Math 441
Assignment #2
Due Friday Sept 25, 2014
1. Read the Case Study for Vista Properties which is on the website. It is bit hard to understand
in parts but clear enough when you think about it. Obey the boss!
a) What LP model do you propose (without including the option)? Does the assumption of
linearity raise any doubts about this model? (I obtained an answer of $2,354,616. for the
objective function)
b) Explain why the Option should be exercised. Thus compare the solution without the option
from (a) with the solution with option exercised.
c) What would it cost to allocate 250 square feet for an ice cream stand for Mr. Wassar’s
nephew?
d) What is the benefit/cost of the individual store maximum constraints? In other words, why
are they present and how do they affect the overall profitability.
2. Integrality Gap.
One can consider two problems
max c · x
Ax ≤ b
x ≥0
max c · x
Ax ≤ b
x ≥0
xi ∈ Z for i ∈ I
Let z ∗ denote the optimal value of c · x for the LP on the left. Let z int denote the optimal value
c · x for the mixed integer LP on the right. Explain why z ∗ ≥ z int . We say that the integrality gap
is z ∗ − z int . It is in general difficult to solve the mixed integer LP on the right so we are happy to
have a small inegrality gap.
In general we report the integrality gap as I have described but sometimes it is better to express
it as a fraction of z ∗ (or indeed z int ) since the relative error is often of most interest.
a) For the hiring/firing problem as given consider changing the problem so that hiring costs
$ 600 and firing costs $ 3000. Determine the integrality gap in this case (you are in the
fortunate position of considering a mixed integer LP which is reatively easy to solve) and also
report it as a fraction of z ∗ .
b) Consider solving the new problem where we have workers on regular time producing 200
rather than 20 units per month. The other data will be the same as for the original problem
(don’t incorporate the changes proposed in a)) This will involve editing your input file. Offer
an explanation why the inegrality gap might increase for this new problem. Determine the
integrality gap by running the LP and the mixed integer LP. Also report it as a fraction of
z∗.