
terv
al
prop
agat
ion
and
cutti
ng p
lane
s
GE
7 S
ep 0
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lide
31
•B
ranc
hing
on
var
iabl
es,
with
pru
ning
bas
ed o
n bo
unds
.
•P
ropa
gatio
n ba
sed
on:
•In
terv
al p
ropa
gatio
n.
•C
uttin
g pl
anes
(kn
apsa
ck c
uts
).
•R
elax
atio
n ba
sed
on li
near
pro
gram
min
g.
Thi
s ex
ampl
e ill
ustr
ates
:
GE
7 S
ep 0
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lide
32
Fre
ight
Tra
nsfe
r
•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
GE
7 S
ep 0
7 S
lide
33
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
GE
7 S
ep 0
7 S
lide
34
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 m
etac
onst
rain
t“k
now
s” w
hich
in
fere
nce
and
rela
xatio
n te
chni
ques
to
use
.
GE
7 S
ep 0
7 S
lide
35
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
Dom
ain
met
acon
stra
int
“kno
ws”
how
to
bra
nch
GE
7 S
ep 0
7 S
lide
36
++
++
++
≥+
++
≤∈
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
Bou
nds
prop
agat
ion
−⋅
−⋅
−⋅
≥=
1
425
34
33
31
7x
GE
7 S
ep 0
7 S
lide
37
++
++
++
≥+
++
≤∈
∈
12
34
12
34
12
34
12
34
min
90
6050
40
75
43
42
8
{1,2
,3},
,,
{0,1
,2,3
}
xx
xx
xx
xx
xx
xx
xx
xx
Bou
nds
prop
agat
ion
−⋅
−⋅
−⋅
≥=
1
425
34
33
31
7x
Red
uced
do
mai
n
GE
7 S
ep 0
7 S
lide
38
++
++
++
≥+
++
≤≤
≤≥
12
34
12
34
12
34
1
min
90
6050
40
75
43
42
8
03,
1i
xx
xx
xx
xx
xx
xx
xx
Con
tinuo
us r
elax
atio
n
Rep
lace
dom
ains
w
ith b
ound
s
Thi
s is
a li
near
pro
gram
min
g pr
oble
m, w
hich
is e
asy
to s
olve
.
Its o
ptim
al v
alue
pro
vide
s a
low
er b
ound
on
optim
al
valu
e of
orig
inal
pro
blem
.
GE
7 S
ep 0
7 S
lide
39
++
++
++
≥+
++
≤≤
≤≥
12
34
12
34
12
34
1
min
90
6050
40
75
43
42
8
03,
1i
xx
xx
xx
xx
xx
xx
xx
Cut
ting
plan
es (v
alid
ineq
ualit
ies)
We
can
crea
te a
tigh
ter
rela
xatio
n (la
rger
min
imum
va
lue)
with
the
addi
tion
of c
uttin
g pl
anes
.
GE
7 S
ep 0
7 S
lide
40
++
++
++
≥+
++
≤≤
≤≥
12
34
12
34
12
34
1
min
90
6050
40
75
43
42
8
03,
1i
xx
xx
xx
xx
xx
xx
xx
Cut
ting
plan
es (v
alid
ineq
ualit
ies)
All
feas
ible
sol
utio
ns o
f the
or
igin
al p
robl
em s
atis
fy a
cu
tting
pla
ne (
i.e.,
it is
val
id).
But
a c
uttin
g pl
ane
may
ex
clud
e (“
cut o
ff”)
sol
utio
ns o
f th
e co
ntin
uous
rel
axat
ion.
Cut
ting
plan
e
Fea
sibl
e so
lutio
ns
Con
tinuo
us
rela
xatio
n
GE
7 S
ep 0
7 S
lide
41
++
++
++
≥+
++
≤≤
≤≥
12
34
12
34
12
34
1
min
90
6050
40
75
43
42
8
03,
1i
xx
xx
xx
xx
xx
xx
xx
Cut
ting
plan
es (v
alid
ineq
ualit
ies)
{1,2
} is
a p
acki
ng
…be
caus
e 7x
1+
5x 2
alon
e ca
nnot
sat
isfy
the
ineq
ualit
y,
even
with
x1
= x
2=
3.
GE
7 S
ep 0
7 S
lide
42
++
++
++
≥+
++
≤≤
≤≥
12
34
12
34
12
34
1
min
90
6050
40
75
43
42
8
03,
1i
xx
xx
xx
xx
xx
xx
xx
Cut
ting
plan
es (v
alid
ineq
ualit
ies)
{1,2
} is
a p
acki
ng
So,
+≥
−⋅
+⋅
34
43
42(7
35
3)
xx
GE
7 S
ep 0
7 S
lide
43
++
++
++
≥+
++
≤≤
≤≥
12
34
12
34
12
34
1
min
90
6050
40
75
43
42
8
03,
1i
xx
xx
xx
xx
xx
xx
xx
Cut
ting
plan
es (v
alid
ineq
ualit
ies)
{1,2
} is
a p
acki
ng
{}
−⋅
+⋅
+≥
=
3
4
42(7
35
3)
2m
ax4,
3x
x
So,
+≥
−⋅
+⋅
34
43
42(7
35
3)
xx
whi
ch im
plie
s
Kna
psac
k cu
t
GE
7 S
ep 0
7 S
lide
44
++
++
++
≥+
++
≤≤
≤≥
12
34
12
34
12
34
1
min
90
6050
40
75
43
42
8
03,
1i
xx
xx
xx
xx
xx
xx
xx
Cut
ting
plan
es (v
alid
ineq
ualit
ies)
Max
imal
Pac
king
sK
naps
ack
cuts
{1,2
}x 3
+ x
4≥
2
{1,3
}x 2
+ x
4≥
2
{1,4
}x 2
+ x
3≥
3
Kna
psac
k cu
ts c
orre
spon
ding
to n
onm
axim
al
pack
ings
can
be
nonr
edun
dant
.
GE
7 S
ep 0
7 S
lide
45
++
++
++
≥+ +
≥+
≥+
≥++
≤≤
≤≥
12
34
12
34
1 342
3
2
1
23
4
4
min
90
6050
40
75
43
42
8
03,
1
2 2 3
i
xx
xx
xx
xx
xx
x
xx
xx
xx
x
xx
Con
tinuo
us r
elax
atio
n w
ith c
uts
Opt
imal
val
ue o
f 523
.3 is
a lo
wer
bou
nd o
n op
timal
val
ue
of o
rigin
al p
robl
em.
Kna
psac
k cu
ts
GE
7 S
ep 0
7 S
lide
46
Bra
nch-
infe
r-an
d-re
lax
tree
Pro
paga
te b
ound
s an
d so
lve
rela
xatio
n of
or
igin
al p
robl
em.
x 1∈
{ 1
23}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
2⅓,3
,2⅔
,0)
valu
e =
523⅓
GE
7 S
ep 0
7 S
lide
47
Bra
nch
on a
va
riabl
e w
ith
noni
nteg
ral v
alue
in
the
rela
xatio
n.
x 1∈
{ 1
23}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
2⅓,3
,2⅔
,0)
valu
e =
523⅓
x 1∈
{1,2
}x 1
= 3
Bra
nch-
infe
r-an
d-re
lax
tree
GE
7 S
ep 0
7 S
lide
48
Pro
paga
te b
ound
s an
d so
lve
rela
xatio
n.
Sin
ce r
elax
atio
n is
infe
asib
le,
back
trac
k.
x 1∈
{ 1
23}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
2⅓,3
,2⅔
,0)
valu
e =
523⅓
x 1∈
{ 1
2 }
x 2∈
{
23}
x 3∈
{ 1
23}
x 4∈
{ 1
23}
infe
asib
lere
laxa
tion
x 1∈
{1,2
}x 1
= 3
Bra
nch-
infe
r-an
d-re
lax
tree
GE
7 S
ep 0
7 S
lide
49
Pro
paga
te b
ound
s an
d so
lve
rela
xatio
n.
Bra
nch
on
noni
nteg
ral
varia
ble.
x 1∈
{ 1
23}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
2⅓,3
,2⅔
,0)
valu
e =
523⅓
x 1∈
{ 1
2 }
x 2∈
{
23}
x 3∈
{ 1
23}
x 4∈
{ 1
23}
infe
asib
lere
laxa
tion
x 1∈
{
3}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
3,2.
6,2,
0)va
lue
= 5
26
x 1∈
{1,2
}x 1
= 3
x 2∈
{0,1
,2}x 2
= 3
Bra
nch-
infe
r-an
d-re
lax
tree
GE
7 S
ep 0
7 S
lide
50
Bra
nch
agai
n.
x 1∈
{ 1
23}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
2⅓,3
,2⅔
,0)
valu
e =
523⅓
x 1∈
{ 1
2 }
x 2∈
{
23}
x 3∈
{ 1
23}
x 4∈
{ 1
23}
infe
asib
lere
laxa
tion
x 1∈
{
3}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
3,2.
6,2,
0)va
lue
= 5
26
x 1∈
{
3}
x 2∈
{012
}x 3
∈{
123
}x 4
∈{0
123}
x =
(3,
2,2¾
,0)
valu
e =
527
½
x 1∈
{1,2
}x 1
= 3
x 2∈
{0,1
,2}x 2
= 3
x 3∈
{1,2
}x 3
= 3
Bra
nch-
infe
r-an
d-re
lax
tree
GE
7 S
ep 0
7 S
lide
51
Sol
utio
n of
re
laxa
tion
is in
tegr
al a
nd
ther
efor
e fe
asib
le
in th
e or
igin
al
prob
lem
.
Thi
s be
com
es th
e in
cum
bent
so
lutio
n.
x 1∈
{ 1
23}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
2⅓,3
,2⅔
,0)
valu
e =
523⅓
x 1∈
{ 1
2 }
x 2∈
{
23}
x 3∈
{ 1
23}
x 4∈
{ 1
23}
infe
asib
lere
laxa
tion
x 1∈
{
3}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
3,2.
6,2,
0)va
lue
= 5
26
x 1∈
{
3}
x 2∈
{012
}x 3
∈{
123
}x 4
∈{0
123}
x =
(3,
2,2¾
,0)
valu
e =
527
½
x 1∈
{
3}
x 2∈
{ 1
2 }
x 3∈
{ 1
2 }
x 4∈
{ 1
23}
x =
(3,
2,2,
1)va
lue
= 5
30fe
asib
le s
olut
ionx 1
∈{1
,2}
x 1=
3
x 2∈
{0,1
,2}x 2
= 3
x 3∈
{1,2
}x 3
= 3
Bra
nch-
infe
r-an
d-re
lax
tree
GE
7 S
ep 0
7 S
lide
52
Sol
utio
n is
no
nint
egra
l, bu
t w
e ca
n ba
cktr
ack
beca
use
valu
e of
re
laxa
tion
is
no b
ette
r th
an
incu
mbe
nt s
olut
ion.
x 1∈
{ 1
23}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
2⅓,3
,2⅔
,0)
valu
e =
523⅓
x 1∈
{ 1
2 }
x 2∈
{
23}
x 3∈
{ 1
23}
x 4∈
{ 1
23}
infe
asib
lere
laxa
tion
x 1∈
{
3}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
3,2.
6,2,
0)va
lue
= 5
26
x 1∈
{
3}
x 2∈
{012
}x 3
∈{
123
}x 4
∈{0
123}
x =
(3,
2,2¾
,0)
valu
e =
527
½
x 1∈
{
3}
x 2∈
{ 1
2 }
x 3∈
{ 1
2 }
x 4∈
{ 1
23}
x =
(3,
2,2,
1)va
lue
= 5
30fe
asib
le s
olut
ion
x 1∈
{
3}
x 2∈
{012
}x 3
∈{
3
}x 4
∈{0
12 }
x =
(3,
1½,3
,½)
valu
e =
530
back
trac
kdu
e to
bou
nd
x 1∈
{1,2
}x 1
= 3
x 2∈
{0,1
,2}x 2
= 3
x 3∈
{1,2
}x 3
= 3
Bra
nch-
infe
r-an
d-re
lax
tree
GE
7 S
ep 0
7 S
lide
53
Ano
ther
fea
sibl
e so
lutio
n fo
und.
No
bette
r th
an
incu
mbe
nt s
olut
ion,
w
hich
is o
ptim
al
beca
use
sear
ch
has
finis
hed.
x 1∈
{ 1
23}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
2⅓,3
,2⅔
,0)
valu
e =
523⅓
x 1∈
{ 1
2 }
x 2∈
{
23}
x 3∈
{ 1
23}
x 4∈
{ 1
23}
infe
asib
lere
laxa
tion
x 1∈
{
3}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
3,2.
6,2,
0)va
lue
= 5
26
x 1∈
{
3}
x 2∈
{012
}x 3
∈{
123
}x 4
∈{0
123}
x =
(3,
2,2¾
,0)
valu
e =
527
½
x 1∈
{
3}
x 2∈
{
3}
x 3∈
{012
}x 4
∈{0
12 }
x =
(3,
3,0,
2)va
lue
= 5
30fe
asib
le s
olut
ion
x 1∈
{
3}
x 2∈
{ 1
2 }
x 3∈
{ 1
2 }
x 4∈
{ 1
23}
x =
(3,
2,2,
1)va
lue
= 5
30fe
asib
le s
olut
ion
x 1∈
{
3}
x 2∈
{012
}x 3
∈{
3
}x 4
∈{0
12 }
x =
(3,
1½,3
,½)
valu
e =
530
back
trac
kdu
e to
bou
nd
x 1∈
{1,2
}x 1
= 3
x 2∈
{0,1
,2}x 2
= 3
x 3∈
{1,2
}x 3
= 3
Bra
nch-
infe
r-an
d-re
lax
tree
GE
7 S
ep 0
7 S
lide
54
Two
optim
al
solu
tions
fou
nd.
In g
ener
al, n
ot a
ll op
timal
sol
utio
ns
are
foun
d,
x 1∈
{ 1
23}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
2⅓,3
,2⅔
,0)
valu
e =
523⅓
x 1∈
{ 1
2 }
x 2∈
{
23}
x 3∈
{ 1
23}
x 4∈
{ 1
23}
infe
asib
lere
laxa
tion
x 1∈
{
3}
x 2∈
{012
3}x 3
∈{0
123}
x 4∈
{012
3}x
= (
3,2.
6,2,
0)va
lue
= 5
26
x 1∈
{
3}
x 2∈
{012
}x 3
∈{
123
}x 4
∈{0
123}
x =
(3,
2,2¾
,0)
valu
e =
527
½
x 1∈ ∈∈∈
{
3}
x 2∈ ∈∈∈
{
3}
x 3∈ ∈∈∈
{012
}x 4
∈ ∈∈∈{0
12 }
x =
(3,3
,0,2
)va
lue
= 53
0op
timal
sol
utio
n
x 1∈ ∈∈∈
{
3}
x 2∈ ∈∈∈
{ 1
2 }
x 3∈ ∈∈∈
{ 1
2 }
x 4∈ ∈∈∈
{ 1
23}
x =
(3,2
,2,1
)va
lue
= 53
0op
timal
sol
utio
n
x 1∈
{
3}
x 2∈
{012
}x 3
∈{
3
}x 4
∈{0
12 }
x =
(3,
1½,3
,½)
valu
e =
530
back
trac
kdu
e to
bou
nd
x 1∈
{1,2
}x 1
= 3
x 2∈
{0,1
,2}x 2
= 3
x 3∈
{1,2
}x 3
= 3
Bra
nch-
infe
r-an
d-re
lax
tree
GE
7 S
ep 0
7 S
lide
55
Oth
er ty
pes
of c
uttin
g pl
anes
•Li
fted
0-1
knap
sack
ineq
ualit
ies
•C
lique
ineq
ualit
ies
•G
omor
y cu
ts
•M
ixed
inte
ger
roun
ding
cut
s
•D
isju
nctiv
e cu
ts
•S
peci
aliz
ed c
uts
–F
low
cut
s (f
ixed
cha
rge
netw
ork
flow
pro
blem
)–
Com
b in
equa
litie
s (t
rave
ling
sale
sman
pro
blem
)–
Man
y, m
any
othe
rs
GE
7 S
ep 0
7 S
lide
56
Exa
mpl
e: P
rodu
ct C
onfig
urat
ion
Bra
nch-
and-
boun
d se
arch
with
pro
paga
tion
and
rela
xatio
n of
var
iabl
e in
dice
s.
Fro
m: T
hors
tein
sson
and
Otto
sson
(20
01)
GE
7 S
ep 0
7 S
lide
57
•T
his
exam
ple
illus
trat
es:
•P
ropa
gatio
n of
var
iabl
e in
dice
s.
–V
aria
ble
inde
x is
con
vert
ed to
a s
peci
ally
str
uctu
red
elem
ent c
onst
rain
t.
–S
peci
ally
str
uctu
red
filte
ring
for
elem
ent.
–V
alid
kna
psac
kcu
ts a
re d
eriv
ed a
nd p
ropa
gate
d.
GE
7 S
ep 0
7 S
lide
58
•T
his
exam
ple
illus
trat
es:
•P
ropa
gatio
n of
var
iabl
e in
dice
s.
–V
aria
ble
inde
x is
con
vert
ed to
a s
peci
ally
str
uctu
red
elem
ent c
onst
rain
t.
–S
peci
ally
str
uctu
red
filte
ring
for
elem
ent.
–V
alid
kna
psac
kcu
ts a
re d
eriv
ed a
nd p
ropa
gate
d.
•R
elax
atio
n of
var
iabl
e in
dice
s.–
Ele
men
t is
inte
rpre
ted
as a
dis
junc
tion
of li
near
sys
tem
s.
–C
onve
x hu
ll re
laxa
tion
for
disj
unct
ion
is u
sed.
GE
7 S
ep 0
7 S
lide
59
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
GE
7 S
ep 0
7 S
lide
60
Pro
blem
dat
a
GE
7 S
ep 0
7 S
lide
61
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
Mod
el o
f the
pro
blem
Am
ount
of a
ttrib
ute
jpr
oduc
ed b
y ty
pe t
iof
com
pone
nt i
t i is
a v
aria
ble
inde
x
Uni
t cos
t of p
rodu
cing
at
trib
ute
j
GE
7 S
ep 0
7 S
lide
62
min
, al
l
, al
l
i
jj
j
ji
ijtik
jj
j
cv
vq
Aj
Lv
Uj
= ≤≤
∑ ∑
Line
ar in
equa
lity
met
acon
stra
int
Mod
el o
f the
pro
blem
GE
7 S
ep 0
7 S
lide
63
min
, al
l
, al
l
i
jj
j
ji
ijtik
jj
j
cv
vq
Aj
Lv
Uj
= ≤≤
∑ ∑In
dexe
d lin
ear
met
acon
stra
int
Mod
el o
f the
pro
blem
GE
7 S
ep 0
7 S
lide
64
To
solv
e it:
•B
ranc
h on
dom
ains
of t
ian
d q i
.
•P
ropa
gate
ele
men
tcon
stra
ints
and
bou
nds
on v
j.
–D
eriv
e an
d pr
opag
ate
knap
sack
cuts
.
•R
elax
ele
men
t.
–C
onve
x hu
ll re
laxa
tion
for
disj
unct
ion.
GE
7 S
ep 0
7 S
lide
65
min
, al
l
, al
l
i
jj
j
ji
ijtik
jj
j
cv
vq
Aj
Lv
Uj
= ≤≤
∑ ∑
Pro
paga
tion
Thi
s is
pro
paga
ted
in th
e us
ual w
ay
GE
7 S
ep 0
7 S
lide
66
Thi
s is
rew
ritte
n as
Pro
paga
tion
min
, al
l
, al
l
i
jj
j
ji
ijtik
jj
j
cv
vq
Aj
Lv
Uj
= ≤≤
∑ ∑T
his
is p
ropa
gate
d in
the
usua
l way
()
1
, al
l
elem
ent
,(,
,,
),,
all
,
ji
i
ii
iji
ijni
vz
j
tq
Aq
Az
ij
=∑
…
GE
7 S
ep 0
7 S
lide
67
Thi
sca
n be
pro
paga
ted
by
(a)
usin
g sp
ecia
lized
filt
ers
for
elem
entc
onst
rain
ts o
f thi
s fo
rm…
Pro
paga
tion
()
1
, al
l
elem
ent
,(,
,,
),,
all
,
ji
i
ii
iji
ijni
vz
j
tq
Aq
Az
ij
=∑
…
GE
7 S
ep 0
7 S
lide
68
Thi
sis
pro
paga
ted
by
(a)
usin
g sp
ecia
lized
filt
ers
for
elem
entc
onst
rain
ts o
f thi
s fo
rm,
(b)
addi
ng k
naps
ack
cuts
for
the
valid
ineq
ualit
ies:
is c
urre
nt
dom
ain
of v
j
Pro
paga
tion
()
1
, al
l
elem
ent
,(,
,,
),,
all
,
ji
i
ii
iji
ijni
vz
j
tq
Aq
Az
ij
=∑
…
{}
{}
max
, al
l
min
, al
l
t i t i
jijk
ik
Di
ijki
jk
Di
Aq
vj
Aq
vj
∈ ∈
≥ ≤
∑ ∑
[,
]j
jv
van
d (c
) pr
opag
atin
g th
e kn
apsa
ck c
uts.
GE
7 S
ep 0
7 S
lide
69
Thi
s is
rel
axed
as
jj
jv
vv
≤≤
min
, al
l
, al
l
i
jj
j
ji
ijtik
jj
j
cv
vq
Aj
Lv
Uj
= ≤≤
∑ ∑
Rel
axat
ion
GE
7 S
ep 0
7 S
lide
70
Thi
s is
rel
axed
by
rela
this
an
d ad
ding
the
knap
sack
cut
s.
Thi
s is
rel
axed
as
jj
jv
vv
≤≤
min
, al
l
, al
l
i
jj
j
ji
ijtik
jj
j
cv
vq
Aj
Lv
Uj
= ≤≤
∑ ∑
Rel
axat
ion
()
1
, al
l
elem
ent
,(,
,,
),,
all
,
ji
i
ii
iji
ijni
vz
j
tq
Aq
Az
ij
=∑
…
GE
7 S
ep 0
7 S
lide
71
Thi
s is
rel
axed
by
writ
ing
each
ele
men
tcon
stra
int a
s a
disj
unct
ion
of li
near
sys
tem
s an
d w
ritin
g a
conv
ex h
ull
rela
xatio
n of
the
disj
unct
ion:
()
1
, al
l
elem
ent
,(,
,,
),,
all
,
ji
i
ii
iji
ijni
vz
j
tq
Aq
Az
ij
=∑
…
,
tt
ii
iijk
iki
ikk
Dk
D
zA
q∈
∈
==
∑∑
Rel
axat
ion
GE
7 S
ep 0
7 S
lide
72
So
the
follo
win
g LP
rel
axat
ion
is s
olve
d at
eac
h no
de
of th
e se
arch
tree
to o
btai
n a
low
er b
ound
:
{}
{}
min
, al
l
, al
l
, al
l
, al
l
knap
sack
cut
s fo
r m
ax,
all
knap
sack
cut
s fo
r m
in,
all
0, a
ll ,
t i
t i
t i t i
jj
j
jijk
iki
kD
jik
kD
jj
j
ii
i
ijki
jk
Di
ijki
jk
Di
ik
cv
vA
qj
i
vv
vj
qi
Aq
vj
Aq
vj
qi
k
∈
∈
∈ ∈
= = ≤≤
≤≤
≥ ≤
≥
∑ ∑∑
∑
∑ ∑
Rel
axat
ion
GE
7 S
ep 0
7 S
lide
73
Afte
r pr
opag
atio
n, t
he s
olut
ion
of th
e re
laxa
tion
is
feas
ible
at t
he r
oot n
ode.
No
bran
chin
g ne
eded
.
{}
{}
min
, al
l
, al
l
, al
l
, al
l
knap
sack
cut
s fo
r m
ax,
all
knap
sack
cut
s fo
r m
in,
all
0, a
ll ,
t i
t i
t i t i
jj
j
jijk
iki
kD
jik
kD
jj
j
ii
i
ijki
jk
Di
ijki
jk
Di
ik
cv
vA
qj
i
vv
vj
qi
Aq
vj
Aq
vj
qi
k
∈
∈
∈ ∈
= = ≤≤
≤≤
≥ ≤
≥
∑ ∑∑
∑
∑ ∑
Sol
utio
n of
the
exam
ple
q 1, q
1C=
1→
t 1 =
C
q 2, q
2A=
2 →
t 2=
A
q 3, q
3B=
3 →
t 3=
B
GE
7 S
ep 0
7 S
lide
74
Afte
r pr
opag
atio
n, t
he s
olut
ion
of th
e re
laxa
tion
is
feas
ible
at t
he r
oot n
ode.
No
bran
chin
g ne
eded
.
{}
{}
min
, al
l
, al
l
, al
l
, al
l
knap
sack
cut
s fo
r m
ax,
all
knap
sack
cut
s fo
r m
in,
all
0, a
ll ,
t i
t i
t i t i
jj
j
jijk
iki
kD
jik
kD
jj
j
ii
i
ijki
jk
Di
ijki
jk
Di
ik
cv
vA
qj
i
vv
vj
qi
Aq
vj
Aq
vj
qi
k
∈
∈
∈ ∈
= = ≤≤
≤≤
≥ ≤
≥
∑ ∑∑
∑
∑ ∑
Sol
utio
n of
the
exam
ple
q 1, q
1C=
1→
t 1 =
C
q 2, q
2A=
2 →
t 2=
A
q 3, q
3B=
3 →
t 3=
B
q 1is
inte
gral
, on
ly o
ne q
1kis
pos
itive
GE
7 S
ep 0
7 S
lide
75
Afte
r pr
opag
atio
n, t
he s
olut
ion
of th
e re
laxa
tion
is
feas
ible
at t
he r
oot n
ode.
No
bran
chin
g ne
eded
.
Sol
utio
n of
the
exam
ple
q 1, q
1C=
1→
t 1 =
C
q 2, q
2A=
2 →
t 2=
A
q 3, q
3B=
3 →
t 3=
B
Mem
ory
B
Mem
ory
B
Mem
ory
B Pow
ersu
pply
C
Per
sona
l com
pute
r Dis
k dr
ive
AD
isk
driv
e A
GE
7 S
ep 0
7 S
lide
76
Com
puta
tiona
l Res
ults
(O
ttoss
on &
Tho
rste
inss
on)
0.010.1110100
1000
8x10
16x2
020
x24
20x3
0
Pro
ble
m
Seconds
CP
LEX
CLP
Hyb
rid
GE
7 S
ep 0
7 S
lide
77
Exa
mpl
e: C
ontin
uous
Glo
bal O
ptim
izat
ion
Bra
nchi
ng o
n co
ntin
uous
var
iabl
es
with
La
gran
gean
-bas
ed b
ound
s pr
opag
atio
n an
d lin
ear
or c
onve
x no
nlin
ear
rela
xatio
n
GE
7 S
ep 0
7 S
lide
78
•B
ranc
hing
by
spl
ittin
g bo
xes.
•P
ropa
gatio
n ba
sed
on:
•In
terv
al a
rithm
etic
.
•La
gran
ge m
ultip
liers
.
•Li
near
rel
axat
ion
usin
g M
cCor
mic
k fa
ctor
izat
ion.
•T
he b
est c
ontin
uous
glo
bal s
olve
rs (
e.g.
, BA
RO
N)
com
bine
th
ese
tecn
niqu
es f
rom
CP
and
OR
.
Thi
s ex
ampl
e ill
ustr
ates
:
GE
7 S
ep 0
7 S
lide
79
it ca
n be
sho
wn
that
if c
onst
rain
t iis
tigh
t,*
*(
)i
i
Uv
gx
λ−≤
Whe
re U
is a
n up
per
boun
d on
the
optim
al v
alue
.
min
()
()
0
fx
gx
xS≥
∈S
uppo
sing
has
optim
al v
alue
v*,
an
d ea
ch c
onst
rain
t iha
s La
gran
ge m
ultip
lier
λ i*,
Pro
paga
tion
usin
g La
gran
ge m
ultip
liers
GE
7 S
ep 0
7 S
lide
80
it ca
n be
sho
wn
that
if c
onst
rain
t iis
tigh
t,*
*(
)i
i
Uv
gx
λ−≤
Thi
s in
equa
lity
can
be p
ropa
gate
d.
min
()
()
0
fx
gx
xS≥
∈S
uppo
sing
has
optim
al v
alue
v*,
an
d ea
ch c
onst
rain
t iha
s La
gran
ge m
ultip
lier
λ i*,
Pro
paga
tion
usin
g La
gran
ge m
ultip
liers
GE
7 S
ep 0
7 S
lide
81
it ca
n be
sho
wn
that
if c
onst
rain
t iis
tigh
t,*
*(
)i
i
Uv
gx
λ−≤
Thi
s in
equa
lity
can
be p
ropa
gate
d.
Whe
n ap
plie
d to
bou
nds
on v
aria
bles
, th
is te
chni
que
yiel
ds b
ound
s pr
opag
atio
n.
Whe
n th
e re
laxa
tion
is li
near
, thi
s be
com
es re
duce
d co
st v
aria
ble
fixin
g (w
idel
y us
ed in
inte
ger
prog
ram
min
g)
min
()
()
0
fx
gx
xS≥
∈S
uppo
sing
has
optim
al v
alue
v*,
an
d ea
ch c
onst
rain
t iha
s La
gran
ge m
ultip
lier
λ i*,
Pro
paga
tion
usin
g La
gran
ge m
ultip
liers
GE
7 S
ep 0
7 S
lide
82
Fea
sibl
e se
t
Glo
bal o
ptim
um
Loca
l opt
imum
x 1
x 2
The
pro
blem
12
12
12
12
max
41
22
[0,1
],
[0,2
]
xx
xx
xx
xx
+=
+≤
∈∈
GE
7 S
ep 0
7 S
lide
83
The
Pro
blem
12
12
12
12
max
41
22
[0,1
],
[0,2
]
xx
xx
xx
xx
+=
+≤
∈∈
Line
ar in
equa
lity
met
acon
stra
int
GE
7 S
ep 0
7 S
lide
84
The
Pro
blem
12
12
12
12
max
41
22
[0,1
],
[0,2
]
xx
xx
xx
xx
+=
+≤
∈∈
Bili
near
(in
)equ
ality
m
etac
onst
rain
t
GE
7 S
ep 0
7 S
lide
85
To s
olve
it:
•S
earc
h: s
plit
inte
rval
dom
ains
of x
1,x 2
.
–E
ach
node
of s
earc
h tr
ee is
a p
robl
em r
estr
ictio
n.
•P
ropa
gatio
n:In
terv
al p
ropa
gatio
n, d
omai
n fil
terin
g.
–U
se L
agra
nge
mul
tiplie
rs
to in
fer
valid
ineq
ualit
y fo
r pr
opag
atio
n.
•R
elax
atio
n:U
se fu
nctio
n fa
ctor
izat
ion
to o
btai
n lin
ear
cont
inuo
us r
elax
atio
n.
GE
7 S
ep 0
7 S
lide
86
Inte
rval
pro
paga
tion
Pro
paga
te in
terv
als
[0,1
], [0
,2]
thro
ugh
cons
trai
nts
to o
btai
n [1
/8,7
/8],
[1/4
,7/4
]
x 1
x 2
GE
7 S
ep 0
7 S
lide
87
Rel
axat
ion
(fun
ctio
n fa
ctor
izat
ion)
Fac
tor
com
plex
func
tions
into
ele
men
tary
fun
ctio
ns th
at h
ave
know
n lin
ear
rela
xatio
ns.
Writ
e 4x
1x2
= 1
as
4y=
1 w
here
y=
x1x
2.
Thi
s fa
ctor
s 4x
1x2
into
line
ar f
unct
ion
4yan
d bi
linea
r fu
nctio
n x 1
x 2.
Line
ar f
unct
ion
4yis
its
own
linea
r re
laxa
tion.
GE
7 S
ep 0
7 S
lide
88
whe
re d
omai
n of
xjis
[,
]j
jx
x
Rel
axat
ion
(fun
ctio
n fa
ctor
izat
ion)
Fac
tor
com
plex
func
tions
into
ele
men
tary
fun
ctio
ns th
at h
ave
know
n lin
ear
rela
xatio
ns.
Writ
e 4x
1x2
= 1
as
4y=
1 w
here
y=
x1x
2.
Thi
s fa
ctor
s 4x
1x2
into
line
ar f
unct
ion
4yan
d bi
linea
r fu
nctio
n x 1
x 2.
Line
ar f
unct
ion
4yis
its
own
linea
r re
laxa
tion.
Bili
near
fun
ctio
n y
= x
1x2
has
rela
xatio
n:
21
12
12
21
12
12
21
12
12
21
12
12
xx
xx
xx
yx
xx
xx
x
xx
xx
xx
yx
xx
xx
x
+−
≤≤
+−
+−
≤≤
+−
GE
7 S
ep 0
7 S
lide
89
The
line
ar r
elax
atio
n be
com
es:
Rel
axat
ion
(fun
ctio
n fa
ctor
izat
ion)
12
12
21
12
12
21
12
12
21
12
12
21
12
12
min
41
22 ,
1,
2j
jj
xx
y xx
xx
xx
xx
yx
xx
xx
x
xx
xx
xx
yx
xx
xx
x
xx
xj
+= +
≤+
−≤
≤+
−+
−≤
≤+
−≤
≤=
GE
7 S
ep 0
7 S
lide
90
Sol
ve li
near
rel
axat
ion.
x 1
x 2
Rel
axat
ion
(fun
ctio
n fa
ctor
izat
ion)
GE
7 S
ep 0
7 S
lide
91
x 1
x 2
Sin
ce s
olut
ion
is in
feas
ible
, sp
lit a
n in
terv
al a
nd b
ranc
h.
Sol
ve li
near
rel
axat
ion.
Rel
axat
ion
(fun
ctio
n fa
ctor
izat
ion)
2[1
,1.7
5]x
∈
2[0
.25,
1]x
∈
GE
7 S
ep 0
7 S
lide
92
x 1
x 2
x 1
x 2
2[1
,1.7
5]x
∈2
[0.2
5,1]
x∈
GE
7 S
ep 0
7 S
lide
93
Sol
utio
n of
re
laxa
tion
is
feas
ible
, va
lue
= 1
.25
Thi
s be
com
es
incu
mbe
nt
solu
tion
x 1
x 2
x 1
x 2
2[1
,1.7
5]x
∈2
[0.2
5,1]
x∈
GE
7 S
ep 0
7 S
lide
94
Sol
utio
n of
re
laxa
tion
is
feas
ible
, va
lue
= 1
.25
Thi
s be
com
es
incu
mbe
nt
solu
tion
x 1
x 2
x 1
x 2S
olut
ion
of
rela
xatio
n is
no
t qui
te
feas
ible
, va
lue
= 1
.854
Als
o us
e La
gran
ge
mul
tiplie
rs fo
r do
mai
n fil
terin
g…
2[1
,1.7
5]x
∈2
[0.2
5,1]
x∈
GE
7 S
ep 0
7 S
lide
95
12
12
21
12
12
21
12
12
21
12
12
21
12
12
min
41
22 ,
1,
2j
jj
xx
y xx
xx
xx
xx
yx
xx
xx
x
xx
xx
xx
yx
xx
xx
x
xx
xj
+= +
≤+
−≤
≤+
−+
−≤
≤+
−≤
≤=
Ass
ocia
ted
Lagr
ange
m
ultip
lier i
n so
lutio
n of
re
laxa
tion
is λ
2=
1.1
Rel
axat
ion
(fun
ctio
n fa
ctor
izat
ion)
GE
7 S
ep 0
7 S
lide
96
Thi
s yi
elds
a v
alid
ineq
ualit
y fo
r pr
opag
atio
n:
Ass
ocia
ted
Lagr
ange
m
ultip
lier i
n so
lutio
n of
re
laxa
tion
is λ
2=
1.1
12
1.85
41.
252
21.
451
1.1
xx
−+
≥−
=
Rel
axat
ion
(fun
ctio
n fa
ctor
izat
ion)
Val
ue o
f re
laxa
tion
Lagr
ange
mul
tiplie
r
Val
ue o
f inc
umbe
nt
solu
tion
12
12
21
12
12
21
12
12
21
12
12
21
12
12
min
41
22 ,
1,
2j
jj
xx
y xx
xx
xx
xx
yx
xx
xx
x
xx
xx
xx
yx
xx
xx
x
xx
xj
+= +
≤+
−≤
≤+
−+
−≤
≤+
−≤
≤=
GE
7 S
ep 0
7 S
lide
97
Exa
mpl
e: A
irlin
e C
rew
Sch
edul
ing
Bra
nch
and
pric
ein
whi
ch a
line
ar r
elax
atio
n of
an
MIL
P is
sol
ved
by C
P-b
ased
col
umn
gene
ratio
n
Fro
m: F
ahle
et a
l. (2
002)
GE
7 S
ep 0
7 S
lide
98
•O
vera
ll m
ixed
inte
ger
prog
ram
min
g fr
amew
ork.
•Li
near
rel
axat
ion
solv
ed b
y C
P-b
ased
col
umn
gene
ratio
n.
Thi
s ex
ampl
e ill
ustr
ates
:
GE
7 S
ep 0
7 S
lide
99
Sol
ving
an
LP b
y co
lum
n ge
nera
tion
Sup
pose
the
LP r
elax
atio
n of
an
inte
ger
prog
ram
min
g pr
oble
m h
as a
hug
e nu
mbe
r of
va
riabl
es:
min
0cx
Ax
b
x
=≥
GE
7 S
ep 0
7 S
lide
100
Sol
ving
an
LP b
y co
lum
n ge
nera
tion
Sup
pose
the
LP r
elax
atio
n of
an
inte
ger
prog
ram
min
g pr
oble
m h
as a
hug
e nu
mbe
r of
va
riabl
es:
min
0cx
Ax
b
x
=≥
We
will
sol
ve a
res
tric
ted
mas
ter
prob
lem
, w
hich
has
a s
mal
l sub
set o
f the
var
iabl
es:
()
min
0
jj
jJ
jj
jJ j
cx
Ax
b
x
λ∈
∈
=
≥
∑
∑C
olum
n jo
f A
GE
7 S
ep 0
7 S
lide
101
Sol
ving
an
LP b
y co
lum
n ge
nera
tion
Sup
pose
the
LP r
elax
atio
n of
an
inte
ger
prog
ram
min
g pr
oble
m h
as a
hug
e nu
mbe
r of
va
riabl
es:
min
0cx
Ax
b
x
=≥
We
will
sol
ve a
res
tric
ted
mas
ter
prob
lem
, w
hich
has
a s
mal
l sub
set o
f the
var
iabl
es:
()
min
0
jj
jJ
jj
jJ j
cx
Ax
b
x
λ∈
∈
=
≥
∑
∑C
olum
n jo
f A
Add
ing
x kto
the
prob
lem
wou
ld im
prov
e th
e so
lutio
n if
x kha
s a
nega
tive
redu
ced
cost
:0
kk
kr
cAλ
=−
<
Row
vec
tor
of d
ual (
Lagr
ange
) m
ultip
liers
GE
7 S
ep 0
7 S
lide
102
Add
ing
x kto
the
prob
lem
wou
ld im
prov
e th
e so
lutio
n if
x kha
s a
nega
tive
redu
ced
cost
:0
kk
kr
cAλ
=−
<
Col
umn
gene
ratio
n
Com
putin
g th
e re
duce
d co
st o
f xk
is k
now
n as
pric
ing
x k.
min is
a c
olum
n of
yc
y
yA
λ−
So
we
solv
e th
e pr
icin
g pr
oble
m:
Cos
t of c
olum
n y
GE
7 S
ep 0
7 S
lide
103
Add
ing
x kto
the
prob
lem
wou
ld im
prov
e th
e so
lutio
n if
x kha
s a
nega
tive
redu
ced
cost
:0
kk
kr
cAλ
=−
<
Col
umn
gene
ratio
n
Com
putin
g th
e re
duce
d co
st o
f xk
is k
now
n as
pric
ing
x k.
min is
a c
olum
n of
yc
y
yA
λ−
Thi
s ca
n of
ten
be s
olve
d by
CP
.
We
hope
to fi
nd a
n op
timal
sol
utio
n be
fore
gen
erat
ing
too
man
y co
lum
ns.
So
we
solv
e th
e pr
icin
g pr
oble
m:
Cos
t of c
olum
n y
GE
7 S
ep 0
7 S
lide
104
Airl
ine
Cre
w S
ched
ulin
g
Flig
ht d
ata
Sta
rt
time
Fin
ish
time
A r
oste
r is
the
sequ
ence
of f
light
s as
sign
ed to
a
sing
le c
rew
mem
ber.
The
gap
bet
wee
n tw
o co
nsec
utiv
e fli
ghts
in a
ro
ster
mus
t be
from
2 to
3 h
ours
. To
tal f
light
tim
e fo
r a
rost
er m
ust b
e be
twee
n 6
and
10
hour
s.
For
exa
mpl
e,
fligh
t 1 c
anno
t im
med
iate
ly p
rece
de 6
fli
ght 4
can
not i
mm
edia
tely
pre
cede
5.
The
pos
sibl
e ro
ster
s ar
e:
(1,3
,5),
(1,
4,6)
, (2,
3,5)
, (2,
4,6)
We
wan
t to
assi
gn c
rew
mem
bers
to fl
ight
s to
min
imiz
e co
st w
hile
cov
erin
g th
e fli
ghts
and
obs
ervi
ng c
ompl
ex
wor
k ru
les.
GE
7 S
ep 0
7 S
lide
105
Airl
ine
Cre
w S
ched
ulin
g
The
re a
re 2
cre
w m
embe
rs, a
nd th
e po
ssib
le r
oste
rs a
re:
1
2
3
4(1
,3,5
), (
1,4,
6), (
2,3,
5), (
2,4,
6)
The
LP
rel
axat
ion
of th
e pr
oble
m is
:
= 1
if w
e as
sign
cre
w m
embe
r 1
to r
oste
r 2,
= 0
oth
erw
ise.
Cos
t of a
ssig
ning
cre
w m
embe
r 1
to r
oste
r 2
Eac
h cr
ew m
embe
r is
ass
igne
d to
ex
actly
1 r
oste
r.
Eac
h fli
ght i
s as
sign
ed a
t lea
st 1
cr
ew m
embe
r.
GE
7 S
ep 0
7 S
lide
106
Airl
ine
Cre
w S
ched
ulin
g
The
re a
re 2
cre
w m
embe
rs, a
nd th
e po
ssib
le r
oste
rs a
re:
1
2
3
4(1
,3,5
), (
1,4,
6), (
2,3,
5), (
2,4,
6)
The
LP
rel
axat
ion
of th
e pr
oble
m is
:
= 1
if w
e as
sign
cre
w m
embe
r 1
to r
oste
r 2,
= 0
oth
erw
ise.
Cos
t of a
ssig
ning
cre
w m
embe
r 1
to r
oste
r 2
Eac
h cr
ew m
embe
r is
ass
igne
d to
ex
actly
1 r
oste
r.
Eac
h fli
ght i
s as
sign
ed a
t lea
st 1
cr
ew m
embe
r.
Ros
ters
that
cov
er fl
ight
1.
GE
7 S
ep 0
7 S
lide
107
Airl
ine
Cre
w S
ched
ulin
g
The
re a
re 2
cre
w m
embe
rs, a
nd th
e po
ssib
le r
oste
rs a
re:
1
2
3
4(1
,3,5
), (
1,4,
6), (
2,3,
5), (
2,4,
6)
The
LP
rel
axat
ion
of th
e pr
oble
m is
:
= 1
if w
e as
sign
cre
w m
embe
r 1
to r
oste
r 2,
= 0
oth
erw
ise.
Cos
t of a
ssig
ning
cre
w m
embe
r 1
to r
oste
r 2
Eac
h cr
ew m
embe
r is
ass
igne
d to
ex
actly
1 r
oste
r.
Eac
h fli
ght i
s as
sign
ed a
t lea
st 1
cr
ew m
embe
r.
Ros
ters
that
cov
er fl
ight
2.
GE
7 S
ep 0
7 S
lide
108
Airl
ine
Cre
w S
ched
ulin
g
The
re a
re 2
cre
w m
embe
rs, a
nd th
e po
ssib
le r
oste
rs a
re:
1
2
3
4(1
,3,5
), (
1,4,
6), (
2,3,
5), (
2,4,
6)
The
LP
rel
axat
ion
of th
e pr
oble
m is
:
= 1
if w
e as
sign
cre
w m
embe
r 1
to r
oste
r 2,
= 0
oth
erw
ise.
Cos
t of a
ssig
ning
cre
w m
embe
r 1
to r
oste
r 2
Eac
h cr
ew m
embe
r is
ass
igne
d to
ex
actly
1 r
oste
r.
Eac
h fli
ght i
s as
sign
ed a
t lea
st 1
cr
ew m
embe
r.
Ros
ters
that
cov
er fl
ight
3.
GE
7 S
ep 0
7 S
lide
109
Airl
ine
Cre
w S
ched
ulin
g
The
re a
re 2
cre
w m
embe
rs, a
nd th
e po
ssib
le r
oste
rs a
re:
1
2
3
4(1
,3,5
), (
1,4,
6), (
2,3,
5), (
2,4,
6)
The
LP
rel
axat
ion
of th
e pr
oble
m is
:
= 1
if w
e as
sign
cre
w m
embe
r 1
to r
oste
r 2,
= 0
oth
erw
ise.
Cos
t of a
ssig
ning
cre
w m
embe
r 1
to r
oste
r 2
Eac
h cr
ew m
embe
r is
ass
igne
d to
ex
actly
1 r
oste
r.
Eac
h fli
ght i
s as
sign
ed a
t lea
st 1
cr
ew m
embe
r.
Ros
ters
that
cov
er fl
ight
4.
GE
7 S
ep 0
7 S
lide
110
Airl
ine
Cre
w S
ched
ulin
g
The
re a
re 2
cre
w m
embe
rs, a
nd th
e po
ssib
le r
oste
rs a
re:
1
2
3
4(1
,3,5
), (
1,4,
6), (
2,3,
5), (
2,4,
6)
The
LP
rel
axat
ion
of th
e pr
oble
m is
:
= 1
if w
e as
sign
cre
w m
embe
r 1
to r
oste
r 2,
= 0
oth
erw
ise.
Cos
t of a
ssig
ning
cre
w m
embe
r 1
to r
oste
r 2
Eac
h cr
ew m
embe
r is
ass
igne
d to
ex
actly
1 r
oste
r.
Eac
h fli
ght i
s as
sign
ed a
t lea
st 1
cr
ew m
embe
r.
Ros
ters
that
cov
er fl
ight
5.
GE
7 S
ep 0
7 S
lide
111
Airl
ine
Cre
w S
ched
ulin
g
The
re a
re 2
cre
w m
embe
rs, a
nd th
e po
ssib
le r
oste
rs a
re:
1
2
3
4(1
,3,5
), (
1,4,
6), (
2,3,
5), (
2,4,
6)
The
LP
rel
axat
ion
of th
e pr
oble
m is
:
= 1
if w
e as
sign
cre
w m
embe
r 1
to r
oste
r 2,
= 0
oth
erw
ise.
Cos
t of a
ssig
ning
cre
w m
embe
r 1
to r
oste
r 2
Eac
h cr
ew m
embe
r is
ass
igne
d to
ex
actly
1 r
oste
r.
Eac
h fli
ght i
s as
sign
ed a
t lea
st 1
cr
ew m
embe
r.
Ros
ters
that
cov
er fl
ight
6.
GE
7 S
ep 0
7 S
lide
112
Airl
ine
Cre
w S
ched
ulin
g
The
re a
re 2
cre
w m
embe
rs, a
nd th
e po
ssib
le r
oste
rs a
re:
1
2
3
4(1
,3,5
), (
1,4,
6), (
2,3,
5), (
2,4,
6)
The
LP
rel
axat
ion
of th
e pr
oble
m is
:
= 1
if w
e as
sign
cre
w m
embe
r 1
to r
oste
r 2,
= 0
oth
erw
ise.
Cos
t c12
of a
ssig
ning
cre
w m
embe
r 1 to
ros
ter
2
Eac
h cr
ew m
embe
r is
ass
igne
d to
ex
actly
1 r
oste
r.
Eac
h fli
ght i
s as
sign
ed a
t lea
st 1
cr
ew m
embe
r.
In a
rea
l pro
blem
, the
re c
an b
e m
illio
nsof
ros
ters
.
GE
7 S
ep 0
7 S
lide
113
Airl
ine
Cre
w S
ched
ulin
g
We
star
t by
solv
ing
the
prob
lem
with
a s
ubse
t of
the
colu
mns
:O
ptim
al
dual
so
lutio
n
u 1 u 2 v 1 v 2 v 3 v 4 v 5 v 6
GE
7 S
ep 0
7 S
lide
114
Airl
ine
Cre
w S
ched
ulin
g
We
star
t by
solv
ing
the
prob
lem
with
a s
ubse
t of
the
colu
mns
:
Dua
l va
riabl
es
u 1 u 2 v 1 v 2 v 3 v 4 v 5 v 6
GE
7 S
ep 0
7 S
lide
115
Airl
ine
Cre
w S
ched
ulin
g
We
star
t by
solv
ing
the
prob
lem
with
a s
ubse
t of
the
colu
mns
:
The
red
uced
cos
t of a
n ex
clud
ed r
oste
r k
for
crew
mem
ber i
is
in r
oste
r k
iki
jj
cu
v−
−∑
We
will
for
mul
ate
the
pric
ing
prob
lem
as
a sh
orte
st p
ath
prob
lem
.
Dua
l va
riabl
es
u 1 u 2 v 1 v 2 v 3 v 4 v 5 v 6
GE
7 S
ep 0
7 S
lide
116
Pric
ing
prob
lem
2
Cre
w
mem
ber
1
Cre
w
mem
ber
2
GE
7 S
ep 0
7 S
lide
117
Pric
ing
prob
lem
Eac
h s-
t pat
h co
rres
pond
s to
a r
oste
r, pr
ovid
ed t
he fl
ight
tim
e is
with
in b
ound
s.
2
Cre
w
mem
ber
1
Cre
w
mem
ber
2
GE
7 S
ep 0
7 S
lide
118
Pric
ing
prob
lem
Cos
t of f
light
3 if
it im
med
iate
ly f
ollo
ws
fligh
t 1, o
ffset
by
dual
mul
tiplie
r fo
r fli
ght 1
2
Cre
w
mem
ber
1
Cre
w
mem
ber
2
GE
7 S
ep 0
7 S
lide
119
Pric
ing
prob
lem
Cos
t of t
rans
ferr
ing
from
hom
e to
flig
ht 1
, of
fset
by
dual
mul
tiplie
r fo
r cr
ew m
embe
r 1
Dua
l mul
tiplie
r om
itted
to b
reak
sy
mm
etry
2
Cre
w
mem
ber
1
Cre
w
mem
ber
2
GE
7 S
ep 0
7 S
lide
120
Pric
ing
prob
lem
Leng
th o
f a p
ath
is r
educ
ed c
ost o
f the
co
rres
pond
ing
rost
er.
2
Cre
w
mem
ber
1
Cre
w
mem
ber
2
GE
7 S
ep 0
7 S
lide
121
Cre
w
mem
ber
1
Cre
w
mem
ber
2
Pric
ing
prob
lem
Arc
leng
ths
usin
g du
al s
olut
ion
of L
P
rela
xatio
n
−10
52
2
0
3
4
56
−1
05
22
-9
3
4
56
−1
2
GE
7 S
ep 0
7 S
lide
122
Cre
w
mem
ber
1
Cre
w
mem
ber
2
Pric
ing
prob
lem
Sol
utio
n of
sho
rtes
t pat
h pr
oble
ms
−10
52
2
0
3
4
56
−1
05
22
-9
3
4
56
−1
2
Red
uced
cos
t = −
1A
dd x
12to
pro
blem
.
Red
uced
cos
t = −
2A
dd x
23to
pro
blem
.A
fter
x 12
and
x 23
are
adde
d to
the
prob
lem
, no
rem
aini
ng v
aria
ble
has
nega
tive
redu
ced
cost
.
GE
7 S
ep 0
7 S
lide
123
Pric
ing
prob
lem
The
sho
rtes
t pat
h pr
oble
m c
anno
t be
solv
ed b
y tr
aditi
onal
sho
rtes
t pa
th a
lgor
ithm
s, d
ue to
the
boun
ds o
n to
tal p
ath
leng
th.
It ca
nbe
sol
ved
by C
P:
()
{}
min
max
Pat
h(,
,),
all
fligh
ts
fligh
ts,
0,
all
i
ii
jj
jX
ii
Xz
Gi
Tf
sT
Xz
i∈
≤−
≤
⊂<
∑
Set
of f
light
s as
sign
ed to
cre
w
mem
ber i
Pat
h le
ngth
Gra
ph
Pat
hgl
obal
con
stra
int
Set
sum
glob
al c
onst
rain
t
Dur
atio
n of
flig
ht j
GE
7 S
ep 0
7 S
lide
124
Exa
mpl
e: M
achi
ne S
ched
ulin
g
Con
stra
int-
dire
cted
sea
rch
usin
g lo
gic-
base
d B
ende
rs d
ecom
posi
tion
GE
7 S
ep 0
7 S
lide
125
•C
onst
rain
t-ba
sed
sear
ch
as B
ende
rs d
ecom
posi
tion
•N
ogoo
ds a
re B
ende
rs c
uts
.
•S
olut
ion
of th
e m
aste
r pr
oble
m
by M
ILP
.
•Allo
cate
jobs
to m
achi
nes.
•S
olut
ion
of th
e su
bpro
blem
by
CP
.
•S
ched
ule
jobs
on
each
mac
hine
Thi
s ex
ampl
e ill
ustr
ates
:
GE
7 S
ep 0
7 S
lide
126
Con
stra
int-
dire
cted
sea
rch
in w
hich
th
e m
aste
r pr
oble
m c
onta
ins
a fix
ed
set o
f var
iabl
es x
.
Ben
ders
dec
ompo
sitio
n
App
lied
to p
robl
ems
of
the
form
min
(,
)
(,
) ,x
y
fx
y
Sx
y
xD
yD
∈∈
Whe
n x
is fi
xed
to s
ome
valu
e, th
e re
sulti
ng
subp
robl
emis
muc
h ea
sier
:
min
(,
)
(,
) yfx
y
Sx
y
yD
∈
…pe
rhap
s be
caus
e it
deco
uple
s in
to
smal
ler
prob
lem
s.
GE
7 S
ep 0
7 S
lide
127
Con
stra
int-
dire
cted
sea
rch
in w
hich
th
e m
aste
r pr
oble
m c
onta
ins
a fix
ed
set o
f var
iabl
es x
.
Ben
ders
dec
ompo
sitio
n
App
lied
to p
robl
ems
of
the
form
min
(,
)
(,
) ,x
y
fx
y
Sx
y
xD
yD
∈∈
Whe
n x
is fi
xed
to s
ome
valu
e, th
e re
sulti
ng
subp
robl
emis
muc
h ea
sier
:
min
(,
)
(,
) yfx
y
Sx
y
yD
∈
…pe
rhap
s be
caus
e it
deco
uple
s in
to
smal
ler
prob
lem
s.
Nog
oods
are
Ben
ders
cut
san
d ex
clud
e so
lutio
ns n
o be
tter
than
x.
The
Ben
ders
cut
is o
btai
ned
by s
olvi
ng t
he in
fere
nce
dual
of
the
subp
robl
em (
clas
sica
lly,
the
linea
r pr
ogra
mm
ing
dual
).
GE
7 S
ep 0
7 S
lide
128
Mac
hine
Sch
edul
ing
•Ass
ign
5 jo
bs to
2 m
achi
nes
(A a
nd B
), a
nd s
ched
ule
the
mac
hine
s as
sign
ed to
eac
h m
achi
ne w
ithin
tim
e w
indo
ws.
•T
he o
bjec
tive
is to
min
imiz
e m
akes
pan
.
•Ass
ign
the
jobs
in th
e m
aste
r pr
oble
m, t
o be
sol
ved
by M
ILP
.
•S
ched
ule
the
jobs
in th
e su
bpro
blem
, to
be s
olve
d by
CP
.
Tim
e la
pse
betw
een
star
t of f
irst j
ob a
nd
end
of la
st jo
b.
GE
7 S
ep 0
7 S
lide
129
Mac
hine
Sch
edul
ing
Job
Dat
aO
nce
jobs
are
ass
igne
d, w
e ca
n m
inim
ize
over
all m
akes
pan
by
min
imiz
ing
mak
espa
n on
eac
h m
achi
ne in
divi
dual
ly.
So
the
subp
robl
em d
ecou
ples
.
Mac
hine
A
Mac
hine
B
GE
7 S
ep 0
7 S
lide
130
Mac
hine
Sch
edul
ing
Job
Dat
aO
nce
jobs
are
ass
igne
d, w
e ca
n m
inim
ize
over
all m
akes
pan
by
min
imiz
ing
mak
espa
n on
eac
h m
achi
ne in
divi
dual
ly.
So
the
subp
robl
em d
ecou
ples
.
Min
imum
mak
espa
n sc
hedu
le f
or jo
bs 1
, 2, 3
, 5
on m
achi
ne A
GE
7 S
ep 0
7 S
lide
131
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
of jo
b j Tim
e w
indo
ws
Jobs
can
not o
verla
p
The
pro
blem
is
GE
7 S
ep 0
7 S
lide
132
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
is
Inde
xed
linea
r m
etac
onst
rain
t
GE
7 S
ep 0
7 S
lide
133
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
is
Spe
cial
ly-s
truc
ture
d in
dexe
d lin
ear
met
acon
stra
int
Dis
junc
tive
sche
dulin
g m
etac
onst
rain
t
GE
7 S
ep 0
7 S
lide
134
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
of jo
b j Tim
e w
indo
ws
Jobs
can
not o
verla
p
The
pro
blem
is
For
a fi
xed
assi
gnm
ent
th
e su
bpro
blem
on
each
mac
hine
iis
()
min
, al
l w
ith
, al
l w
ith
disj
unct
ive
(),
()
j
j
jx
jj
jj
jx
jj
jj
ijj
M
Ms
pj
xi
rs
dp
jx
i
sx
ip
xi
≥+
=
≤≤
−=
==
x
GE
7 S
ep 0
7 S
lide
135
Ben
ders
cut
s
Sup
pose
we
assi
gn jo
bs 1
,2,3
,5 to
mac
hine
A in
iter
atio
n k.
We
can
prov
e th
at 1
0 is
the
optim
al m
akes
pan
by p
rovi
ng th
at th
e sc
hedu
le is
infe
asib
le w
ith m
akes
pan
9.
Edg
e fin
ding
de
rives
infe
asib
ility
by
reas
onin
g on
ly w
ith jo
bs 2
,3,5
. S
o th
ese
jobs
alo
ne c
reat
e a
min
imum
mak
espa
n of
10.
So
we
have
a B
ende
rs c
ut2
34
1
10if
()
0ot
herw
ise
k
xx
xA
vB
x+
==
=
≥=
GE
7 S
ep 0
7 S
lide
136
Ben
ders
cut
s
We
wan
t the
mas
ter
prob
lem
to b
e an
MIL
P, w
hich
is g
ood
for
assi
gnm
ent p
robl
ems.
So
we
writ
e th
e B
ende
rs c
ut2
34
1
10if
()
0ot
herw
ise
k
xx
xA
vB
x+
==
=
≥=
Usi
ng 0
-1 v
aria
bles
:(
)2
35
102
0A
AA
vx
xx
v
≥+
+−
≥=
1 if
job
5 is
as
sign
ed to
m
achi
ne A
GE
7 S
ep 0
7 S
lide
137
Mas
ter
prob
lem
The
mas
ter
prob
lem
is a
n M
ILP
:
{}
5
1
5
1
55
13
23
5
4
min
10, e
tc.
10, e
tc.
, 2
, etc
.,
,
v10
(2)
8
0,1
Aj
Aj
j
Bj
Bj
j
ijij
ijij
jj
AA
A
B
ij
v
px
px
vp
xv
px
iA
B
xx
x
vx
x= =
==
≤ ≤
≥≥
+=
≥+
+−
≥ ∈
∑ ∑
∑∑
Con
stra
ints
der
ived
fro
m ti
me
win
dow
s
Con
stra
ints
der
ived
fro
m r
elea
se ti
mes
Ben
ders
cut
from
mac
hine
A
Ben
ders
cut
from
mac
hine
B
GE
7 S
ep 0
7 S
lide
138
Com
puta
tiona
l Res
ults
(Ja
in &
Gro
ssm
ann)
0.010.1110100
1000
1000
0
1000
00
12
34
5
Pro
ble
m s
ize
SecondsM
ILP
CP
OP
L
Hyb
rid
Pro
blem
siz
es
(jobs
, mac
hine
s)1
-(3
,2)
2 -
(7,3
)3
-(1
2,3)
4 -
(15,
5)5
-(2
0,5)
Eac
h da
ta p
oint
re
pres
ents
an
aver
age
of 2
inst
ance
s
MIL
P a
nd C
P ra
n ou
t of
mem
ory
on 1
of t
he
larg
est i
nsta
nces
GE
7 S
ep 0
7 S
lide
139
Enh
ance
men
t Usi
ng “
Bra
nch
and
Che
ck”
(Tho
rste
inss
on)
Com
puta
tion
times
in s
econ
ds.
Pro
blem
s ha
ve 3
0 jo
bs, 7
mac
hine
s.
020406080100
120
140
12
34
5
Pro
ble
m
Seconds
Hyb
rid
Bra
nch
& c
heck
GE
7 S
ep 0
7 S
lide
140
Str
onge
r B
ende
rs c
uts
If al
l rel
ease
tim
es a
re th
e sa
me,
we
can
stre
ngth
en th
e B
ende
rs c
uts.
We
are
now
usi
ng th
e cu
t 1
ik
ikij
ikj
J
vM
xJ
∈
≥−
+
∑
Min
mak
espa
n on
mac
hine
iin
iter
atio
n k
Set
of j
obs
assi
gned
to
mac
hine
iin
ite
ratio
n k
A s
tron
ger
cut p
rovi
des
a us
eful
bou
nd e
ven
if on
ly s
ome
of th
e jo
bs in
J i
kar
e as
sign
ed to
mac
hine
i:(1
)ik
ikij
ijj
J
vM
xp
∈
≥−
−∑
The
se r
esul
ts c
an b
e ge
nera
lized
to
cum
ulat
ive
sche
dulin
g.
GE
7 S
ep 0
7 S
lide
141
•T
hese
are
cho
sen
beca
use:
•T
hey
illus
trat
e ho
w s
ched
ulin
g in
tera
cts
with
ot
her
aspe
cts
of s
uppl
y ch
ain.
•And
thus
how
CP
can
inte
ract
with
oth
er
met
hods
.
•S
ince
they
wer
e pa
rt o
f a g
over
nmen
t (E
U)
supp
orte
d pr
ojec
t (LI
SC
OS
), a
fair
amou
nt o
f de
tail
was
rel
ease
d to
pub
lic.
•All
wer
e so
lved
with
hel
p of
Das
h’s
Mos
el s
yste
m.
Thr
ee s
ucce
ss s
torie
s
GE
7 S
ep 0
7 S
lide
142
Man
ufac
ture
of p
olyp
ropy
lene
s in
3 s
tage
spo
lym
eriz
atio
n
inte
rmed
iate
st
orag
e
extr
usio
n
Pro
cess
sch
edul
ing
and
lot s
izin
g at
BA
SF
GE
7 S
ep 0
7 S
lide
143
•M
anua
l pla
nnin
g (o
ld m
etho
d)
•R
equi
red
3 da
ys
•Li
mite
d fle
xibi
lity
and
qual
ity c
ontr
ol
•24
/7 c
ontin
uous
pro
duct
ion
•V
aria
ble
batc
h si
ze.
•S
eque
nce-
depe
nden
t cha
ngeo
ver
times
.
Pro
cess
sch
edul
ing
and
lot s
izin
g at
BA
SF
GE
7 S
ep 0
7 S
lide
144
•In
term
edia
te s
tora
ge
•Li
mite
d ca
paci
ty
•O
ne p
rodu
ct p
er s
ilo
•E
xtru
sion
•P
rodu
ctio
n ra
te d
epen
ds o
n pr
oduc
t and
m
achi
ne
Pro
cess
sch
edul
ing
and
lot s
izin
g at
BA
SF
GE
7 S
ep 0
7 S
lide
145
•T
hree
pro
blem
s in
one
•Lo
t siz
ing
–ba
sed
on c
usto
mer
dem
and
fore
cast
s
•Ass
ignm
ent –
put e
ach
batc
h on
a
part
icul
ar m
achi
ne
•S
eque
ncin
g –
deci
de th
e or
der
in w
hich
ea
ch m
achi
ne p
roce
sses
bat
ches
ass
igne
d to
it
Pro
cess
sch
edul
ing
and
lot s
izin
g at
BA
SF
GE
7 S
ep 0
7 S
lide
146
•T
he p
robl
ems
are
inte
rdep
ende
nt
•Lo
t siz
ing
depe
nds
on a
ssig
nmen
t, si
nce
mac
hine
s ru
n at
diff
eren
t spe
eds
•Ass
ignm
ent d
epen
ds o
n se
quen
cing
, due
to
res
tric
tions
on
chan
geov
ers
•S
eque
ncin
g de
pend
s on
lot s
izin
g, d
ue to
lim
ited
inte
rmed
iate
sto
rage
Pro
cess
sch
edul
ing
and
lot s
izin
g at
BA
SF
GE
7 S
ep 0
7 S
lide
147
•S
olve
the
prob
lem
s si
mul
tane
ousl
y
•L
ot s
izin
g:so
lve
with
MIP
(us
ing
XP
RE
SS
-MP
)
•A
ssig
nmen
t:so
lve
with
MIP
•Se
quen
cing
:so
lve
with
CP
(us
ing
CH
IP)
•T
he M
IP a
nd C
P a
re li
nked
mat
hem
atic
ally
.
•U
se lo
gic-
base
d B
ende
rs d
ecom
posi
tion,
de
velo
ped
only
in th
e la
st fe
w y
ears
.
Pro
cess
sch
edul
ing
and
lot s
izin
g at
BA
SF
GE
7 S
ep 0
7 S
lide
148
Sam
ple
sche
dule
, illu
stra
ted
with
Vis
ual S
ched
uler
(A
viS
/3)
Sou
rce
: B
AS
F
GE
7 S
ep 0
7 S
lide
149
•B
enef
its
•O
ptim
al s
olut
ion
obta
ined
in 1
0 m
ins.
•E
ntire
pla
nnin
g pr
oces
s (d
ata
gath
erin
g,
etc.
) re
quire
s a
few
hou
rs.
•M
ore
flexi
bilit
y
•F
aste
r re
spon
se to
cus
tom
ers
•B
ette
r qu
ality
con
trol
Pro
cess
sch
edul
ing
and
lot s
izin
g at
BA
SF
GE
7 S
ep 0
7 S
lide
150
•Tw
o pr
oble
ms
to s
olve
sim
ulta
neou
sly
•Lo
t siz
ing
•M
achi
ne s
ched
ulin
g
•F
ocus
on
solv
ent-
base
d pa
ints
, for
whi
ch
ther
e ar
e fe
wer
sta
ges.
•B
arbo
t is
a P
ortu
gues
e pa
int m
anuf
actu
rer.
Sev
eral
mac
hine
s of
eac
h ty
pe
Pai
nt p
rodu
ctio
n at
Bar
bot
GE
7 S
ep 0
7 S
lide
151
•S
olut
ion
met
hod
sim
ilar
to B
AS
F c
ase
(MIP
+ C
P).
•B
enef
its
•O
ptim
al s
olut
ion
obta
ined
in a
few
min
utes
fo
r 20
mac
hine
s an
d 80
pro
duct
s.
•P
rodu
ct s
hort
ages
elim
inat
ed.
•10
% in
crea
se in
out
put.
•F
ewer
cle
anup
mat
eria
ls.
•C
usto
mer
lead
tim
e re
duce
d.
Pai
nt p
rodu
ctio
n at
Bar
bot
GE
7 S
ep 0
7 S
lide
152
•T
he P
euge
ot 2
06 c
an b
e m
anuf
actu
red
with
12,
000
optio
n co
mbi
natio
ns.
•P
lann
ing
horiz
on is
5 d
ays
Pro
duct
ion
line
sequ
enci
ng a
t Peu
geot
-Citr
oën
GE
7 S
ep 0
7 S
lide
153
•E
ach
car
pass
es th
roug
h 3
shop
s.
•O
bjec
tives
•G
roup
sim
ilar
cars
(e.
g. in
pai
nt s
hop)
.
•R
educ
e se
tups
.
•B
alan
ce w
ork
stat
ion
load
s.
Pro
duct
ion
line
sequ
enci
ng a
t Peu
geot
-Citr
oën
GE
7 S
ep 0
7 S
lide
154
•S
peci
al c
onst
rain
ts
•C
ars
with
a s
un r
oof s
houl
d be
gr
oupe
d to
geth
er in
ass
embl
y.
•Air-
cond
ition
ed c
ars
shou
ld n
ot b
e as
sem
bled
con
secu
tivel
y.
•E
tc.
Pro
duct
ion
line
sequ
enci
ng a
t Peu
geot
-Citr
oën
GE
7 S
ep 0
7 S
lide
155
•P
robl
em h
as tw
o pa
rts
•D
eter
min
e nu
mbe
r of
car
s of
eac
h ty
pe
assi
gned
to e
ach
line
on e
ach
day.
•D
eter
min
e se
quen
cing
for
eac
h lin
e on
ea
ch d
ay.
•P
robl
ems
are
solv
ed s
imul
tane
ousl
y.
•Aga
in b
y M
IP +
CP.
Pro
duct
ion
line
sequ
enci
ng a
t Peu
geot
-Citr
oën
GE
7 S
ep 0
7 S
lide
156
Sam
ple
sche
dule
Sou
rce:
Peu
geot
/Citr
oën
GE
7 S
ep 0
7 S
lide
157
•B
enef
its
•G
reat
er a
bilit
y to
bal
ance
suc
h in
com
patib
le b
enef
its a
s fe
wer
se
tups
and
fast
er c
usto
mer
ser
vice
.
•B
ette
r sc
hedu
les.
Pro
duct
ion
line
sequ
enci
ng a
t Peu
geot
-Citr
oën
GE
7 S
ep 0
7 S
lide
158
A c
lass
ic p
rodu
ctio
n se
quen
cing
pro
blem
Sou
rce:
Peu
geot
/Citr
oën
Line
bal
anci
ng a
t Peu
geot
-Citr
oën
GE
7 S
ep 0
7 S
lide
159
•O
bjec
tive
•E
qual
ize
load
at w
ork
stat
ions
.
•K
eep
each
wor
ker
on o
ne s
ide
of th
e ca
r
•C
onst
rain
ts
•P
rece
denc
e co
nstr
aint
s be
twee
n so
me
oper
atio
ns.
•E
rgon
omic
req
uire
men
ts.
•R
ight
equ
ipm
ent a
t sta
tions
(e.
g. a
ir so
cket
)
Line
bal
anci
ng a
t Peu
geot
-Citr
oën
GE
7 S
ep 0
7 S
lide
160
•S
olut
ion
agai
n ob
tain
ed b
y an
inte
grat
ed m
etho
d.
•M
IP: o
btai
n so
lutio
n w
ithou
t reg
ard
to
prec
eden
ce c
onst
rain
ts.
•C
P: R
esch
edul
e to
enf
orce
pre
cede
nce
cons
trai
nts.
•T
he tw
o m
etho
ds in
tera
ct.
Line
bal
anci
ng a
t Peu
geot
-Citr
oën
GE
7 S
ep 0
7 S
lide
161
Sou
rce:
Peu
geot
/Citr
oën
GE
7 S
ep 0
7 S
lide
162
•B
enef
its
•B
ette
r eq
ualiz
atio
n of
load
.
•S
ome
stat
ions
cou
ld b
e cl
osed
, red
ucin
g la
bor.
•Im
prov
emen
ts n
eede
d
•R
educ
e tr
acks
ide
clut
ter.
•E
qual
ize
spac
e re
quire
men
ts.
•K
eep
wor
kers
on
one
side
of c
ar.
Line
bal
anci
ng a
t Peu
geot
-Citr
oën
GE
7 S
ep 0
7 S
lide
163
Bac
kgro
und
Rea
ding
Muc
h of
this
talk
is b
ased
on:
•J.
N. H
ooke
r, In
tegr
ated
Met
hods
for
Opt
imiz
atio
n, S
prin
ger
(200
7).
•J.
N. H
ooke
r, O
pera
tions
res
earc
h m
etho
ds in
con
stra
int
prog
ram
min
g, in
F. R
ossi
, P. v
an B
eek
and
T. W
alsh
, eds
., H
andb
ook
of C
onst
rain
t Pro
gram
min
g, E
lsev
ier
(200
6) 5
27-5
70.
Cite
d re
fere
nces
:•
T. F
ahle
, U. J
unke
r, S
. E. K
aris
h, N
. Koh
n, M
. Sel
lman
n, B
. V
aabe
n. C
onst
rain
t pro
gram
min
g ba
sed
colu
mn
gene
ratio
n fo
r cr
ew
assi
gnm
ent,
Jour
nal o
f Heu
ristic
s 8
(200
2) 5
9-81
•
E. T
hors
tein
sson
and
G. O
ttoss
on, L
inea
r re
laxa
tion
and
redu
ced-
cost
bas
ed p
ropa
gatio
n of
con
tinuo
us v
aria
ble
subs
crip
ts, A
nnal
s of
O
pera
tions
Res
earc
h 11
5 (2
001)
15-
29.
GE
7 S
ep 0
7 S
lide
164
