美国数学建模MCM1985题目汇总.docx

1、美国数学建模MCM1985题目汇总1985Problem A Animal Populations Choose a fish or mammal for which appropriate data are available to model it accurately. Model the animals natural interactions with its environment by expressing population levels of different groups in terms of the significant parameters of the env

2、ironment. 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 food or space limitations that are supported by the data. Consider the value of the various quantities involved, the numb

3、er 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 population size and time that optimizes the value of the harvest over a long period of time. Check that the policy optimizes th

4、at value over a realistic range of environmental conditions.Problem B Strategic Reserve ManagementCobalt, which is not produced in the US, is essential to a number of industries. (Defense accounted for 17% of the cobalt production in 1979.) Most cobalt comes from central Africa, a politically unstab

5、le 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 stockpile 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

6、 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. You will need to consider such questions as: How big should the stockpile be? At what rate should it be acquired? What is

7、 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? At what price is it reasonable to sell the metal? How should it be allocated? 1986Problem A Hydrographic Data The table

8、below gives the depth Z of water in feet for 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)?Problem B

9、 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, fire, and police services. Figure 1 indicates the demand figure omitted, or number of emer

10、gencies 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 block in the N-S direction and 20 seconds in the E-W direction. Your task is to locate the two

11、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 uniformly distributed on the streets bordering each block and that the facilities may be located anywh

12、ere on the streets. MCM1987A题 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 in the past. The salt is brought into and removed from the domes by driving front-end loaders up

13、 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 too close to the edge of the salt pile, the salt might shift, and the loader could be thrown

14、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 maximum height of 15 ft. Construct a mathematical model for this situation and find a recommended maximum height for salt i

15、n the domes. 盐的存贮美国中西部一个州把冬天用来洒在马路上的盐存贮在一个球顶仓库里大约有15年了。图87A-1表示在过去15年中盐是怎么存贮的*通过驾驶铲斗车在由盐铺成的坡道上进出仓里并利用铲斗车上的铲子把盐装进仓里或从仓里取出来。 最近，一个小组确定这种做法是不安全的。如果铲斗车太靠近盐堆的顶端，盐就要滑动，而铲斗车就耍翻到为加固仓库而筑的拥壁上去。小组建议， 如果盐堆是用铲斗车堆起来的，那么盐堆的最高高度不要超过15英尺。对这种情况建立一个数学模型并求得在仓库中的盐堆的最大高度。图中仓高50英尺，拥壁 高4英尺，仓的外直径103英尺，门的净空高l 9英尺9英寸，铲斗车高10英尺

16、9英寸。 Problem 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 realize that squeezing as many cars into the lot as possible leads to right-angle parking with the cars

17、 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 vehicles, the owner might then have to hire expert drivers for “valet parking”. On the other hand, most d

18、rivers 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 in the lot, leading to less revenue for the parking lot owner.MCM1988A题 The Drug Runner ProblemTwo

19、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 within 2 degrees. The signal came from a region of active drug exchange, and it is inferred that there

20、 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 degree angle direction. The helicopters speed is three times the speed of the boat. The helicopter will

21、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 optimal search method for the helicopter. Use a 95% confidence level in your calculations. 确定毒品走私船的位置相

22、距5.43哩的监听站收听到一个短暂的无线电讯号。收听到讯早的时候测向仪分别定位在 111和119处见图88A-1)，测向仪的精度为2，该讯号来自一个毒品交换活跃的地方，据推测该处有一只机动船正等着有人来取毒品。当时正 值黄昏、无风、无潮流。一架小型直升飞机离开监听站的简易机场并能精确地沿111角方向飞行。直升飞机的飞行速度是走私船的三倍。在离船500英尺时 船上能听到直升飞机的声音。直升飞机只有一种侦察仪器 -探照订。在200英尺远的地方探照灯只能照明半径为25英尺的圆域。1. 说明飞行员能找到正等着的毒品船的(最小)区域。 2. 研究一种直升飞机的最佳搜索方法。 在你的计算中要有95的精度。

23、 本题是由加州Claremont McKenna学院的J.A.Ferling提供的。这是一个分类(分组问题) 的修正简化形式。原问题和现在简化的问题都还没有一种已知的最化解法。 Problem Packing Railroad Flatcars Two 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). Tabl

24、e 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 o

25、f 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.MCM1989A题 The Midge Classification ProblemTwo species of midges, Af and Apf, have bee

26、n 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. Given a midge that you know is species Af or Apf, how would you go about classifying it? Apply your m

27、ethod to three specimens with (antenna, wing) lengths (1.24,1.80),(1.28,1.84),(1.40,2.04). 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? 蠓的分类两种蠓Af和Apf己由生物学家W.L.Grongan和W.W.Wirth(198

28、1年)根据它们的触角长 度和翼长加以区分(见图89A-1)，9只Af蠓用圆圈标记，6只Apf蠓用黑点标记。根据给出的触角长度和翼长识别出一只标本是Af还是Apf是重要的。 1. 给定一只Af或者Apf族的蝶，你如何正确地区分它属于哪一族? 2. 将你的方法用于触角长和翼长分别为(1.24，1.80)、(1.28，1.84)、(1.40，2.04)的三个标本。 3. 设Af是宝贵的传粉益虫，Apf是某种疾病的载体，是否应该修改你的分类方法，若需修改，怎么改? Problem Aircraft QueuingA common procedure at airports is to assign a

29、ircraft (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 the following information for each A/C: The

30、 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 there are seven types of A/C with passeng

31、er 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. MCM1991B题 The Steiner Tree ProblemThe cost for a communication line between two stations is proportional to the length of the line. The

32、 cost for conventional minimal spanning trees of a set of stations can often be cut by introducing phantom stations and then constructing a new Steiner tree. This device allows costs to be cut by up to 13.4% (= 1- sqrt(3/4). Moreover, a network with n stations never requires more than n-2 points to construct the cheapest Steiner tree. Two simple cases are shown in Figure 1. For local networks, it often is necessary to use rectilinear or checker-board distances, instead of straight Euclidean lines. Distances in this metric are