MthSc810 – Mathematical Programming
Fall 2011, Solutions to Homework #6
Problem 1.
Implement, in AMPL, a solution to the following problem (it might
be useful in the next midterm):
A classroom can be thought of as a set of
n
twodimensional vectors, in
dicating each a
seat
, i.e., position in the classroom’s
x
and
y
coordinates.
Let us assume that, as happens in M104, these
n
points are
fixed
(i.e.,
they are given as an input and are not a variable that we want to find)
as the tables can’t be moved. Each seat is then represented by a vector
(
a
i
,b
i
), for
i
= 1
,
2
...,n
.
Given
k <n
students, where should they sit? We seek a subset of cardi
nality
k
of the set
N
of
n
seats, or a vector of
n
variables, each variable
x
i
tells us whether seat
i
has a student or not.
In order to avoid cheating, we want to determine the placement that
maximizes the minimum (Euclidean) distance between pairs of seats
where both seats have a student. In other words, in every assignment
S
⊂
N
=
{
1
,
2
...,n
}
a pair of students will have minimum distance.
This minimum distance is a function
d
(
S
). We want to find the assign
ment
S
that maximizes
d
(
S
).
Then solve it for M104, on four instances where
k
is 10, 15, 20, and 25 re
spectively. Approximate the position of each seat in M104. Finally, represent
graphically the choices for
k
= 10 and
k
= 15.
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 Fall '08
 Staff
 Math, Vectors, Optimization, M104

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