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2000-2013_mcm美国大学生数学建模竞赛原版题目.doc
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2000-2013_mcm美国大学生数学建模竞赛原版题目
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2000 Mathematical Contest in Modeling
The Problems
Problem A: Air traffic Control
Problem B: Radio Channel Assignments
Problem A Air traffic Control
Dedicated to the memory of Dr. Robert Machol, former chief scientist of the Federal Aviation
Agency
To improve safety and reduce air traffic controller workload, the Federal Aviation Agency
(FAA) is considering adding software to the air traffic control system that would automatically
detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at
the FAA has posed the following problems.
Requirement A: Given two airplanes flying in space, when should the air traffic controller
consider the objects to be too close and to require intervention?
Requirement B: An airspace sector is the section of three-dimensional airspace that one air
traffic controller controls. Given any airspace sector, how do we measure how complex it is
from an air traffic workload perspective? To what extent is complexity determined by the
number of aircraft simultaneously passing through that sector (1) at any one instant? (2)
during any given interval of time?(3) during a particular time of day? How does the number
of potential conflicts arising during those periods affect complexity?
Does the presence of additional software tools to automatically predict conflicts and alert the
controller reduce or add to this complexity?
In addition to the guidelines for your report, write a summary (no more than two pages) that
the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your
conclusions.
Problem B
Radio Channel Assignments
We seek to model the assignment of radio channels to a symmetric network of transmitter
locations over a large planar area, so as to avoid interference. One basic approach is to
partition the region into regular hexagons in a grid (honeycomb-style), as shown in Figure 1,
where a transmitter is located at the center of each hexagon.
Figure 1
An interval of the frequency spectrum is to be allotted for transmitter frequencies. The
interval will be divided into regularly spaced channels, which we represent by integers 1, 2,
3, ... . Each transmitter will be assigned one positive integer channel. The same channel can
be used at many locations, provided that interference from nearby transmitters is avoided.
Our goal is to minimize the width of the interval in the frequency spectrum that is needed to
assign channels subject to some constraints. This is achieved with the concept of a span. The
span is the minimum, over all assignments satisfying the constraints, of the largest channel
used at any location. It is not required that every channel smaller than the span be used in
an assignment that attains the span.
Let
s
be the length of a side of one of the hexagons. We concentrate on the case that there
are two levels of interference.
Requirement A: There are several constraints on frequency assignments. First, no two
transmitters within distance 4s of each other can be given the same channel. Second, due to
spectral spreading, transmitters within distance 2
s
of each other must not be given the same
or adjacent channels: Their channels must differ by at least 2. Under these constraints, what
can we say about the span in,
Requirement B: Repeat Requirement A, assuming the grid in the example spreads
arbitrarily far in all directions.
Requirement C: Repeat Requirements A and B, except assume now more generally that
channels for transmitters within distance 2s differ by at least some given integer
k
, while
those at distance at most 4s must still differ by at least one. What can we say about the span
and about efficient strategies for designing assignments, as a function of
k
?
Requirement D: Consider generalizations of the problem, such as several levels of
interference or irregular transmitter placements. What other factors may be important to
consider?
Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining
your findings.
2001 Mathematical Contest in Modeling
The Problems
Problem A: Choosing a Bicycle Wheel
Problem B: Escaping a Hurricane's Wrath (An Ill Wind...)
Problem A: Choosing a Bicycle Wheel
Cyclists have different types of wheels they can use on their bicycles. The two basic types of
wheels are those constructed using wire spokes and those constructed of a solid disk (see
Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid
wheel is never used on the front for a road race but can be used on the rear of the bike.
Professional cyclists look at a racecourse and make an educated guess as to what kind of
wheels should be used. The decision is based on the number and steepness of the hills, the
weather, wind speed, the competition, and other considerations. The director sportif of your
favorite team would like to have a better system in place and has asked your team for
information to help determine what kind of wheel should be used for a given course.
Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the
right.
The director sportif needs specific information to help make a decision and has asked your
team to accomplish the tasks listed below. For each of the tasks assume that the same
spoked wheel will always be used on the front but there is a choice of wheels for the rear.
� Task 1. Provide a table giving the wind speed at which the power required for a solid
rear wheel is less than for a spoked rear wheel. The table should include the wind
speeds for different road grades starting from zero percent to ten percent in one
percent increments. (Road grade is defined to be the ratio of the total rise of a hill
divided by the length of the road. If the hill is viewed as a triangle, the grade is the
sine of the angle at the bottom of the hill.) A rider starts at the bottom of the hill at a
speed of 45 kph, and the deceleration of the rider is proportional to the road grade. A
rider will lose about 8 kph for a five percent grade over 100 meters.
� Task 2. Provide an example of how the table could be used for a specific time trial
course.
� Task 3. Determine if the table is an adequate means for deciding on the wheel
configuration and offer other suggestions as to how to make this decision.
Problem B: Escaping a Hurricane's Wrath (An Ill Wind...)
Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in
1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which
is the principal route going inland from Charleston to the relatively safe haven of Columbia in
the center of the state. What is normally an easy two-hour drive took up to 18 hours to
complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north
and spared the state this time, but the public outcry is forcing state officials to find ways to
avoid a repeat of this traffic nightmare.
The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so
that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston
to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the
South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading
inland from Myrtle Beach and Hilton Head is also planned.
A simplified map of South Carolina is shown. Charleston has approximately 500,000 people,
Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along
the rest of the coastal strip. (More accurate data, if sought, are widely available.)
The interstates have two lanes of traffic in each direction except in the metropolitan areas
where they have three. Columbia, another metro area of around 500,000 people, does not
have sufficient hotel space to accommodate the evacuees (including some coming from
farther north by other routes), so some traffic continues outbound on I-26 towards
Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. In 1999, traffic leaving
Columbia going northwest was moving only very slowly. Construct a model for the problem to
investigate what strategies may reduce the congestion observed in 1999. Here are the
questions that need to be addressed:
1. Under what conditions does the plan for turning the two coastal-bound lanes of I-26
into two lanes of Columbia-bound traffic, essentially turning the entire I-26 into
one-way traffic, significantly improve evacuation traffic flow?
2. In 1999, the simultaneous evacuation of the state's entire coastal region was ordered.
Would the evacuation traffic flow improve under an alternative strategy that staggers
the evacuation, perhaps county-by-county over some time period consistent with the
pattern of how hurricanes affect the coast?
3. Several smaller highways besides I-26 extend inland from the coast. Under what
conditions would it improve evacuation flow to turn around traffic on these?
4. What effect would it have on evacuation flow to establish more temporary shelters in
Columbia, to reduce the traffic leaving Columbia?
5. In 1999, many families leaving the coast brought along their boats, campers, and
motor homes. Many drove all of their cars. Under what conditions should there be
restrictions on vehicle types or numbers of vehicles brought in order to guarantee
timely evacuation?
6. It has been suggested that in 1999 some of the coastal residents of Georgia and
Florida, who were fleeing the earlier predicted landfalls of Hurricane Floyd to the
south, came up I-95 and compounded the traffic problems. How big an impact can
they have on the evacuation traffic flow?
Clearly identify what measures of performance are used to compare
strategies. Required: Prepare a short newspaper article, not to exceed two
pages, explaining the results and conclusions of your study to the public.
Clearly identify what measures of performance are used to compare strategies.
Required: Prepare a short newspaper article, not to exceed two pages, explaining the
results and conclusions of your study to the public.
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