1985~2015年美国大学生数学建模竞赛题目集锦.docx

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1、19852015年美国大学生数学建模竞赛题目集锦目录1985 MCM A: Animal Populations31985 MCM B: Strategic Reserve Management31986 MCM A: Hydrographic Data41986 MCM B: Emergency-Facilities Location41987 MCM A: The Salt Storage Problem51987 MCM B: Parking Lot Design51988 MCM A: The Drug Runner Problem51988 MCM B: Packing Railro

2、ad Flatcars61989 MCM A: The Midge Classification Problem61989 MCM B: Aircraft Queueing61990 MCM A: The Brain-Drug Problem61990 MCM B: Snowplow Routing71991 MCM A: Water Tank Flow81991 MCM B: The Steiner Tree Problem81992 MCM A: Air-Traffic-Control Radar Power81992 MCM B: Emergency Power Restoration9

3、1993 MCM A: Optimal Composting101993 MCM B: Coal-Tipple Operations111994 MCM A: Concrete Slab Floors111994 MCM B: Network Design121995 MCM A: Helix Construction131995 MCM B: Faculty Compensation131996 MCM A: Submarine Tracking131996 MCM B: Paper Judging131997 MCM A: The Velociraptor Problem141997 MC

4、M B: Mix Well for Fruitful Discussions151998 MCM A: MRI Scanners161998 MCM B: Grade Inflation171999 MCM A: Deep Impact171999 MCM B: Unlawful Assembly182000 MCM A: Air Traffic Control182000 MCM B: Radio Channel Assignments192001 MCM A: Choosing a Bicycle Wheel202001 MCM B: Escaping a Hurricanes Wrath

5、 (An Ill Wind.)212002 MCM A: Wind and Waterspray232002 MCM B: Airline Overbooking232003 MCM A: The Stunt Person242003 MCM B: Gamma Knife Treatment Planning242004 MCM A: Are Fingerprints Unique?252004 MCM B: A Faster QuickPass System252005 MCM A: Flood Planning262005 MCM B: Tollbooths262006 MCM A: Po

6、sitioning and Moving Sprinkler Systems for Irrigation272006 MCM B: Wheel Chair Access at Airports282007 MCM A: Gerrymandering292007 MCM B: The Airplane Seating Problem292008 MCM A: Take a Bath302008 MCM B: Creating Sudoku Puzzles302009 MCM A: Designing a Traffic Circle302009 MCM B: Energy and the Ce

7、ll Phone302010 MCM A: The Sweet Spot322010 MCM B: Criminology322011 MCM A: Snowboard Course332011 MCM B: Repeater Coordination332012 MCM A: The Leaves of a Tree332012 MCM B: Camping along the Big Long River342013 MCM A: The Ultimate Brownie Pan342013 MCM B: Water, Water, Everywhere352014 MCM A: The

8、Keep-Right-Except-To-Pass Rule352014 MCM B: College Coaching Legends362015 MCM A: Eradicating Ebola362015 MCM B: Searching for a lost plane361985 MCM A: Animal PopulationsChoose a fish or mammal for which appropriate data are available to model it accurately. Model the animals natural interactions w

9、ith its environment by expressing population levels of different groups in terms of the significant parameters of the environment. Then adjust the model to account for harvesting in a form consistent with the actual method by which the animal is harvested. Include any outside constraints imposed by

10、food or space limitations that are supported by the data.Consider the value of the various quantities involved, the number harvested, and the population size itself, in order to devise a numerical quantity that represents the overall value of the harvest. Find a harvesting policy in terms of populat

11、ion size and time that optimizes the value of the harvest over a long period of time. Check that the policy optimizes that value over a realistic range of environmental conditions.1985 MCM B: Strategic Reserve ManagementCobalt, which is not produced in the US, is essential to a number of industries.

12、 (Defense accounted for 17% of the cobalt production in 1979.) Most cobalt comes from central Africa, a politically unstable region. The Strategic and Critical Materials Stockpiling Act of 1946 requires a cobalt reserve that will carry the US through a three-year war. The government built up a stock

13、pile in the 1950s, sold most of it off in the early 1970s, and then decided to build it up again in the late 1970s, with a stockpile goal of 85.4 million pounds. About half of this stockpile had been acquired by 1982.Build a mathematical model for managing a stockpile of the strategic metal cobalt.

14、You will need to consider such questions as: How big should the stockpile be? At what rate should it be acquired? What is a reasonable price to pay for the metal?You will also want to consider such questions as: At what point should the stockpile be drawn down? At what rate should it be drawn down?

15、At what price is it reasonable to sell the metal? How should it be allocated?Useful Information on CobaltThe government has projected a need ot 25 million pounds of cobalt in 1985.The U.S. has about 100 million pounds of proven cobalt deposits. Production becomes economically feasible when the price

16、 reaches $22/lb (as occurred in 1981). It takes four years to get operations rolling, and thsn six million pounds per year can be produced.In 1980, 1.2 million pounds of cobalt were recycled, 7% of total consumption.1986 MCM A: Hydrographic DataThe table below gives the depth Z of water in feet for

17、surface points with rectangular coordinates X, Y in yards table of 14 data points omitted. The depth measurements were taken at low tide. Your ship has a draft of five feet. What region should you avoid within the rectangle (75,200) x (-50, 150)?XYZ129.07.54140.0141.58108.528.0688.0147.08185.522.561

18、95.0137.58105.585.58157.5-6.59107.5-81.0977.03.08162.0-66.59162.084.04117.5-35.591986 MCM B: Emergency-Facilities LocationThe township of Rio Rancho has hitherto not had its own emergency facilities. It has secured funds to erect two emergency facilities in 1986, each of which will combine ambulance

19、, fire, and police services. Figure 1 indicates the demand figure omitted, or number of emergencies per square block, for 1985. The “L” region in the north is an obstacle, while the rectangle in the south is a part with shallow pond. It takes an emergency vehicle an average of 15 seconds to go one b

20、lock in the N-S direction and 20 seconds in the E-W direction. Your task is to locate the two facilities so as to minimize the total response time. Assume that the demand is concentrated at the center of the block and that the facilities will be located on corners. Assume that the demand is uniforml

21、y distributed on the streets bordering each block and that the facilities may be located anywhere on the streets.1987 MCM A: The Salt Storage ProblemFor approximately 15 years, a Midwestern state has stored salt used on roads in the winter in circular domes. Figure 1 shows how salt has been stored i

22、n the past. The salt is brought into and removed from the domes by driving front-end loaders up ramps of salt leading into the domes. The salt is piled 25 to 30 ft high, using the buckets on the front-end loaders.Recently, a panel determined that this practice is unsafe. If the front-end loader gets

23、 too close to the edge of the salt pile, the salt might shift, and the loader could be thrown against the retaining walls that reinforce the dome. The panel recommended that if the salt is to be piled with the use of the loaders, then the piles should be restricted to a matimum height of 15 ft.Const

24、ruct a mathematical model for this situation and find a recommended maximum height for salt in the domes.1987 MCM B: Parking Lot DesignThe owner of a paved, 100 by 200 , corner parking lot in a New England town hires you to design the layout, that is, to design how the lines are to be painted.You re

25、alize that squeezing as many cars into the lot as possible leads to right-angle parking with the cars aligned side by side. However, inexperienced drivers have difficulty parking their cars this way, which can give rise to expensive insurance claims. To reduce the likelihood of damage to parked vehi

26、cles, the owner might then have to hire expert drivers for valet parking.On the other hand, most drivers seem to have little difficulty in parking in one attempt if there is a large enough turning radius from the access lane. Of course, the wider the access lane, the fewer cars can be accommodated i

27、n the lot, leading to less revenue for the parking lot owner.1988 MCM A: The Drug Runner ProblemTwo listening posts 5.43 miles apart pick up a brief radio signal. The sensing devices were oriented at 110 degrees and 119 degrees, respectively, when the signal was detected; and they are accurate to wi

28、thin 2 degrees. The signal came from a region of active drug exchange, and it is inferred that there is a powerboat waiting for someone to pick up drugs. it is dusk, the weather is calm, and there are no currents. A small helicopter leaves from Post 1 and is able to fly accurately along the 110 degr

29、ee angle direction. The helicopters speed is three times the speed of the boat. The helicopter will be heard when it gets within 500 ft of the boat. This helicopter has only one detection device, a searchlight. At 200 ft, it can just illuminate a circular region with a radius of 25 ft. Develop an op

30、timal search method for the helicopter. Use a 95% confidence level in your calculations.1988 MCM B: Packing Railroad FlatcarsTwo railroad flatcars are to be loaded with seven types of packing crates. The crates have the same width and height but varying thickness (t, in cm) and weight (w, in kg). Ta

31、ble 1 gives, for each crate, the thickness, weight, and number available table omitted. Each car has 10.2 meters of length available for packing the crates (like slices of toast) and can carry up to 40 metric tons. There is a special constraint on the total number of C_5, C_6, and C_7 crates because

32、 of a subsequent local trucking restriction: The total space (thickness) occupied by these crates must not exceed 302.7 cm. Load the two flatcars (see Figure 1) so as to minimize the wasted floor space figure omitted.1989 MCM A: The Midge Classification ProblemTwo species of midges, Af and Apf, have

33、 been identified by biologists Grogan and Wirth on the basis of antenna and wing length (see Figure 1). It is important to be able to classify a specimen as Af of Apf, given the antenna and wing length.1. Given a midge that you know is species Af or Apf, how would you go about classifying it?2. Appl

34、y your method to three specimens with (antenna, wing) lengths (1.24,1.80),(1.28,1.84),(1.40,2.04).3. Assume that the species is a valuable pollinator and species Apf is a carrier of a debilitating disease. Would you modify your classification scheme and if so, how?1989 MCM B: Aircraft QueueingA comm

35、on procedure at airports is to assign aircraft (A/C) to runways on a first-come-first-served basis. That is, as soon as an A/C is ready to leave the gate (“push-back”), the pilot calls ground control and is added to the queue. Suppose that a control tower has access to a fast online database with th

36、e following information for each A/C: the time it is scheduled for pushback; the time it actually pushes back; the number of passengers who are scheduled to make a connection at the next stop, as well as the time to make that connection; and the schedule time of arrival at its next stop Assume that

37、there are seven types of A/C with passenger capacities varying from 100 to 400 in steps of 50. Develop and analyze a mathematical model that takes into account both the travelers and airlines satisfaction.1990 MCM A: The Brain-Drug ProblemResearches on brain disorders test the effects of the new med

38、ical drugs for example, dopamine against Parkinsons disease with intracerebral injections. To this end, they must estimate the size and the sape of the spatial distribution of the drug after the injection, in order to estimate accurately the region of the brain that the drug has affected.The researc

39、h data consist of the measurements of the amounts of drug in each of 50 cylindrical tissue samples (see Figure 1 and Table 1). Each cylinder has length 0.76 mm and diameter 0.66 mm. The centers of the parallel cylinders lie on a grid with mesh 1mm X 0.76mm X 1mm, so that the sylinders touch one anot

40、her on their circular bases but not along their sides, as shown in the accompanying figure. The injection was made near the center of the cylinder with the highest scintillation count. Naturally, one expects that there is a drug also between the cylinders and outside the region covered by the sample

41、s.Estimate the distribution in the region affected by the drug.One unit represents a scintillation count, or 4.753e-13 mole of dopamine. For example, the table shows that the middle rear sylinder contails 28353 units.Table 1. Amounts of drug in each of 50 cylindrical tissue samples.Rear vertical sec

42、tion164442132041418848070221441151583522091230272835313138681789212602092111731727213130337651715453Front vertical section16332443224316671240556098104823221371553119742478533544411431149603182301294206110362581881990 MCM B: Snowplow RoutingThe solid lines of the map (see Figure 1) represent paved t

43、wo-lane county roads in a snow removal district in Wicomico County, Maryland figure omitted. The broken lines are state highways. After a snowfall, two plow-trucks are dispatched from a garage that is about 4 miles west of each of the two points (*) marked on the map. Find an efficient way to use th

44、e two trucks to sweep snow from the county roads. The trucks may use the state highways to access the county roads. Assume that the trucks neither break down nor get stuck and that the road intersections require no special plowing techniques.1991 MCM A: Water Tank FlowSome state water-right agencies

45、 require from communities data on the rate of water use, in gallons per hour, and the total amount of water used each day. Many communities do not have equipment to measure the flow of water in or out of the municipal tank. Instead, they can measure only the level of water in the tank, within 0.5% a

46、ccuracy, every hour. More importantly, whenever the level in the tank drops below some minimum level L, a pump fills the tank up to the maximum level, H; however, there is no measurement of the pump flow either. Thus, one cannot readily relate the level in the tank to the amount of water used while

47、the pump is working, which occurs once or twice per day, for a couple of hours each time. Estimate the flow out of the tank f(t) at all times, even when the pump is working, and estimate the total amount of water used during the day. Table 1 gives real data, from an actual small town, for one day ta

48、ble omitted. The table gives the time, in, since the first measurement, and the level of water in the tank, in hundredths of a foot. For example, after 3316 seconds, the depth of water in the tank reached 31.10 feet. The tank is a vertical circular cylinder, with a height of 40 feet and a diameter of 57 feet. Usually, the pump starts filling the tank when t