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IE 335 Operations Research  Optimization
Solutions to Homework 4
Fall 2011
Problem 15
Let
x
ijg
represent the number of grade
g
students in neighborhood
i
assigned to school
j
for
i
= 1
,...,I
j
= 1
,...,J
, and
g
= 1
,...,G
min
G
X
g
=1
J
X
j
=1
I
X
i
=1
d
ij
x
ijg
(1)
s.t.
I
X
i
=1
x
ijg
≤
C
jg
for
j
= 1
,...,J
and
g
= 1
,...,G
(2)
J
X
j
=1
x
ijg
=
S
ig
for
i
= 1
,...,I
and
g
= 1
,...,G
(3)
x
ijg
≥
0
for
i
= 1
,...,I
,
j
= 1
,...,J
and
g
= 1
,...,G
(4)
The objective function (1) tries to minimize the weighted sum of the number of students of each grade,
neighborhood and school weighted by the distance of the neighborhood from the school. Constraint (2)
ensures that the total number of students do not exceed the capacity of a grade in a school and constraint
(3) ensures that
all
students are assigned to schools.
Problem 16
Let’s use the decision variables given in the hint:
x
i
= fraction of item
i
allocated to Captain Hook
i
= 1
,...,
4;
y
i
= fraction of item
i
allocated to Captain Sparrow
i
= 1
,...,
4
.
If we allocate the items according to the decision variables above, then we can write the total value points
of Captain Hook as
40
x
1
+ 10
x
2
+ 10
x
3
+ 40
x
4
.
Similarly, we can write the total value points of Captain Sparrow as
30
y
1
+ 20
y
2
+ 20
y
3
+ 30
y
4
.
Then, the following optimization model with a maximin objective determines how to allocate the treasure
between Captain Hook and Captain Sparrow, in a way that maximizes the minimum total value points of
any captain:
max
min
{
40
x
1
+ 10
x
2
+ 10
x
3
+ 40
x
4
,
30
y
1
+ 20
y
2
+ 20
y
3
+ 30
y
4
}
(5)
s.t.
x
i
+
y
i
≤
1
for
i
= 1
,
2
,
3
,
4
,
(6)
x
i
≥
0
for
i
= 1
,
2
,
3
,
4
,
(7)
y
i
≥
0
for
i
= 1
,
2
,
3
,
4
.
(8)
1
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View Full DocumentConstraint (6) ensures that at most 100% of each item is allocated to the captains. Constraints (7) and (8)
ensure that the fractions are nonnegative.
We can convert this optimization model into a linear program, using the technique we discussed in
Lecture 9. Let
f
be an auxiliary decision variable that represents the minimum total value points of any
captain. Then the above optimization model is equivalent to the following linear program:
max
f
s.t.
f
≤
40
x
1
+ 10
x
2
+ 10
x
3
+ 40
x
4
,
f
≤
30
y
1
+ 20
y
2
+ 20
y
3
+ 30
y
4
,
x
i
+
y
i
≤
1
for
i
= 1
,
2
,
3
,
4
,
x
i
≥
0
for
i
= 1
,
2
,
3
,
4
,
y
i
≥
0
for
i
= 1
,
2
,
3
,
4
.
Problem 17
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 Fall '08
 Jeanphilippe,R
 Operations Research, Optimization

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