健壮的意思-sadly怎么读
2014 MCM Problems
PROBLEM A: The
Keep-Right-Except-To-Pass Rule
In countries
where driving automobiles on the right is the rule
(that is, USA,
China and most other countries
except for Great Britain, Australia, and some
former British colonies), multi-lane freeways
often employ
a rule that
requires drivers
to drive in the right-most lane unless they are
passing another vehicle, in which case they
move one lane to the left,
pass, and return to
their former travel lane.
Build and
analyze a mathematical model to analyze the
performance of this rule
in light and heavy
traffic. You may wish to examine tradeoffs between
traffic
flow and safety, the role of under- or
over-posted speed limits (that is, speed
limits that are too low or too high), andor
other factors that may not be
explicitly
called out in this problem statement. Is this rule
effective in promoting
better traffic flow? If
not, suggest and analyze alternatives (to include
possibly
no rule of this kind at all) that
might promote greater traffic flow, safety, andor
other factors that you deem important.
In
countries where driving automobiles on the left is
the norm, argue whether
or not your solution
can be carried over with a simple change of
orientation, or
would additional requirements
be needed.
Lastly, the rule as stated above
relies upon human judgment for compliance. If
vehicle transportation on the same roadway was
fully under the control of an
intelligent
system – either part of the road network or
imbedded in the design
of all vehicles using
the roadway – to what extent would this change the
results
of your earlier analysis?
PROBLEM
B: College Coaching Legends
Sports
Illustrated
, a magazine for sports
enthusiasts, is looking for the ―best all
time
college coach‖ male or female for the previous
century. Build a
mathematical model to choose
the
best
college coach or coaches (past or
present) from among either male or female
coaches in such sports as college
hockey or
field hockey, football, baseball or softball,
basketball, or soccer. Does
it make a
difference which time line horizon that you use in
your analysis, i.e.,
does coaching in 1913
differ from coaching in 2013? Clearly articulate
your
metrics for assessment. Discuss how your
model can be applied in general
across both
genders and all possible sports. Present your
model’s top 5 coaches
in each of 3 different
sports.
In addition to the MCM format and
requirements, prepare a 1-2 page article
for
Sports Illustrated
that explains your
results and includes a non-technical
explanation of your mathematical model that
sports fans will understand.
2013 MCM
Problems
PROBLEM A: The Ultimate Brownie
Pan
When baking in a rectangular pan
heat is concentrated in the 4 corners and the
product gets overcooked at the corners (and to
a lesser extent at the edges). In
a round pan
the heat is distributed evenly over the entire
outer edge and the
product is not overcooked
at the edges. However, since most ovens are
rectangular in shape using round pans is not
efficient with respect to using the
space in
an oven. Develop a model to show the distribution
of heat across the
outer edge of a pan for
pans of different shapes - rectangular to circular
and
other shapes in between.
Assume
1. A width to length ratio of
W
L for
the oven which is rectangular in shape.
2.
Each pan must have an area of
A
.
3.
Initially two racks in the oven, evenly spaced.
Develop a model that can be used to
select the best type of pan (shape) under
the
following conditions:
1. Maximize number of
pans that can fit in the oven (N)
2. Maximize
even distribution of heat (H) for the pan
3.
Optimize a combination of conditions (1) and (2)
where weights p and (1-
p
)
are
assigned to illustrate how the results vary with
different values
of
WL
and
p
.
In addition to your MCM formatted
solution, prepare a one to two page
advertising sheet for the new Brownie Gourmet
Magazine highlighting your
design and results.
PROBLEM B: Water, Water, Everywhere
Fresh
water is the limiting constraint for development
in much of the world.
Build a mathematical
model for determining an effective, feasible, and
cost-efficient water strategy for 2013 to meet
the projected water needs of
[pick one country
from the list below] in 2025, and identify the
best water
strategy. In particular, your
mathematical model must address storage and
movement; de-salinization; and conservation.
If possible, use your model to
discuss the
economic, physical, and environmental implications
of your strategy.
Provide a non-technical
position paper to governmental leadership
outlining
your approach, its feasibility and
costs, and why it is the ―best water strategy
choice.‖
Countries: United States, China,
Russia, Egypt, or Saudi Arabia
2012 MCM
Problems
PROBLEM A: The Leaves of a Tree
weight of the leaves (or for that
matter any other parts of the tree)? How might
one classify leaves? Build a mathematical
model to describe and classify leaves.
Consider and answer the following:
? Why
do leaves have the various shapes that they have?
? Do the shapes ―minimize‖ overlapping
individual shadows that are cast, so as
to
maximize exposure? Does the distribution of leaves
within the ―volume‖ of
the tree and its
branches effect the shape?
? Speaking of
profiles, is leaf shape (general characteristics)
related to tree
profilebranching structure?
? How would you estimate the leaf mass of
a tree? Is there a correlation
between the
leaf mass and the size characteristics of the tree
(height, mass,
volume defined by the profile)?
In addition to your one page summary sheet
prepare a one page letter to an
editor of a
scientific journal outlining your key findings.
PROBLEM B: Camping along the Big
Long River
Visitors to the Big Long River (225
miles) can enjoy scenic views and exciting
white water rapids. The river is inaccessible
to hikers, so the only way to enjoy
it is to
take a river trip that requires several days of
camping. River trips all start
at First Launch
and exit the river at Final Exit, 225 miles
downstream.
Passengers take either oar-
powered rubber rafts, which travel on average 4
mph or motorized boats, which travel on
average 8 mph. The trips range from 6
to 18
nights of camping on the river, start to finish..
The government agency
responsible for managing
this river wants every trip to enjoy a wilderness
experience, with minimal contact with other
groups of boats on the river.
Currently,
X
trips travel down the Big Long River
each year during a six month
period (the rest
of the year it is too cold for river trips). There
are
Y
camp sites
on the Big Long
River, distributed fairly uniformly throughout the
river corridor.
Given the rise in popularity
of river rafting, the park managers have been
asked
to allow more trips to travel down the
river. They want to determine how they
might
schedule an optimal mix of trips, of varying
duration (measured in nights
on the river) and
propulsion (motor or oar) that will utilize the
campsites in the
best way possible. In
other words, how many more boat trips could be
added to
the Big Long River’s rafting season?
The river managers have hired you to
advise
them on ways in which to develop the best schedule
and on ways in
which to determine the carrying
capacity of the river, remembering that no two
sets of campers can occupy the same site at
the same time. In addition to your
one page
summary sheet, prepare a one page memo to the
managers of the
river describing your key
findings.
2011 MCM Problems
PROBLEM
A: Snowboard Course
Determine the shape
of a snowboard course (currently known as a
―halfpipe‖)
to maximize the production of
―vertical air‖ by a skilled snowboarder.
Tailor the shape to optimize other
possible requirements, such as maximum
twist
in the air.
What tradeoffs may be
required to develop a ―practical‖ course?
PROBLEM B: Repeater Coordination
The
VHF radio spectrum involves line-of-sight
transmission and reception. This
limitation
can be overcome by ―repeaters,‖ which pick up weak
signals, amplify
them, and retransmit them on
a different frequency. Thus, using a repeater,
low-power users (such as mobile stations) can
communicate with one another
in situations
where direct user-to-user contact would not be
possible. However,
repeaters can interfere
with one another unless they are far enough apart
or
transmit on sufficiently separated
frequencies.
In addition to geographical
separation, the ―continuous tone-coded squelch
system‖ (CTCSS), sometimes nicknamed ―private
line‖ (PL), technology can be
used to mitigate
interference problems. This system associates to
each
repeater a separate subaudible tone that
is transmitted by all users who wish to
communicate through that repeater. The
repeater responds only to received
signals
with its specific PL tone. With this system, two
nearby repeaters can
share the same frequency
pair (for receive and transmit); so more repeaters
(and hence more users) can be accommodated in
a particular area.
For a circular
flat area of radius 40 miles radius, determine the
minimum
number of repeaters necessary to
accommodate 1,000 simultaneous users.
Assume
that the spectrum available is 145 to 148 MHz, the
transmitter
frequency in a repeater is either
600 kHz above or 600 kHz below the receiver
frequency, and there are 54 different PL tones
available.
How does your solution change
if there are 10,000 users?
Discuss the
case where there might be defects in line-of-sight
propagation
caused by mountainous areas.
2010 MCM Problems
PROBLEM A: The Sweet Spot
Explain the
―sweet spot‖ on a baseball bat.
Every hitter knows that there is a spot on the
fat part of a baseball bat where maximum power
is transferred to the ball when hit. Why isn’t
this spot at the end of the bat? A simple
explanation based on torque might seem to
identify the end of the bat as the sweet spot, but
this is known to be empirically incorrect.
Develop a model that helps explain this empirical
finding.
Some players believe
that ―corking‖ a bat (hollowing out a cylinder in
the head of the bat and
filling it with cork
or rubber, then replacing a wood cap) enhances the
―sweet spot‖ effect.
Augment your model to
confirm or deny this effect. Does this explain why
Major League
Baseball prohibits ―corking‖?
Does the material out of which the bat
is constructed matter? That is, does this model
predict
different behavior for wood (usually
ash) or metal (usually aluminum) bats? Is this why
Major
League Baseball prohibits metal
bats?
PROBLEM B: Criminology
In
1981 Peter Sutcliffe was convicted of thirteen
murders and subjecting a number of other
people to vicious attacks. One of the methods
used to narrow the search for Mr. Sutcliffe was to
find a ―center of mass‖ of the locations of
the attacks. In the end, the suspect happened to
live
in the same town predicted by this
technique. Since that time, a number of more
sophisticated
techniques have been developed
to determine the ―geographical profile‖ of a
suspected serial
criminal based on the
locations of the crimes.
Your team
has been asked by a local police agency to develop
a method to aid in their
investigations of
serial criminals. The approach that you develop
should make use of at least
two different
schemes to generate a geographical profile. You
should develop a technique to
combine the
results of the different schemes and generate a
useful prediction for law
enforcement
officers. The prediction should provide some kind
of estimate or guidance about
possible
locations of the next crime based on the time and
locations of the past crime scenes. If
you
make use of any other evidence in your estimate,
you must provide specific details about
how
you incorporate the extra information. Your method
should also provide some kind of
estimate
about how reliable the estimate will be in a given
situation, including appropriate
warnings.
In addition to the required one-page
summary, your report should include an additional
two-page executive summary. The executive
summary should provide a broad overview of the
potential issues. It should provide an
overview of your approach and describe situations
when it
is an appropriate tool and situations
in which it is not an appropriate tool. The
executive
summary will be read by a chief of
police and should include technical details
appropriate to the
intended audience
2009 MCM Problems
PROBLEM A:
Designing a Traffic Circle
Many
cities and communities have traffic circles—from
large ones with many
lanes in the circle (such
as at the Arc de Triomphe in Paris and the Victory
Monument in Bangkok) to small ones with one or
two lanes in the circle. Some
of these traffic
circles position a stop sign or a yield sign on
every incoming road
that gives priority to
traffic already in the circle; some position a
yield sign in
the circle at each incoming road
to give priority to incoming traffic; and some
position a traffic light on each incoming road
(with no right turn allowed on a
red light).
Other designs may also be possible.
The
goal of this problem is to use a model to
determine how best to control
traffic flow in,
around, and out of a circle. State clearly the
objective(s) you use
in your model for making
the optimal choice as well as the factors that
affect
this choice. Include a Technical
Summary of not more than two double-spaced
pages that explains to a Traffic Engineer how
to use your model to help choose
the
appropriate flow-control method for any specific
traffic circle. That is,
summarize the
conditions under which each type of traffic-
control method
should be used. When traffic
lights are recommended, explain a method for
determining how many seconds each light
should remain green (which may
vary according
to the time of day and other factors). Illustrate
how your model
works with specific examples.
PROBLEM B: Energy and the Cell Phone
This question involves the ―energy‖
consequences of the cell phone revolution. Cell
phone
usage is mushrooming, and many people
are using cell phones and giving up their landline
telephones. What is the consequence of this in
terms of electricity use? Every cell phone comes
with a battery and a recharger.
Requirement 1
Consider the
current US, a country of about 300 million people.
Estimate from available data
the number
H
of households, with
m
members
each, that in the past were serviced by landlines.
Now, suppose that all the landlines are
replaced by cell phones; that is, each of the
m
members
of the household has a cell
phone. Model the consequences of this change for
electricity
utilization in the current US,
both during the transition and during the steady
state. The analysis
should take into account
the need for charging the batteries of the cell
phones, as well as the
fact that cell phones
do not last as long as landline phones (for
example, the cell phones get
lost and break).
Requirement 2
Consider
a second ―Pseudo US‖—a country of about 300
million people with about the same
economic
status as the current US. However, this emerging
country has neither landlines nor
cell phones.
What is the optimal way of providing phone service
to this country from an energy
perspective? Of
course, cell phones have many social consequences
and uses that landline
phones do not allow. A
discussion of the broad and hidden consequences of
having only
landlines, only cell phones, or a
mixture of the two is welcomed.
Requirement 3
Cell phones
periodically need to be recharged. However, many
people always keep their
recharger plugged in.
Additionally, many people charge their phones
every night, whether they
need to be recharged
or not. Model the energy costs of this wasteful
practice for a Pseudo US
based upon your
answer to Requirement 2. Assume that the Pseudo US
supplies electricity
from oil. Interpret your
results in terms of barrels of oil.
Requirement 4
Estimates vary on
the amount of energy that is used by various
recharger types (TV, DVR,
computer
peripherals, and so forth) when left plugged in
but not charging the device. Use
accurate data to model the energy
wasted by the current US in terms of barrels of
oil per day.
Requirement 5
Now consider population and economic growth
over the next 50 years. How might a typical
Pseudo US grow? For each 10 years for the next
50 years, predict the energy needs for
providing phone service based upon your
analysis in the first three requirements. Again,
assume electricity is provided from oil.
Interpret your predictions in term of barrels of
oil.
2008 MCM Problems
PROBLEM A: Take a Bath
Consider
the effects on land from the melting of the north
polar ice cap due to the predicted
increase in
global temperatures. Specifically, model the
effects on the coast of Florida every ten
years for the next 50 years due to the
melting, with particular attention given to large
metropolitan areas. Propose appropriate
responses to deal with this. A careful discussion
of the
data used is an important part of the
answer.
PROBLEM B: Creating Sudoku
Puzzles
Develop an algorithm to construct
Sudoku puzzles of varying difficulty. Develop
metrics to
define a difficulty level. The
algorithm and metrics should be extensible to a
varying number of
difficulty levels. You
should illustrate the algorithm with at least 4
difficulty levels. Your
algorithm should
guarantee a unique solution. Analyze the
complexity of your algorithm. Your
objective
should be to minimize the complexity of the
algorithm and meet the above
requirements.
2007 MCM Problems
PROBLEM A: 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 relative 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
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
exercise to create the
state must contain
the same population. The definition of
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.
PROBLEM 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.
Note: The 2 page executive summary is to be
included IN ADDITION to the reports required by
the contest guidelines.
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:http:
2006 MCM Problems
PROBLEM A: Positioning and Moving Sprinkler
Systems for Irrigation
There are a
wide variety of techniques available for
irrigating a field. The technologies range
from advanced drip systems to periodic
flooding. One of the systems that is used on
smaller
ranches is the use of
sprinkler
heads are put in place across fields, and they are
moved by hand at periodic intervals
to insure
that the whole field receives an adequate amount
of water. This type of irrigation
system is
cheaper and easier to maintain than other systems.
It is also flexible, allowing for use
on a
wide variety of fields and crops. The disadvantage
is that it requires a great deal of time
and
effort to move and set up the equipment at regular
intervals.
Given that this type of
irrigation system is to be used, how can it be
configured to minimize the
amount of time
required to irrigate a field that is 80 meters by
30 meters? For this task you are
asked to find
an algorithm to determine how to irrigate the
rectangular field that minimizes the
amount of
time required by a rancher to maintain the
irrigation system. One pipe set is used in
the
field. You should determine the number of
sprinklers and the spacing between sprinklers,
and you should find a schedule to move the
pipes, including where to move them.
A pipe set consists of a number of pipes that
can be connected together in a straight line. Each
pipe has a 10 cm inner diameter with rotating
spray nozzles that have a 0.6 cm inner diameter.
When put together the resulting pipe is 20
meters long. At the water source, the pressure is
420 Kilo- Pascal's and has a flow rate of 150
liters per minute. No part of the field should
receive more than 0.75 cm per hour of water,
and each part of the field should receive at least
2 centimeters of water every 4 days. The total
amount of water should be applied as uniformly
as possible
PROBLEM B: Wheel
Chair Access at Airports
One of the
frustrations with air travel is the need to fly
through multiple airports, and each stop
generally requires each traveler to change to
a different airplane. This can be especially
difficult
for people who are not able to
easily walk to a different flight's waiting area.
One of the ways
that an airline can make the
transition easier is to provide a wheel chair and
an escort to those
people who ask for help. It
is generally known well in advance which
passengers require help,
but it is not
uncommon to receive notice when a passenger first
registers at the airport. In rare
instances an
airline may not receive notice from a passenger
until just prior to landing.
Airlines are
under constant pressure to keep their costs down.
Wheel chairs wear out and are
expensive and
require maintenance. There is also a cost for
making the escorts available.
Moreover, wheel
chairs and their escorts must be constantly moved
around the airport so that
they are available
to people when their flight lands. In some large
airports the time required to
move
across the airport is nontrivial. The wheel chairs
must be stored somewhere, but space is
expensive and severely limited in an airport
terminal. Also, wheel chairs left in high traffic
areas
represent a liability risk as people try
to move around them. Finally, one of the biggest
costs is
the cost of holding a plane if
someone must wait for an escort and becomes late
for their flight.
The latter cost is
especially troubling because it can affect the
airline's average flight delay
which can lead
to fewer ticket sales as potential customers may
choose to avoid an airline.
Epsilon Airlines
has decided to ask a third party to help them
obtain a detailed analysis of the
issues and
costs of keeping and maintaining wheel chairs and
escorts available for passengers.
The airline
needs to find a way to schedule the movement of
wheel chairs throughout each day
in a cost
effective way. They also need to find and define
the costs for budget planning in both
the
short and long term.
Epsilon Airlines has
asked your consultant group to put together a bid
to help them solve their
problem. Your bid
should include an overview and analysis of the
situation to help them decide
if you fully
understand their problem. They require a detailed
description of an algorithm that
you would
like to implement which can determine where the
escorts and wheel chairs should be
and how
they should move throughout each day. The goal is
to keep the total costs as low as
possible.
Your bid is one of many that the airline will
consider. You must make a strong case as
to
why your solution is the best and show that it
will be able to handle a wide range of airports
under a variety of circumstances.
Your bid
should also include examples of how the algorithm
would work for a large (at least 4
concourses), a medium (at least two
concourses), and a small airport (one concourse)
under
high and low traffic loads. You should
determine all potential costs and balance their
respective
weights. Finally, as populations
begin to include a higher percentage of older
people who have
more time to travel but may
require more aid, your report should include
projections of
potential costs and needs in
the future with recommendations to meet future
needs.
2005 MCM Problems
PROBLEM A: Flood Planning
Lake
Murray in central South Carolina is formed by a
large earthen dam, which
was completed in 1930
for power production. Model the flooding
downstream
in the event there is a
catastrophic earthquake that breaches the dam.
Two particular questions:
Rawls Creek
is a year-round stream that flows into the Saluda
River a short
distance downriver from the dam.
How much flooding will occur in Rawls Creek
from a dam failure, and how far back will it
extend?
Could the flood be so
massive downstream that water would reach up to
the
S.C. State Capitol Building, which is on a
hill overlooking the Congaree River?
PROBLEM B: Tollbooths
Heavily-traveled
toll roads such as the Garden State Parkway ,
Interstate 95,
and so forth, are multi-lane
divided highways that are interrupted at intervals
by toll plazas. Because collecting tolls is
usually unpopular, it is desirable to
minimize
motorist annoyance by limiting the amount of
traffic disruption
caused by the toll plazas.
Commonly, a much larger number of tollbooths is
provided than the number of travel lanes
entering the toll plaza. Upon entering
the
toll plaza, the flow of vehicles fans out to the
larger number of tollbooths,
and when leaving
the toll plaza, the flow of vehicles is required
to squeeze back
down to a number of travel
lanes equal to the number of travel lanes before
the
toll plaza. Consequently, when traffic is
heavy, congestion increases upon
departure
from the toll plaza. When traffic is very heavy,
congestion also builds
at the entry to the
toll plaza because of the time required for each
vehicle to
pay the toll.
Make a
model to help you determine the optimal number of
tollbooths to deploy
in a barrier-toll plaza.
Explicitly consider the scenario where there is
exactly one
tollbooth per incoming travel
lane. Under what conditions is this more or less
effective than the current practice? Note that
the definition of
to you to determine.
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