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