Lecture #35 Prof. John W. Sutherland April 12, 2006 · 2018. 8. 1. · © John W. Sutherland...
34
© John W. Sutherland Service Processes & Systems Dept. of Mechanical Engineering - Engineering Mechanics Michigan Technological University Lecture #35 Prof. John W. Sutherland April 12, 2006
Transcript of Lecture #35 Prof. John W. Sutherland April 12, 2006 · 2018. 8. 1. · © John W. Sutherland...
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Lectu
re #
35
Pro
f. J
oh
n W
. S
uth
erl
an
d
Ap
ril 12, 2006
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
�T
his
mo
del d
eals
wit
h f
ind
ing
th
e a
m
inim
um
co
st
pla
n f
or
tran
sp
ort
ing
a
sin
gle
co
mm
od
ity f
rom
a n
um
ber
of
so
urc
es (
e.g
., f
acto
ries)
to a
nu
mb
er
of
desti
nati
on
s (
e.g
., w
are
ho
uses).
�T
he m
od
el can
be e
xte
nd
ed
in
a d
irect
man
ner
to c
over
sit
uati
on
s in
are
as
rela
ted
to
in
ven
tory
co
ntr
ol,
em
plo
ym
en
t sch
ed
ulin
g, p
ers
on
nel
assig
nm
en
t, c
ash
flo
w, etc
.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
�T
he
da
ta o
f th
e m
od
el
inc
lud
e:
�L
evel
of
su
pp
ly a
t each
so
urc
e a
nd
am
ou
nt
of
dem
an
d a
t each
desti
nati
on
�T
he u
nit
tran
sp
ort
ati
on
co
st
of
the c
om
mo
dit
y
fro
m e
ach
so
urc
e t
o e
ach
desti
nati
on
�S
inc
e t
he
re i
s o
nly
on
e c
om
mo
dit
y,
a
de
sti
na
tio
n c
an
re
ce
ive
its
de
ma
nd
fro
m o
ne
o
r m
ore
so
urc
es
. T
he
ob
jec
tiv
e i
s t
o
de
term
ine
th
e a
mo
un
t to
be
sh
ipp
ed
fro
m
ea
ch
so
urc
e t
o e
ac
h d
es
tin
ati
on
su
ch
th
at
the
to
tal
tra
ns
po
rta
tio
n c
os
t is
min
imiz
e.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
�T
he
ba
sic
as
su
mp
tio
n o
f th
e m
od
el
is t
ha
t th
e t
ran
sp
ort
ati
on
co
st
on
a g
ive
n r
ou
te i
s
dir
ec
tly p
rop
ort
ion
al
to t
he
nu
mb
er
of
un
its
tr
an
sp
ort
ed
.�
Th
is f
igu
re d
ep
icts
th
e t
ran
sp
ort
ati
on
mo
de
l a
s a
ne
two
rk w
ith
m s
ou
rce
s a
nd
n
de
sti
na
tio
ns
.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
�T
he s
ou
rces a
nd
desti
nati
on
s a
re
rep
resen
ted
by n
od
es. T
he a
rc jo
inin
g a
so
urc
e a
nd
a d
esti
nati
on
rep
resen
ts t
he
rou
te t
hro
ug
h w
hic
h t
he c
om
mo
dit
y is
tran
sp
ort
ed
. T
he a
mo
un
t o
f su
pp
ly a
t so
urc
e i is a
ian
d t
he d
em
an
d a
t d
esti
nati
on
j is b
j. T
he u
nit
tr
an
sp
ort
ati
on
co
st
betw
een
so
urc
e i
an
d d
esti
nati
on
j is c
ij.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
�x
ijre
pre
sen
ts t
he a
mo
un
t tr
an
sp
ort
ed
fr
om
so
urc
e i t
o d
esti
nati
on
j;
then
th
e
LP
mo
del re
pre
sen
tin
g t
he
tran
sp
ort
ati
on
pro
ble
m is g
iven
as:
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
�A
sli
gh
tly t
we
ak
ed
mo
de
l v
ers
ion
re
qu
ire
s
tha
t th
e t
ota
l s
up
ply
eq
ua
l th
e t
ota
l d
em
an
d.
�W
he
n t
he
to
tal
su
pp
ly e
qu
als
th
e t
ota
l d
em
an
d,
the
re
su
ltin
g f
orm
ula
tio
n i
s c
all
ed
a
ba
lan
ce
d t
ran
sp
ort
ati
on
mo
de
l.�
It d
iffe
rs f
rom
th
e f
irs
t o
ne
on
ly i
n t
he
fa
ct
tha
t a
ll in
eq
ua
lity
co
ns
tra
ints
are
vie
we
d a
s
eq
ua
liti
es
:
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
Exa
mp
le
�T
he M
G A
uto
Co
mp
an
y h
as p
lan
ts in
L
.A., D
etr
oit
, an
d N
ew
Orl
ean
s. It
s
majo
r d
istr
ibu
tio
n c
en
ters
are
lo
cate
d
in D
en
ver
an
d M
iam
i. T
he c
ap
acit
ies o
f th
e 3
pla
nts
du
rin
g t
he n
ext
qu
art
er
are
1000, 1500, an
d 1
200 c
ars
. T
he
qu
art
erl
y d
em
an
ds a
t th
e t
wo
d
istr
ibu
tio
n c
en
ters
are
2300 a
nd
1400
cars
. T
he t
rain
tra
nsp
ort
ati
on
co
st
per
car
per
mile is a
pp
rox. 8 c
en
ts.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
Exa
mp
le
�M
ileag
e c
hart
:
�C
osts
(c
ij):
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
ST
D T
ran
sp
ort
ati
on
Mo
del E
xam
ple
�U
sin
g n
um
eri
c c
od
es t
o r
ep
resen
t th
e
pla
nts
an
d d
ist.
cen
ters
, w
e let
xij
rep
resen
t th
e n
um
ber
of
cars
tr
an
sp
ort
ed
fro
m s
ou
rce i
to
desti
nati
on
j.
�S
ince t
he t
ota
l su
pp
ly
(=1000+
1500+
1200=
3700)
hap
pen
s t
o
eq
ual th
e t
ota
l d
em
an
d
(=2300+
1400=
3700),
th
e r
esu
ltin
g
tran
sp
ort
ati
on
mo
del is
bala
nced
.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
Exa
mp
le
�H
en
ce t
he f
ollo
win
g L
P m
od
el
pre
sen
tin
g t
he p
rob
lem
has a
ll
eq
uality
co
nstr
ain
ts:
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
Exa
mp
le
�A
no
the
r w
ay f
or
rep
res
en
tin
g t
he
tr
an
sp
ort
ati
on
m
od
el
is t
o u
se
th
e
tra
ns
po
rta
tio
n
tab
lea
u –
a
ma
trix
wh
ere
th
e
co
st
ele
me
nts
cij
are
giv
en
in
th
e
no
rth
ea
st
co
rne
r
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
po
rta
tio
n M
od
el
Exa
mp
le
�S
olu
tio
n t
o t
he m
od
el:
1000
0
1300
200
01200
0 0
200
1300
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Th
e A
ssig
nm
en
t M
od
el
�C
on
sid
er
the s
itu
ati
on
of
assig
nin
g m
jo
bs (
or
wo
rkers
) to
n m
ach
ines. A
jo
b i
(=1,2
,…,m
) w
hen
assig
ned
to
mach
ine j
(=1,2
,…,n
) in
cu
rs a
co
st
cij. T
he
ob
jecti
ve is t
o a
ssig
n t
he jo
bs t
o t
he
mach
ines (
on
e jo
b p
er
mach
ine)
at
the
least
tota
l co
st.
�T
he f
orm
ula
tio
n o
f th
is p
rob
lem
may
be
reg
ard
ed
as a
sp
ecia
l case o
f th
e
tran
sp
ort
ati
on
mo
del.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Th
e A
ssig
nm
en
t M
od
el
�H
ere
jo
bs r
ep
res
en
t “so
urc
es”
an
d m
ach
ines
rep
res
en
t “d
esti
nati
on
s.
�T
he s
up
ply
avail
ab
le a
t each
so
urc
e i
s 1
; th
at
is, a
i=1
for
all i.
Sim
ilarl
y,
the d
em
an
d r
eq
uir
ed
at
each
d
esti
nati
on
is 1
; th
at
is, b
j=1 f
or
all j. T
he c
ost
of
“tr
an
sp
ort
ing
”(a
ssig
nin
g)
job
i t
o m
ach
ine j is c
ij.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Th
e A
ssig
nm
en
t M
od
el
�B
efo
re m
od
el
can
be s
olv
ed
by t
ran
sp
ort
ati
on
te
ch
niq
ue, n
ecess
ary
to
bala
nce p
rob
lem
by a
dd
ing
fi
cti
tio
us jo
bs/m
ach
ines,
dep
en
din
g o
n w
heth
er
m <
n o
r m
> n
.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Th
e A
ssig
nm
en
t M
od
el -
Exam
ple
�C
on
sid
er:
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Th
e A
ssig
nm
en
t M
od
el -
Exam
ple
�T
he
op
tim
al
so
luti
on
of
the
as
sig
nm
en
t m
od
el
rem
ain
s t
he
sa
me
if
a c
on
sta
nt
is
ad
de
d o
r s
ub
tra
cte
d t
o a
ny r
ow
or
co
lum
n o
f th
e c
os
t m
atr
ix.
�O
ne
ca
n c
rea
te a
ne
w c
’ ijm
atr
ix w
ith
ze
ro
en
trie
s,
an
d i
f th
es
e z
ero
ele
me
nts
or
a
su
bs
et
the
reo
f c
on
sti
tute
a f
ea
sib
le s
olu
tio
n,
this
fe
as
ible
so
luti
on
is
op
tim
al
--b
ec
au
se
c
os
ts c
an
no
t b
e n
eg
ati
ve
.�
In p
rev
iou
s t
ab
le z
ero
ele
me
nts
are
cre
ate
d
by s
ub
tra
cti
ng
sm
all
es
t e
lem
en
t in
ea
ch
ro
w
(co
lum
n)
fro
m t
he
co
rre
sp
on
din
g r
ow
(c
olu
mn
).
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Th
e A
ssig
nm
en
t M
od
el -
Exam
ple
�If
on
e c
on
sid
ers
th
e
row
s f
irs
t, t
he
ne
w c
’ ij–
ma
trix
is
:
�T
his
la
st
ma
trix
ca
n
be
ma
de
to
in
clu
de
m
ore
ze
ros
by
su
btr
ac
tin
g q
3=
2 f
rom
th
e t
hir
d c
olu
mn
:
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Th
e A
ssig
nm
en
t M
od
el -
Exam
ple
�T
he s
qu
are
s in
th
e last
tab
le g
ive t
he
feasib
le (
an
d h
en
ce o
pti
mal)
assig
nm
en
t (1
,1),
(2,3
), a
nd
(3,2
),
co
sti
ng
5+
12+
13=
30
�U
nfo
rtu
nate
ly, it
is n
ot
alw
ays p
ossib
le
to o
bta
in a
feasib
le a
ssig
nm
en
t u
sin
g
this
meth
od
. F
urt
her
rule
s a
re t
hu
s
req
uir
ed
to
fin
d t
he o
pti
mal so
luti
on
:
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Assig
nm
en
t M
od
el
–E
xam
ple
2
�N
ow
, c
arr
yin
g
ou
t th
e s
am
e
init
ial
ste
ps,
on
e g
ets
:
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Assig
nm
en
t M
od
el
–E
xam
ple
2
�F
ea
sib
le a
ss
ign
me
nt
to t
he
ze
ro e
lem
en
ts
is n
ot
po
ss
ible
in
th
is
ca
se
.�
We
ma
y d
raw
a
min
imu
m n
um
be
r o
f li
ne
s t
hro
ug
h s
om
e
of
the
ro
ws
an
d
co
lum
ns
su
ch
th
at
all
th
e z
ero
s a
re c
ros
se
d
ou
t.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Assig
nm
en
t M
od
el
–E
xam
ple
2
�T
he n
ext
ste
p is t
o s
ele
ct
the s
mall
est
un
cro
ss
ed
-ou
t ele
men
t, t
his
ele
men
t is
su
btr
acte
d f
rom
every
u
ncro
ss
ed
-ou
t ele
men
t an
d a
dd
ed
to
every
ele
men
t at
the in
ters
ecti
on
of
two
lin
es
�T
his
giv
es t
he o
pti
mal
assig
nm
en
t (1
,1),
(2,3
),(3
,2),
an
d (
4,4
). T
he c
orr
es
po
nd
ing
to
tal co
st
is
1+
10+
5+
5=
21
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
sh
ipm
en
t P
rob
lem
�T
ran
ssh
ipm
en
t p
rob
lem
s a
re t
ran
sp
ort
ati
on
p
rob
lem
s in
wh
ich
a s
hip
men
t m
ay m
ove
th
rou
gh
in
term
ed
iate
no
des (
tran
ssh
ipm
en
t n
od
es)
befo
re
reach
ing
a p
art
icu
lar
desti
nati
on
no
de.
�T
ran
ssh
ipm
en
t p
rob
lem
s c
an
be c
on
vert
ed
to
larg
er
tran
sp
ort
ati
on
pro
ble
ms a
nd
so
lved
by a
sp
ecia
l tr
an
sp
ort
ati
on
pro
gra
m.
�T
ran
ssh
ipm
en
t p
rob
lem
s c
an
als
o b
e s
olv
ed
by
gen
era
l p
urp
ose lin
ear
pro
gra
mm
ing
co
des.
�T
he n
etw
ork
rep
res
en
tati
on
fo
r a t
ran
ssh
ipm
en
t p
rob
lem
wit
h t
wo
so
urc
es,
thre
e i
nte
rmed
iate
no
des,
an
d t
wo
desti
nati
on
s i
s s
ho
wn
on
th
e n
ext
slid
e.
�A
nd
ers
on
/Sw
een
ey/W
illiam
s:
slid
e c
on
ten
t –
J. S
. L
ou
cks
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
sh
ipm
en
t N
etw
ork
222
333 444 555
666 777
111
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Inte
rmed
iate
No
des
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rmed
iate
No
des
So
urc
esS
ou
rces
Des
tin
atio
ns
Des
tin
atio
ns
ss22
Dem
and
Dem
and
Su
pp
lyS
up
ply
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Tra
ns
sh
ipm
en
t P
rob
lem
�L
inear
Pro
gra
mm
ing
Fo
rmu
lati
on
xij
rep
res
en
ts t
he s
hip
men
t fr
om
no
de i
to n
od
e j
Min
Σ
ΣΣ
ΣΣ
ΣΣ
Σc
ijx
iji j
s.t
. Σ ΣΣΣ
xij
<s
ifo
r each
ori
gin
ij Σ ΣΣΣ
xik
-Σ ΣΣΣ
xkj
= 0 fo
r each
in
term
ed
iate
ij
no
de k
Σ ΣΣΣx
ij=
d
jfo
r each
desti
nati
on
ji
xij
>0 f
or
all i
an
d j
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Exa
mp
le:
Zero
nS
helv
ing
�T
he N
ort
hsid
ean
d S
ou
thsid
e f
acil
itie
s o
f Z
ero
nIn
du
str
ies s
up
ply
th
ree f
irm
s (
Zro
x,
Hew
es,
Ro
ckri
te)
wit
h c
usto
miz
ed
sh
elv
ing
fo
r it
s o
ffic
es.
Zero
n's
two
facilit
ies b
oth
o
rder
sh
elv
ing
pro
du
ct
fro
m s
am
e t
wo
m
an
ufa
ctu
rers
, A
rno
ld M
an
ufa
ctu
rers
an
d
Su
pers
helf
, In
c.
�C
urr
en
tly w
eekly
de
man
ds b
y t
he u
sers
are
50 f
or
Zro
x,
60 f
or
Hew
es, an
d 4
0 f
or
Ro
ckri
te. B
oth
Arn
old
an
d S
up
ers
helf
can
su
pp
ly a
t m
ost
75 u
nit
s t
o its
cu
sto
mers
.
�A
dd
itio
nal d
ata
is s
ho
wn
on
th
e n
ext
slid
e.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Ex
am
ple
:
Ex
am
ple
: Z
ero
nZ
ero
nS
he
lvin
gS
he
lvin
g
Bec
ause
of
lon
g s
tan
din
g c
on
trac
ts b
ased
on
pas
t B
ecau
se o
f lo
ng
sta
nd
ing
co
ntr
acts
bas
ed o
n p
ast
ord
ers,
un
it c
ost
s fr
om
th
e m
anu
fact
ure
rs t
o t
he
ord
ers,
un
it c
ost
s fr
om
th
e m
anu
fact
ure
rs t
o t
he
sup
pli
ers
are:
sup
pli
ers
are:
Zer
on
Zer
on
NNZ
ero
nZ
ero
nSS
Arn
old
5
Arn
old
5
88
Su
per
shel
fS
up
ersh
elf
7
47
4
Th
e co
sts
to i
nst
all
the
shel
vin
g a
t th
e v
ario
us
loca
tio
ns
Th
e co
sts
to i
nst
all
the
shel
vin
g a
t th
e v
ario
us
loca
tio
ns
are:
are:
Zro
xZ
rox
Hew
esH
ewes
Ro
ckri
teR
ock
rite
Th
om
as
1
5
8
Th
om
as
1
5
8
Was
hb
urn
3
4
4W
ash
bu
rn
3
4
4
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Exam
ple
: Z
ero
nS
he
lvin
g N
etw
ork
AR
NO
LD
WA
SH
BU
RN
ZR
OX
HE
WE
S
7575 7575
5050 6060 4040
55 88 77 44
1155
88 33
44
44
Arn
old
Arn
old
Sup
erS
uper
She
lfS
helf
Hew
esH
ewes
Zro
xZ
rox
Zer
onZ
eron NN
Zer
onZ
eron SS
Roc
kR
ock --
Rite
Rite
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Exa
mp
le:
Zero
nS
helv
ing
�L
inear
Pro
gra
mm
ing
Fo
rmu
lati
on
�D
ecis
ion
Vari
ab
les D
efi
ned
xij
= a
mo
un
t sh
ipp
ed
fro
m m
an
ufa
ctu
rer
ito
su
pp
lier
j
xjk
= a
mo
un
t sh
ipp
ed
fro
m s
up
plier
jto
cu
sto
mer
k
wh
ere
i
= 1
(A
rno
ld),
2 (
Su
pers
helf
)
j=
3 (
Zero
nN
), 4
(Z
ero
nS
)
k=
5 (
Zro
x),
6 (
Hew
es),
7 (
Ro
ckri
te)
�O
bje
cti
ve F
un
cti
on
Defi
ned
Min
imiz
e O
vera
ll S
hip
pin
g C
osts
:
Min
5
x13
+ 8
x14
+ 7
x23
+ 4
x24
+ 1
x35
+ 5
x36
+ 8
x37
+ 3
x45 +
4x
46
+ 4
x47
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Exa
mp
le:
Zero
nS
helv
ing
�C
on
str
ain
ts D
efi
ned
Am
ou
nt
Ou
t o
f A
rno
ld:
x
13
+ x
14
<75
Am
ou
nt
Ou
t o
f S
up
ers
helf
: x
23
+ x
24
<75
Am
ou
nt
Th
rou
gh
Zero
nN
: x
13
+ x
23
-x
35
-x
36
-x
37
= 0
Am
ou
nt
Th
rou
gh
Zero
nS
: x
14
+ x
24
-x
45
-x
46
-x
47
= 0
Am
ou
nt
Into
Zro
x:
x
35
+ x
45
= 50
Am
ou
nt
Into
Hew
es:
x
36
+ x
46
= 60
Am
ou
nt
Into
Ro
ckri
te:
x
37
+ x
47
= 40
No
n-n
eg
ati
vit
y o
f V
ari
ab
les:
xij
>0, fo
r all i
an
d j.
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Exam
ple
: Z
ero
nS
helv
ing
--
So
luti
on
AR
NO
LD
WA
SH
BU
RN
ZR
OX
HE
WE
S
7575 7575
5050 6060 4040
55 88 77 44
1155
88 3344
44
Arn
old
Arn
old
Sup
erS
uper
She
lfS
helf
Hew
esH
ewes
Zro
xZ
rox
Zer
onZ
eron NN
Zer
onZ
eron SS
Roc
kR
ock --
Rite
Rite
75
75
75
75
50
50 2
525
35
35
40
40
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Th
e T
ran
ssh
ipm
en
t –
MG
Mo
de
l
�In
term
s o
f th
e t
ran
ssh
ipm
en
t m
od
el, w
e h
ave 5
so
urc
es a
nd
5
desti
nati
on
s:
©Jo
hn
W. S
uth
erl
an
d
Serv
ice P
rocesses &
Syste
ms
Dep
t. o
f M
ech
an
ical E
ng
ineeri
ng
-E
ng
ineeri
ng
Mech
an
ics
Mic
hig
an
Tech
no
log
ical
Un
ivers
ity
Op
tim
al
So
luti
on
�T
ran
ss
hip
pin
g t
ak
es
pla
ce
be
twe
en
De
tro
it
an
d t
he
Ne
w O
rle
an
s p
lan
ts,
wh
ere
20
0 c
ars
a
re s
hip
pe
d f
rom
De
tro
it t
o M
iam
i v
ia N
ew
O
rle
an
s.
All
oth
er
sh
ipm
en
ts a
re d
ire
ct
fro
m
the
pla
nts
to
th
e d
istr
ibu
tio
n c
en
ters
.
