數(shù)學(xué)建模mcm1985-2005年題目匯總

1、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 environment. Then ad

2、just 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 number harvested, and

3、 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 that value over a r

4、ealistic 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 unstable region. The St

5、rategic 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 stockpile goal o

6、f 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 a reasonable pri

7、ce 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 below gives the d

8、epth 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 Emergency-Facili

9、ties LocationThe township of 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 anywhere on the streets. MCM19

10、87A題 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 ramps of salt leading in

11、to 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 against the retaining wal

12、ls 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 in the domes. 鹽的存貯美國中西部一個州

13、把冬天用來灑在馬路上的鹽存貯在一個球頂倉庫里大約有15年了。圖87A-1表示在過去15年中鹽是怎么存貯的*通過駕駛鏟斗車在由鹽鋪成的坡道上進(jìn)出倉里并利用鏟斗車上的鏟子把鹽裝進(jìn)倉里或從倉里取出來。 最近，一個小組確定這種做法是不安全的。如果鏟斗車太靠近鹽堆的頂端，鹽就要滑動，而鏟斗車就耍翻到為加固倉庫而筑的擁壁上去。小組建議， 如果鹽堆是用鏟斗車堆起來的，那么鹽堆的最高高度不要超過15英尺。對這種情況建立一個數(shù)學(xué)模型并求得在倉庫中的鹽堆的最大高度。圖中倉高50英尺，擁壁 高4英尺，倉的外直徑103英尺，門的凈空高l 9英尺9英寸，鏟斗車高10英尺Problem B Parking Lot Des

14、ignThe 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 aligned side by side. However

15、, 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 drivers seem to have little dif

16、ficulty 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題 The Drug Runner ProblemTwo listening posts 5.43 miles apa

17、rt 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 is a powerboat waiting for so

18、meone 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 be heard when it gets within 5

19、00 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. 確定毒品走私船的位置相距5.43哩的監(jiān)聽站收聽到一個短暫的無線電訊號。收聽到訊早的

20、時候測向儀分別定位在 111和119處見圖88A-1)，測向儀的精度為2，該訊號來自一個毒品交換活躍的地方，據(jù)推測該處有一只機(jī)動船正等著有人來取毒品。當(dāng)時正 值黃昏、無風(fēng)、無潮流。一架小型直升飛機(jī)離開監(jiān)聽站的簡易機(jī)場并能精確地沿111角方向飛行。直升飛機(jī)的飛行速度是走私船的三倍。在離船500英尺時 船上能聽到直升飛機(jī)的聲音。直升飛機(jī)只有一種偵察儀器 -探照訂。在200英尺遠(yuǎn)的地方探照燈只能照明半徑為25英尺的圓域。說明飛行員能找到正等著的毒品船的(最小)區(qū)域。 研究一種直升飛機(jī)的最佳搜索方法。 在你的計算中要有95的精度。 本題是由加州Claremont McKenna學(xué)院的J.A.Ferli

21、ng提供的。這是一個分類(分組問題) 的修正簡化形式。原問題和現(xiàn)在簡化的問題都還沒有一種已知的最化解法。 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). Table 1 gives, for each crate, the thick

22、ness, 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 of a subsequent local trucking restri

23、ction: 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題 The Midge Classification ProblemTwo species of midges, Af and Apf, have been identified by biologists Grogan an

24、d 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 method to three specimens with (anten

25、na, 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? 蠓的分類兩種蠓Af和Apf己由生物學(xué)家W.L.Grongan和W.W.Wirth(1981年)根據(jù)它們的觸角長 度和翼長加以區(qū)分(見圖89A-1)，9只Af蠓用

26、圓圈標(biāo)記，6只Apf蠓用黑點(diǎn)標(biāo)記。根據(jù)給出的觸角長度和翼長識別出一只標(biāo)本是Af還是Apf是重要的。 給定一只Af或者Apf族的蝶，你如何正確地區(qū)分它屬于哪一族? 將你的方法用于觸角長和翼長分別為(1.24，1.80)、(1.28，1.84)、(1.40，2.04)的三個標(biāo)本。 設(shè)Af是寶貴的傳粉益蟲，Apf是某種疾病的載體，是否應(yīng)該修改你的分類方法，若需修改，怎么改? Problem Aircraft QueuingA common procedure at airports is to assign aircraft (A/C) to runways on a first-come-firs

27、t-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 time it is scheduled for pushback; The time

28、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 passenger capacities varying from 100 to 400 in step

29、s of 50. Develop and analyze a mathematical model that takes into account both the travelers and airlines satisfaction. MCM1991B題 The Steiner Tree ProblemThe cost for a communication line between two stations is proportional to the length of the line. The cost for conventional minimal spanning trees

30、 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 simp

31、le 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 computed as shown in Figure 2. Suppose you wish to design a minimum costs spanning tree for a local network with

32、9 stations. Their rectangular coordinates are: a(0,15), b(5,20), c(16,24), d(20,20), e(33,25), f(23,11), g(35,7), h(25,0) i(10,3). You are restricted to using rectilinear lines. Moreover, all phantom stations must be located at lattice points (i.e., the coordinates must be integers). The cost for ea

33、ch line is its length. Find a minimal cost tree for the network. Suppose each stations has a cost w*d(3/2), where d=degree of the station. If w=1.2, find a minimal cost tree. Try to generalize this problem 通訊網(wǎng)絡(luò)的極小生成樹兩個通訊站間通訊線路的費(fèi)用與線路的長度成正比。通過引入若干個“虛設(shè)站”并構(gòu)造一個新的 Steiner樹就可以降低由一組站生成Nf自統(tǒng)的極小生成樹所需的費(fèi)用。用這種方法可

34、降低費(fèi)用多達(dá)13.4%(1-sqrt(3)/2). 而且為構(gòu)造一個有n個站的網(wǎng)絡(luò)的費(fèi)用最低的Steiner樹絕不需要多于(n-2)個虛設(shè)站。下面是兩個簡單的例子。 對于局部網(wǎng)絡(luò)而言，有必要用直折線距離或“棋盤”距離來代替歐氏直線距離。假定你希望設(shè)計一個有9個站的局部網(wǎng)絡(luò)的最低造價生成樹。這9個站的直角坐標(biāo)是:a(0,15), b(5,20), c(16,24), d(20,20), e(33,25), f(23,11), g(35,7), h(25,0), i(10,3)限定你只能用直線，而且所有的虛設(shè)站必須位于格點(diǎn)上(即其坐標(biāo)是整數(shù))。每條直線段的造價是其長度值。 求該網(wǎng)絡(luò)的一個極小費(fèi)用樹。

35、 假定每個站的費(fèi)用為d3/2w, 其中d通訊站助度，若w=1.2，求極小費(fèi)用樹。 試推廣本問題。 本題是由馬里蘭州沙里斯勃菜州立大學(xué)數(shù)學(xué)科學(xué)系B.A.Fusaro(他也是MCM的Director)提供的，他是受啟發(fā)于Cipra.Barry AEuclidean geometry alive and well in the computerage SIAM New5，v.24(1991)，no.1，16-17，19Problem Water Tank FlowSome state water-right agencies require from communities data on the

36、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% accuracy, every hour. More importantly,

37、 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 the pump is working, which occurs once

38、 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 table omitted. The table gives the time,

39、 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 fi

40、lling the tank when the level drops to about 27.00 feet, and the pump stops when the level rises back to about 35.50 feet. 1992Problem A Air-Traffic-Control Radar PowerYou are to determine the power to be radiated by an air-traffic-control radar at a major metropolitan airport. The airport authority

41、 wants to minimize the power of the radar consistent with safety and cost. The authority is constrained to operate with its existing antennae and receiver circuitry. The only option that they are considering is upgrading the transmitter circuits to make the radar more powerful. The question that you

42、 are to answer is what power (in watts) must be released by the radar to ensure detection of standard passenger aircraft at a distance of 100 kilometers.Problem B Emergency Power RestorationPower companies serving coastal regions must have emergency response systems for power outages due to storms.

43、Such systems require the input of data that allow the time and cost required for restoration to be estimated and the value of the outage judged by objective criteria. In the past, Hypothetical Electric Company (HECO) has been criticized in the media for its lack of a prioritization scheme. You are a

44、 consultant to HECO power company. HECO possesses a computerized database with real time access to service calls that currently require the following information: several further paragraphs of details omitted. HECO has hired you to develop the objective criteria and schedule the work for the storm r

45、estoration requirements in Table 1 table omitted. HECO has asked for a technical report for their purposes and an executive summary in laymens terms that can be presented to the media. Further, they would like recommendations for the future. To determine your prioritized scheduling system, you will

46、have to make additional assumptions. Deal those assumptions. In the future, you may desire additional data. If so, detail the information desired.1993Problem A Optimal CompostingAn environmentally conscious institutional cafeteria is recycling customers uneaten food into compost by means of microorg

47、anisms. Each day, the cafeteria blends the leftover food into a slurry, mixes the slurry with crisp salad wastes from the kitchen and a small amount of shredded newspaper, and feeds the resulting mixture to a culture of fungi and soil bacteria, which digest slurry, greens, and papers into usable com

48、post. The crisp green provide pockets of oxygen for the fungi culture, and the paper absorbs excess humidity. At times, however, the fungi culture is unable or unwilling to digest as much of the leftovers as customers leave; the cafeteria does not blame the chef for the fungi cultures lack of appeti

49、te. Also, the cafeteria has received offers for the purchase of large quantities of it compost. Therefore, the cafeteria is investigating ways to increase its production of compost. Since it cannot yet afford to build a new composting facility, the cafeteria seeks methods to accelerate the fungi cul

50、tures activity, for instance, by optimizing the fungi cultures environment (currently held at about 120 F and 100% humidity), or by optimizing the composition of the moisture fed to the fungi culture, or both. Determine whether any relation exists between the proportions of slurry, greens, and paper

51、 in the mixture fed to the fungi culture, and the rate at which the fungi culture composts the mixture. if no relation exists, state so. otherwise, determine what proportions would accelerate the fungi cultures activity. In addition to the technical report following the format prescribed in the cont

52、est instructions, provide a oneno technical recommendation for implementation for the cafeteria manager. Table 1 shows the composition of various mixtures in pounds of each ingredient kept in separate bins, and the time that it took the fungi to culture to compost the mixtures, from the date fed to

53、the date completely composted table omitted.Problem B Coal-Tipple OperationsThe Aspen-Boulder Coal Company runs a loading facility consisting of a large coal tipple. When the coal trains arrive, they are loaded from the tipple. The standard coal train takes 3 hours to load, and the tipples capacity

54、is 1.5 standard trainloads of coal. Each day, the railroad sends three standard trains to the loading facility, and they arrive at any time between 5 A.M. and 8 P.M. local time. Each of the trains has three engines. If a train arrives and sits idle while waiting to be loaded, the railroad charges a

55、special fee, called a demurrage. The fee is $5,000 per engine per hour. In addition, a high-capacity train arrives once a week every Thursday between 11 A.M. and 1 P.M. This special train has five engines and holds twice as much coal as a standard train. An empty tipple can be loaded directly from t

56、he mine to its capacity in six hours by a single loading crew. This crew (and its associated equipment) cost $9,000 per hour. A second crew can be called out to increase the loading rate by conducting an additional tipple-loading operation at the cost of $12,000 per hour. Because of safety requireme

57、nts, during tipple loading no trains can be loaded. Whenever train loading is interrupted to load the tipple, demurrage charges are in effect. The management of the Coal Company has asked you to determine the expected annual costs of this tipples loading operations. Your analysis should include the

58、following considerations: How often should the second crew be called out? What are the expected monthly demurrage costs? If the standard trains could be scheduled to arrive at precise times, what daily schedule would minimize loading costs? Would a third tipple-loading crew at $12,000 per hour reduc

59、e annual operations costs? Can this tipple support a fourth standard train every day?1994Problem A Concrete Slab FloorsThe U.S. Dept. of Housing and Urban Development (HUD) is considering constructing dwellings of various sizes, ranging from individual houses to large apartment complexes. A principa

60、l concern is to minimize recurring costs to occupants, especially the costs of heating and cooling. The region in which the construction is to take place is temperate, with a moderate variation in temperature throughout the year. Through special construction techniques, HUD engineers can build dwell