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2007 MCM A: Gerrymandering
Gerrymandering
The
United
States
Constitution
provides
that
the
House
of
Representatives
shall
be
composed
of
some
number
(currently 435) of individuals who are
elected from each state in proportion to the
state’s population r
elative to that
of the country as a whole. While this
provides a way of determining how many
representatives each state will have, it says
nothing
about
how
the
district
represented
by
a
particular
representative
shall
be
determined
geographically.
This
oversight
has led to egregious (at least some
people think so, usually not the incumbent)
district shapes that look “unnatural”
by some standards.
Hence the following question: Suppose
you were given the opportunity to draw
congressional districts for a state. How
would you do so as a purely “baseline”
exercise to create the “simplest” shapes for all
the districts in a state? The
rules
include only that each district in the state must
contain the same population. The definition of
“simple” is up
to you; but you need to
make a convincing argument to voters in the state
that your solution is fair. As an application
of your method, draw geographically
simple congressional districts for the state of
New York.
2007 MCM B: The
Airplane Seating Problem
Airlines are free to seat passengers
waiting to board an aircraft in any order
whatsoever. It has become customary to
seat
passengers
with
special
needs
first,
followed
by
first-class
passengers
(who
sit
at
the
front
of
the
plane).
Then
coach
and business-class
passengers are seated by groups of rows, beginning
with the row at the back of the plane and
proceeding
forward.
Apart from consideration
of the passengers’ wait time, from the airline’s
point of view, time is money, and boarding
time is best minimized. The plane makes
money for the airline only when it is in motion,
and long boarding times limit the
number of trips that a plane can make
in a day.
The development of larger
planes, such as the Airbus A380 (800 passengers),
accentuate the problem of minimizing boarding
(and deboarding) time.
Devise
and
compare
procedures
for
boarding
and
deboarding
planes
with
varying
numbers
of
passengers:
small
(85
–
210),
midsize
(210
–
330), and
large (450
–
800).
Prepare
an
executive
summary,
not to
exceed two single-
spaced
pages,
in
which
you
set
out
your
conclusions
to an
audience
of airline
executives, gate agents, and flight crews.
An article appeared in
the NY Times Nov 14, 2006 addressing procedures
currently being followed and the importance to
the
airline
of
finding
better
solutions.
The
article
can
be
seen
at:
/2006/11/14/business/
2007 ICM: Organ Transplant:
The Kidney Exchange Problem
Transplant
Network:
Despite
the
continuing
and
dramatic
advances
in
medicine
and
health
technology,
the
demand
for
organs
for
transplantation
drastically
exceeds
the
number
of
donors.
To
help
this
situation,
US
Congress
passed
the
National Organ
Transplant
Act
in
1984,
establishing
the
Organ
Procurement
and
Transplantation
Network
(OPTN)
to
match
organ
donors
to
patients
with organ needs.
Even with all this organizational technology and
service in place, there are nearly 94,000
transplant
candidates in the US waiting
for an organ transplant and this number is
predicted to exceed 100,000 very soon. The average
wait time exceeds three
years
—
double that in some
areas, such as large cities. Organs for transplant
are obtained either
from a cadaver
queue or from living donors. The keys for the
effective use of the cadaver queue are cooperation
and good
communication throughout the
network. The good news is that the system is
functioning and more and more donors (alive and
deceased) are identified and used each
year with record numbers of transplants taking
place every month. The bad news is
that
the candidate list grows longer and longer. Some
people think that the current system with both
regional and national
aspects
is
headed
for
collapse
with
consequential
failures
for
some
of
the
neediest
patients.
Moreover,
fundamental
questions
remain:
Can
this
network
be
improved
and
how
do
we
improve
the
effectiveness
of
a
complex
network
like
OPTN?
Different
countries
have different
processes and policies, which of these work best?
What is the future status of the current system?
Task 1:
For a
beginning reference, read the OPTN Website
(
) with its policy
descriptions and data banks
(
/data
and
/latestData/
). Build a
mathematical model for the
generic US
transplant network(s). This model must be able to
give insight into the following: Where are the
potential
bottlenecks
for
efficient
organ
matching?
If
more
resources
were
available
for
improving
the
efficiency
of
the
donor-matching
process,
where and how could they be used? Would this
network function better if it was divided into
smaller networks (for
instance at the
state level)? And finally, can you make the system
more effective by saving and prolonging more
lives? If
so, suggest policy changes
and modify your model to reflect these
improvements.
Task
2:
Investigate the transplantation
policies used in a country other than the US. By
modifying your model from Task
1,
determine if the US policy be would improved by
implementing the procedures used in the other
country. As members of
an expert
analysis team (knowledge of public health issues
and network science) hired by Congress to perform
a study of
these questions, write a
one-page report to Congress addressing the
questions and issues of Task 1 and the information
and
possible improvements you have
discovered from your research of the different
country’s policies. Be sure to reference
how you used your models from Task 1 to
help address the issues.
Focusing on Kidney Exchange: Kidneys filter blood,
remove waste, make hormones, and produce urine.
Kidney failure can
be caused by many
different diseases and conditions. People with
end-stage kidney disease face death, dialysis (at
over
$$60,000/yr), or the hope for a
kidney transplant. A transplant can come from the
cadavers of an individual who agreed to
donate organs after death or from a
live donor. In the US, about 68,000 patients are
waiting for a kidney from a deceased
donor, while each year only 10,000 are
transplanted from cadavers and 6,000 from living
individuals (usually relatives of
the
patients).
Hence
the
median
wait
for
a
matching
kidney
is
three <
/p>
years
—
unfortunatel
y,
some
needy
patients
do
not
survive
long enough to
receive a kidney.