Facility Location and Network
Design Models
Facility Location Assumptions
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Based on graph G=(V,E)
Demand nodes, I  V, are known and fixed
Set J  V of possible facility locations
Note: I and J may have common nodes
Edges (i,j)E indicate it is possible for
demand at node i(j) to be directly satisfied if
facility is located at node j(i)
• “distance” between nodes i and j is dij
Set Covering Location Problem
min
x
jJ
j
s.t.
x
jNi
j
 1,
x j  {0,1},
iI
jJ
Maximal Covering Location Problem
max
h y
i
iI
i
s.t.
x
jNi
x
jJ
j
j
 yi ,
iI
 p,
x j  {0,1},
jJ
yi  {0,1},
iI
p-Center Problem
min W
s.t.
y
jJ
x
jJ
ij
 1,
j
 p,
iI
yij  x j ,
i  I, j  J
d
iI
jJ
ij
yij  W ,
x j  {0,1},
jJ
yij  {0,1},
i  I, j  J
p-Median Problem
min
hd
iI jJ
i
ij
yij
s.t.
y
jJ
x
jJ
ij
 1,
j
 p,
iI
yij  x j ,
i  I, j  J
x j  {0,1},
jJ
yij  {0,1},
i  I, j  J
Fixed-Charge Location Problem
min
f x
j
jJ
j
   hi dij yij
iI jJ
s.t.
y
jJ
x
jJ
ij
 1,
j
 p,
h y
iI
i
ij
 Cjxj ,
iI
jJ
x j  {0,1},
jJ
yij  {0,1},
i  I, j  J
Local p-Median Problem


min  hi   cij yij 
iI
 jNi

s.t.
x
jJ
j
 p,
yij  x j ,
i  I , j  Ni


hi   yij   T

iI
 jNi 
 yij  1,
iI
x j , yij  {0,1},
i  I , j  Ni
jNi
Network Flow Model
min

( i , j )A
cij xij
s.t.

j:( i , j )A
xij 

j:( j ,i )A
x ji  bi , i  V
0  xij  uij for each (i, j )  A
Network Design Problem
min

cij xij 
xij 

( i , j ) A

( i , j )A
fij yij
s.t.

j:( i , j ) A
j:( j ,i ) A
x ji  bi , i  V
0  xij  uij yij ,
(i, j )  A
yij  {0,1},
(i, j )  A