integrating constraint programming and mathematical programming...
TRANSCRIPT
-
Inte
gra
ting
Con
stra
int P
rogr
am
min
g a
nd
Ma
the
ma
tica
l Pro
gra
mm
ing
John
Hoo
ker
Ca
rneg
ie M
ello
n U
niv
ers
ity
Uni
vers
ity o
f Yor
k, J
une
2003
-
The
se s
lides
are
ava
ilabl
e at
http
://b
a.gs
ia.c
mu.
edu/
jnh
g
-
A S
imp
le E
xam
ple
Inte
gra
ting
CP
an
d M
PB
ran
ch I
nfe
r an
d R
elax
Dec
om
po
sitio
nR
ecen
t S
ucc
ess
Sto
ries
Rel
axat
ion
Put
ting
it To
geth
erS
urve
ys a
nd T
utor
ials
-
gP
rogr
amm
ing
Pro
gram
min
≠
•C
onst
rain
t pro
gra
mm
ing
is r
elat
ed to
com
pute
r pr
ogra
mm
ing
.
•M
athe
mat
ical
pro
gra
mm
ing
has
noth
ing
to d
o w
ith
com
pute
r pr
ogra
mm
ing.
•“P
rogr
amm
ing”
his
toric
ally
ref
ers
to lo
gist
ics
plan
s (G
eorg
e D
antz
ig’s
firs
t app
licat
ion)
.
•M
P is
pur
ely
decl
arat
ive.
-
A S
impl
e E
xam
ple
Sol
utio
n by
Con
stra
int P
rogr
amm
ing
Sol
utio
n by
Int
eger
Pro
gram
min
gS
olut
ion
by a
Hyb
rid M
etho
d
-
Th
e P
rob
lem }
4,,1{
},
,{
diff
eren
t-
all
30
25
3su
bje
ct t
o
48
5m
in
32
1
32
1
32
1
…∈
≥+
++
+
jx
xx
xxx
x
xx
x
We
will
illu
stra
te h
ow s
earc
h, in
fere
nce
and
rela
xatio
n m
ay b
e co
mbi
ned
to s
olve
this
pro
blem
by:
•co
nstr
aint
pro
gram
min
g
•in
tege
r pr
ogra
mm
ing
•a
hybr
id a
ppro
ach
-
So
lve
as a
co
nst
rain
t pro
gra
mm
ing
pro
ble
m
}4,
,1{}
,,
{di
ffere
nt-
all
302
53
48
5
32
1
32
1
32
1
…∈
≥+
+≤
++
jx
xx
xxx
xz
xx
x
Sta
rt w
ith z
= ∞
.W
ill d
ecre
ase
as fe
asib
le
solu
tions
are
fou
nd.
Sea
rch:
Dom
ain
split
ting
Infe
ren
ce:
Dom
ain
redu
ctio
n R
ela
xatio
n:C
onst
rain
t sto
re (
set o
f cu
rren
t var
iabl
e do
mai
ns)
Co
nst
rain
t sto
re ca
n be
vie
wed
as
cons
istin
g of
in-d
omai
n co
nstr
aint
s xj∈
Dj,
whi
ch f
orm
a r
elax
atio
n of
the
prob
lem
.
Glo
bal c
onst
rain
t
-
Dom
ain
redu
ctio
n fo
r in
equa
litie
s
•B
ound
s pr
opag
atio
n on
302
53
48
5
32
1
32
1
≥+
+≤
++
xx
xz
xx
x
impl
ies
For
exa
mpl
e,30
25
33
21
≥+
+x
xx
25
812
305
23
303
12
=−
−≥
−−
≥x
xx
So
the
dom
ain
of x 2
is r
educ
ed t
o {2
,3,4
}.
-
Dom
ain
redu
ctio
n fo
r al
l-diff
eren
t (e.g
., R
ég
in)
•M
aint
ain
hype
rarc
con
sist
ency
on
},
,{
diffe
rent
-al
l3
21
xx
x
Sup
pose
for
exam
ple:
Dom
ain
of x 1
Dom
ain
of x 2
Dom
ain
of x 3
12
12
1234
The
n on
e ca
n re
duce
th
e do
mai
ns:
12
12
34
•In
gen
eral
, so
lve
a m
axim
um c
ardi
nalit
y m
atch
ing
prob
lem
and
app
ly a
theo
rem
of
Ber
ge
-
z =
∞1234
234
1234
34
23
234
234
12
infe
asib
lex
= (
3,4
,1)
valu
e =
51
z =
∞z
= 5
2
Domain of x1
Domain of x2
Domain of x3
D2=
{2,3
}D
2={4
}
Dom
ain
of x 2
D2=
{2}
D2=
{3}
D1=
{3}
D1=
{2}
infe
asib
lex
= (
4,3
,2)
valu
e =
52
-
So
lve
as a
n in
teg
er p
rog
ram
min
g p
rob
lem
Sea
rch:
Bra
nch
on v
aria
bles
with
frac
tiona
l val
ues
in s
olut
ion
of
cont
inuo
us r
elax
atio
n.In
fere
nce
: G
ener
ate
cutt
ing
plan
es (
cove
ring
ineq
ualit
ies)
.R
ela
xatio
n:C
ontin
uous
(LP
) re
laxa
tion.
-
Rew
rite
prob
lem
usi
ng in
tege
r pr
ogra
mm
ing
mod
el:
Let y
ijbe
1 if
x i=
j, 0
oth
erw
ise.
ji
y
jy
iy
ijy
x
xx
x
xx
x iji
ij
jijj
iji
, al
l},1,0{
4,,1
,1
3,2,1,1
3,2,1,
302
53
subj
ect t
o
48
5m
in
3 14
1
5
1
32
1
32
1 ∈
=≤
==
==
≥+
++
+
∑∑
∑
==
=
…
-
Co
ntin
uou
s re
laxa
tion
ji
yx
xx
xx
xx
xx
jy
iy
ijy
x
xx
x
xx
x
ij
iij
jijj
iji
, a
ll1
,0
8
445
4,,1
,1
3,2,1,1
3,2,1,
17
42
4su
bje
ct to
53
4m
in
32
1
32
31
213 14 1
4 1
32
1
32
1
≤≤
≥+
+≥
+≥
+≥
+
=≤
==
==
≥+
++
+
∑∑
∑
==
=
…
Rel
ax in
tegr
ality
Cov
erin
g in
equa
litie
s
-
Bra
nch
an
d b
ou
nd
(B
ran
ch a
nd
rel
ax)
The
incu
mb
en
t so
lutio
nis th
e be
st fe
asib
le s
olut
ion
foun
d so
far.
At e
ach
node
of t
he b
ranc
hing
tre
e:
•If
The
re is
no
need
to
bran
ch f
urth
er.
•N
o fe
asib
le s
olut
ion
in th
at s
ubtr
ee c
an b
e be
tter
th
an th
e in
cum
bent
sol
utio
n.
•U
se S
OS
-1 b
ranc
hing
.
Opt
imal
va
lue
of
rela
xatio
n ≥
Valu
e of
in
cum
bent
so
lutio
n
-
5.4
90
00
1
2/12/1
00
2/12/1
00
=
=
z
y
y 12 =
1y 1
3 =
1y 1
4 =
1
2.5
00
01
0
8.00
02.0
01
00
=
=
z
y
50
00
2/1
2/1
01
00
10
00
=
=
z
y
y 11 =
1
Infe
as.
Infe
as.
Infe
as.
z =
54
z=
51
Infe
as.
z =
52
50
02/1
02/1
10
00
00
10
=
=
z
y
Infe
as.
4.50
00
10
09.0
01.0
10
00
=
=
z
y
8.50
01
00
15/13
00
15/2
00
10
=
=
z
y
Infe
as.
Infe
as.
Infe
as.
Infe
as.
-
So
lve
usi
ng
a h
ybri
d a
pp
roac
h
Sea
rch
:
•B
ranc
h on
frac
tiona
l var
iabl
es in
sol
utio
n of
re
laxa
tion.
•D
rop
co
nstr
aint
s w
ith y ij’s
. T
his
mak
es r
elax
atio
n to
o
larg
e w
itho
ut m
uch
imp
rove
men
t in
qua
lity.
•If
vari
able
s ar
e al
l int
egra
l, b
ranc
h b
y sp
littin
g d
om
ain.
•U
se b
ranc
h an
d bo
und.
Infe
ren
ce:
•U
se b
ound
s pr
opag
atio
n fo
r al
l ine
qual
ities
.•
Mai
ntai
n hy
pera
rc c
onsi
sten
cy f
or a
ll-di
ffere
nt
cons
trai
nts.
-
Rela
xatio
n:
•P
ut k
naps
ack
cons
trai
nt in
LP
.
•P
ut c
over
ing
ineq
ualit
ies
base
d on
kn
apsa
ck/a
ll-di
ffere
nt in
to L
P.
-
}4,
,1{
8
445
},
,{
diffe
rent
-a
ll
302
53
s.t.
48
5m
in
32
1
32
31
21
32
1
32
1
32
1
…∈
≥+
+≥
+≥
+≥
+
≥+
+≤
++
jx
xx
x
xx
xx
xx
xx
xxx
x
zx
xx
Mo
del
for
hyb
rid
ap
pro
ach
Cov
erin
g in
equa
litie
s
Gen
erat
e an
d
pro
pag
ate
cove
ring
in
equa
litie
s at
ea
ch n
od
e o
f se
arch
tree
-
z =
∞1 2 3 4
2 3 4
1 2 3 4
3 4
2 3
2 23 4
x=
(3
,4,1
)va
lue
= 5
1
z =
52
x=
(2
,4,3
)va
lue
= 5
4
x=
(3
.7,3
,2)
valu
e =
50
.3
x=
(3
.5,3
.5,1
)va
lue
= 4
9.5
x 2 =
3x 2
=4
2 3
4 4
1 2 3
x 1=
2x 1
=3
x=
(2
,4,2
)va
lue
= 5
0
x=
(4
,3,2
)va
lue
= 5
2in
feas
ible
x 1=
3x 1
=4
-
Inte
gra
ting
CP
and
MP
Mot
ivat
ion
Two
Inte
grat
ion
Sch
emes
-
Mo
tivat
ion
to In
teg
rate
CP
an
d M
P
•In
fere
nce
+ r
elax
atio
n.
•C
P’s
infe
renc
e te
chni
ques
tend
to b
e ef
fect
ive
whe
n co
nstr
aint
s co
ntai
n fe
w v
aria
bles
.
•M
isle
adin
g to
say
CP
is e
ffect
ive
on “
high
ly
cons
trai
ned”
pro
blem
s.
•M
P’s
rel
axat
ion
tech
niqu
es te
nd to
be
effe
ctiv
e w
hen
cons
trai
nts
or o
bjec
tive
func
tion
cont
ain
man
y va
riabl
es.
•F
or e
xam
ple,
cos
t and
pro
fit.
-
•“H
oriz
onta
l” +
“ve
rtic
al”
stru
ctur
e.
•C
P’s
idea
of
glob
al c
onst
rain
t exp
loits
str
uctu
re
with
in a
pro
blem
(ho
rizon
tal s
truc
ture
).
•M
P’s
foc
us o
n sp
ecia
l cla
sses
of p
robl
ems
is
usef
ul fo
r so
lvin
g re
laxa
tions
or
subp
robl
ems
(ver
tical
str
uctu
re).
Mo
tivat
ion
to In
teg
rate
CP
an
d M
P
-
•P
roce
dura
l + d
ecla
rativ
e.
•P
arts
of t
he p
robl
em a
re b
est e
xpre
ssed
in M
P’s
de
clar
ativ
e (s
olve
r-in
depe
nden
t) m
anne
r.
•O
ther
par
ts b
enef
it fr
om s
earc
h di
rect
ions
pro
vide
d by
use
r.
Mo
tivat
ion
to In
teg
rate
CP
an
d M
P
-
Inte
gra
tion
Sch
emes
Rec
ent w
ork
can
be b
road
ly s
een
as u
sing
two
inte
grat
ive
idea
s: •B
ran
ch-in
fer-
an
d-r
ela
x -V
iew
CP
and
MP
met
hods
as
spec
ial c
ases
of a
bra
nch-
infe
r-an
d-re
lax
met
hod.
•D
eco
mp
osi
tion
-Dec
ompo
se p
robl
ems
into
a C
P p
art
and
an M
P p
art,
perh
aps
usin
g a
Ben
ders
sch
eme.
-
Bra
nch
-infe
r-an
d-r
elax
•E
xist
ing
CP
and
MP
com
bine
bra
nchi
ng,
infe
renc
e an
d re
laxa
tion.
•B
ran
chin
g –
enum
erat
e so
lutio
ns b
y br
anch
ing
on
varia
bles
or
viol
ated
con
stra
ints
.
•In
fere
nce
–de
duce
new
con
stra
ints
•C
P:d
omai
n re
duct
ion
•M
P: c
uttin
g pl
anes
.
•R
ela
xatio
n –
rem
ove
som
e co
nstr
aint
s be
fore
so
lvin
g
•C
P: t
he c
onst
rain
t sto
re (
varia
ble
dom
ains
)
•M
P: c
ontin
uous
rel
axat
ion
-
Dec
om
po
sitio
n
•S
ome
prob
lem
s ca
n be
dec
ompo
sed
into
a m
aste
r pr
oble
m a
nd s
ubpr
oble
m.
•M
aste
r pr
oble
m s
earc
hes
over
som
e of
the
varia
bles
.
•F
or e
ach
sett
ing
of th
ese
varia
bles
, sub
prob
lem
so
lves
the
prob
lem
ove
r th
e re
mai
ning
var
iabl
es.
•O
ne s
chem
e is
a g
ener
aliz
ed B
ende
rs
deco
mpo
sitio
n.
•C
P is
nat
ural
for
subp
robl
em, w
hich
can
be
seen
as
an in
fere
nce
(dua
l) pr
oble
m.
-
Bra
nch
Infe
r a
nd R
ela
x
A S
econ
d E
xam
ple:
Dis
cret
e Lo
t S
izin
g
-
Dis
cret
e Lo
t Siz
ing
•M
anuf
actu
re a
t mos
t one
pro
duct
eac
h da
y.
•W
hen
man
ufac
turin
g st
arts
, it m
ay c
ontin
ue s
ever
al d
ays.
•S
witc
hing
to a
noth
er p
rodu
ct in
curs
a c
ost.
•T
here
is a
cer
tain
dem
and
for
each
pro
duct
on
each
day
.
•P
rodu
cts
are
stoc
kpile
d to
mee
t dem
and
betw
een
man
ufac
turin
g ru
ns.
•M
inim
ize
inve
ntor
y co
st +
cha
ngeo
ver
cost
.
-
Dis
cret
e lo
t siz
ing
t=
1
2
3
4
5
6
7
8
AB
A
y t=
A
A
A
B
B
0
A
0
job 0 =
dum
my
job
-
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
ijtit
itiit
itit
jtijt
ti
ijt
jtt
iijt
ti
it
itit
ti
itit
itit
itt
iiti
jijt
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
IP m
odel
(Wo
lsey)
-
Mo
del
ing
var
iab
le in
dic
es w
ith
ele
men
t
To im
plem
ent v
aria
bly
inde
xed
cons
tant
Rep
lace
with
z an
d ad
d co
nstr
aint
whi
ch s
ets z
= a
y
()z
aa
yn),
,,
(,e
lem
en
t1…
ya
ya
To im
plem
ent v
aria
bly
inde
xed
varia
ble
Rep
lace
with
z an
d ad
d co
nstr
aint
whi
ch p
osts
the
cons
trai
nt
z =
xy.
The
re a
re s
trai
ghtfo
rwar
d fil
terin
g al
gorit
hms
for
ele
men
t.
() z
xx
yn),
,,
(,el
emen
t1…
yx
yx
-
()
()
ti
xi
yt
is
Cx
ti
sd
xs
tq
v
ts
hu
vu
itt
itit
itit
itt
i
yy
t
iit
itt
tt
tt
, al
l ,
0,
all
,0,
0,
all
,
all
,
al
l ,
s.t.
)(
min
1,
1
=→
≠≥
≤≤
+=
+≥≥
+
−−
∑
∑st
ock
leve
l
chan
geov
er c
ost
tota
l inv
ento
ry +
cha
ngeo
ver
cost
inve
ntor
y ba
lanc
e
daily
pro
duct
ion
Hyb
rid m
odel
-
()
()
ti
xi
yt
is
Cx
ti
sd
xs
tq
v
ts
hu
vu
itt
itit
itit
itt
i
yy
t
iit
itt
tt
tt
, al
l ,
0,
all
,0,
0,
all
,
all
,
al
l ,
s.t.
)(
min
1,
1
=→
≠≥
≤≤
+=
+≥≥
+
−−
∑
∑G
ener
ate
ineq
ualit
ies
to p
ut
into
rel
axat
ion
(to
be d
iscu
ssed
)
Put
into
rel
axat
ion
App
ly c
onst
rain
t pro
paga
tion
to e
very
thin
g
To c
reat
e re
laxa
tion:
-
So
lutio
n
Sea
rch
:Dom
ain
split
ting,
bra
nch
and
boun
d us
ing
rela
xatio
n of
sel
ecte
d co
nstr
aint
s.
Infe
ren
ce:
Dom
ain
redu
ctio
n an
d co
nstr
aint
pro
paga
tion.
Cha
ract
eris
tics:
•C
ondi
tiona
l con
stra
ints
impo
se c
onse
quen
t whe
n an
tece
dent
bec
omes
true
in th
e co
urse
of b
ranc
hing
.
•R
elax
atio
n is
som
ewha
t wea
ker
than
in IP
bec
ause
logi
cal
cons
trai
nts
are
not a
ll re
laxe
d.
•B
ut L
P r
elax
atio
ns a
re m
uch
smal
ler-
-qua
drat
ic r
athe
r th
an c
ubic
siz
e.
•D
omai
n re
duct
ion
help
s pr
une
tree
.
-
De
com
posi
tion
Idea
Beh
ind
Ben
ders
Dec
ompo
sitio
nLo
gic
Circ
uit V
erifi
catio
nM
achi
ne S
ched
ulin
g
-
Idea
Beh
ind
Ben
der
s D
eco
mp
osi
tion
“Lea
rn f
rom
on
e’s
mis
take
s.”
•D
istin
guis
h pr
imar
y va
riabl
es fr
om s
econ
dary
var
iabl
es.
•S
earc
h ov
er p
rimar
y va
riabl
es (
ma
ster
pro
ble
m).
•F
or e
ach
tria
l val
ue o
f pr
imar
y va
riabl
es, s
olve
pro
blem
ov
er s
econ
dary
var
iabl
es (
sub
pro
ble
m).
•C
an b
e vi
ewed
as
solv
ing
a su
bpro
blem
to g
ener
ate
Ben
ders
cu
tsor
“nog
oods
.”
•Add
the
Ben
ders
cut
to th
e m
aste
r pr
oble
m to
req
uire
nex
t so
lutio
n to
be
bett
er th
an la
st, a
nd r
e-so
lve.
•C
an a
lso
be v
iew
ed a
s pr
ojec
ting
prob
lem
ont
o pr
imar
y va
riabl
es.
-
Log
ic c
ircu
it ve
rific
atio
n(J
NH
, Y
an
)
Logi
c ci
rcui
ts A
and
B a
re e
quiv
alen
t whe
n th
e fo
llow
ing
circ
uit i
s a
taut
olog
y:
A B
x 1 x 2 x 3
≡ ≡
and
inpu
ts
The
circ
uit i
s a
taut
olog
y if
the
outp
ut o
ver
all 0
-1 in
puts
is 1
.
-
x 1 x 2 x 3
inpu
ts
and
or and
not
not
or or
not
not
and
y 1 y 3y 2
y 4 y 5
y 6
For
inst
ance
, che
ck w
heth
er t
his
circ
uit i
s a
taut
olog
y:
The
sub
prob
lem
is to
find
whe
ther
the
out
put c
an b
e 0
whe
n th
e in
put x
is fi
xed
to a
giv
en v
alue
.
But
sin
ce x
dete
rmin
es th
e ou
tput
of t
he c
ircui
t, th
e su
bpro
blem
is e
asy:
just
com
pute
the
outp
ut.
-
x 1 x 2 x 3
and
or and
not
not
or or
not
not
and
y 1 y 3y 2
y 4 y 5
y 6
For
exa
mpl
e, le
t x =
(1,
0,1)
.
1
1
11
10
0
0 1
To c
onst
ruct
a B
ende
rs c
ut, i
dent
ify w
hich
sub
sets
of t
he
inpu
ts a
re s
uffic
ient
to g
ener
ate
an o
utpu
t of 1
.
For
inst
ance
,
su
ffice
s.)1,0(
),
(3
2=
xx
-
x 1 x 2 x 3
and
or and
not
not
or or
not
not
and
y 1 y 3y 2
y 4 y 5
y 6
1
1
11
10
0
0 1
For
thi
s, it
suf
fices
th
at y 4
= 1
and
y 5 =
1.
For
thi
s, it
suf
fices
th
at y 2
= 0
.
For
thi
s, it
suf
fices
th
at y 2
= 0
.
For
thi
s, it
suf
fices
th
at x 2
= 0
and
x 3 =
1.
So,
Ben
ders
cut
is
32
xx
¬∨
-
Now
sol
ve th
e m
aste
r pr
oble
m 32
xx
¬∨
One
sol
utio
n is
)0,0,0(
),
,(
32
1=
xx
x
Thi
s pr
oduc
es o
utpu
t 0, w
hich
sho
ws
the
circ
uit i
s no
t a
taut
olog
y.
Not
e: T
his
is a
ctua
lly a
cas
e of
cla
ssic
al B
ende
rs.
The
su
bpro
blem
can
be
writ
ten
as a
n LP
(a
Hor
n-S
AT p
robl
em).
-
Mac
hin
e sc
hed
ulin
g
Ass
ign
each
job
to o
ne m
achi
ne s
o as
to p
roce
ss a
ll jo
bs a
t m
inim
um c
ost.
Mac
hine
s ru
n at
diff
eren
t sp
eeds
and
incu
r di
ffere
nt c
osts
per
job.
Eac
h jo
b ha
s a
rele
ase
date
and
a d
ue
date
.
•In
thi
s pr
oble
m, t
he m
aste
r pr
oble
m a
ssig
ns jo
bs to
mac
hine
s.
The
sub
prob
lem
sch
edul
es jo
bs a
ssig
ned
to e
ach
mac
hine
.
•C
lass
ical
mix
ed in
tege
r pr
ogra
mm
ing
solv
es th
e m
aste
r pr
oble
m.
•C
onst
rain
t pro
gram
min
g so
lves
the
subp
robl
em, a
1-m
achi
ne
sche
dulin
g pr
oble
m w
ith ti
me
win
dow
s.
•T
his
prov
ides
a g
ener
al f
ram
ewor
k fo
r co
mbi
ning
mix
ed
inte
ger
prog
ram
min
g an
d co
nstr
aint
pro
gram
min
g.
-
Mo
del
ing
reso
urc
e-co
nstr
aine
d sc
hed
ulin
g w
ith
cum
ula
tive
Jobs
1,2
,3 c
onsu
me
3 un
its o
f res
ourc
es.
Jobs
4,5
con
sum
e 2
units
.M
axim
um L
= 7
units
of
reso
urce
s av
aila
ble.
Min
mak
espa
n =
8
L
1 23
4 5re
sour
ces
02
1=
=t
t0
54
3=
==
tt
t ()7
),2,2,3,3,3(),5,5,3,3,3(
),,..
.,(
cum
ula
tive
51
tt
Sta
rt ti
mes
Dur
atio
nsR
esou
rce
cons
umpt
ion
L
-
()
ie
ix
Di
xt
jS
Dt
jR
t
C
jij
jjj
jx
j
jjj
jx
j
j
a
ll,
1,),
|(
),|
(cu
mu
lativ
e
a
ll,
a
ll,
s.t.
min
==
≤+≥
∑
A m
odel
for
the
mac
hine
sch
edul
ing
prob
lem
:
Rel
ease
dat
e fo
r jo
b j
Cos
t of
assi
gnin
g m
achi
ne
x jto
job
j
Mac
hine
ass
igne
d to
job j
Sta
rt ti
mes
of j
obs
assi
gned
to
mac
hine
i
Sta
rt ti
me
for
job j
Job
dura
tion
Dea
dlin
e
Res
ourc
e co
nsum
ptio
n =
1 fo
r ea
ch
job
-
For
a g
iven
set
of a
ssig
nmen
ts
t
he s
ubpr
oble
m is
the
set o
f 1-
mac
hine
pro
blem
s,x
()
ie
ix
Di
xt
jij
jj
a
ll,
1,),
|(
),|
(cu
mu
lativ
e=
=
Fea
sibi
lity
of e
ach
prob
lem
is c
heck
ed b
y co
nstr
aint
pr
ogra
mm
ing.
-
Sup
pose
ther
e is
no
feas
ible
sch
edul
e fo
r m
achi
ne
. T
hen
jobs
ca
nnot
all
be a
ssig
ned
to m
achi
ne
.
Sup
pose
in fa
ct th
at s
ome
subs
et
of th
ese
jobs
ca
nnot
be
assi
gned
to
mac
hine
.
The
n w
e ha
ve a
Ben
ders
cu
t
}|
{i
xj
j=
)(
so
me
for
x
Jj
ix
ij
∈≠
)(x
J i
i i
i
-
Kk
ix
Jj
ix
jS
Dt
jR
t
C
ki
j
jj
xj
jjj
jx
j
j
,,1
, al
l ),
(
som
efo
r
al
l,
al
l,
s.t.
min
…=
∈≠
≤+≥
∑
Thi
s yi
elds
the
mas
ter
prob
lem
,
Thi
s pr
oble
m c
an b
e w
ritte
n as
a m
ixed
0-1
pro
blem
:
-
}1,0{
al
l
},{
min
}{
max
,,1
, al
l ,1
)1(
al
l ,1
al
l,
al
l,
s.t.
min
∈
−≤
=≥
−≥
≤+≥
∑∑∑
∑
∑
= ij
jj
jj
ijj
ij
ix
jij
iij
ji
ijij
j
jjij
ijij
y
iR
Sy
D
Kk
iy
jy
jS
yD
t
jR
t
yC
k j
…
Valid
co
nstr
aint
ad
ded
to
impr
ove
perf
orm
ance
-
Com
puta
tiona
l Res
ults
(J
ain
& G
ross
ma
nn
)
Co
mp
uta
tio
nal R
esu
lts
0.0
1
0.1110
100
1000
10000
100000
12
34
5
Pro
ble
m s
ize
Seconds
MIL
P
CP
OP
L
Benders
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
-
An
En
han
cem
ent:
Bra
nch
an
d C
hec
k(J
NH
, T
ho
rste
inss
on
)
•G
ener
ate
a B
ende
rs c
ut w
hene
ver
a fe
asib
le s
olut
ion
i
s fo
und
in th
e m
aste
r pr
oble
m tr
ee s
earc
h.
•K
eep
the
cuts
(es
sent
ially
nog
oods
) in
the
prob
lem
for
the
rem
aind
er o
f the
tree
sea
rch.
•S
olve
the
mas
ter
prob
lem
onl
y on
ce b
ut c
ontin
ually
upd
ate
it. •T
his
was
app
lied
to th
e m
achi
ne s
ched
ulin
g pr
oble
m
desc
ribed
ear
lier.
x
-
Enh
ance
men
t Usi
ng “
Bra
nch
and
Che
ck”
(Th
ors
tein
sso
n)
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
-
Re
cent
Suc
cess
Sto
ries
Pro
duct
Con
figur
atio
nP
roce
ss S
ched
ulin
g at
BA
SF
Pai
nt P
rodu
ctio
n at
Bar
bot
Pro
duct
ion
Line
Seq
uenc
ing
at P
euge
ot/C
itroë
nLi
ne B
alan
cing
at P
euge
ot/C
itroë
n
-
Exa
mp
le o
f F
aste
r S
olu
tion
:P
rod
uct
co
nfig
ura
tion
•F
ind
optim
al s
elec
tion
of c
ompo
nent
s to
mak
e up
a p
rodu
ct,
subj
ect t
o co
nfig
urat
ion
cons
trai
nts.
•U
se c
ontin
uous
rel
axat
ion
of e
lem
ent c
onst
rain
ts a
nd r
educ
ed
cost
pro
paga
tion.
-
Com
puta
tiona
l Res
ults
(O
tto
sso
n &
Th
ors
tein
sso
n)
0.010.1110100
1000
8x10
16x2
020
x24
20x3
0
Pro
ble
m
Seconds
CP
LEX
CLP
Hyb
rid
-
Pro
cess
Sch
edu
ling
an
d
Lot S
izin
g a
t B
AS
F
Man
ufac
ture
of
poly
prop
ylen
es in
3 s
tage
s
poly
mer
izat
ion
inte
rmed
iate
st
orag
e
extr
usio
n
-
Pro
cess
Sch
edu
ling
and
Lo
t Siz
ing
at B
AS
F
•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
•Va
riabl
e ba
tch
size
.
•S
eque
nce-
depe
nden
t ch
ange
over
tim
es.
-
Pro
cess
Sch
edu
ling
and
Lo
t Siz
ing
at B
AS
F
•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
edu
ling
and
Lo
t Siz
ing
at B
AS
F
•T
hree
pro
blem
s in
one
•Lo
t si
zing
–ba
sed
on c
usto
mer
dem
and
fore
cast
s
•Ass
ignm
ent
–pu
t eac
h ba
tch
on a
par
ticul
ar
mac
hine
•S
eque
ncin
g –
deci
de t
he o
rder
in w
hich
eac
h m
achi
ne p
roce
sses
bat
ches
ass
igne
d to
it
-
Pro
cess
Sch
edu
ling
and
Lo
t Siz
ing
at B
AS
F
•T
he p
robl
ems
are
inte
rdep
ende
nt
•Lo
t si
zing
dep
ends
on
assi
gnm
ent,
sinc
e m
achi
nes
run
at d
iffer
ent
spee
ds
•Ass
ignm
ent
depe
nds
on s
eque
ncin
g, d
ue to
re
stric
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
edu
ling
and
Lo
t Siz
ing
at B
AS
F
•S
olve
the
prob
lem
s si
mul
tane
ousl
y
•L
ot s
izin
g:s
olve
with
MIP
(us
ing
XP
RE
SS
-MP
)
•A
ssig
nm
en
t:sol
ve w
ith M
IP
•S
eq
uen
cin
g: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
.
-
Sam
ple
sche
dule
, ill
ustr
ated
with
Vis
ual S
ched
uler
(A
viS
/3)
Sou
rce
: B
AS
F
-
Pro
cess
Sch
edu
ling
and
Lo
t Siz
ing
at B
AS
F
•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, e
tc.)
re
quire
s a
few
hou
rs.
•M
ore
flexi
bilit
y
•F
aste
r re
spon
se t
o cu
stom
ers
•B
ette
r qu
ality
con
trol
-
Pai
nt P
rod
uct
ion
at B
arb
ot
•Tw
o pr
oble
ms
to s
olve
sim
ulta
neou
sly
•Lo
t si
zing
•M
achi
ne s
ched
ulin
g
•F
ocus
on
solv
ent-
base
d pa
ints
, for
w
hich
ther
e ar
e fe
wer
sta
ges.
•B
arbo
t is
a P
ortu
gues
e pa
int
man
ufac
ture
r.
Sev
eral
mac
hine
s o
f ea
ch ty
pe
-
Pai
nt P
rod
uct
ion
at B
arb
ot
•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
for
20 m
achi
nes
and
80 p
rodu
cts.
•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.
-
Pro
du
ctio
n L
ine
Seq
uen
cin
g
at P
eug
eot/C
itro
ën
•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
du
ctio
n Li
ne
Seq
uen
cing
at P
eug
eot/C
itroë
n
•E
ach
car
pass
es t
hrou
gh 3
sho
ps.
•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
du
ctio
n Li
ne
Seq
uen
cing
at P
eug
eot/C
itroë
n
•S
peci
al c
onst
rain
ts
•C
ars
with
a s
un r
oof
shou
ld b
e 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
du
ctio
n Li
ne
Seq
uen
cing
at P
eug
eot/C
itroë
n
•P
robl
em h
as t
wo
part
s
•D
eter
min
e nu
mbe
r of
car
s of
eac
h ty
pe
assi
gned
to
each
line
on
each
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
.
-
Sam
ple
sche
dule
Sou
rce
: P
eug
eot
/Citr
oën
-
Pro
du
ctio
n Li
ne
Seq
uen
cing
at P
eug
eot/C
itroë
n
•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
set
ups
and
fast
er c
usto
mer
ser
vice
.
•B
ette
r sc
hedu
les.
-
Lin
e B
alan
cin
g a
t P
eug
eot/C
itro
ën
A c
lass
ic p
rodu
ctio
n se
quen
cing
pro
blem
Sou
rce
: P
eug
eot
/Citr
oën
-
Lin
e B
alan
cing
at P
eug
eot/C
itroë
n
•O
bjec
tive
•E
qual
ize
load
at w
ork
stat
ions
.
•K
eep
each
wor
ker
on o
ne s
ide
of t
he c
ar
•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
)
-
Lin
e B
alan
cing
at P
eug
eot/C
itroë
n
•S
olut
ion
agai
n ob
tain
ed b
y a
hybr
id m
etho
d.
•M
IP:
obta
in s
olut
ion
with
out r
egar
d to
pr
eced
ence
con
stra
ints
.
•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.
-
Sou
rce
: P
eug
eot
/Citr
oën
-
Lin
e B
alan
cing
at P
eug
eot/C
itroë
n
•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.
-
Re
laxa
tion
Rel
axin
g all-
diff
ere
nt
Rel
axin
g ele
men
tR
elax
ing c
ycle
(TS
P)
Rel
axin
g cu
mu
lativ
eR
elax
ing
a di
sjun
ctio
n of
line
ar s
yste
ms
Lagr
ange
an r
elax
atio
n
-
Use
s o
f R
elax
atio
n
•S
olve
a r
elax
atio
n of
the
pro
blem
res
tric
tion
at e
ach
node
of
the
sear
ch t
ree.
Thi
s pr
ovid
es a
bou
nd fo
r th
e br
anch
-and
-bou
nd p
roce
ss.
•In
a d
ecom
posi
tion
appr
oach
, pl
ace
a re
laxa
tion
of t
he
subp
robl
em in
to th
e m
aste
r pr
oble
m.
-
Ob
tain
ing
a R
elax
atio
n
•O
R h
as a
wel
l-dev
elop
ed te
chno
logy
for
fin
ding
po
lyhe
dral
rel
axat
ions
for
disc
rete
con
stra
ints
(e
.g.,
cutt
ing
plan
es).
•R
elax
atio
ns c
an b
e de
velo
ped
for
glob
al c
onst
rain
ts,
such
as a
ll-d
iffere
nt,
ele
men
t, cu
mu
lativ
e.
•D
isju
nctiv
e re
laxa
tions
are
ver
y us
eful
(f
or d
isju
nctio
ns o
f lin
ear
or n
onlin
ear
syst
ems)
.
-
Rel
axat
ion
of a
lldiff
ere
nt
{}n
x
xx
j
n
,,1
),
,(
alld
iff1
…
…
∈C
onve
x hu
ll re
laxa
tion,
whi
ch is
the
stro
nges
t po
ssib
le li
near
re
laxa
tion (
JNH
, W
illia
ms
& Y
an
):
{}
∑∑ ∈=<
⊆+
≥
+=
Jj
j
n jj
nJ
nJ
JJ
x
nn
x
||
with
,
,1
all
),1|
(|||
)1(
211
21
…
For
n =
4:
1,
,,
3,3
,3,3
,3,3
6,6
,6,6
10
43
21
43
42
32
41
31
21
43
24
31
42
13
21
43
21
≥≥
+≥
+≥
+≥
+≥
+≥
+≥
++
≥+
+≥
++
≥+
+=
++
+
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
-
Rel
axat
ion
of e
lem
en
t
To im
plem
ent v
aria
bly
inde
xed
cons
tant
Rep
lace
with
z an
d ad
d co
nstr
aint
(
)za
ay
n),
,,
(,e
lem
en
t1…
Con
vex
hull
rela
xatio
n of
ele
men
t con
stra
int i
s si
mpl
y
}{
ma
x}
{m
inj
Dj
jD
ja
za
yy
∈∈
≤≤
Cur
rent
dom
ain
ofy
ya
ya
-
If 0
≤x j
≤m
0fo
r ea
ch j,
ther
e is
a s
impl
e co
nvex
hul
l rel
axat
ion (J
NH
):
()
∑∑
∈∈
≤≤
−−
yy
Dj
jD
jy
jx
zm
Dx
01
Rel
axat
ion
of e
lem
en
t
To im
plem
ent v
aria
bly
inde
xed
varia
ble
Rep
lace
with
z an
d ad
d co
nstr
aint
whi
ch p
osts
con
stra
int
(
)zx
xy
n),
,,
(,e
lem
en
t1…
)(
jD
jx
zy
=∈∨
If 0
≤x j
≤m
jfo
r ea
ch j,
anot
her
rela
xatio
n is
∑
∑
∑
∑
∈
∈
∈
∈−
+
≤≤
+−
y
y
y
y
Dj
j
Dj
yjj
Dj
j
Dj
yjj
mDmx
z
mDmx
1
1
1
1
yx
yx
-
Exa
mp
le:
x y, w
here
Dy=
{1,
2,3}
and
50
40
30
321
≤≤
≤≤
≤≤
xxx
Rep
lace
x y w
ith z
and
elem
ent(y
,(x1,
x 2,x
3),z
)
Rel
axat
ion:
47
12
04
71
24
71
54
72
04
71
20
47
12
47
15
47
20
32
13
21
32
13
21
10
++
+≤
≤−
++
++
≤≤
−+
+x
xx
zx
xx
xx
xz
xx
x
-
Rel
axat
ion
of c
ycle
Use
cla
ssic
al c
uttin
g pl
anes
for
trav
elin
g sa
lesm
an p
robl
em:
),
,cy
cle(
sub
ject
to
min
1n
jjy
yy
cj
…
∑
Vis
it ea
ch c
ity
exac
tly o
nce
in a
si
ngle
tou
r
Dis
tanc
e fr
om c
ity j
to c
ity y
j
y j=
city
imm
edia
tely
fo
llow
ing
city
j
Can
als
o w
rite:
),
,d
iffer
ent(
-al
lsu
bje
ct t
o
min
1
1
n
jy
y
yy
cj
j
…
∑+
y j=
jth
city
in to
ur
-
Rel
axat
ion
of c
um
ula
tive
(JN
H,
Ya
n)
Whe
re t =
(t 1
,…,t n
)ar
e jo
b st
art t
imes
d=
(d 1
,…,d
n)ar
e jo
b du
ratio
nsr
= (
r 1,…
,rn)
are
reso
urce
con
sum
ptio
n ra
tes
Lis
max
imum
tota
l res
ourc
e co
nsum
ptio
n ra
tea
= (
a 1,…
,an)
are
earli
est s
tart
tim
es
()
Lr
dt
,,
,cu
mul
ativ
e
-
One
can
con
stru
ct a
rel
axat
ion
cons
istin
g of
the
follo
win
g va
lid c
uts.
If so
me
subs
et o
f job
s {j1,
…,j k
} ar
e id
entic
al (
sam
e re
leas
e tim
e a 0
, dur
atio
n d0,
and
res
ourc
e co
nsum
ptio
n ra
te
r 0),
then
[] 0
210
)1(
2)1
(1
dQ
Pk
Pa
Pt
tkj
j+
−+
+≥
++⋯
is a
val
id c
ut a
nd is
face
t-de
finin
g if
ther
e ar
e no
dea
dlin
es,
whe
re1
,0
−
=
=Qk
PrL
Q
-
The
follo
win
g cu
t is
valid
for
any
subs
et o
f job
s {j
1,…
,j k}
i
k i
ij
jd
Lri
kt
tk
∑ =
−
+−
≥+
+1
2121)
(1⋯
Whe
re th
e jo
bs a
re o
rder
ed b
y no
ndec
reas
ing
r jd j
.
Ana
logo
us c
uts
can
be b
ased
on
dead
lines
.
-
Exa
mp
le:
Con
side
r pr
oble
m w
ith fo
llow
ing
min
imum
mak
espa
n so
lutio
n (a
ll re
leas
e tim
es =
0):
Min
mak
espa
n =
8
L
1 23
4 5
time
reso
urce
s
0
6
23
3
5,5
,3,3
,3s.
t.
min
765
43
21
745
43
2
145
43
21
32
1
54
32
1
≥≥
++
++
≥+
++
≥+
++
≥+
++
++
++
≥ jtt
tt
tt
tt
tt
tt
tt
tt
t
tt
tt
tzz
Rel
axat
ion:
Res
ultin
g bo
und:
17
.5m
akes
pan
≥=
z
Fac
et d
efin
ing
-
Rel
axin
g D
isju
nct
ion
s o
f Lin
ear
Sys
tem
s
()
kk
kb
xA
≤∨
(Ele
men
tis a
spe
cial
cas
e.)
Con
vex
hull
rela
xatio
n (B
ala
s).
0
1
al
l,
≥
=
=≤
∑
∑
kkkk
kk
kk
y
y
xx
ky
bx
Ak
Add
ition
al v
aria
bles
nee
ded.
Can
be
exte
nded
to n
onlin
ear
syst
ems
(Stu
bb
s &
Meh
rotr
a)
-
“Big
M”
rela
xatio
n
0
1
al
l),
1(
≥
=−
−≤
∑ kkk
kk
k
y
y
ky
Mb
xA
k
Whe
re (
taki
ng t
he m
ax in
eac
h ro
w):
k ik i
k ik i
xk
k ib
kk
bA
xA
M−
≠
≤=
}'
al
l,
|{
max
max
''
Thi
s si
mpl
ifies
for
a di
sjun
ctio
n of
ineq
ualit
ies
whe
re 0
≤x j
≤m
j (B
ea
um
on
t):(
)k
kK k
bx
a≤
=∨ 1
∑∑
==
−+
≤
K k
kkK k
kkK
Mbx
Ma
11
1w
here
{}
jj
k jk
ma
M∑
=,0
ma
x
-
Exa
mp
le:
≤=∨
≤=∨
=1
080
ma
chin
e
larg
e
550
ma
chin
e
sma
ll
0
ma
chin
e
no
xz
xzx
Out
put o
f mac
hine
Fix
ed c
ost o
f mac
hine
Con
vex
hull
rela
xatio
n:
0,
110
5
80
50 3
2
32
32
32 ≥≤
++
≤+
≥
yy
yy
yy
x
yy
z
x
z
Big
-M r
elax
atio
n:
0,
1
8050
55
51
0
10
10 3
2
32
32
3
2
32 ≥≤
+≥≥+
≤−
≤+
≤
yy
yy
yz
yz
yx
yx
yy
x
x
z
-
Put
ting
It To
geth
er
Ele
men
ts o
f a
Gen
eral
Sch
eme
Pro
cess
ing
Net
wor
k D
esig
nB
ende
rs D
ecom
posi
tion
-
Ele
men
ts o
f a G
ener
al S
chem
e
•M
odel
con
sist
s of
•d
ecl
ara
tion
win
do
w (var
iabl
es,
initi
al d
omai
ns)
•re
laxa
tion
win
do
ws(i
nitia
lize
rela
xatio
ns &
sol
vers)
•co
nst
rain
t win
do
ws (e
ach
with
its
own
synt
ax)
•o
bje
ctiv
e f
un
ctio
n (opt
iona
l)
•se
arc
h w
ind
ow(
invo
kes
prop
agat
ion,
bra
nchi
ng,
rela
xatio
n, e
tc.)
•B
asic
alg
orith
m s
earc
hes
over
pro
blem
res
tric
tions
, dra
win
g in
fere
nces
and
sol
ving
rel
axat
ions
for
each
.
-
Ele
men
ts o
f a G
ener
al S
chem
e
•R
elax
atio
ns m
ay in
clud
e:
•C
onst
rain
t sto
re (
with
dom
ains
)
•Li
near
pro
gram
min
g re
laxa
tion,
etc.
•T
he r
elax
atio
ns li
nk th
e w
indo
ws.
•P
ropa
gatio
n (e
.g.,
thro
ugh
cons
trai
nt s
tore
).
•S
earc
h de
cisi
ons
(e.g
., no
nint
egra
l sol
utio
ns o
f lin
ear
rela
xatio
n).
-
Ele
men
ts o
f a G
ener
al S
chem
e
•C
onst
rain
ts in
voke
spe
cial
ized
infe
renc
e an
d re
laxa
tion
proc
edur
es t
hat e
xplo
it th
eir
stru
ctur
e. F
or e
xam
ple,
they
•R
educ
e do
mai
ns (
in-d
omai
n co
nstr
aint
s ad
ded
to
cons
trai
nt s
tore
).
•Add
con
stra
ints
to o
rigin
al p
robl
ems
(e.g
. cu
ttin
g pl
anes
, lo
gica
l inf
eren
ces,
nog
oods
)
•Add
cut
ting
plan
es to
line
ar r
elax
atio
n (e
.g.,
Gom
ory
cuts
).
•Add
spe
cial
ized
rel
axat
ions
to li
near
rel
axat
ion
(e.g
., re
laxa
tions
for e
lem
en
t, c
um
ula
tive, etc.
)
-
Ele
men
ts o
f a G
ener
al S
chem
e
•A g
ener
ic a
lgor
ithm
:
•P
roce
ss c
onst
rain
ts.
•In
fer
new
co
nstr
aint
s, r
educ
e d
om
ains
& p
rop
agat
e,
gene
rate
re
laxa
tions
.
•S
olve
rel
axat
ions
.
•C
heck
fo
r em
pty
do
mai
ns,
solv
e LP
, et
c.
•C
ontin
ue s
earc
h (r
ecur
sive
ly).
•C
reat
e ne
w p
rob
lem
res
tric
tions
if d
esir
ed (
e.g,
ne
w t
ree
bra
nche
s).
•S
elec
t pro
ble
m r
estr
ictio
n to
exp
lore
nex
t (e
.g.,
b
ackt
rack
or
mo
ve d
eep
er in
the
tree
).
-
Exa
mp
le:
Pro
cess
ing
Net
wo
rk D
esig
n
•F
ind
optim
al d
esig
n of
pro
cess
ing
netw
ork.
•A “
supe
rstr
uctu
re”
(larg
est
poss
ible
net
wor
k) is
giv
en,
but n
ot a
ll pr
oces
sing
uni
ts a
re n
eede
d.
•In
tern
al u
nits
gen
erat
e ne
gativ
e pr
ofit.
•O
utpu
t uni
ts g
ener
ate
posi
tive
prof
it.
•In
stal
latio
n of
uni
ts in
curs
fixe
d co
sts.
•O
bjec
tive
is to
max
imiz
e ne
t pro
fit.
-
Sam
ple
Pro
cess
ing
Su
per
stru
ctu
re
Uni
t 1
Uni
t 2
Uni
t 3
Uni
t 4
Uni
t 5
Uni
t 6
Out
puts
in fi
xed
prop
ortio
n
-
Dec
lara
tion
Win
do
w
u i∈
[0,c
i]
flow
thro
ugh
unit i
x ij ∈
[0,c
ij]
flow
on
arc
(i,j)
z i∈
[0,∞
] fix
ed c
ost o
f uni
t i
y i ∈
Di=
{tr
ue,fa
lse}
pres
ence
or
abse
nce
of u
nit
i
-
Ob
ject
ive
Fu
nct
ion
Win
do
w
)(
max
ii
ii
zur
−∑
Net
rev
enue
gen
erat
ed b
y un
it i p
er u
nit f
low
-
Rel
axat
ion
Win
do
w
Typ
e:C
onst
rain
t sto
re,
cons
istin
g of
var
iabl
e do
mai
ns.
Ob
ject
ive f
un
ctio
n:N
one.
So
lver:
Non
e.
-
Rel
axat
ion
Win
do
w
Typ
e:Li
near
pro
gram
min
g.
Ob
ject
ive f
un
ctio
n:S
ame
as o
rigin
al p
robl
em.
So
lver:
LP
sol
ver.
-
Co
nst
rain
t Win
do
w
Typ
e:Li
near
(in
)equ
aliti
es.
Ax
+ B
u =
b(f
low
bal
ance
equ
atio
ns)
Infe
ren
ce:B
ound
s co
nsis
tenc
y m
aint
enan
ce.
Rela
xatio
n: Add
red
uced
bou
nds
to c
onst
rain
t sto
re.
Rela
xatio
n: Add
equ
atio
ns to
LP
rel
axat
ion.
-
Co
nst
rain
t Win
do
w
Typ
e:D
isju
nctio
n of
line
ar in
equa
litie
s.
Infe
ren
ce:N
one.
Rela
xatio
n: Add
Bea
umon
t’s p
roje
cted
big
-M
rela
xatio
n to
LP
.
≤¬∨
≥
0i
i
ii
iu
yd
zy
-
Co
nst
rain
t Win
do
w
Typ
e:P
ropo
sitio
nal l
ogic
.
Don
’t-be
-stu
pid
cons
trai
nts:
Infe
ren
ce:R
esol
utio
n (a
dd r
esol
vent
s to
con
stra
int s
et).
Rela
xatio
n: Add
red
uced
dom
ains
of y i’s
to c
onst
rain
t sto
re.
Rela
xatio
n (
op
tion
al):
Add
0-1
ineq
ualit
ies
repr
esen
ting
prop
ositi
ons
to L
P.
)(
)(
)(
)(
)(
)(
32
61
3
32
56
2
32
45
42
65
31
2
43
32
1
yy
yy
yy
yy
yy
yy
yy
yy
yy
yy
yy
yy
yy
∨→
→∨
→→
∨→
∨→
∨→
→→
∨→
-
Sea
rch
Win
do
w
Pro
cedu
re B
andB
sear
ch(P,
R,S
,Net
Bra
nch)
(can
ned
bran
ch &
bou
nd s
earc
h us
ing
Net
Bra
nch
as
bran
chin
g ru
le)
-
Use
r-D
efin
ed W
ind
ow
Pro
cedu
re N
etB
ranc
h(P,
R,S
,i)
Let i
be a
uni
t for
whi
ch u i
> 0
and
z i<
d i.
If i=
1 th
en c
reat
e P’f
rom
P b
y le
ttin
g D
i=
{T
}an
d re
turn
P’.
If i=
2 th
en c
reat
e P’f
rom
P b
y le
ttin
g D
i=
{F
}an
d re
turn
P’.
-
Ben
der
s D
eco
mp
osi
tion
•B
ende
rs is
a s
peci
al c
ase
of th
e ge
nera
l fra
mew
ork.
•T
he B
ende
rs s
ubpr
oble
ms
are
prob
lem
res
tric
tions
ove
r w
hich
the
sear
ch is
con
duct
ed.
•B
ende
rs c
uts
are
gene
rate
d co
nstr
aint
s.
•T
he M
aste
r pr
oble
m is
the
rela
xatio
n.
•T
he s
olut
ion
of t
he r
elax
atio
n de
term
ines
whi
ch
subp
robl
em to
sol
ve n
ext.
-
•A.
Bo
ckm
ayr
and
J. H
oo
ker,
Co
nst
rain
t pro
gram
min
g, in
K. A
ard
al,
G.
Nem
hau
ser
and
R. W
eism
ante
l, ed
s.,
Ha
nd
bo
ok
of D
iscr
ete
Op
timiz
atio
n, N
ort
h-H
olla
nd
, to
ap
pea
r.
•S
. Hei
pck
e, Co
mb
ined
Mo
delli
ng
an
d P
rob
lem
So
lvin
g in
Ma
them
ati
cal
Pro
gra
mm
ing
an
d C
on
stra
int P
rog
ram
min
g, P
hD
thes
is,
Un
iver
sity
of B
uck
ingh
am,
19
99
.•
J. H
oo
ker,
Lo
gic,
op
timiz
atio
n a
nd
co
nst
rain
t pro
gra
mm
ing,
INF
OR
MS
Jo
urn
al o
n
Co
mp
utin
g 14
(20
02
) 2
95
-32
1.
•J.
Ho
oke
r, Lo
gic
-Ba
sed
Meth
od
s fo
r O
ptim
iza
tion
: Co
mb
inin
g O
pt
imiz
atio
n a
nd
C
on
stra
int S
atis
fact
ion, W
iley,
20
00
.•
M. M
ilan
o, I
nte
grat
ion
of O
R a
nd
AI
con
stra
int-
based
tech
niq
ues
for
com
bin
ato
rial
op
timiz
atio
n, h
ttp
://w
ww
-lia.
dei
s.u
nib
o.it
/Sta
ff/M
ich
elaM
ilan
o/t
uto
rialIJ
CA
I20
01.
pdf
Su
rvey
s/T
uto
rials
on
Hyb
rid M
eth
od
s