美国数学建模比赛历年试题Word文件下载.docx

1、determine how many boxes to usedetermine how the boxes will be stackeddetermine if any modifications to the boxes would helpgeneralize to different bined weights （stunt person & motorcycle） and different jump heights Note that, in Tomorrow Never Dies, the James Bond character on a motorcycle jumps o

2、ver a helicopter. PROBLEM B: Gamma Knife Treatment PlanningStereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tis

3、sue. Three modalities are monly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators. The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a h

4、eavy helmet. All 201 beams simultaneously intersect at the isocenter, resulting in a spherical （approximately） dose distribution at the effective dose levels. Irradiating the isocenter to deliver dose is termed a “shot.” Shots can be represented as different spheres. Four interchangeable outer colli

5、mator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. For a target volume larger than one shot, multiple shots can be used to cover the entire target. In practice, most target volumes are treated with 1 to 15 shots. The target volume i

6、s a bounded, three-dimensional digital image that usually consists of millions of points. The goal of radiosurgery is to deplete tumor cells while preserving normal structures. Since there are physical limitations and biological uncertainties involved in this therapy process, a treatment plan needs

7、to account for all those limitations and uncertainties. In general, an optimal treatment plan is designed to meet the following requirements.1.Minimize the dose gradient across the target volume. 2.Match specified isodose contours to the target volumes. 3.Match specified dose-volume constraints of t

8、he target and critical organ. 4.Minimize the integral dose to the entire volume of normal tissues or organs. 5.Constrain dose to specified normal tissue points below tolerance doses. 6.Minimize the maximum dose to critical volumes. In gamma unit treatment planning, we have the following constraints:

9、 1.Prohibit shots from protruding outside the target. 2.Prohibit shots from overlapping （to avoid hot spots）. 3.Cover the target volume with effective dosage as much as possible. But at least 90% of the target volume must be covered by shots. 4.Use as few shots as possible. Your tasks are to formula

10、te the optimal treatment planning for a gamma knife unit as a sphere-packing problem, and propose an algorithm to find a solution. While designing your algorithm, you must keep in mind that your algorithm must be reasonably efficient. 2002 Contest ProblemsProblem AAuthors: Tjalling YpmaTitle: Wind a

11、nd WatersprayAn ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer （which measures wind speed an

12、d direction） located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray f

13、alls outside the pool area.Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.Problem B Bill Fox and Rich West Airline OverbookingYoure all packed and ready to go on a trip to visit your best friend

14、in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat.Historically, airlines know that only a certain percentage of passengers who have made reservat

15、ions on a particular flight will actually take that flight. Consequently, most airlines overbook-that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bum

16、ped and thus unable to take the flight for which they had reservations.Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.Consider the overbooking issue in

17、light of the current situation:Less flights by airlines from point A to point BHeightened security at and around airportsPassengers fearLoss of billions of dollars in revenue by airlines to dateBuild a mathematical model that examines the effects that different overbooking schemes have on the revenu

18、e received by an airline pany in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the panys revenue is maximized. Insure that your model reflects the issues above, and consider alternatives for handling bumped p

19、assengers. Additionally, write a short memorandum to the airlines CEO summarizing your findings and analysis.MCM2000Problem A Air traffic ControlTo improve safety and reduce air traffic controller workload, the Federal Aviation Agency （FAA） is considering adding software to the air traffic control s

20、ystem that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA r traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA h

21、as posed the following problemsRequirement A: Given two airplanes flying in space, when should the air traffic controller ld 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

22、 one air traffic controller controls. Given any airspace sector, how we measure how plex it is from an air traffic workload perspective? To what extent is plexity determined by the number of we measure how plex it is from an air traffic workload perspective? To what extent is plexity determined by t

23、he number of aircraft simultaneously passing through that sector （1） at any one instant? （2） During any given interval of time? （3） During particular time of day? How does the number of potential conflicts arising during those periods affect plexity?Does the presence of additional software tools to

24、automatically predict conflicts and alert the controller reduce or add to this plexity?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 conclusionsProblem B Radio Channel Ass

25、ignmentsWe 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 （honeyb-style）, as shown in Figure 1, where a transmitter is

26、located at the center of each hexagon. 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 sam

27、e 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

28、 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 tha

29、t there are two levels of interference There are several constraints on frequency assignments. First, no two transmitters within distance of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjace

30、nt channels: Their channels must differ by at least 2. Under these constraints, what can we say about the span in, 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 chann

31、els for transmitters within distance differ by at least some given integer k, while those at distance at most 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 irregul