a framework for combining solution methodspublic.tepper.cmu.edu/jnh/havana2003.pdf · machine...
TRANSCRIPT
A F
ram
ew
ork
for
Com
bini
ng S
olut
ion
Me
thod
s
John
Hoo
ker
Ca
rne
gie
Me
llon
Un
ive
rsity
Inte
rnat
iona
l Con
fere
nce
on O
pera
tions
Res
earc
hH
ava
na
, Se
pte
mb
er
20
03
A S
imp
le K
nap
sack
Exa
mp
leIn
teg
ratin
g C
P a
nd
MP
, an
d L
oca
l Sea
rch
Bra
nch
In
fer
and
Rel
axD
eco
mp
osi
tion
Rec
ent
Su
cces
s S
tori
es
A S
impl
e K
naps
ack
Exa
mpl
e
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 ij
iij
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,
MP,
and
L
oca
l Se
arc
h
Mot
ivat
ion
Inte
grat
ion
Sch
emes
Mo
tivat
ion
to In
teg
rate
CP
an
d M
P
•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 c
onst
rain
ed”
prob
lem
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.
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: Vie
w C
P a
nd M
P m
etho
ds a
s sp
ecia
l cas
es o
f a b
ranc
h-in
fer-
and-
rela
x m
etho
d.
•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.
•G
en
era
l sch
em
e: The
se a
re s
peci
al c
ases
of a
gen
eral
sc
hem
e fo
r in
tegr
atin
g C
P, M
P a
nd lo
cal s
earc
h.
Bra
nch
-infe
r-an
d-r
elax
•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
sol
ving
•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.
Gen
eral
Sch
eme
•D
ecom
posi
tion,
Bra
nch-
infe
r-an
d-re
lax,
and
loca
l se
arch
are
spe
cial
cas
es o
f a g
ener
al s
chem
e:
•E
num
erat
e pr
oble
m r
estr
ictio
ns.
•Le
af n
odes
of t
ree.
•B
ende
rs s
ubpr
oble
ms
•Lo
cal s
earc
h ne
ighb
orho
ods.
•S
olve
rel
axat
ion
of e
ach
rest
rictio
n.
•LP
rel
axat
ion.
•B
ende
rs m
aste
r pr
oble
m.
•R
elax
atio
n of
nei
ghbo
rhoo
d se
arch
Gen
eral
Sch
eme
•S
olut
ion
of r
elax
atio
n gu
ides
sea
rch.
•F
ract
ions
in L
P s
olut
ion
help
det
erm
ine
bran
chin
g.
•S
olut
ion
of B
ende
rs m
aste
r pr
oble
m d
eter
min
es
next
sub
prob
lem
.
•S
olut
ion
of n
eigh
borh
ood
sear
ch is
cen
ter
of
next
nei
ghbo
rhoo
d.
Bra
nch
Infe
r a
nd R
ela
x
The
kna
psac
k ex
ampl
e ill
ustr
ated
this
.
De
com
posi
tion
Idea
Beh
ind
Ben
ders
Dec
ompo
sitio
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.
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
()
iL
ix
Ri
xD
ix
t
jS
Dt
jR
t
C
ij
ijj
ijj
j
jj
xj
jjj
jx
j
j
a
ll,
),|
(),
|(
),|
(cu
mul
ativ
e
a
ll,
all
,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 fo
r ea
ch jo
b
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
()
iL
ix
Ri
xD
ix
ti
jij
jij
jj
al
l,
),|
(),
|(
),|
(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
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ix
jS
Dt
jR
t
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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
)
Sub
pro
ble
ms
are
sim
ple
1-m
achi
ne s
ched
ulin
g p
rob
lem
s
Co
mp
uta
tio
nal R
esu
lts
0.0
1
0.11
10
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
Min C
ost, 2
Mac
hines
Loga
rithmi
c sca
le
0.010.1110100
1000
1000
0
510
1520
25
Jobs
Sec
CP MILP
Hybr
id
= c
om
pu
tatio
n te
rmin
ated
Com
puta
tiona
l Res
ults
(J
NH
)S
ubp
rob
lem
s ar
e fu
ll re
sour
ce-c
ons
trai
ned
sch
edul
ing
pro
ble
ms
Min
Cost
, 3 M
achi
nes
Loga
rithm
ic sc
ale
0.010.1110100
1000
1000
0
57
911
1315
17
Jobs
Sec
CP MILP
Hybr
id
Min C
ost, 4
Mac
hines
Loga
rithmi
c sca
le
0.010.1110100
1000
1000
0
57
911
1315
17
Jobs
Sec
CP MILP
Hybri
d
Min M
akes
pan,
2 Mac
hines
Loga
rithmi
c sca
le
0.010.1110100
1000
57
911
1315
17
Jobs
Sec
CP Hybri
d
Min M
akes
pan,
3 M
achin
esLo
garit
hmic
scale
0.010.1110100
1000
1000
0
510
1520
Jobs
Sec
CP Hybr
id
Min
Mak
espa
n, 4
Mac
hine
sLo
garith
mic
scale
0.010.1110100
1000
1000
0
510
1520
2530
Jobs
Sec
CP Hybr
id
An
En
han
cem
ent:
Bra
nch
an
d C
hec
k(J
NH
, T
ho
rste
inss
on
)
•S
olve
the
mas
ter
prob
lem
onl
y on
ce b
ut c
ontin
ually
upd
ate
it w
ith B
ende
rs c
uts
whe
n fe
asib
le s
olut
ions
are
fou
nd.
•T
his
was
app
lied
to th
e m
achi
ne s
ched
ulin
g pr
oble
m
desc
ribed
ear
lier.
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
cess
Sch
edul
ing
at B
AS
FP
aint
Pro
duct
ion
at B
arbo
tP
rodu
ctio
n Li
ne S
eque
ncin
g at
Peu
geot
/Citr
oën
Line
Bal
anci
ng a
t Peu
geot
/Citr
oën
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.