IEOR 162, Spring 2011
Suggested Solution to Homework 04
Problem 1
Solution 1.
First, we label the two refinery at Los Angeles and Chicago as refinery 1 and 2 and the two
distribution points at Houston and New York City as distribution points 1 and 2.
Then let the decision
variables be
x
i
= million barrels of capacity created for refinery
i
,
i
= 1
,
2, and
y
ij
= million barrels of oil shipped from refinery
i
to distribution point
j
,
i
= 1
,
2,
j
= 1
,
2.
For parameters, let
P
ij
be the profit (in thousands) per million barrels of oil shipped from refinery
i
to
distribution point
j
,
C
i
be the unit cost (in thousands) of expanding capacity for one million barrel in
refinery
i
,
K
i
be the current capacity (in million barrel) in refinery
i
, and
D
j
be the demand size (in million
barrels) at distribution point
j
for all
i
= 1
,
2 and
j
= 1
,
2:
P
=
•
20
15
18
17
‚
,
C
=
•
120
150
‚
,
K
=
•
2
3
‚
,
D
=
£
5
5
/
.
With the definitions of variables and parameters, we formulate the problem as
max
10
2
X
i
=1
2
X
j
=1
P
ij
y
ij

2
X
i
=1
C
i
x
i
s.t.
2
X
i
=1
y
ij
≤
D
j
∀
j
= 1
,
2
2
X
j
=1
y
ij
≤
K
i
+
x
i
∀
i
= 1
,
2
x
i
, y
ij
≥
0
∀
i
= 1
,
2
, j
= 1
,
2
.
The objective function consists of two parts, the 10year total profit and the onetime expansion cost. The
first constraint ensures that the total sales at each distribution point is at most the demand size. The second
constraint ensures that the total production quantity at each refinery does not excess the (postexpansion)
capacity. The last constraint is the nonnegativity constraint.
Solution 2.
We may alternatively define, for
i
= 1
,
2 and
j
= 1
,
2,
w
ij
= million barrels of “original” capacity allocated to refinery
i
and distribution point
j
and
z
ij
= million barrels of “additional” capacity allocated to refinery
i
and distribution point
j
.
The formulation is
max
10
2
X
i
=1
2
X
j
=1
P
ij
(
w
ij
+
z
ij
)

2
X
i
=1
C
i
2
X
i
=1
z
ij
s.t.
2
X
i
=1
(
w
ij
+
z
ij
)
≤
D
j
∀
j
= 1
,
2
2
X
j
=1
w
ij
≤
K
i
∀
i
= 1
,
2
w
ij
, z
ij
≥
0
∀
i
= 1
,
2
, j
= 1
,
2
.
The two formulations are equivalent. To see this, note that
x
i
=
∑
2
j
=1
z
ij
and
y
ij
=
z
ij
+
w
ij
.