In some cases, the waste generated by the production of material at a facility must be
disposed of at special waste disposal locations. We need to identify which waste
facilities to open, given a capacity for each waste unit and per-unit transportation
costs for shipping waste from a plant facility to a waste facility. We can model this
problem by extending our facility location models. Suppose we have the network
shown on the next slide and let K={4, 9, 10} be possible locations for our waste
disposal facilities. At each plant jJ, the amount of waste material produced is
proportional (with proportionality constant 0<j<1) to the amount of goods produced;
for example, if plant 5 produces 500 units, then it also produces 500 units of waste.
These proportionality constants are
3 ,5 ,6 ,7 ,8    0.1,0.15,0.2,0.125,0.175
The one-time fixed cost associated with opening a waste disposal facility at kK is
f
and their capacities are
W
4
, f9W , f10W    5000, 7000, 6000 
C
W
4
, C9W , C10W   100,140,120 
The cost of shipping one unit of waste from plant jJ to waste disposal unit kK is
$10djk, where djk is the shortest distance from j to k, while the cost of shipping one
unit of demand from plant j to customer i is $20dij. Assuming that a customer can
receive its demand from multiple plants and that the waste from a plant can be shipped
to multiple disposal locations, determine how to satisfy customer demands and waste
removal requirements at minimum cost. Where are the plants and waste disposal units?
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
min
f j x j  20 hi dij yij   f kW xkW  10  d jk w jk

jJ
iI jJ
kK
jJ kK
s.t.
y
jJ
1
ij
w
kK
jk
i  I
(2)
  j  hi yij j  J
(3)
iI
yij  x j
i  I , j  J
(4)
j  J
(5)
k  K
(6)
x j , xkW  {0,1}
j  J , k  K
(7)
y ij , w jk  0
i  I , j  J , k  K (8,9)
h y
iI
i
ij
 Cjxj
W W
w

C
 jk k xk
jJ
Solution
• Optimal cost is $42,813.80
• Optimal solution opens plants in all possible
locations. Plant 3 services customer 3, plant 5
serves customers 2 and 5, plant 6 serves
customers 1, 6, and 9, plant 7 serves customer 7,
and plant 8 serves customers 4, 8, and 10.
• Optimal solution opens waste disposal facilities at
locations 4 and 10, with location 4 handling waste
from plants 3, 5, and 7, and location 10 handling
waste from plants 6 and 8.