数学建模美国赛历年试题
美赛历年赛题
美赛历年赛题
美国数学建模竞赛（MCM/ICM）自1985年创办以来已有35年的历史，每年都会发布三个模型问题供参赛选手在限定时间内进行研究和解答。
经过不断发展和完善，MCM/ICM成为了世界范围内最具影响力的数学建模竞赛之一。
以下是MCM/ICM历年来的一些典型赛题：
1985年 MCM A题：研究在给定经济情况下，如何规划BMW公司未来的生产计划及车型。
1987年 MCM A题：在地球上一个非常均匀的平面，建立一个小型城市，考虑各种环境因素如何影响城市的设施和功能。
1991年 MCM D题：分析社会上性别和种族歧视。
1997年 MCM C题：分析为什么珊瑚礁的污染问题比林区污染问题显得更为严重。
2002年 MCM A题：研究货轮舱位的装载问题，最大化收益同时保证船上货物负荷均衡。
2006年 MCM A题：建立模型研究地球大气环境中的水循环，探究人类活动对水循环的影响。
2010年 MCM A题：分析美国电力网络的可靠性，研究如何在自然灾害和人为故障的情况下使电力网络正常运作。
2014年 MCM A题：分析对于Fermi问题和经济增长的数学建模，探究经济增长的限制因素和未来发展趋势。
2018年 MCM A题：研究美国国家公园的野生动植物种类和数量变化，确定如何平衡保护野生动植物和国家公园的多个目的。
从这些题目中可以看出，MCM/ICM的竞赛内容涵盖了众多领域，如管理学、环保、气象、物流、生物学等等。
这不仅考验了参赛选手的数学建模水平，更需要他们具备良好的跨学科素养。
正是这种多学科交叉融合的特性，使得MCM/ICM成为了培养未来数学、理工科人才的重要平台之一。
历年美国大学生数学建模竞赛试题MCM.翻译版doc
1985 年美国大学生数学建模竞赛MCM 试题1985年MCM:动物种群选择适宜的鱼类和哺乳动物数据准确模型。
模型动物的自然表达人口水平与环境相互作用的不同群体的环境的重要参数,然后调整账户获取表单模型符合实际的动物提取的方法。
包括任何食物或限制以外的空间限制,得到数据的支持。
考虑所涉及的各种数量的价值,收获数量和人口规模本身,为了设计一个数字量代表的整体价值收获。
找到一个收集政策的人口规模和时间优化的价值收获在很长一段时间。
检查政策优化价值在现实的环境条件。
1985年MCM B:战略储藏管理钴、不产生在美国,许多行业至关重要。
(国防占17%的钴生产。
1979年)钴大局部来自非洲中部,一个政治上不稳定的地区。
1946年的战略和关键材料储藏法案需要钴储藏,将美国政府通过一项为期三年的战争。
建立了库存在1950年代,出售大局部在1970年代初,然后决定在1970年代末建立起来,与8540万磅。
大约一半的库存目标的储藏已经在1982年收购了。
建立一个数学模型来管理储藏的战略金属钴。
你需要考虑这样的问题:库存应该有多大?以什么速度应该被收购?一个合理的代价是什么金属?你也要考虑这样的问题:什么时候库存应该画下来吗?以什么速度应该是画下来吗?在金属价格是合理出售什么?它应该如何分配?有用的信息在钴政府方案在2500万年需要2500万磅的钴。
美国大约有1亿磅的钴矿床。
生产变得经济可行当价格到达22美元/磅(如发生在1981年)。
要花四年滚动操作,和thsn六百万英镑每年可以生产。
1980年,120万磅的钴回收,总消费的7%。
1986 年美国大学生数学建模竞赛MCM 试题1986年MCM A:水文数据下表给出了Z的水深度尺外表点的直角坐标X,Y在码(14数据点表省略)。
深度测量在退潮。
你的船有一个五英尺的草案。
你应该防止什么地区的矩形(75200)X(-50、150)?1986年MCM B:Emergency-Facilities位置迄今为止,力拓的乡牧场没有自己的应急设施。
MCM美国大学生数学建模比赛2000-2011年题目
2000 Mathemat ical Contest in ModelingThe ProblemsProblem A: Air traffic ControlProblem B: Radio Channel AssignmentsProblem A Air traffic ControlDedicated to the memory of Dr. Robert Machol, former chief scientist of the Federal Aviation AgencyTo 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 BRadio Channel AssignmentsWe 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 1An 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 2s 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.2001Problem A: Choosing a Bicycle WheelCyclists 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 dis k (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 theright.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 s poked 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 windspeeds for different road grades starting from zero percent to ten percent in onepercent increments. (Road grade is defined to be the ratio of the total rise of a hilldivided 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 riderwill 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 a nd 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 i nland 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-26into two lanes of Columbia-bound traffic, essentially turning the entire I-26 intoone-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 thepattern of how hurricanes affect the coast?3.Several smaller highways besides I-26 extend inland from the coast. Under whatconditions 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 inColumbia, to reduce the traffic leaving Columbia?5.In 1999, many families leaving the coast brought along their boats, campers, andmotor homes. Many drove all of their cars. Under what conditions should there berestrictions on vehicle types or numbers of vehicles brought in order to guaranteetimely 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 tocompare strategies. Required: Prepare a short newspaper article, not to exceed twopages, 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.2002 Mathemat ical Contest in ModelingThe ProblemsProblem AAuthors: Tjalling YpmaTit le: Wind and 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 and 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 falls 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 BAuthors: Bill Fox and Rich WestTit le: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend 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 reservations 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 bumped 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 light of the current situa tion: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 revenue received by an airline company 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 company's revenue is maximized. Insure that your model reflects the issues above, and consider alternatives for handling "bumped" passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.2003 MCM ProblemsPROBLEM A: The Stunt PersonAn exciting action scene in a m ovie is going to be filmed, and you are the stunt coordinator! A stunt person on a m otorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. You need to protect the stunt person, and also use relatively few cardboard boxes (lower cost, not seen by cam era, etc.).Your job is to:∙determine what size boxes to use∙determine how many boxes to use∙determine how the boxes will be stacked∙determine if any modifications to the boxes would help∙generalize to different combined weights (stunt person & motorcycle) and different jump heightsNote that, in "Tomorrow Never Dies", the Jam es Bond character on a m otorcycle jumps over a helicopter.PROBLEM B: G amma Knife Treat ment PlanningStereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, sm all intracranial 3D brain tum or without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. Three modalities are commonly used in this area; they are the gamma knife unit, heavy charged particle beam s, and external high-energy photon beams from linear accelerators.The gamma knife unit delivers a single high dose of ionizing radiation emanating from201 cobalt-60 unit sources through a heavy 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 diff erent spheres. Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14,and 18 mm are available for irradiating different size volumes. For a target volum e larger than one shot, m ultiple shots can be used to cover the entire t arget. In practice, m ost target volum es are treated with 1 to 15 shots. The target volum e is a bounded, three-dimensional digital image that usually consists of m illions of points.The goal of radiosurgery is to deplete tum or cells while preserving norma l structures. Since there are physical limitations and biological uncertainties involved in this therapy process, a treatm ent plan needs to account for all those limitations and uncertainties. In general, an optimal treat m ent plan is designed to m eet 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 the 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 treatm ent planning, we have the following constraints: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% ofthe target volume must be covered by shots.e as few shots as possible.Your tasks are to formulate the optim al treat m ent 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.2003 ICM ProblemPROBLEM C:To view and print problem C, you will need to have the Adobe Acrobat Reader installed in your Web browser. Downloading and installing acrobat is simple, safe, and only takes a few minutes. Download Acrobat Here.2004 MCM ProblemsPROBLEM A: Are Fingerprints Unique?It is a commonplace belief that the thumbprint of every human who has ever lived is different. Develop and analyze a model that will allow you to assess the probability that this is true. Compare the odds (that you found in this problem) of misidentification by fingerprint evidence against the odds of misidentification by DNA evidence.PROBLEM B: A Faster QuickPass System"QuickPass" systems are increasingly appearing to reduce people's time waiting in line, whether it is at tollbooths, amusement parks, or elsewhere. Consider the design of a QuickPass system for an amusement park. The amusement park has experimented by offering QuickPasses for several popular rides as a test. The idea is that for certain popular rides you can go to a kiosk near that ride and insert your daily park entrance ticket, and out will come a slip that states that you can return to that ride at a specific time later. For example, you insert your daily park entrance ticket at 1:15 pm, and the QuickPass states that you can come back between 3:30 and 4:30 pm when you can use your slip to enter a second, and presumably much shorter, line that will get you to the ride faster. To prevent people from obtaining QuickPasses for several rides at once, the QuickPass machines allow you to have only one active QuickPass at a time.You have been hired as one of several competing consultants to improve the operation of QuickPass. Customers have been complaining about some anomalies in the test system. For example, customers observed that in one instance QuickPasses were being offered for a return time as long as 4 hours later. A short time later on the same ride, the QuickPasses were given for times only an hour or so later. In some instances, the lines for people with Quickpasses are nearly as long and slow as the regular lines.The problem then is to propose and test schemes for issuing QuickPasses in order to increase people's enjoyment of the amusement park. Part of the problem is to determine what criteria to use in evaluating alternative schemes. Include in your report a non-technical summary for amusement park executives who must choose between alternatives from competing consultants.2005 MCM ProblemsPROBLEM A: Flood PlanningLake Murray in central South Carolina is formed by a large earthen dam, which was completed in1930 for power production. Model the flooding downstream in the event there is a catastrophic earthquake that breaches the dam.Two particular questions:Rawls Creek is a year-round stream that flows into the Saluda River a short distance downriver from the dam. How much flooding will occur in Rawls Creek from a dam failure, and how far back will it extend?Could the flood be so massive downstream that water would reach up to the S.C. State Capitol Building, which is on a hill overlooking the Congaree River?PROBLEM B: TollboothsHeavily-traveled toll roads such as the Garden State Parkway , Interstate 95, and so forth, are multi-lane divided highways that are interrupted at intervals by toll plazas. Because collecting tolls is usually unpopular, it is desirable to minimize motorist annoyance by limiting the amount of traffic disruption caused by the toll plazas. Commonly, a much larger number of tollbooths is provided than the number of travel lanes entering the toll plaza. Upon entering the toll plaza, the flow of vehicles fans out to the larger number of tollbooths, and when leaving the toll plaza, the flow of vehicles is required to squeeze back down to a number of travel lanes equal to the number of travel lanes before the toll plaza. Consequently, when traffic is heavy, congestion increases upon departure from the toll plaza. When traffic is very heavy, congestion also builds at the entry to the toll plaza because of the time required for each vehicle to pay the toll.Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane. Under what conditions is this more or less effective than the current practice? Note that the definition of "optimal" is up to you to determine.2006 MCM ProblemsPROBLEM A: Posit ioning and Moving Sprinkler Systems for Irrigat ionThere are a wide variety of techniques available for irrigating a field. The technologies range from advanced drip systems to periodic flooding. One of the systems that is used on smaller ranches is the use of "hand move" irrigation systems. Lightweight aluminum pipes with sprinkler heads are put in place across fields, and they are moved by hand at periodic intervals to insure that the whole field receives an adequate amount of water. This type of irrigation sys tem is cheaper and easier to maintain than other systems. It is also flexible, allowing for use on a wide variety of fields and crops. The disadvantage is that it requires a great deal of time and effort to move and set up the equipment at regular intervals.Given that this type of irrigation system is to be used, how can it be configured to minimize the amount of time required to irrigate a field that is 80 meters by 30 meters? For this task you are asked to find an algorithm to determine how to irrigate the rectangular field that minimizes the amount of time required by a rancher to maintain the irrigation system. One pipe set is used in the field. Y ou should determine the number of sprinklers and the spacing between sprinklers, and you should find a sch edule to move the pipes, including where to move them.A pipe set consists of a number of pipes that can be connected together in a straight line. Each pipe has a 10 cm inner diameter with rotating spray nozzles that have a 0.6 cm inner diameter. When pu t together the resulting pipe is 20 meters long. At the water source, the pressure is 420 Kilo- Pascal’s and has a flow rate of 150 liters per minute. No part of the field should receive more than 0.75 cm per hour of water, and each part of the field should receive at least 2 centimeters of water every 4 days. The total amount of water should be applied as uniformly as possiblePROBLEM B: Wheel Chair Access at AirportsOne of the frustrations with air travel is the need to fly through multiple airports, and each stop generally requires each traveler to change to a different airplane. This can be especially difficult for people who are not able to easily walk to a different flight's waiting area. One of the ways that an airline can make the transition easier is to provide a wheel chair and an escort to those people who ask for help. It is generally known well in advance which passengers require help, but it is not uncommon to receive notice when a passenger first registers at the airport. In rare instances an airline may not receive notice from a passenger until just prior to landing.Airlines are under constant pressure to keep their costs down. Wheel chairs wear out and are expensive and require maintenance. There is also a cost for making the escorts available. Moreover, wheel chairs and their escorts must be constantly moved around the airport so that they are available to people when their flight lands. In some large airports the time required to move across the airport is nontrivial. The wheel chairs must be stored somewhere, but space is expensive and severely limited in an airport terminal. Also, wheel chairs left in high traffic areas represent a liability risk as people try to move around them. Finally, one of the biggest costs is the cost of holding a plane if someone must wait for an escort and becomes late for their flight. The latter cost is especially troubling because it can affect the airline's average flight delay which can lead to fewer ticket sales as potential customers may choose to avoid an airline.Epsilon Airlines has decided to ask a third party to help them obtain a detailed analysis of the issues and costs of keeping and maintaining wheel chairs and escorts available for passengers. The airline needs to find a way to schedule the movement of wheel chairs throughout each day in a cost effective way. They also need to find and define the costs for budget planning in both the short and long term.Epsilon Airlines has asked your consultant group to put together a bid to help them solve their problem. Your bid should include an overview and analysis of the situation to help them decide if you fully understand their problem. They require a detailed description of an algorithm that you would like to implement which can determine where the escorts and wheel chairs should be and how they should move throughout each day. The goal is to keep the total costs as low as possible. Your bid is one of many that the airline will consider. You must make a strong case as to why your solution is the best and show that it will be able to handle a wide range of airports under a variety of circumstances.Your bid should also include examples of how the algorithm would work for a large (at least 4 concourses), a medium (at least two concourses), and a small airport (one concourse) under high and low traffic loads. You should determine all potential costs and balance their respective weights. Finally, as populations begin to include a higher percentage of older people who have more time to travel but may require more aid, your report should include projections of potential costs and needs in the future with recommendations to meet future needs.2007 MCM ProblemsPROBLEM A: G errymanderingThe United States Constitution provides that the House of Representatives shall be composed of some number (currently 435) of individuals who are elected from each state in proportion to the state’s population relative to that of the country as a whole. While this provides a way of determining how many representatives each state will have, it says nothing about how the district represented by a particular representative shall be determined geographically. This oversight has led to egregious (at least some people think so, usually not the incumbent) district shapes that look “un natural” by some standards.Hence the following question: Suppose you were given the opportunity to draw congressional districts for a state. How would you do so as a purely “baseline” exercise to create the “simplest” shapes for all the districts in a state? The rules include only that each district in the state must contain the same population. The definition of “simple” is up to you; but you need to make a convincing argument to voters in the state that your solution is fair. As an application of your method, draw geographically simple congressional districts for the state of New Y ork.PROBLEM B: The Airplane Seat ing ProblemAirlines are free to seat passengers waiting to board an aircraft in any order whatsoever. It has become customary to seat passengers with special needs first, followed by first-class passengers (who sit at the front of the plane). Then coach and business-class passengers are seated by groups of rows, beginning with the row at the back of the plane and proceeding forward.Apart from consideration of the passengers’ wait time, from the airline’s point of view, time is money, and boarding time is best minimized. The plane makes money for the airline only when it is in motion, and long boarding times limit the number of trips that a plane can make in a day.The development of larger planes, such as the Airbus A380 (800 passengers), accentuate the problem of minimizing boarding (and deboarding) time.Devise and compare procedures for boarding and deboarding planes with varying numbers of passengers: small (85–210), midsize (210–330), and large (450–800).Prepare an executive summary, not to exceed two single-spaced pages, in which you set out your conclusions to an audience of airline executives, gate agents, and flight crews.Note: The 2 page executive summary is to be included IN ADDITION to the reports required by the contest guidelines.An article appeared in the NY Times Nov 14, 2006 addressing procedures currently being followed and the importance to the airline of finding better solutions. The article can be seen at: http://travel2.nyt /2006/11/14/business/14boarding.ht ml2008 MCM ProblemsPROBLEM A: Take a Bat hConsider the effects on land from the melting of the north polar ice cap due to the predicted increase in global temperatures. Specifically, model the effects on the coast of Florida every ten years for the next 50 years due to the melting, with particular attention given to large metropolitan areas. Propose appropriate responses to deal with this. A careful discussion of the data used is an important part of the answer.PROBLEM B: Creat ing Sudoku PuzzlesDevelop an algorithm to construct Sudoku puzzles of varying difficulty. Develop metrics to define a difficulty level. The algorithm and metrics should be extensible to a varying number of difficulty levels. You should illustrate the algorithm with at least 4 difficulty levels. Your algorithm should guarantee a unique solution. Analyze the complexity of your algorithm. Your objective should be to minimize the complexity of the algorithm and meet the above requirements.2009 MCM Problems。
1996-1997 年美国大学生数模竞赛题
1996 美国大学生数模竞赛题Problem AThe world's oceans contain an ambient noise field. Seismic disturbances,surface shipping, and marine mammals are sources that, in different frequency ranges, contri bute to this field. We wish to consider how this ambient noise might be used to detect large moving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, developa method for detecting the presence of a moving submarine, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed freqency and amplitude.Problem BWhen determining the winner of a competition like the Mathematical Contest in Modeling, ther e are generally a large number of papers to judge. Let's say there are P=100 papers. A group of J judges is collected to accomplish the judging. Funding for the contest constains both the number of judges that can be obtained and amount of time that they can judge. For eample if P=100, then J=8 is typical.Ideally, each judge would read paper and rank-order them, but there are toomany papers for this. Instead, there will be a number of screening rounds in which each judg e will read some number of papers and give them scores. Then some selection scheme is used t o reduce the number of papers underconsideration: If the papers are rank-ordered, then the bottom 30% that each judge rank-orde rs could be rejected. Alternatively, if the judges do not rank-order, but instead give them numerical score (say, from 1 to 100),then all papers below some cut-off level could be rejected.The new pool of papers is then passed back to the judges, and the process is repeated. A con cern is then the total number of papers that judge reads must be substantially less than P.The process is stopped when there are only W papers left. There are the winners. Typically f or P=100, W=3.Your task is to determine a selection scheme, using a combination ofrank-ordering, numerical scoring, and other methods, by which the final Wpapers will include only papers from among the "best" 2W papers. (By "best",we assume that t here is an absolute rank-ordering to which all judges would agree.) For example, the top thr ee papers. Among all such methods, the one that required each judge to read the least number of papers is desired.Note the possibility of systematic bias in a numerical scoring scheme. Forexample, for a specific collection of papers, one judge could average 70points, while another could average 80 points. How would you scale your scheme to accommodat e for changes in the contest parameters (P, J, and W)?1997 年美国大学生数模竞赛题Problem A: The Velociraptor ProblemThe Velociraptor, Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists think that it wa s a very tenacious hunter, and may have hunted in pairs or larger packs. Unfortunately, ther e is no way to observe its hunting behavior in the wild as can be done with modern mammalian predators. A group of paleontologists has approached your team and asked for help in modeli ng the hunting behavior of the velociraptor.They hope to compare your results with field data reported by biologists studying the behavi ors of lions, tigers, and similar predatory animals.The average adult velociraptor was 3 meters long with a hip height of 0.5meters and an approximate mass of 45 Kg. It is estimated that the animalcould run extremely fast, at speeds of 60 km/hr., for about 15 seconds.After the initial burst of speed, the animal needed to stop and recover from a buildup of lactic acid in its muscles.Suppose that Velociraptor prey on Thescelosaurus neglectus, a herbivorous biped approximately the same size as the Velociraptor. A biomechanical analysis of a fossilized thescelosaurus indicates that if could run at a speed of about 50km.hr. for long periods of time.Part1Assuming the velociraptor is a solitary hunter, design a mathematical model that describes a hunting strategy for a single velociraptor stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Assume that the thecelosaurus can always detect the velociraptor when in comes within 15 meters, but may detect the predator at even greater ranges (up to 50 meters) depending upon the habitat and weather conditions. Additionally, due to its physical structure and strength, the velociraptor has a limited turning radius when running at full speed. This radius is estimated to be three times the animal's hip height. On the other hand, the thescelosaurus is extremely agile and has a turning radius of 0.5 meters. Part 2Assuming more realistically that the velociraptor hunted in pairs, design a new model that describes a hunting strategy for two velociraptors stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Use the other assumptions and limitations given in Part 1.Problem B: Mix Well For Fruitful DiscussionsSmall group meetings for the discussion of important issues, particularly long-rang planning, are gaining popularity. It is believed that large groups discourage productive discussion and that a dominant personalitywill usually control and direct the discussion. Thus, in corporate board meetings the board will meet in small groups to discuss issues before meeting as a whole. These smaller groups still run risk of control by a dominant personality. In an attempt to reduce this danger it is common to schedule several sessions with a different mix of people in each group.A meeting of an Tostal Corporation will be attended by 29 Board Members of which nine are in-horse members(i.e., corporate employees). The meeting is to be an all-day affair with three sessions scheduled for the morning and four for the afternoon. Each session will take 45 minutes, beginning on the hour from 9:00 A.M. to 4:00 P.M., with lunch scheduled at noon. Each morning session will consist of six discussion group with each discussion group led by one of the corporation's six senior officers. None of these of officers are board members. Thus each senior officer will lead three different discussion groups. The sessions will consist of only four discussion groups.The president of the corporation wants a list of board-member assignmentsto discussion group for each of seven sessions. The assignments should achieve as much of a mix of members as much as possible. The ideal assignment would have each board member with each other board member in a discussion group the same number of times while minimizing common membership of groups for the different sessions.The assignments should also satisfy the following criteria:1.For the morning sessions, no board member should be in the same senior officer's discussion group twice.2.No discussion group should contain a disproportionate number of in-house members.Give a list of assignments for members 1-9 and 10-29 and officers 1-6. Indicate how well the criteria in the precious paragraphs are met. Since it is possible that some board members will cancel at the last minute or that some not scheduled will show up, an algorithm that the secretary could use to adjust the assignments with an user to make assignments for future meetings involving different levels of participation for each type of attendee.。
1997年美国大学生数学建模竞赛题目
1997Problem A: The Velociraptor ProblemThe velociraptor, Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists think that it was a very tenacious hunter and may that hunted in pairs or larger packs. Unfortunately, there is no way to observe its hunting behavior in the wild, as can be done with modern mammalian predators. A group of paleontologists has approached your team and asked for help in modeling the hunting behavior of the velociraptor. They hope to compare your results with field data reported by biologists studying the behaviors of lions, tigers, and similar predatory animals.The average adult velociraptor was 3 m long with a hip height of 0.5 m and an approxiate mass of 45 kg. It is estimated that the animal could run extremely fast, at speeds of 60km/hr, for about 15 sec. After the initial burst of speed, the animal needed to stop and recover from a buildup of lactic acid in its muscles.Suppose that velociraptor preyed on Thescelosaurus neglectus, a herbivorous biped approximately the same size as the velociraptor. A biomechanical analysis of a fossilized thescelosaurus indicates that it could run at a speed of about 50 km/hr for long periods of time.Part 1Assuming the velociraptor is a solitary hunter, design a mathematical model that describes a hunting strategy for a single velociraptor stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Assume that the thescelosaurus can always detect the velociraptor when it comes within 15m,but may detect the predator at even greater ranges (up to 50m) depending upon the habitat and weather conditions. Additionally, due to its physical structure and strength, the velociraptor has a limited turning radius when running at full speed. This radius is estimated to be three times the animal's hip height. On the other hand, the thescelosaurus is extremely agile and has a turning radius of 0.5 m.Part 2Assuming more realistically that the velociraptor hunted in pairs, design a new model that describes a hunting strategy for two velociraptors stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Use the other assumptions and limitations given in Part 1.Problem B: Mix Well For Fruitful DiscussionsSmall group meeting for the discussion of important issues, particularly long-range planning ,are gaining popularity. It is believed that large groups discourage productive discussion and that a dominant personality will usually control and direct the discussion. Thus, in corporate board meetings, the board will meet in small groups to discuss issues before meeting as a whole. These smaller groups still run the risk of control by a dominantpersonality. In an attempt to reduce this danger, it is common to schedule several sessions with a different mix of people in each group.A meeting of An Tostal Corporation will be attended by 29 board members of which nine are in-house members (i.e., corporate employees). The meeting is to be an ally-day affair with three sessions scheduled for the morning and four for the afternoon. Each session will take 45 minutes, beginning on the hour from 9:00 A. M. To 4:00 P. M., with lunch scheduled at noon. Each morning session will consist of six discussion groups with each discussion group led by one of the corporation's six senior officers. None of these officers is a board member. Thus, each senior officer will lead three different discussion groups. The senior officers will not be involved in the afternoon sessions, and each of these sessions will consist of only four different discussion groups.The president of the corporation wants a list of board-member assignments to discussion groups for each of the seven sessions. The assignments should achieve as much of a mix of the members as possible. The ideal assignment would have each board member with each other board member in a discussion group the same number of times while minimizing common membership of groups for the different sessions. The assignments should also satisfy the following criteria:1．For the morning sessions, no board member should be in the same senior officer's discussion group twice.2．No discussion group should contain a disproportionate number of in-house members.Give a list of assignments for members 1-9 and 10-29 and officers 1-6. Indicate how well the criteria in the previous paragraphs are met. Since it is possible that some board members will cancel at the last minute or that some not scheduled will show up, an algorithm that the secretary could use to adjust the assignments with an hour's notice would be appreciated. It would be ideal if the algorithm could also be used to make assignments for future meetings involving different levels of participation for each type of attendee.。
美国数学建模竞赛题目(1985--2009年)
美国数学建模竞赛题目1985年：A题：动物群体的管理B题：战略物资储备的管理问题1986年：A题：海底地型测量问题B题：应急设施的优化选址问题1987年：A题：堆盐问题（盐堆稳定性问题）B题：停车场安排问题1988年：A题：确定毒品走私船位置B题：平板列车车厢的优化装载1989年：A题：蠓虫识别问题；最佳分类与隔离B题：飞机排队模型1990年：A题：脑中多巴胺的分布B题：铲雪车的路径与效率问题1991年：A题：估计水塔的水流量B题：通信网络费用问题1992年：A题：雷达系统的功率与设计式样B题：紧急修复系统的研制1993年：A题：堆肥问题B题：煤炭装卸场的最优操作1994年：A题：保温房屋设计问题B题：计算机网络的最小接通时间1996年：A题：大型水下物体的探测B题：快速遴选优胜者问题1997年：A题：恐龙捕食问题B题：会议混合安排问题1998年：A题：MRI图象处理问题B题：分数贬值问题1999年：A题：小星体撞击地球问题B题：公用设施的合法容量问题C题：确定环境污染的物质、位置、数量和时间的问题2000年：A题：空间交通管制B题：无线电信道分配C题：大象群落的兴衰2001年：A题：选择自行车车轮B题：逃避飓风怒吼C题：我们的水系-不确定的前景2002年：A题：风和喷水池B题：航空公司超员订票C题：如果我们过分扫荡自己的土地，将会失去各种各样的蜥蜴。
2003年：A题：特技演员B题：Gamma刀治疗方案C题：航空行李的扫描对策2004年：A题：指纹是独一无二的吗？B题：更快的快通系统C题：安全与否？2005年：A题：flood planningB题：tollboothsC题: Nonrenewable Resources2006年：A题：Positioning and Moving SprinklerSystems for IrrigationB题：Wheel Chair Access at AirportsC题：Trade-offs in the fight againstHIV/AIDS2007年：A题：GerrymanderingB题：The Airplane Seating ProblemC题：Organ Transplant: The Kidney Exchange Problem2008年：A题：Take a BathB题：Creating Sudoku PuzzlesC题：Finding the Good in Health Care Systems2009年：A题：Designing a Traffic CircleB题：Energy and the Cell PhoneC题：Creating Food Systems: Re-Balancing Human-Influenced Ecosystems。
1985~美国大学生数学建模竞赛题目集锦
1985~2015年美国大学生数学建模竞赛题目集锦目录1985 MCM A: Animal Populations (3)1985 MCM B: Strategic Reserve Management (3)1986 MCM A: Hydrographic Data (4)1986 MCM B: Emergency-Facilities Location (4)1987 MCM A: The Salt Storage Problem (5)1987 MCM B: Parking Lot Design (5)1988 MCM A: The Drug Runner Problem (5)1988 MCM B: Packing Railroad Flatcars (6)1989 MCM A: The Midge Classification Problem (6)1989 MCM B: Aircraft Queueing (6)1990 MCM A: The Brain-Drug Problem (6)1990 MCM B: Snowplow Routing (7)1991 MCM A: Water Tank Flow (8)1991 MCM B: The Steiner Tree Problem (8)1992 MCM A: Air-Traffic-Control Radar Power (8)1992 MCM B: Emergency Power Restoration (9)1993 MCM A: Optimal Composting (10)1993 MCM B: Coal-Tipple Operations (11)1994 MCM A: Concrete Slab Floors (11)1994 MCM B: Network Design (12)1995 MCM A: Helix Construction (13)1995 MCM B: Faculty Compensation (13)1996 MCM A: Submarine Tracking (13)1996 MCM B: Paper Judging (13)1997 MCM A: The Velociraptor Problem (14)1997 MCM B: Mix Well for Fruitful Discussions (15)1998 MCM A: MRI Scanners (16)1998 MCM B: Grade Inflation (17)1999 MCM A: Deep Impact (17)1999 MCM B: Unlawful Assembly (18)2000 MCM A: Air Traffic Control (18)2000 MCM B: Radio Channel Assignments (19)2001 MCM A: Choosing a Bicycle Wheel (20)2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...). (21)2002 MCM A: Wind and Waterspray (23)2002 MCM B: Airline Overbooking (23)2003 MCM A: The Stunt Person (24)2003 MCM B: Gamma Knife Treatment Planning (24)2004 MCM A: Are Fingerprints Unique? (25)2004 MCM B: A Faster QuickPass System (25)2005 MCM A: Flood Planning (26)2005 MCM B: Tollbooths (26)2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation (27)2006 MCM B: Wheel Chair Access at Airports (28)2007 MCM A: Gerrymandering (29)2007 MCM B: The Airplane Seating Problem (29)2008 MCM A: Take a Bath (30)2008 MCM B: Creating Sudoku Puzzles (30)2009 MCM A: Designing a Traffic Circle (30)2009 MCM B: Energy and the Cell Phone (30)2010 MCM A: The Sweet Spot (32)2010 MCM B: Criminology (32)2011 MCM A: Snowboard Course (33)2011 MCM B: Repeater Coordination (33)2012 MCM A: The Leaves of a Tree (33)2012 MCM B: Camping along the Big Long River (34)2013 MCM A: The Ultimate Brownie Pan (34)2013 MCM B: Water, Water, Everywhere (35)2014 MCM A: The Keep-Right-Except-To-Pass Rule (35)2014 MCM B: College Coaching Legends (35)2015 MCM A: Eradicating Ebola (35)2015 MCM B: Searching for a lost plane (35)1985 MCM A: Animal PopulationsChoose a fish or mammal for which appropriate data are available to model it accurately. Model the animal's natural interactions with 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 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 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 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. (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 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 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 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?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 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 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)?1986 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, 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 block 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 uniformly 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 in 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 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.Construct 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 realize 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 vehicles, 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 in 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 within 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 degree angle direction. The helicopter's 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 optimal 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). Table 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 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 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. Apply 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 debilitatingdisease. Would you modify your classification scheme and if so, how?1989 MCM B: Aircraft QueueingA common 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 the 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 there are seven types of A/C with passenger capacities varying from 100 to 400 in steps of 50. Develop and analyze amathematical 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 medical drugs – for example, dopamine against Parkinson's 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 research 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 another 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 samples.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 sectionFront vertical section1990 MCM B: Snowplow RoutingThe solid lines of the map (see Figure 1) represent paved two-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 the 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 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% accuracy, 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 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[ table 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 the level drops to about 27.00 feet, and the pump stops when the level rises back to about 35.50 feet.1991 MCM B: 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 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 computed as shown in Figure 2.Suppose you wish to design a minimum costs spanning tree for a local network with 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 each line is its length.1. Find a minimal cost tree for the network.2. Suppose each stations has a cost w*d^(3/2), where d=degree of the station. If w=1.2, find aminimal cost tree.3. Try to generalize this problem1992 MCM 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 wants to minimize the power of the radar consistent with safety andcost. 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 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.1992 MCM B: Emergency Power RestorationPower companies serving coastal regions must have emergency response systems for power outages due to storms. 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 consultant to HECO power company. HECO possesses a computerized database with real time access to service calls that currently require the following information:▪time of report,▪type of requestor,▪estimated number of people affected, and▪location (x,y).Cre sites are located at coordinates (0,0) and (40,40), where x and y are in miles. The region serviced by HECO is within -65 < x < 60 and -50 < y < 50. The region is largely metropolitan with an excellent road network. Crews must return to their dispatch site only at the beginning and end of shift. Company policy requires that no work be initiated until the storm leaves the area, unless the facility is a commuter railroad or hospital, which may be processed immediately if crews are available.HECO has hired you to develop the objective criteria and schedule the work for the storm restoration requirements listed in Table 1 using their work force described in Table 2. Note that the first call was received at 4:20 A.M. and that the storm left the area at 6:00 A.M. Also note that many outages were not reported until much later in the day.HECO has asked for a technical report for their purposes and an “executive summary” in laymen's terms that can be presented to the media. Further, they would like recommendations for the future. To determine your prioritized scheduling system, you will have to make additional assumptions. Detail those assumptions. In the future, you may desire additional data. If so, detail the information desired.Table 1. Storm restoration requirements. (table incomplete)Table 2. Crew descriptions.1993 MCM A: Optimal CompostingAn environmentally conscious institutional cafeteria is recycling customers' uneaten food into compost by means of microorganisms. 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 compost. 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 culture's lack of appetite. 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 culture's activity, for instance, by optimizing the fungiculture's 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 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 culture's activity. In addition to the technical report following the format prescribed in the contest instructions, provide a one-page nontechnical 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 the date completely composted [table omitted].1993 MCM 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 tipple's capacity 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 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 the 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 requirements, 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 tipple's loading operations. Your analysis should include the 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 reduce annual operations costs?▪Can this tipple support a fourth standard train every day?1994 MCM 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 principal concern is to minimize recurring costs to occupants, especially the costs of heating and cooling. The region inwhich 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 dwellings that do not need to rely on convection- that is, there is no need to rely on opening doors or windows to assist in temperature variation. The dwellings will be single-story, with concrete slab floors as the only foundation. You have been hired as a consultant to analyze the temperature variation in the concrete slab floor to determine if the temperature averaged over the floor surface can be maintained within a prescribed comfort zone throughout the year. If so, what size/shape of slabs will permit this?Part 1, Floor Temperature: Consider the temperature variation in a concrete slab given that the ambient temperature varies daily within the ranges given Table 1. Assume that the high occurs at noon and the low at midnight. Determine if slabs can be designed to maintain a temperature averaged over the floor surface within the prescribed comfort zone considering radiation only. Initially, assume that the heat transfer into the dwelling is through the exposed perimeter of the slab and that the top and bottom of the slabs are insulated. Comment on the appropriateness and sensitivity of these assumptions. If you cannot find a solution that satisfies Table 1, can you find designs that satisfy a Table 1 that you propose?Part 2, Building Temperature: Analyze the practicality of the initial assumptions and extend the analysis to temperature variation within the single-story dwelling. Can the house be kept within the comfort zone?Part 3, Cost of Construction: Suggest a design that considers HUD's objective of reducing or eliminating heating and cooling costs, considering construction restrictions and costs.1994 MCM B: Network DesignIn your company, information is shared among departments on a daily basis. This information includes the previous day's sales statistics and current production guidance. It is important to get this information out as quickly as possible. [Network diagram (with 5 nodes and 7 capacitated edges) omitted.]We are interested in scheduling transfers in an optimal way to minimize the total time it takes to complete them all. This minimum total time is called the makespan. Consider the three following situations for your company: [Three more network diagrams (on roughly 20 nodes each) omitted.]1995 MCM A: Helix ConstructionA small biotechnological company must design, prove, program and test a mathematical algorithm to locate “in real time” all the intersections of a helix and a plane in general positions in space. Design, justify, program and test a method to compute all the intersections of a plane and a helix, both in general positions (at any locations and with any orientations) in space. A segment of the helix may represent, for example, a helicoidal suspension spring or a piece of tubing in a chemical or medical apparatus. Theoretical justification of the proposed algorithm is necessary to verify the solution from several points of view, for instance, through mathematical proofs of parts of the algorithm, and through tests of the final program with known examples. Such documentation and tests will be required by government agencies for medical use.1995 MCM B: Faculty CompensationAluacha Balaclava College, and undergraduate facility, has just hired a new Provost whose first priority is the institution of a fair and reasonable faculty-compensation plan. She has hired your consulting team to design a compensation system that reflects the following circumstances and principles: [Three paragraphs of details omitted] Design a new pay system, first withoutcost-of-living increases. Incorporate cost-of-living increases, and then finally, design a transition process for current faculty that will move all salaries towards your system without reducing anyone's salary. The Provost requires a detailed compensation system plan for implementation, as well as a brief, clear, executive summary outlining the model, its assumptions, strengths, weaknesses and expected results, which she can present to the Board and faculty. [A detailed table of current salaries is omitted.]1996 MCM A: Submarine TrackingThe world's oceans contain an ambient noise field. Seismic disturbances, surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large maving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, develop a method for detecting the presence of a moving submarine, its speed, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed frequency and amplitude.1996 MCM B: Paper JudgingWhen determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge. Let's say there are P=100 papers. A group of J judges is collected to accomplish the judging. Funding for the contest contrains both the number of judges that can be obtained and the amount of time they can judge. For example if P=100, then J=8 is typical.。
美赛历年题目_pdf
马剑整理历年美国大学生数学建模赛题目录MCM85问题-A 动物群体的管理 (3)MCM85问题-B 战购物资储备的管理 (3)MCM86问题-A 水道测量数据 (4)MCM86问题-B 应急设施的位置 (4)MCM87问题-A 盐的存贮 (5)MCM87问题-B 停车场 (5)MCM88问题-A 确定毒品走私船的位置 (5)MCM88问题-B 两辆铁路平板车的装货问题 (6)MCM89问题-A 蠓的分类 (6)MCM89问题-B 飞机排队 (6)MCM90-A 药物在脑内的分布 (6)MCM90问题-B 扫雪问题 (7)MCM91问题-B 通讯网络的极小生成树 (7)MCM 91问题-A 估计水塔的水流量 (7)MCM92问题-A 空中交通控制雷达的功率问题 (7)MCM 92问题-B 应急电力修复系统的修复计划 (7)MCM93问题-A 加速餐厅剩菜堆肥的生成 (8)MCM93问题-B 倒煤台的操作方案 (8)MCM94问题-A 住宅的保温 (9)MCM 94问题-B 计算机网络的最短传输时间 (9)MCM-95问题-A 单一螺旋线 (10)MCM95题-B A1uacha Balaclava学院 (10)MCM96问题-A 噪音场中潜艇的探测 (11)MCM96问题-B 竞赛评判问题 (11)MCM97问题-A Velociraptor(疾走龙属)问题 (11)MCM97问题-B为取得富有成果的讨论怎样搭配与会成员 (12)MCM98问题-A 磁共振成像扫描仪 (12)MCM98问题-B 成绩给分的通胀 (13)MCM99问题-A 大碰撞 (13)MCM99问题-B “非法”聚会 (14)MCM2000问题-A空间交通管制 (14)MCM2000问题-B: 无线电信道分配 (14)MCM2001问题- A: 选择自行车车轮 (15)MCM2001问题-B 逃避飓风怒吼（一场恶风...） .. (15)MCM2001问题-C我们的水系-不确定的前景 (16)MCM2002问题-A风和喷水池 (16)MCM2002问题-B航空公司超员订票 (16)MCM2002问题-C (16)MCM2003问题-A: 特技演员 (18)MCM2003问题-B: Gamma刀治疗方案 (18)MCM2003问题-C航空行李的扫描对策 (19)MCM2004问题-A：指纹是独一无二的吗？ (19)MCM2004问题-B：更快的快通系统 (19)MCM2004问题-C安全与否？ (19)MCM2005问题A.水灾计划 (19)MCM2005B.Tollbooths (19)MCM2005问题C：不可再生的资源 (20)MCM2006问题A: 用于灌溉的自动洒水器的安置和移动调度 (20)MCM2006问题B: 通过机场的轮椅 (20)MCM2006问题C : 抗击艾滋病的协调 (21)MCM2007问题B ：飞机就座问题 (24)MCM2007问题C：器官移植：肾交换问题 (24)MCM2008问题A:给大陆洗个澡 (28)MCM2008问题B：建立数独拼图游戏 (28)MCM85问题-A 动物群体的管理在一个资源有限，即有限的食物、空间、水等等的环境里发现天然存在的动物群体。
历年美国大学生数学建模竞赛试题
PROBLEM A: The Keep-Right-Except-To-Pass RuleIn countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employ a rule that requires drivers to drive in the right-most lane unless they arepassing another vehicle, in which case they move one lane to the left,pass, and return to their former travel lane.Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wish to examine tradeoffs between traffic flow and safety, the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement. Is this rule effective in promoting better traffic flow? If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flow, safety, and/or other factors that you deem important.In countries where driving automobiles on the left is the norm, argue whether or not your solution can be carried over with a simple change of orientation, or would additional requirements be needed.Lastly, the rule as stated above relies upon human judgment for compliance. If vehicle transportation on the same roadway was fully under the control of an intelligent system – either part of the road network or imbedded in the design of all vehicles using the roadway – to what extent would this change the results of your earlier analysis?PROBLEM B: College Coaching LegendsSports Illustrated, a magazine for sports enthusiasts, is looking for the ―best all time college coach‖ male or female for the previous century. Build a mathematical model to choose the best college coach or coaches (past or present) from among either male or female coaches in such sports as college hockey or field hockey, football, baseball or softball, basketball, or soccer. Does it make a difference which time line horizon that you use in your analysis, i.e., does coaching in 1913 differ from coaching in 2013? Clearly articulate your metrics for assessment. Discuss how your model can be applied in general across both genders and all possible sports. Present your model’s top 5 coaches in each of 3 different sports.In addition to the MCM format and requirements, prepare a 1-2 page article for Sports Illustrated that explains your results and includes a non-technical explanation of your mathematical model that sports fans will understand.PROBLEM A: The Ultimate Brownie PanWhen baking in a rectangular pan heat is concentrated in the 4 corners and the product gets overcooked at the corners (and to a lesser extent at the edges). In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges. However, since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven. Develop a model to show the distribution of heat across the outer edge of a pan for pans of different shapes - rectangular to circular and other shapes in between.Assume1. A width to length ratio of W/L for the oven which is rectangular in shape.2. Each pan must have an area of A.3. Initially two racks in the oven, evenly spaced.Develop a model that can be used to select the best type of pan (shape) under the following conditions:1. Maximize number of pans that can fit in the oven (N)2. Maximize even distribution of heat (H) for the pan3. Optimize a combination of conditions (1) and (2) where weights p and (1- p) are assigned to illustrate how the results vary with different valuesof W/L and p.In addition to your MCM formatted solution, prepare a one to two page advertising sheet for the new Brownie Gourmet Magazine highlighting your design and results.PROBLEM B: Water, Water, EverywhereFresh water is the limiting constraint for development in much of the world. Build a mathematical model for determining an effective, feasible, andcost-efficient water strategy for 2013 to meet the projected water needs of [pick one country from the list below] in 2025, and identify the best water strategy. In particular, your mathematical model must address storage and movement; de-salinization; and conservation. If possible, use your model to discuss the economic, physical, and environmental implications of your strategy. Provide a non-technical position paper to governmental leadership outlining your approach, its feasibility and costs, and why it is the ―best water strategy choice.‖Countries: United States, China, Russia, Egypt, or Saudi ArabiaPROBLEM A: The Leaves of a Tree"How much do the leaves on a tree weigh?" How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical model to describe and classify leaves. Consider and answer the following:• Why do leaves have the various shapes that they have?• Do the shapes ―minimize‖ overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the ―volume‖ of the tree and its branches effect the shape?• Speaking of profiles, is leaf shape (general characteristics) related to tree profile/branching structure?• How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)?In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings.PROBLEM B: Camping along the Big Long RiverVisitors to the Big Long River (225 miles) can enjoy scenic views and exciting white water rapids. The river is inaccessible to hikers, so the only way to enjoy it is to take a river trip that requires several days of camping. River trips all start at First Launch and exit the river at Final Exit, 225 miles downstream. Passengers take either oar- powered rubber rafts, which travel on average 4 mph or motorized boats, which travel on average 8 mph. The trips range from 6 to 18 nights of camping on the river, start to finish.. The government agency responsible for managing this river wants every trip to enjoy a wilderness experience, with minimal contact with other groups of boats on the river. Currently, X trips travel down the Big Long River each year during a six month period (the rest of the year it is too cold for river trips). There are Y camp sites on the Big Long River, distributed fairly uniformly throughout the river corridor. Given the rise in popularity of river rafting, the park managers have been asked to allow more trips to travel down the river. They want to determine how they might schedule an optimal mix of trips, of varying duration (measured in nights on the river) and propulsion (motor or oar) that will utilize the campsites in thebest way possible. In other words, how many more boat trips could be added to the Big Long River’s rafting season? The river managers have hired you to advise them on ways in which to develop the best schedule and on ways in which to determine the carrying capacity of the river, remembering that no two sets of campers can occupy the same site at the same time. In addition to your one page summary sheet, prepare a one page memo to the managers of the river describing your key findings.2011 MCM ProblemsPROBLEM A: Snowboard CourseDetermine the shape of a snowboard course (currently known as a ―halfpipe‖) to maximize the production of ―vertical air‖ by a skilled snowboarder. "Vertical air" is the maximum vertical distance above the edge of the halfpipe. Tailor the shape to optimize other possible requirements, such as maximum twist in the air.What tradeoffs may be required to develop a ―practical‖ course?PROBLEM B: Repeater CoordinationThe VHF radio spectrum involves line-of-sight transmission and reception. This limitation can be overcome by ―repeaters,‖ which pick up weak signals, amplify them, and retransmit them on a different frequency. Thus, using a repeater, low-power users (such as mobile stations) can communicate with one another in situations where direct user-to-user contact would not be possible. However, repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.In addition to geographical separation, the ―continuous tone-coded squelch system‖ (CTCSS), sometimes nicknamed ―private line‖ (PL), technology can be used to mitigate interference problems. This system associates to each repeater a separate subaudible tone that is transmitted by all users who wish to communicate through that repeater. The repeater responds only to received signals with its specific PL tone. With this system, two nearby repeaters can share the same frequency pair (for receive and transmit); so more repeaters (and hence more users) can be accommodated in a particular area.For a circular flat area of radius 40 miles radius, determine the minimum number of repeaters necessary to accommodate 1,000 simultaneous users. Assume that the spectrum available is 145 to 148 MHz, the transmitter frequency in a repeater is either 600 kHz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available.How does your solution change if there are 10,000 users?Discuss the case where there might be defects in line-of-sight propagation caused by mountainous areas.2010 MCM ProblemsPROBLEM A: The Sweet SpotExplain the ―sweet spot‖ on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some playe rs believe that ―corking‖ a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the ―sweet spot‖ effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits ―corking‖?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?PROBLEM B: CriminologyIn 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other people to vicious attacks. One of the methods used to narrow the search for Mr. Sutcliffe was to find a ―center of mass‖ of the locations of the attacks. In the end, the suspect happened to live in the same town predicted by this technique. Since that time, a number of more sophisticated techniques have been developed to determine the ―geographical profile‖ of a suspected serialcriminal based on the locations of the crimes.Your team has been asked by a local police agency to develop a method to aid in their investigations of serial criminals. The approach that you develop should make use of at least two different schemes to generate a geographical profile. You should develop a technique to combine the results of the different schemes and generate a useful prediction for law enforcement officers. The prediction should provide some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes. If you make use of any other evidence in your estimate, you must provide specific details about how you incorporate the extra information. Your method should also provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings.In addition to the required one-page summary, your report should include an additionaltwo-page executive summary. The executive summary should provide a broad overview of the potential issues. It should provide an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool. The executive summary will be read by a chief of police and should include technical details appropriate to the intended audience2009 MCM ProblemsPROBLEM A: Designing a Traffic CircleMany cities and communities have traffic circles—from large ones with many lanes in the circle (such as at the Arc de Triomphe in Paris and the Victory Monument in Bangkok) to small ones with one or two lanes in the circle. Some of these traffic circles position a stop sign or a yield sign on every incoming road that gives priority to traffic already in the circle; some position a yield sign in the circle at each incoming road to give priority to incoming traffic; and some position a traffic light on each incoming road (with no right turn allowed on a red light). Other designs may also be possible.The goal of this problem is to use a model to determine how best to control traffic flow in, around, and out of a circle. State clearly the objective(s) you use in your model for making the optimal choice as well as the factors that affect this choice. Include a Technical Summary of not more than two double-spaced pages that explains to a Traffic Engineer how to use your model to help choose the appropriate flow-control method for any specific traffic circle. That is, summarize the conditions under which each type of traffic-control method should be used. When traffic lights are recommended, explain a method fordetermining how many seconds each light should remain green (which may vary according to the time of day and other factors). Illustrate how your model works with specific examples.PROBLEM B: Energy and the Cell PhoneThis question involves the ―energy‖ consequences of the cell phone revolution. Cell phone usage is mushrooming, and many people are using cell phones and giving up their landline telephones. What is the consequence of this in terms of electricity use? Every cell phone comes with a battery and a recharger.Requirement 1Consider the current US, a country of about 300 million people. Estimate from available data the number H of households, with m members each, that in the past were serviced by landlines. Now, suppose that all the landlines are replaced by cell phones; that is, each of the m members of the household has a cell phone. Model the consequences of this change for electricity utilization in the current US, both during the transition and during the steady state. The analysis should take into account the need for charging the batteries of the cell phones, as well as the fact that cell phones do not last as long as landline phones (for example, the cell phones get lost and break).Requirement 2Consider a second ―Pseudo US‖—a country of about 300 million people with about the same economic status as the current US. However, this emerging country has neither landlines nor cell phones. What is the optimal way of providing phone service to this country from an energy perspective? Of course, cell phones have many social consequences and uses that landline phones do not allow. A discussion of the broad and hidden consequences of having only landlines, only cell phones, or a mixture of the two is welcomed.Requirement 3Cell phones periodically need to be recharged. However, many people always keep their recharger plugged in. Additionally, many people charge their phones every night, whether they need to be recharged or not. Model the energy costs of this wasteful practice for a Pseudo US based upon your answer to Requirement 2. Assume that the Pseudo US supplies electricity from oil. Interpret your results in terms of barrels of oil.Requirement 4Estimates vary on the amount of energy that is used by various recharger types (TV, DVR, computer peripherals, and so forth) when left plugged in but not charging the device. Useaccurate data to model the energy wasted by the current US in terms of barrels of oil per day. Requirement 5Now consider population and economic growth over the next 50 years. How might a typical Pseudo US grow? For each 10 years for the next 50 years, predict the energy needs for providing phone service based upon your analysis in the first three requirements. Again, assume electricity is provided from oil. Interpret your predictions in term of barrels of oil.2008 MCM ProblemsPROBLEM A: Take a BathConsider the effects on land from the melting of the north polar ice cap due to the predicted increase in global temperatures. Specifically, model the effects on the coast of Florida every ten years for the next 50 years due to the melting, with particular attention given to large metropolitan areas. Propose appropriate responses to deal with this. A careful discussion of the data used is an important part of the answer.PROBLEM B: Creating Sudoku PuzzlesDevelop an algorithm to construct Sudoku puzzles of varying difficulty. Develop metrics to define a difficulty level. The algorithm and metrics should be extensible to a varying number of difficulty levels. You should illustrate the algorithm with at least 4 difficulty levels. Your algorithm should guarantee a unique solution. Analyze the complexity of your algorithm. Your objective should be to minimize the complexity of the algorithm and meet the above requirements.2007 MCM ProblemsPROBLEM A: GerrymanderingThe United States Constitution provides that the House of Representatives shall be composed of some number (currently 435) of individuals who are elected from each state in proportion to the state's population relative to that of the country as a whole. While this provides a way ofdetermining how many representatives each state will have, it says nothing about how the district represented by a particular representative shall be determined geographically. This oversight has led to egregious (at least some people think so, usually not the incumbent) district shapes that look "unnatural" by some standards.Hence the following question: Suppose you were given the opportunity to draw congressional districts for a state. How would you do so as a purely "baseline" exercise to create the "simplest" shapes for all the districts in a state? The rules include only that each district in the state must contain the same population. The definition of "simple" is up to you; but you need to make a convincing argument to voters in the state that your solution is fair. As an application of your method, draw geographically simple congressional districts for the state of New York.PROBLEM B: The Airplane Seating ProblemAirlines are free to seat passengers waiting to board an aircraft in any order whatsoever. It has become customary to seat passengers with special needs first, followed by first-class passengers (who sit at the front of the plane). Then coach and business-class passengers are seated by groups of rows, beginning with the row at the back of the plane and proceeding forward.Apart from consideration of the passengers' wait time, from the airline's point of view, time is money, and boarding time is best minimized. The plane makes money for the airline only when it is in motion, and long boarding times limit the number of trips that a plane can make in a day.The development of larger planes, such as the Airbus A380 (800 passengers), accentuate the problem of minimizing boarding (and deboarding) time.Devise and compare procedures for boarding and deboarding planes with varying numbers of passengers: small (85-210), midsize (210-330), and large (450-800).Prepare an executive summary, not to exceed two single-spaced pages, in which you set out your conclusions to an audience of airline executives, gate agents, and flight crews.Note: The 2 page executive summary is to be included IN ADDITION to the reports required by the contest guidelines.An article appeared in the NY Times Nov 14, 2006 addressing procedures currently being followed and the importance to the airline of finding better solutions. The article can be seen at:/2006/11/14/business/14boarding.html2006 MCM ProblemsPROBLEM A: Positioning and Moving Sprinkler Systems for IrrigationThere are a wide variety of techniques available for irrigating a field. The technologies range from advanced drip systems to periodic flooding. One of the systems that is used on smaller ranches is the use of "hand move" irrigation systems. Lightweight aluminum pipes with sprinkler heads are put in place across fields, and they are moved by hand at periodic intervals to insure that the whole field receives an adequate amount of water. This type of irrigation system is cheaper and easier to maintain than other systems. It is also flexible, allowing for use on a wide variety of fields and crops. The disadvantage is that it requires a great deal of time and effort to move and set up the equipment at regular intervals.Given that this type of irrigation system is to be used, how can it be configured to minimize the amount of time required to irrigate a field that is 80 meters by 30 meters? For this task you are asked to find an algorithm to determine how to irrigate the rectangular field that minimizes the amount of time required by a rancher to maintain the irrigation system. One pipe set is used in the field. You should determine the number of sprinklers and the spacing between sprinklers, and you should find a schedule to move the pipes, including where to move them.A pipe set consists of a number of pipes that can be connected together in a straight line. Each pipe has a 10 cm inner diameter with rotating spray nozzles that have a 0.6 cm inner diameter. When put together the resulting pipe is 20 meters long. At the water source, the pressure is 420 Kilo- Pascal's and has a flow rate of 150 liters per minute. No part of the field should receive more than 0.75 cm per hour of water, and each part of the field should receive at least 2 centimeters of water every 4 days. The total amount of water should be applied as uniformly as possiblePROBLEM B: Wheel Chair Access at AirportsOne of the frustrations with air travel is the need to fly through multiple airports, and each stop generally requires each traveler to change to a different airplane. This can be especially difficult for people who are not able to easily walk to a different flight's waiting area. One of the ways that an airline can make the transition easier is to provide a wheel chair and an escort to those people who ask for help. It is generally known well in advance which passengers require help, but it is not uncommon to receive notice when a passenger first registers at the airport. In rare instances an airline may not receive notice from a passenger until just prior to landing.Airlines are under constant pressure to keep their costs down. Wheel chairs wear out and are expensive and require maintenance. There is also a cost for making the escorts available. Moreover, wheel chairs and their escorts must be constantly moved around the airport so that they are available to people when their flight lands. In some large airports the time required tomove across the airport is nontrivial. The wheel chairs must be stored somewhere, but space is expensive and severely limited in an airport terminal. Also, wheel chairs left in high traffic areas represent a liability risk as people try to move around them. Finally, one of the biggest costs is the cost of holding a plane if someone must wait for an escort and becomes late for their flight. The latter cost is especially troubling because it can affect the airline's average flight delay which can lead to fewer ticket sales as potential customers may choose to avoid an airline.Epsilon Airlines has decided to ask a third party to help them obtain a detailed analysis of the issues and costs of keeping and maintaining wheel chairs and escorts available for passengers. The airline needs to find a way to schedule the movement of wheel chairs throughout each day in a cost effective way. They also need to find and define the costs for budget planning in both the short and long term.Epsilon Airlines has asked your consultant group to put together a bid to help them solve their problem. Your bid should include an overview and analysis of the situation to help them decide if you fully understand their problem. They require a detailed description of an algorithm that you would like to implement which can determine where the escorts and wheel chairs should be and how they should move throughout each day. The goal is to keep the total costs as low as possible. Your bid is one of many that the airline will consider. You must make a strong case as to why your solution is the best and show that it will be able to handle a wide range of airports under a variety of circumstances.Your bid should also include examples of how the algorithm would work for a large (at least 4 concourses), a medium (at least two concourses), and a small airport (one concourse) under high and low traffic loads. You should determine all potential costs and balance their respective weights. Finally, as populations begin to include a higher percentage of older people who have more time to travel but may require more aid, your report should include projections of potential costs and needs in the future with recommendations to meet future needs.2005 MCM ProblemsPROBLEM A: Flood PlanningLake Murray in central South Carolina is formed by a large earthen dam, which was completed in 1930 for power production. Model the flooding downstream in the event there is a catastrophic earthquake that breaches the dam.Two particular questions:Rawls Creek is a year-round stream that flows into the Saluda River a short distance downriver from the dam. How much flooding will occur in Rawls Creek from a dam failure, and how far back will it extend?Could the flood be so massive downstream that water would reach up to the S.C. State Capitol Building, which is on a hill overlooking the Congaree River?PROBLEM B: TollboothsHeavily-traveled toll roads such as the Garden State Parkway , Interstate 95, and so forth, are multi-lane divided highways that are interrupted at intervals by toll plazas. Because collecting tolls is usually unpopular, it is desirable to minimize motorist annoyance by limiting the amount of traffic disruption caused by the toll plazas. Commonly, a much larger number of tollbooths is provided than the number of travel lanes entering the toll plaza. Upon entering the toll plaza, the flow of vehicles fans out to the larger number of tollbooths, and when leaving the toll plaza, the flow of vehicles is required to squeeze back down to a number of travel lanes equal to the number of travel lanes before the toll plaza. Consequently, when traffic is heavy, congestion increases upon departure from the toll plaza. When traffic is very heavy, congestion also builds at the entry to the toll plaza because of the time required for each vehicle to pay the toll.Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane. Under what conditions is this more or less effective than the current practice? Note that the definition of "optimal" is up to you to determine.以上是2005年——2014年美国大学生数学建模竞赛试题更多试题详见/undergraduate/contests/matrix/index.html。
2004、2010、2011、2012美国数学建模数学竞赛试题及翻译、分析
2004年A题题目是研究人的指纹相同导致确认身份时产生错误的可能性和因为DNA相同导致产生错误的可能性。
这道题与生物有关。
2004年，美国提出了“科技展望——生物盾国家纳米技术”，针对微分子进行研究，与人体相关的DNA也就成为焦点。
指纹是区分人的重要参数，它由DNA决定，由于现在自然环境改变，人体的DNA也有可能发生变异，这样引起相同指纹出现也是有可能的。
研究这种变异的可能性有利于刑事案件的侦破，提高政府的防御能力等。
本题目可以采用机理分析法来建模。
可以采用计算机模拟人体染色体中的基因排列进而检验、统计排列结果与指纹形成情况B题题目：现在的快通系统在收费站、娱乐公园和其他的地方，正在越来越频繁的使用，来减少人们排队等候的时间，现在我们考虑为一个娱乐公园所设计的快通系统，在一次测试中，这个公园在几个游客比较多的景点旁都设置的快通系统，这个系统的设计创意是对于那些比较热门的景点，可以到旁边的一个机器，将其门票插入后出来一张纸条，上面写着具体的你可以回来时间段，比如说你把你的门票在1：15插到机子里，系统就会告诉你，你可以在3:30-4:30回来，这个时候队伍就比较短，你可以凭你的纸条加入这个队伍，很快就可以进入景点，为了防止游客同时在几个景点使用这个系统。
系统的机器只允许你一次在一个景点排队等候。
现在你是几个被公园雇佣的互相竞争的一个，你的职责是改善快通系统的运行。
很多游客都在抱怨测试期间系统的异常现象，比如说有一次系统提供的回到景点的时间是4小时以后，但是才过一会儿，在相同景点系统提供的时间只有1小时。
在另外一些时候根据快通系统组织起来的游客的等候队伍，就和普通的队伍一样长一样慢。
现在的问题是要提出并且测试一个模型，这个模型能让快通系统的等候纸条的发放能增加人们在公园的乐趣的目的。
问题的一部分就是首先决定衡量不同模型的标准，在你提交的报告里还要附带一份技术性的总结，以便公园的领导在不同的顾问所提出的模型中选择。
美国(MCM)(ICM)试题(96年-09年)(英文版)
美国大学生数学建模竞赛试题1996 American MCM Problems Problem AThe world's oceans contain an ambient noise field. Seismic disturbances, surface shipping, and marine mammals are sources that, in different frequency ranges,contribute to this field. We wish to consider how this ambient noise might be used to detect large moving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, developa method for detecting the presence of a moving submarine, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed frequency and amplitude.Problem BWhen determining the winner of a competition like the Mathematical Contest inModeling, there are generally a large number of papers to judge. Let's saythere are P=100 papers.A group of J judges is collected to accomplish thejudging. Funding for the contest constains both the number of judges that canbe obtained and amount of time that they can judge. For eample if P=100, thenJ=8 is typical.Ideally, each judge would read paper and rank-order them, but there are toomany papers for this. Instead, there will be a number of screening rounds inwhich each judge will read some number of papers and give them scores. Thensome selection scheme is used to reduce the number of papers under consideration: If the papers are rank-ordered, then the bottom 30% that eachjudge rank-orders could be rejected. Alternatively, if the judges do not rank-order, but instead give them numerical score (say, from 1 to 100),then all papers below some cut-off level could be rejected.The new pool of papers is then passed back to the judges, and the process is repeated.A concern is then the total number of papers that judge reads must besubstantially less than P. The process is stopped when there are only W papersleft. There are the winners. Typically for P=100, W=3.Your task is to determine a selection scheme, using a combination of rank-ordering, numerical scoring, and other methods, by which the final Wpapers will include only papers from among the "best" 2W papers. (By "best",we assume that there is an absolute rank-ordering to which all judges wouldagree.) For example, the top three papers. Among all such methods, the one thatrequired each judge to read the least number of papers is desired.Note the possibility of systematic bias in a numerical scoring scheme. For example, for a specific collection of papers, one judge could average 70points, while another could average 80 points. How would you scale your schemeto accommodate for changes in the contest parameters (P, J, and W)?1997 American MCM ProblemsProblem A The Velociraptor ProblemThe velociraptor,Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists think that it was a very tenacious hunter, and may have hunted in pairs or largerpacks .Unfortunately, there is no way to observe its hunting behavior in the wild as can be done with modern mammalian predators. A group of paleontologists has approached your team and asked for help in modeling the hunting behavior of the velociraptor. They hope to compare your results with field data reported by biologists studying the behaviors of lions, tigers, and similar predatory animals.The average adult velociraptor was 3 meters long with a hip height of 0.5 meters and an approximate mass of 45 kg. It is estimated that the animal could run extremely fast at speed of 60 km/hr for about 15 seconds. After the initial burst of speed ,the animal needed to stop and recover from a buildup of lactic acid in its muscles.Suppose that velociraptor preyed on Thescelosaurus neglectus, a herbivorous biped approximately the same size as the Velociraptor. A biomachanical analysis of a fossilized Thescelosaurus indicates that it could run at a speed of about 50 km/hr. for long period of time.Part1Assuming the velociraptor is a solitary hunter, design a mathematical model that describe a hunting strategy for a single velociraptor stalking and chasing a single Thescelosaurus as well as the evasive strategy of the prey. Assume that the Thescelosaurus can always detect the velociraptor when it comes within 15 meters .but may detect the predator at even greater ranges (up to 50 meters depending upon the habitat and weather conditions. Additionally ,due to its physical structure and strength, the velociraptorhas a limited turning radius when running at full speed. This radius is estimated to be three times the animal's hip height. On the other hand, the Thescelosaurus is extremely agile and has a turning radius of 0.5 meters.Part2Assuming more realistically that the velociraptor hunted in pairs, design a new model that describes a hunting strategy for two velociraptor stalking and chasing a single Thescelosaurus as well as the evasive strategy of the prey. Use the other assumptions and limitations given in Part 1.Problem B Mix Well For Fruitful DiscussionsSmall group meeting for the discussions of important issues, particular long-range planning ,are gaining popularity. It is believed that large groups discourage productive discussion and that a dominant personality will usually control and direct the discussion. Thus ,in corporate board meetings the board will meet in small groups to discuss issues before meeting as a whole, these smaller groups still tun the risk of control by a dominant personality. In an attempt to reduce this danger it is common to schedule several sessions with a different mix of people in each group.A meeting of An Tostal Corporation will be attended by 29 Board Members of which nine are in-house members(i.e., corporate employees).The meeting is to be an all-day affair with three sessions scheduled for the morning and four for the afternoon. Each session will take 45 minutes, beginning on the hour from 9:00 A.M. to 4:00 P.M., with lunch scheduled at noon. Each morning session will consist of six discussion groups with each discussion group led by one of the corporation's six senior officers. None of these officers are board members. Thus each senior officers will not be involved in the afternoon sessions and each of these sessions will consist of only four different discussion groups.The president of the corporation wants a list of board-member assignment to discussion groups for each of the seven sessions. The assignments should achieve as much of a mix of the members as much as possible. The ideal assignment would have each board member in a discussion group the same number of times while minimizing common membership of groups for the different sessions.The assignment should also satisfy the following criteria:1.For the morning sessions ,no board member should be in the same senior officer's discussion group twice.2.No discussion group should contain a disproportionate number of in-house members.Give a list of assignments for members 1-9 and 10-29 and officers 1-6.Indicate how well the criteria in the previous paragraphs are met. Since it is possible that some board members will cancel at the last minute or that some not scheduled will show up, an algorithm that the secretary could use to adjust the assignments with an hour's notice would be appreciated. It would be ideal if the algorithm could also be used to make assignments for future meetings involving different levels of participation for each type of attendee.1998 American MCM ProblemsProblem A MRI ScannersIntroductionIndustrial medical diagnostic machines known as Magnetic Resonance Imager (MRI) scan a three-dimensional object such as a brain, and deliver their results in the form of a three-dimensional array of pixel. Each pixel consists of one number indicating a color or a shade of gray that encodes a measure of water concentration in a small region of the scanned object at the location of the pixel .For instance,0 can picture high water concentration in black (ventricles, blood vessels),128 can picture a medium water concentration in gray(brain nuclei and gray matter),and 255 can picture a low water density in white (liquid-rich white matter consisting of myelinated axons).Such MRI scanners also include facilities to picture on a screen any horizontal or vertical slice through the three-dimensional array (slices are parallel to any of the three Cartesian coordinate axes ).Algorithms for picturing slices through oblique planes ,however ,are proprietary .Current algorithms are limited in terms of the angles and parameter options available ;are implemented only on heavily used dedicated workstations ;lack input capabilities for marking points in the picture before slicing; and tend to blur and "feather out" sharp boundaries between the original pixels.A more faithful, flexible algorithm implemented on a personal computer would be useful.(1)for planning minimally invasive treatments,(2)for calibrating the MRI machines,(3)for investigating structures oriented obliquely in space, such as post-mortem tissue sections in a animal research,(4)for enabling cross-sections at any angle through a brain atlas consisting (4)for enabling cross-sections at any angle through a brain atlas consistingof black-and-white line drawingTo design such an algorithm, one can access the value and locations of the pixels, but not the initial data gathered by the scanners.ProblemDesign and test an algorithm that produces sections of three-dimensional arrays by planes in any orientation in space, preserving the original gray-scale value as closely as possible.Data SetsThe typical data set consists of a three-dimensional array A of numbers A(i,j,k) which indicates the density A(i,j,k) of the object at the location (x,y,z)i,j,k. Typically A(i,j,k) can range from 0 to 255.In most applications the data set is quite large.Teams should design data sets to test and demonstrate their algorithms. The data sets should reflect conditions likely Teams should design data sets to test and demonstrate their algorithms. The data sets should reflect conditions likely to be of diagnostic interest. Teams should also characterize data sets the limit the effectiveness of their algorithms.SummaryThe algorithm must produce a picture of the slice of the three-dimensional array by a plane in space. The plane can have any orientation and any location in space.(The plane can miss some or all data points.)The result of the algorithm should be a model of the density of the scanned object over the selected plane.Problem B Grade InflationBackgroundSome college administrators are concerned about the grading at A Better Class(ABC) college. On average, the faculty at ABC have been giving out high grades(the average grade now given out is an A-),and it is impossible to distinguish between the good and mediocre students .The terms of a very generous scholarship only allow the top 10% of the students to be funded, so a class ranking is required.The dean had the thought of comparing each student to the other students in each class ,and using this information to build up a ranking. For example, if a student obtains an A in a class in which all students obtain an A, then this student is only "average" in this class. On the other hand, if a student obtain the only A in a class, then that student is clearly "above average". Combining information from several classes might allow students to be placed in deciles (top 10%,next 10%,ect.)across the college.ProblemAssuming that the grades given out are(A+,A-,B+,B-,...)can the dean's idea be made to work?Assuming that the grades given out are only (A,B,C,...)can the dean's idea be made to work?Can any other schemes produce a desired ranking?A concern is that the grade in a single class could change many student's deciles. Is this possible?Data SetsTeams should design data sets to test and demonstrate their algorithms. Teams should characterize data sets that limit the effectiveness of their algorithms.Mathematical Contest in Modeling 1999 ProblemsProblem A - Deep ImpactFor some time, the National Aeronautics and Space Administration (NASA) has been considering the consequences of a large asteroid impact on the earth.As part of this effort, your team has been asked to consider the effects of such an impact were the asteroid to land in Antarctica. There are concerns that an impact there could have considerably different consequences than one striking elsewhere on the planet.You are to assume that an asteroid is on the order of 1000 m in diameter, and that it strikes the Antarctic continent directly at the South Pole.Your team has been asked to provide an assessment of the impact of such an asteroid. In particular, NASA would like an estimate of the amount and location of likely human casualties from this impact, an estimate of the damage done to the food production regions in the oceans of the southern hemisphere, and an estimate of possible coastal flooding caused by large-scale melting of the Antarctic polar ice sheet.Problem B - Unlawful AssemblyMany public facilities have signs in rooms used for public gatherings which state that it is "unlawful" for the rooms to be occupied by more than a specified number of people. Presumably, this number is based on the speed with which people in the room could be evacuated from the room's exits in case of an emergency. Similarly, elevators and other facilities often have "maximum capacities" posted.Develop a mathematical model for deciding what number to post on such a sign as being the "lawful capacity". As part of your solution discuss criteria, other than public safety in the case of a fire or other emergency, that might govern the number of people considered "unlawful" to occupy the room (or space). Also, for the model that you construct, consider the differences between a room with movable furniture such as a cafeteria (with tables and chairs), a gymnasium, a public swimming pool, and a lecture hall with a pattern of rows and aisles. You may wish to compare and contrast what might be done for a variety of different environments: elevator, lecture hall, swimming pool, cafeteria, or gymnasium. Gatherings such as rock concerts and soccer tournaments may present special conditions.Apply your model to one or more public facilities at your institution (or neighboring town). Compare your results with the stated capacity, if one is posted. If used, your model is likely to be challenged by parties with interests in increasing the capacity. Write an article for the local newspaper defending your analysis.2000 Mathematical Contest in ModelingProblem A Air traffic ControlDedicated 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 AssignmentsWe 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 1An 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 smallerthan 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 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 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 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 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 (MCM)Problem A: Choosing a Bicycle WheelCyclists have different types of wheels they can use on their bicycles. The two basic typesof 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 fora 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 butthere is a choice of wheels for the rear.Task 1. Provide a table iving the wind peed 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 courseTask 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 landfallof Hurricane Floydin 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 reversalof traffic onI-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 northby 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 restrictionson 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 upI-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.问题 A: 选择自行车车轮骑自行车的人有几种不同类型的车轮可以用在他们的自行车上。
美国大学生数学建模竞赛试题_MCM_Problem_A
In addition to the required one-page summary for your MCM submission, your report must include a one-page non-technical explanation for users of the bathtub that describes your strategy while explaining why it is so difficult to get an evenly maintained temperature throughout the bath water.
Develop a model of the temperature of the bathtub water in ace and time to determine the best strategy the person in the bathtub can adopt to keep the temperature even throughout the bathtub and as close as possible to the initial temperature without wasting too much water.
2016 MCM Problem A A Hot Bath
A person fills a bathtub with hot water from a single faucet and settles into the bathtub to cleanse and relax. Unfortunately, the bathtub is not a spa-style tub with a secondary heating system and circulating jets, but rather a simple water containment vessel. After a while, the bath gets noticeably cooler, so the person adds a constant trickle of hot water from the faucet to reheat the bathing water. The bathtub is designed in such a way that when the tub reaches its capacity, excess water escapes through an overflow drain.
美国数学建模比赛历年试题
1985~2014年美国大学生数学建模竞赛题目集锦目录1985 MCM A: Animal Populations (3)1985 MCM B: Strategic Reserve Management (3)1986 MCM A: Hydrographic Data (4)1986 MCM B: Emergency-Facilities Location (4)1987 MCM A: The Salt Storage Problem (5)1987 MCM B: Parking Lot Design (5)1988 MCM A: The Drug Runner Problem (5)1988 MCM B: Packing Railroad Flatcars (6)1989 MCM A: The Midge Classification Problem (6)1989 MCM B: Aircraft Queueing (6)1990 MCM A: The Brain-Drug Problem (7)1990 MCM B: Snowplow Routing (8)1991 MCM A: Water Tank Flow (8)1991 MCM B: The Steiner Tree Problem (8)1992 MCM A: Air-Traffic-Control Radar Power (9)1992 MCM B: Emergency Power Restoration (9)1993 MCM A: Optimal Composting (11)1993 MCM B: Coal-Tipple Operations (12)1994 MCM A: Concrete Slab Floors (12)1994 MCM B: Network Design (13)1995 MCM A: Helix Construction (14)1995 MCM B: Faculty Compensation (14)1996 MCM A: Submarine Tracking (14)1996 MCM B: Paper Judging (14)1997 MCM A: The Velociraptor Problem (15)1997 MCM B: Mix Well for Fruitful Discussions (16)1998 MCM A: MRI Scanners (17)1998 MCM B: Grade Inflation (18)1999 MCM A: Deep Impact (19)1999 MCM B: Unlawful Assembly (20)2000 MCM A: Air Traffic Control (20)2000 MCM B: Radio Channel Assignments (21)2001 MCM A: Choosing a Bicycle Wheel (22)2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...) .. (23)2002 MCM A: Wind and Waterspray (25)2002 MCM B: Airline Overbooking (26)2003 MCM A: The Stunt Person (26)2003 MCM B: Gamma Knife Treatment Planning (27)2004 MCM A: Are Fingerprints Unique? (28)2004 MCM B: A Faster QuickPass System (28)2005 MCM A: Flood Planning (29)2005 MCM B: Tollbooths (29)2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation.. 29 2006 MCM B: Wheel Chair Access at Airports (30)2007 MCM A: Gerrymandering (31)2007 MCM B: The Airplane Seating Problem (32)2008 MCM A: Take a Bath (32)2008 MCM B: Creating Sudoku Puzzles (33)2009 MCM A: Designing a Traffic Circle (33)2009 MCM B: Energy and the Cell Phone (33)2010 MCM A: The Sweet Spot (35)2010 MCM B: Criminology (35)2011 MCM A: Snowboard Course (36)2011 MCM B: Repeater Coordination (36)2012 MCM A: The Leaves of a Tree (37)2012 MCM B: Camping along the Big Long River (37)2013 MCM A: The Ultimate Brownie Pan (38)2013 MCM B: Water, Water, Everywhere (38)2014 MCM A:The Keep-Right-Except-To-Pass Rule (39)2014 MCM B:College Coaching Legends (39)1985 MCM A: Animal PopulationsChoose a fish or mammal for which appropriate data are available to model it accurately. Model the animal's natural interactions with 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 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 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 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. (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 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 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 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?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 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 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)?The 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 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 block in the N-S direction and 20 seconds in the E-Wdirection. 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 uniformly 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 in 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 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.Construct 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 realize 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 vehicles, 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 in 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 within 2 degrees. The signalcame 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 degree angle direction. The helicopter's 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 optimal 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). Table 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 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 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 goabout classifying it?2. Apply 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 acarrier of a debilitating disease. Would you modify your classificationscheme and if so, how?1989 MCM B: Aircraft QueueingA common procedure at airports is to assign aircraft (A/C) to runways on afirst-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 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 passenger capacities varying from 100 to 400 in steps of50. Develop and analyze a mathematical model that takes into accountboth the travelers' and airlines' satisfaction.1990 MCM A: The Brain-Drug ProblemResearches on brain disorders test the effects of the new medical drugs – for example, dopamine against Parkinson's 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 research 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 another 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 samples.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 sectionThe solid lines of the map (see Figure 1) represent paved two-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 the 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 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% accuracy, 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 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 oneday[ table 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 the level drops to about 27.00 feet, and the pump stops when the level rises back to about 35.50 feet.1991 MCM B: 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 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 to13.4% (= 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 computed as shown in Figure 2.Suppose you wish to design a minimum costs spanning tree for a local network with 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 rectiline ar lines. Moreover, all ―phantom‖ stations must be located at lattice points (i.e., the coordinates must be integers). The cost for each line is its length.1. Find a minimal cost tree for the network.2. Suppose each stations has a cost w*d^(3/2), where d=degree of thestation. If w=1.2, find a minimal cost tree.3. Try to generalize this problem1992 MCM 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 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 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.1992 MCM B: Emergency Power RestorationPower companies serving coastal regions must have emergency response systems for power outages due to storms. 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 consultant to HECO power company. HECO possesses a computerized database with real time access to service calls that currently require the following information:▪time of report,▪type of requestor,▪estimated number of people affected, and▪location (x,y).Cre sites are located at coordinates (0,0) and (40,40), where x and y are in miles. The region serviced by HECO is within -65 < x < 60 and -50 < y < 50. The region is largely metropolitan with an excellent road network. Crews must return to their dispatch site only at the beginning and end of shift. Company policy requires that no work be initiated until the storm leaves the area, unless the facility is a commuter railroad or hospital, which may be processed immediately if crews are available.HECO has hired you to develop the objective criteria and schedule the work for the storm restoration requirements listed in Table 1 using their work force described in Table 2. Note that the first call was received at 4:20 A.M. and that the storm left the area at 6:00 A.M. Also note that many outages were not reported until much later in the day.HECO has asked for a technical rep ort for their purposes and an ―executive summary‖ in laymen's terms that can be presented to the media. Further, they would like recommendations for the future. To determine your prioritized scheduling system, you will have to make additional assumptions. Detail those assumptions. In the future, you may desire additional data. If so, detail the information desired.Table 1. Storm restoration requirements. (table incomplete)An environmentally conscious institutional cafeteria is recycling customers' uneaten food into compost by means of microorganisms. 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 compost. 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 culture's lack of appetite. 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 culture's activity, for instance, by optimizing the fungi culture's 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 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 culture's activity. In addition to the technical report following the format prescribed in the contest instructions, provide a one-page nontechnical recommendation forimplementation 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 the date completely composted [table omitted].1993 MCM 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 tipple's capacity 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 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 the 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 requirements, 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 tipple's loading operations. Your analysis should include the 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 reduce annual operations costs?▪Can this tipple support a fourth standard train every day?1994 MCM 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 principal 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 dwellings that do not need to rely on convection- that is, there is no need to rely on opening doors or windows to assist in temperature variation. The dwellings will be single-story, with concrete slab floors as the only foundation. You have been hired as a consultant to analyze the temperature variation in the concrete slab floor to determine if the temperature averaged over the floor surface can be maintained within a prescribed comfort zone throughout the year. If so, what size/shape of slabs will permit this?Part 1, Floor Temperature: Consider the temperature variation in a concrete slab given that the ambient temperature varies daily within the ranges given Table 1. Assume that the high occurs at noon and the low at midnight. Determine if slabs can be designed to maintain a temperature averaged over the floor surface within the prescribed comfort zone considering radiation only. Initially, assume that the heat transfer into the dwelling is through the exposed perimeter of the slab and that the top and bottom of the slabs are insulated. Comment on the appropriateness and sensitivity of these assumptions. If you cannot find a solution that satisfies Table 1, can you find designs that satisfy a Table 1 that you propose?and extend the analysis to temperature variation within the single-story dwelling. Can the house be kept within the comfort zone?Part 3, Cost of Construction: Suggest a design that considers HUD's objective of reducing or eliminating heating and cooling costs, considering construction restrictions and costs.1994 MCM B: Network DesignIn your company, information is shared among departments on a daily basis. This information includes the previous day's sales statistics and current production guidance. It is important to get this information out as quickly as possible. [Network diagram (with 5 nodes and 7 capacitated edges) omitted.] We are interested in scheduling transfers in an optimal way to minimize the total time it takes to complete them all. This minimum total time is called the makespan. Consider the three following situations for your company: [Three more network diagrams (on roughly 20 nodes each) omitted.]1995 MCM A: Helix ConstructionA small biotechnological company must design, prove, program and test a mathematical algorithm to locate ―in real time‖ all the inter sections of a helix and a plane in general positions in space. Design, justify, program and test a method to compute all the intersections of a plane and a helix, both in general positions (at any locations and with any orientations) in space. A segment of the helix may represent, for example, a helicoidal suspension spring or a piece of tubing in a chemical or medical apparatus. Theoretical justification of the proposed algorithm is necessary to verify the solution from several points of view, for instance, through mathematical proofs of parts of the algorithm, and through tests of the final program with known examples. Such documentation and tests will be required by government agencies for medical use.1995 MCM B: Faculty CompensationAluacha Balaclava College, and undergraduate facility, has just hired a new Provost whose first priority is the institution of a fair and reasonablefaculty-compensation plan. She has hired your consulting team to design a compensation system that reflects the following circumstances and principles: [Three paragraphs of details omitted] Design a new pay system, first without cost-of-living increases. Incorporate cost-of-living increases, and then finally, design a transition process for current faculty that will move all salaries towards your system without reducing anyone's salary. The Provost requires a detailed compensation system plan for implementation, as well as a brief, clear, executive summary outlining the model, its assumptions, strengths, weaknesses and expected results, which she can present to the Board and faculty. [A detailed table of current salaries is omitted.]1996 MCM A: Submarine TrackingThe world's oceans contain an ambient noise field. Seismic disturbances, surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large maving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, develop a method for detecting the presence of a moving submarine, its speed, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed frequency and amplitude.1996 MCM B: Paper JudgingWhen determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge. Let's say there are P=100 papers. A group of J judges is collected to accomplish the。
1986美国大学生数学建模竞赛英文版试题
1986 年美国大学生数学建模竞赛 MCM 试题1986 MCM A: Hydrographic DataThe table 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)?1986 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, 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 block 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 uniformly distributed on the streets bordering each block and that the facilities may be located anywhere on the streets.。
美国大学生数学建模竞赛试题_ICM_Problem_D
As you develop your model and prepare to test it, you will need to assemble a collection of data. Below are just some examples of the types of data you may find useful in this project. Depending on your exact model, some types of data may be very important and others may be entirely irrelevant. In addition to the sample sources provided below, you might want to consider a few important world events throughout history – if some recent big news events, such as the rumors of country-turned-pop singer Taylor Swift’s possible engagement had instead happened in 1860, what percentage of the population would know about it and how quickly; likewise, if an important person was assassinated today, how would that news spread? How might that compare to the news of US President Abraham Lincoln’s assassination?
历年美国数学建模竞赛题目
历年美国数学建模竞赛题目1985 A J 动物群体的常微分方程.pdf1985 A J 动物群体的管理.pdf1985 A O An Effective Method for Harvesting Salmon.pdf1985 A O Grizzly Bears in Yellowstone National Park.pdf1985 A O Population Dynamics of Deer.pdf1985 A O Population Dynamics of The Peruvian Anchovy.pdf1985 B J B题的若干知识.pdf1985 B J 战略物资的存贮管理.rar1985 B O Managing a Cobalt Stockpile.pdf1985 B O The Problem of Managing a Strategic Reserve.pdf1986 A O Contour Interpolation of Random Data.pdf1986 A O Contouring of Hydrographic Data.pdf1986 A O Interpolating a Topographical Map of The Ocean Floor.pdf 1986 A O Spline Analysis of Hydrographic Data.pdf1986 A O 水道测量数据.pdf1986 B J 应急设施位置.pdf1986 B J 应急设施的优化选址问题.pdf1986 B J 应急设施的位置.pdf1987 A J 盐的贮存.pdf1987 A O The Salt Problem—Making a Mountain Out of Molehills.pdf 1988 A J 关于毒品走私船位置问题的数学模型.pdf1988 B J 两辆平板车的装载问题.pdf1988 B J 两辆铁路平板车的装货问题.pdf1988 B O Locating a Drug Runner Miami Vice Style.pdf1989 A J 判别分析和蠓虫分类.pdf1989 A J 蠓的分类.pdf1989 A O Neural-Network Approach to Classification Problems.pdf 1989 B J 飞机起飞的最优次序.pdf1990 A J 扩散问题的偏微分方程模型.pdf1990 A J 精神病用药问题.pdf1990 A J 试题分析.pdf1990 A O Error-Function Diffusion A Dopamine–Fick’s Model.pdf 1990 B J 扫雪问题.pdf1990 B J 扫雪问题的数学模型.pdf1991 A J 估计水箱的水流量.pdf1991 A J 估计水箱的水流量模型.pdf1991 A J 水塔水流量估计.pdf1991 A J 逼近观察数据的一些样条模型.pdf1991 B J 可靠网络中生成树的优化模型.pdf1991 B J 最小Steiner生成树.pdf1991 B J 最小费用斯坦纳树的构造.pdf1991 B O Finding Optimal Steiner Trees.pdf1991 B P 水塔流量估计.rar1992 B J 应急电力修复系统的修复计划.pdf1992 B O Development of an Emergency-Response System.pdf1993 A J 通过数学建模解决混合物转化为有机肥最佳过程问题.pdf1993 A O Coal-Tipple Operations.pdf1993 B J 倒煤台的操作方案.pdf1993 B J 煤车装卸系统的优化操作.PDF1994 A J 房屋隔热经济效益核算.pdf1994 B J 计算机网络的最小接通时间.pdf1994 B J 计算机网络的最短传输时间.pdf1994 B M 信息传递最少用时的数学模型.pdf1994 B O Talking Fast Finding the Makespan of a Communications Network.pdf1995 A C Author’s Commentary The Outstanding Helix Intersections Papers.pdf1995 A JC 单个的螺旋线.pdf1995 A O A Specialized Root-Finding Method for Rapidly Determining the Intersections of a Plane and a Helix.pdf1995 A O Planes and Helices.pdf1995 A O The Single Helix.pdf1995 B H 学院教师的付薪方案.pdf1995 B L 工资调整系统.pdf1995 B L 教员工资分配调整方案.pdf1995 B O How to Keep Your Job as Provost.pdf1995 B O Long-Term and Transient Pay Scale for College Faculty.pdf1995 B O Paying Professors What They’re Worth.pdf1995 B O The World’s Most Complicated Payroll.pdf1996 A J The Outstanding Helix Intersections Papers.pdf1996 A M 利用环境噪声场探测无自噪声潜艇.pdf1996 A O Detection of a Silent Submarine.pdf1996 A O Gone Fishin.pdf1996 A O How to Locate a Submarine.pdf1996 A O Imaging Underwater Objects with Ambient Noise.pdf1996 A P The Outstanding Submarine Location Papers.pdf1996 B J The Outstanding Contest Judging Papers A.pdf1996 B J The Outstanding Contest Judging Papers B.pdf1996 B JC 竞赛择优问题.pdf1996 B JC 竞赛评卷仿真.pdf1996 B M 快速评卷的方案设计.pdf1996 B M 竞赛评判问题.pdf1996 B O Judging a Mathematics Contest.pdf1996 B O Modeling Better Modeling Judges.pdf1996 B O Select the Winners Fast.pdf1996 B O The Inconsistent Judge.pdf1996 B O The Paper Selection Scheme Simulation Analysis.pdf1997 A H 恐龙的追逐捕食模型.pdf1997 A J The Outstanding Velociraptor Papers.pdf1997 A O A Three-Phase Model for Predator–Prey Analysis.pdf1997 A O Lunch on the Run.pdf1997 A O Modeling Optimal Predator and Prey Strategies.pdf1997 A O Pursuit–Evasion Games in the Late Cretaceous.pdf1997 A O The Geometry and the Game Theory of Chases.pdf1997 B J The Outstanding Discussion Groups Papers.pdf1997 B M The Well-Mixed Assignments.pdf1997 B M 有效讨论的最优混合解.pdf1997 B O A Greedy Algorithm for Solving Meeting Mixing Problems.pdf1997 B O An Assignment Model for Fruitful Discussions.pdf1997 B O Meetings Bloody Meetings.pdf1997 B O Using Simulated Annealing.pdf1997 B P The Outstanding Discussion Groups Papers.pdf1998 A J Judge's Commentary The Outstanding Scanner Papers.pdf1998 A M A Quick Algorithm for MRI Problem.pdf1998 A M Image Reconstruction in MRI.pdf1998 A O A Method for Taking Cross Sections of Three-Dimensional Gridded Data.pdf1998 A O A Model for Arbitrary Plane Imaging, or the Brain in Pain Falls Mainly on the Plane.pdf1998 A O A Tricubic Interpolation Algorithm for MRI Image Cross Sections.pdf1998 A O MRI Slice Picturing.pdf1998 A P Proposer's Commentary The Outstanding Scanner Papers.pdf1998 B H Place Students in Deciles Reasonably.pdf1998 B O A Case for Stricter Grading.pdf1998 B O Alternatives to the Grade Point Average for Ranking Students.pdf1998 B O Grade Infation A Systematic Approach to Fair Achievement Indexing.pdf1998 B O Judge's Commentary The Outstanding Grade Inflation Papers.pdf1998 B P Practitioner's Commentary The Outstanding Grade Inflation Papers.pdf1999 A H The Assessment Metheod of Impact.pdf1999 A O Antarctic Asteroid Effects.pdf1999 A O Asteroid Impact at the South Pole A Model-Based Risk Assessment.pdf1999 A O Not an Armageddon.pdf1999 A O The Sky is Falling.pdf1999 B H How to Calculate the Lawful Capacity in the Constraied Condition.pdf1999 B H How to Calculate the Lawful Capcity in the Constrained Condition .pdf1999 B J Judge's Commentary The Outstanding Lawful Capacity Papers.pdf1999 B O Determining the People Capacity of a Structur.pdf1999 B O Don't Panic.pdf1999 B O Hexagonal Unpacking.pdf1999 B O Room Capacity Analysis Using a Pair of Evacuation Models.pdf1999 B O Standing Room Only.pdf2000 A J Judge's Commentary The Outstanding Air Traffic Control Papers.pdf2000 A M Channel Assignment Strategies for Cellular Phone Systems.pdf2000 A M The Model For Measuring Complexity of Air Traffic Control Predicting and Adjusting Path Conflicts.pdf2000 A O Air Traffic Control.pdf2000 A O The Iron Laws of Air Traffic Control.pdf2000 A O The Safe Distance Between Airplanes and the Complexity of an Airspace Sector.pdf 2000 A O You Make the Call Feasibility of Computerized Aircraft Control.pdf2000 B J Author Judge's Commentary The Outstanding Channel Assignment Papers.pdf2000 B O A Channel Assignment Model The Span Without a Face.pdf2000 B O Groovin'with the Big Band(width).pdf2000 B O Radio Channel Assignments.pdf2000 B O Utilize the Limited Frequency Resources Efficiently.pdf2000 B O We're Sorry,You're Outside the Coverage Area.pdf2000 C J Judge's Commentary The Outstanding Elephant Population Papers.pdf2000 C O A Computational Solution for Elephant Overpopulation.pdf2000 C O EigenElephants When Is Enough,Enough.pdf2000 C O Elephant Population A Linear Model.pdf2001 A J Author-Judge's Commentary The Outstanding Bicycle Wheel Papers.pdf2001 A O A Systematic Technique for Optimal Bicycle Wheel Selection.pdf2001 A O Can’t Quite Put Our Finger On It.pdf2001 A O Selection of a Bicycle Wheel Type.pdf2001 A O Spokes or Discs.pdf2001 A P Choosing a Bicycle Wheel.zip2001 B M Strategies for Escaping a Hurricane's Wrath.zip2001 B H Hurricane Evacuation .pdf2001 B J Judge's Commentary The Outstanding Hurricane Evacuation Papers.pdf2001 B M What If Another Floyd Escaping a Hurricane's Wrath.pdf2001 B M When a Hard Wind Blows the Traffic Slows.pdf2001 B O Jammin'with Floyd A Traffic Flow Analysis of South Carolina Hurricane Evacuation.pdf2001 B O Please Move Quickly and Quietly to the Nearest Freeway.pdf2001 B O Project H.E.R.O. Hurricane Evacuation Route Optimization.pdf2001 B O The Crowd Before the Storm.pdf2001 B O Traffic Flow Models and the Evacuation Problempdf.pdf2001 B P 飓风来临的最佳疏散方案.rar2001 C J Judge’s Commentary The Outstanding Zebra Mussel Papers.pdf2001 C O A Multiple Regression Model to Predict Zebra Mussel Population Growth.pdf 2001 C O Identifying Potential Zebra Mussel Colonization.pdf2001 C O Waging War Against the Zebra Mussel.pdf2002 A J Judge’s Commentary The Outstanding Wind and Waterspray Papers.pdf2002 A M Blowin'in the Wind.pdf2002 A M Fountain Spray as a Particle Model.pdf2002 A M Woner Control Beautiful Foutain.rar2002 A O A Foul Weather Fountain.pdf2002 A O Simulating a Fountain.pdf2002 A O The Fountain That Math Built.pdf2002 A O Wind and Waterspray.pdf2002 B H How much to overbook this flight.zip2002 B J Judge’s Commentary The Outstanding Airline Overbooking Papers.pdf2002 B M Whole.rar2002 B O ACE is High.pdf2002 B O Overbooking on Airlines.pdf2002 B O Probabilistically Optimized Airline Overbooking Strategies.pdf2002 B O The Airline Overbooking Problem.pdf2002 B O Things That Go Bump in the Flight.pdf2002 C M If we Scrub our land too much we may lose the LIZARDs.rar2002 C M Life Model of Florida Scrub Lizard.rar2002 C O Cleaning Up the Scrub Saving the Florida Scrub Lizard.pdf2002 C O Where's the Scrub Aye,There's the Rub.pdf2003 A H Shaken, not Stirred.pdf2003 A M The Stunt Person.rar2003 A O Cardboard Comfortable When it comes to Crashing.pdf2003 A O Safe Landings.pdf2003 A O Thinking Outside the Box and Over the Elephant.pdf2003 A O You Too Can Be James Bond.pdf2003 A P Design and Stack the Cardboard Boxes.pdf2003 A P The design of the buffer cardboard boxes.pdf2003 B M Optimization of Stereotactic Radiosurgery Treatment Planning.pdf2003 B O Shelling Tumors with Caution and Wiggles.pdf2003 B P Shelling Procedure and Optimization by Simulated Annealing For Sphere Packing.pdf 2003 C H Aviation Baggage Screening.pdf2003 C H Security Screening at Airport.pdf2003 C H To Screen or Not.pdf2003 C M Aviation Baggage Screening Smart Approach to Screen.rar2003 C P Aviation Baggage Screening.pdf2004 A J Editor's Commentary Fingerprint Identification .pdf2004 A J Judge's Commentary The Outstanding Fingerprints Papers.pdf2004 A J Publisher's Editorial The Good Fight.pdf2004 A M Are Fingerprints Unique.pdf2004 A M Are Fingerprints Unique.rar2004 A M Fe-Fi-Fo Thumb.pdf2004 A O Can't Quite Put Our Finger On It.pdf2004 A O Not Such a Small Whorl After All.pdf2004 A O The Myth of The Myth of Fingerprints.pdf2004 A O Z Rule of Thumb Prints Beat DNA.pdf2004 B H a Faster QuickPass System.pdf2004 B H Magic Regulation Scheme for QuickPass System.pdf2004 B J Editor's Commentary Fingerprint Identification .pdf2004 B J Judges' Commentary The Quick Pass Fusaro Award Paper.pdf2004 B M Virtual Lines in Topoland with these Designs.pdf2004 B O A Myopic Aggregate-Decision Model for Reservation Systems in Amusement Parks.pdf 2004 B O An Adaptive Approach to Virtual Queing.pdf2004 B O Developing Improved Algorithms for QuickPass Systems.pdf2004 B O Developing Improved Algorithms for QuickPass Systems.pdf .pdf2004 B O KalmanQueue An Adaptive Approach to Virtual Queueing.pdf2004 B O Theme-Park Queueing Systems.pdf2004 B O Z Theme Park Simulation with a Nash-Equilibrium-Based Visitor Behavior Model.pdf 2004 B P Make Your Way Faster.pdf2004 B P Optimized QuickPass System.pdf2004 B P You Must Be at Least This Tall to Ride This Paper.pdf2004 C H ?IT Security Keep Hackers and Virus Out.pdf2004 C J Authors' Commentary The Outstanding Information Technology Security Papers.pdf 2004 C J Judge's Commentary The Outstanding Information Technology Security Papers.pdf 2004 C O Catch Thieves Online IT Security.pdf2004 C O Firewalls and Beyond Engineering IT Security.pdf2004 C O It's All About the Bottom Line.pdf2004 C O Making the CIA Work for You.pdf2005 A J Judge's Commentary The Outstanding Flood Planning Papers.pdf2005 A M One Two Step .pdf2005 A O Analysis of Dam Failure in the Saluda River Valley.pdf2005 A O From Lake Murray to a Dam Slurry.pdf2005 A O Through the Breach Modeling Flooding from a Dam Failure in South Carolina.pdf 2005 A O Z Catastrophic Consequences of Earthquake Destruction of the Saluda Dam.pdf 2005 B H For Whom the Booth Tolls .pdf2005 B H Is the Number of Tollbooths Optimal.pdf2005 B H Modeling Toll Plaza Behavior Using.pdf2005 B H Optimal Design of Toll Plaza.pdf2005 B H ?Pass the Plaza more Quickly .pdf2005 B J Judge's Commentary The Outstanding Tollbooths Papers.pdf2005 B M Giving Queueing the Booth.pdf2005 B O A Quasi-Sequential Cellular-Automaton Approach to Traffic Modeling.pdf2005 B O A Single-Car Interaction Model of Traffic for a Highway Toll Plaza.pdf2005 B O For Whom the Booth Tolls.pdf2005 B O Lane Changes and Close Following Troublesome Tollbooth Traffic.pdf2005 B O The Booth Tolls for Thee .pdf2005 B O The Booth Tolls for Thee.pdf2005 B O The Multiple Single Server Queueing System.pdf2005 B O Two Tools for Tollbooth Optimization.pdf2005 C H A Projection of Southeast Alaskan Salmon Populations.pdf2005 C H Between a Rockfish and a Hard Plaice.pdf2005 C H The future of “black gold”.pdf2005 C H When will the oil run out.pdf2005 C J Author's Commentary The Outstanding Exhaustible Resource Papers.pdf2005 C J Editorial Where Else to Publish.pdf2005 C J Judge's Commentary The Outstanding Exhaustible Resource Papers.pdf2005 C O Preventing the Hydrocalypse A Model for Predicting and Managing Worldwide Water Resource.pdf2005 C O The Coming Oil Crisis.pdf2005 C O The Petroleum Armageddon.pdf2006 A H A Simulated Annealing Approach to Irrigation.pdf2006 A H Minimizing Maintenance Cost for Hand-Moved Irrigation Systems.pdf2006 A H On Portable Irrigation Systems .pdf2006 A H Optimal Design of Irrigation Schedule.pdf2006 A J Judge's Commentary The Outstanding Irrigation Problem Papers.pdf2006 A M Optimizing a Handmove Sprinkler System .pdf2006 A M Piping Hot Weather.pdf2006 A M Positioning and Moving Sprinkler Systems for Irrigation.rar2006 A O Fastidious Farmer Algorithms (FFA).pdf2006 A O Fastidious Farmer Algorithms.pdf2006 A O Optimization of Irrigation.pdf2006 A O Z A Schedule for Lazy but Smart Ranchers.pdf2006 A O Z Developing Improved Algorithms for Irrigation Systems.pdf2006 A O Z Optimization of Irrigation.pdf2006 A O Z Sprinkle, Sprinkle, Little Yard.pdf2006 A O Z Sprinkler Systems for Dummies Optimizing a Hand-Moved Sprinkler System.pdf 2006 A P Positioning and Moving Sprinkler Systems for Irrigation .pdf2006 B H The Scheme of the Wheelchair Dispatch and Cost Analysis for Epsilon Airlines.pdf 2006 B H Transfer Suffers NEVER.pdf2006 B J Judges' Commentary The Fusaro Award Wheelchair Paper.pdf2006 B J Special Section on the MCM Judges Commentary The Fusaro Award Wheelchair Paper.pdf 2006 B M Minimal Costs for Serving Disabilities.pdf2006 B M Operational Research for Wheelchair Service Provided by Epsilon Airlines.pdf 2006 B M sly_airport.rar2006 B M When the Model Hits the Runway.pdf2006 B O A Simulation-Driven Approach for a Cost-Efficient Airport Wheelchair Assistance Service.pdf2006 B O Application of Min-Cost Flow to Airline Accessibility Services.pdf2006 B O Z A Simulation-Driven Approach for a Cost-Efficient Airport Wheelchair Assistance Service.pdf2006 B O Z Cost Minimization of Providing a Wheelchair Escort Service.pdf2006 B O Z Minimization of Cost for Transfer Escorts in an Airport Terminal.pdf2006 B O Z Profit Maximizing Allocation of Wheelchairs in a Multi-Concourse Airport.pdf 2006 C H Fighting against AIDS.pdf2006 C H War of the World Fight against AIDS.pdf2006 C J Author's Commentary The Outstanding HIV AIDS Papers.pdf2006 C J HIV The Math..pdf2006 C M AIDS A Global Crisis.pdf2006 C O AIDS Modeling a Global Crisis and Australia.pdf2006 C O Managing the HIV AIDS Pandemic 2006-2055.pdf2006 C O Managing the HIVAIDS Pandemic.pdf2006 C O The Spreading HIV AIDS Problem.pdf2006 C O The United Nations and the Quest for the Holy Grail (of AIDS).pdf2006 C O The United Nations and the Quest for the Holy Grail.pdf2007 A H Genetic Algorithm for Non-Partisan Legislative Districting.pdf2007 A O A Cluster-Theoretic Approach to Political Districting.pdf2007 A O Applying Voronoi Diagrams to the Redistricting Problem.pdf2007 A O When Topologists Are Politicians.pdf2007 B H A Practical Approach to Boarding Deboarding an A380.pdf2007 B H The Airplane Seating Problem 2.pdf2007 B H The Airplane Seating Problem.pdf2007 B H 朱姝（自动化）、朱俊华（自动化）、丁金金（信息与计算科学）.pdf2007 B H 陈侠航（数学与应用数学）何军（测控技术与仪器）杨水生（数学与应用数学）.pdf 2007 B M A Quadrilateral Approach to Congressional Districting.pdf2007 B M An Analysis of the Kidney Transplant Network.pdf2007 B O Boarding at the Speed of Flight.pdf2007 B O Novel Approaches to Airplane Boarding.pdf2007 C C Organ Transplant The Kidney Exchange Problem.pdf2007 C H Kidney Exchange.pdf2007 C H Organ Transplant The Kidney Exchange Problem 2.pdf2007 C H Organ Transplant The Kidney Exchange Problem.pdf2007 C H 王教团（信息与计算科学）周朝卫（信息与计算科学）周龙飞（信息管理与信息系统）.pdf 2007 C J Author's Commentary The Outstanding Kidney Exchange Papers.pdf2007 C J Judges' Commentary The Outstanding Kidney Exchange Papers.pdf2007 C J Write Your Own Contest Entry.pdf2007 C M More Kidney Donors More Lives Can Be Saved.pdf2007 C O Analysis of Kidney Transplant System Using Markov Process Models.pdf2007 C O Optimizing the Effectiveness of Organ Allocation.pdf2007 C P Practitioner's Commentary The Outstanding Kidney Exchange Papers.pdf。
美国数学建模比赛历年试题
2003 MCM ProblemsPROBLEM A: The Stunt PersonAn exciting action scene in a movie is going to be filmed, and you are the stunt coordinator! A stunt person on a motorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. You need to protect the stunt person, and also use relatively few cardboard boxes (lower cost, not seen by camera, etc.).Your job is to:•determine what size boxes to use•determine how many boxes to use•determine how the boxes will be stacked•determine if any modifications to the boxes would help•generalize to different bined weights (stunt person & motorcycle) and different jump heightsNote that, in "Tomorrow Never Dies", the James Bond character on a motorcycle jumps over a helicopter.PROBLEM B: Gamma Knife Treatment PlanningStereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographicallywell-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. 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 heavy 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 collimator 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 is 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 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 the 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: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 thetarget volume must be covered by shots.e as few shots as possible.Your tasks are to formulate 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 and 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 and 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 falls 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 BAuthors: Bill Fox and Rich WestTitle: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend 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 reservations 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 bumped 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 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 revenue 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 pany's revenue is maximized. Insure that your model reflects the issues above, and consider alternatives for handling "bumped" passengers. Additionally, write a short memorandum to the airline's 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 system 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 has 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 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 the 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 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 AssignmentsWe 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 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 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 interferenceRequirement A: 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 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 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 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 findingsMCM2000问题A 空间交通管制为加强安全并减少空中交通指挥员的工作量，联邦航空局(FAA)考虑对空中交通管制系统添加软件，以便自动探测飞行器飞行路线可能的冲突，并提醒指挥员。
2000-2013美国数学建模竞赛(MCM、ICM)历年试题汇总
目录2000 年美国大学生数学建模竞赛MCM、ICM 试题 (3)2000 MCM A: Air Traffic Control (3)2000 MCM B: Radio Channel Assignments (3)2000 ICM: Elephants: When is Enough, Enough? (5)2001 年美国大学生数学建模竞赛MCM、ICM 试题 (7)2001 MCM A: Choosing a Bicycle Wheel (7)2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...). (8)2001 ICM: Our Waterways - An Uncertain Future (10)2002 年美国大学生数学建模竞赛MCM、ICM 试题 (14)2002 MCM A: Wind and Waterspray (14)2002 MCM B: Airline Overbooking (14)2002 ICM: Scrub Lizards (15)2003 年美国大学生数学建模竞赛MCM、ICM 试题 (19)2003 MCM A: The Stunt Person (19)2003 MCM B: Gamma Knife Treatment Planning (19)2003 ICM: Aviation Baggage Screening Strategies: To Screen or Not to Screen, that is the Question (20)2004 年美国大学生数学建模竞赛MCM、ICM 试题 (24)2004 MCM A: Are Fingerprints Unique? (24)2004 MCM B: A Faster QuickPass System (24)2004 ICM: To Be Secure or Not to Be? (24)2005 年美国大学生数学建模竞赛MCM、ICM 试题 (25)2005 MCM A: Flood Planning (25)2005 MCM B: Tollbooths (25)2005 ICM: Nonrenewable Resources (25)2006 年美国大学生数学建模竞赛MCM、ICM 试题 (27)2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation (27)2006 MCM B: Wheel Chair Access at Airports (27)2006 ICM: Trade-offs in the fight against HIV/AIDS (28)2007 年美国大学生数学建模竞赛MCM、ICM 试题 (32)2007 MCM A: Gerrymandering (32)2007 MCM B: The Airplane Seating Problem (32)2007 ICM: Organ Transplant: The Kidney Exchange Problem (33)2008 年美国大学生数学建模竞赛MCM、ICM 试题 (38)2008 MCM A: Take a Bath (38)2008 MCM B: Creating Sudoku Puzzles (38)2008 ICM: Finding the Good in Health Care Systems (38)2009 年美国大学生数学建模竞赛MCM、ICM 试题 (40)2009 MCM A: Designing a Traffic Circle (40)2009 MCM B: Energy and the Cell Phone (40)2009 ICM: Creating Food Systems: Re-Balancing Human-Influenced Ecosystems41 2010年美国大学生数学建模竞赛 MCM、ICM 试题 (42)2010 MCM A: The Sweet Spot (42)2010 MCM B: Criminology (43)2010 ICM: The Great Pacific Ocean Garbage Patch (44)2011年美国大学生数学建模竞赛 MCM、ICM 试题 (45)2011 MCM A: Snowboard Course (45)2011 MCM B: Repeater Coordination (45)2011 ICM: Environmentally and Economically Sound (46)2012年美国大学生数学建模竞赛 MCM、ICM 试题 (48)2012 MCM A: The Leaves of a Tree (48)2012 MCM B: Camping along the Big Long River (50)2012 ICM: Modeling for Crime Busting (51)2013年美国大学生数学建模竞赛 MCM、ICM 试题 (59)2013 MCM A: The Ultimate Brownie Pan (59)2013 MCM B: Water, Water, Everywhere (61)2013 ICM: NetworkModeling of Earth's Health (62)2000 年美国大学生数学建模竞赛MCM、ICM 试题2000 MCM 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 system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analysit 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: And 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 sector1.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.2000 MCM B: Radio Channel AssignmentsWe 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 grix (honeycomb-style), as shown in Figure 1, where a transmitter is located at the center of each hexagon.An interval of the frequency spectrum is to be alloted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1,2,3, … . Each transmitter wil 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 assugn 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 contrainsts on the 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 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these contraints, what can we say about the span in Figure 1?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 cna 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.2000 ICM: Elephants: When is Enough, Enough?“Ultimately, if a habitat is undesirably changed by elephants, then their removal should be considered -even by culling.”National Geographic (Earth Almanac) –December 1999 A large National Park in South Africa contains approximately 11,000 elephants. Management policy requires a healthy environment that can maintain a stable herf of 11,000 elephants. Each year park rangers count the elephant population. During the past 20 years whole herds have been removed to keep the population as close to 11,000 as possible. The process involved shooting (for the most part) and occasionally relocating approximately 600 to 800 elephants per year.Recently, there has been a public outcry against the shooting of these elephants. In addition, it is no longer feasible to relocate even a small population of elephants each year. A contraceptive dart, however, has been developed that can prevent a mature elephant cow from conceiving for a period of two years.Here is some information about eh elephants in the Park:∙There is very little emigration of immigration of elephants.∙The gender ratio is very close to 1:1 and control measures have endeavored to maintain parity.∙The gender ratio of newborn calves is also about 1:1. Twins are born about 1.35% of the time.∙Cows first conceive between the ages of 10 and 12 and produce, on average, a calf every 3.5 years until they reach an age of about 60.Gestation is approximately 22 months.∙The contraceptive dart causes an elephant cow to come into oestrus every month (but not conceiving). Elephants usually have courtship only once in 3.5 years, so the monthly cycle can cause additional stress.∙ A cow can be darted every year without additional detrimental effects. A mature elephant cow will not be able to conceive for 2 years after thelast darting.∙Between 70% and 80% of newborn calves survive to age 1 year.Thereafter, the survival rate is uniform across all ages and is very high(over 95%), until about age 60; it is a good assumption that elephantsdie before reading age 70.There is no hunting and negligible poaching in the Park.The park management has a rough data file of the approximate ages and gender of the elephants they have transported out of the region during the past 2 years. This data is available on website: icm2000data.xls. Unfortunately no data is available for the elephants that have been shot or remain in the Park.Your overall task is to develop and use models to investigate how the contraceptive dart might be used for population control. Specifically:Task 1: Develop and use a model to speculate about the likely survival rate for elephants aged 2 to 60. Also speculate about the current age structure of the elephant population.Task 2: Estimate how many cows would need to be darted each year to keep the population fixed at approximately 11,000 elephants. Show how the uncertainty in the data at your disposal affects your estimate. Comment on any changes in the age structure of the population and how this might affect tourists. (You may want to look ahead about 30-60 years.)Task 3: If it were feasible to relocate between 50 and 300 elephants per year, how would this reduce the number of elephants to be darted? Comment on the trade-off between darting and relocation.Task 4: Some opponents of darting argue that if there were a sudden loss of a large number of elephants (due to disease or uncontrolled poaching), even if darting stopped immediately, the ability of the population to grow again would be seriously impeded. Investigate and respond to this concer.Task 5: The management in the Park is skeptical about modeling. In particular, they argue that a lack of complete data makes a mockery of any attempt to use models to guide their decision. In addition to your technical report, include a carefully crafted report (3-page maximum) written explicitly for the park management that responds to their concerns and provides advice. Also suggest ways to increase the park managers confidence in your model and your conclusions.Task 6: If your model works, other elephant parks in Africa would be interested in using it. Prepare a darting plan for parks of various sizes (300-25,000 elephants), with slightly different survival rates and transportation possibilities.2001 年美国大学生数学建模竞赛MCM、ICM 试题2001 MCM A: Choosing a Bicycle WheelCyclists 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 startingfrom zero percent to ten percent in one percent increments. (Roadgrade is defined to be the ratio of the total rise of a hill divided by thelength 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 proportionalto 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.2001 MCM 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-boundlanes of I-26 into two lanes of Columbia-bound traffic, essentiallyturning the entire I-26 into one-way traffic, significantly improveevacuation traffic flow?2.In 1999, the simultaneous evacuation of the state's entire coastal regionwas ordered. Would the evacuation traffic flow improve under analternative strategy that staggers the evacuation, perhapscounty-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 moretemporary 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 whatconditions should there be restrictions on vehicle types or numbers ofvehicles brought in order to guarantee timely evacuation?6.It has been suggested that in 1999 some of the coastal residents ofGeorgia and Florida, who were fleeing the earlier predicted landfalls ofHurricane Floyd to the south, came up I-95 and compounded the traffic problems. How big an impact can they have on the evacuation trafficflow? Clearly identify what measures of performance are used tocompare 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.2001 ICM: Our Waterways - An Uncertain FutureZebra mussels, Dreissena polymorpha, are small, fingernail-sized, freshwater mollusks unintentionally introduced to North America via ballast water from a transoceanic vessel. Since their introduction in the mid 1980s, they have spread through all of the Great Lakes and to an increasing number of inland waterways in the United States and Canada. Zebra mussels colonize on various surfaces,such as docks, boat hulls, commercial fishing nets, water intake pipes and valves, native mollusks and other zebra mussels. Their only known predators, some diving ducks, freshwater drum, carp, and sturgeon, are not numerous enough to have a significant effect on them. Zebra mussels have significantly impacted the Great Lakes ecosystem and economy. Many communities are trying to control or eliminate these aquatic pests. SOURCE: Great Lakes Sea Grant Network /.Researchers are attempting to identify the environmental variables related to the zebra mussel infestation in North American waterways. The relevant factors that may limit or prevent the spread of the zebra mussel are uncertain. You will have access to some reference data to include listings of several chemicals and substances in the water system that may affect the spread of the zebra mussel throughout waterways. Additionally, you can assume individual zebra mussels grow at a rate of 15 millimeters per year with a life span between 4 - 6 years. The typical mussel can filter 1 liter of water each day.Requirement A: Discuss environmental factors that could influence the spread of zebra mussels.Requirement B: Utilizing the chemical data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAChem1.xls, and the mussel population data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAPopulation 1.xls model the population growth of zebra mussels in Lake A. Be sure to review the Information about the collection of the zebra mussel data. Requirement C: Utilizing additional data on Lake A from another scientist provided at :ap/undergraduate/contests/icm/imagesdata/LakeAChem2.xls and additional mussel population data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAPopulation 2.xls corroborate the reasonableness of your model from Requirement B. As a result of this additional data, adjust your earlier model. Analyze the performance of your model. Discuss the sensitivity of your model. Requirement D: Utilizing the Chemical data from two lakes (Lake B and Lake C) in the United States provided atap/undergraduate/contests/icm/imagesdata/LakeB.xls and ap/undergraduate/contests/icm/imagesdata/LakeC.xls determine if these lakes are vulnerable to the spread of zebra mussels. Discuss your prediction.Requirement E: The community in the vicinity of Lake B (in requirement D) is considering specific policies for the de-icing of roadways near the lake duringthe winter season. Provide guidance to the local government officials regarding a policy on “de-icing agents.”In your guidance include predictions on the long-term impact of de-icing on the zebra mussel population. Requirement F: It has been recommended by a local community in the United States to introduce round goby fish. Zebra mussels are not often eaten by native fish species so they represent a dead end ecologically. However, round gobies greater than 100 mm feed almost exclusively on zebra mussels. Ironically, because of habitat destruction, the goby is endangered in its native habitat of the Black and Caspian Seas in Russia. In addition to your technical report, include a carefully crafted report (3-page maximum) written explicitly for the local community leaders that responds to their recommendation to introduce the round goby. Also suggest ways to help reduce the growth of the mussel within and among waterways.Information about the collection of the zebra mussel dataThe developmental state of the Zebra mussel is categorized by three stages: veligers (larvae), settling juveniles, and adults. Veligers (microscopic zebra mussel larvae) are free-swimming, suspended in the water for one to three weeks, after which they begin searching for a hard surface to attach to and begin their adult life. Looking for zebra mussel veligers is difficult because they are not easily visible by the naked eye. Settled juvenile zebra mussels can be felt on smooth surfaces like boats and motors. An advanced zebra mussel infestation can cover a surface, even forming thick mats sometimes reaching very high densities. The density of juveniles was determined along the lake using three 15×15 cm settling plates. The top plate remained in the water for the entire sampling season (S - seasonal) to estimate seasonal accumulation. The middle and bottom plates are collected after specific periods (A –alternating ) of time denoted by “Lake Days”in the data files.The settling plates are placed under the microscope and all juveniles on the undersides of the plate are counted and densities are reported as juveniles/m^2.2002 年美国大学生数学建模竞赛MCM、ICM 试题2002 MCM A: Wind and 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 and 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 falls 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.2002 MCM B: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend 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 reservations 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 bumped 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 light of the current situation: Less flights by airlines from point A to point B Heightened security at and around airports Passengers' fear Loss of billions of dollars in revenue by airlines to dateBuild a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company 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 company's revenue is maximized. Insure that your model reflects the issues above, andconsider alternatives for handling “bumped”passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.2002 ICM: Scrub LizardsThe Florida scrub lizard is a small, gray or gray-brown lizard that lives throughout upland sandy areas in the Central and Atlantic coast regions of Florida. The Florida Committee on Rare and Endangered Plants classified the scrub lizard as endangered.You will find a fact sheet on the Florida Scrub Lizard at/undergraduate/contests/mcm/contests/2002/problem s/icm2002data/scrublizard.pdfThe long-term survival of the Florida scrub lizard is dependent upon preservation of the proper spatial configuration and size of scrub habitat patches.Task 1: Discuss factors that may contribute to the loss of appropriate habitat for scrub lizards in Florida. What recommendations would you make to the state of Florida to preserve these habitats and discuss obstacles to the implementation of your recommendations?Task 2: Utilize the data provided in Table 1 to estimate the value for Fa (the average fecundity of adult lizards); Sj (the survivorship of juvenile lizards- between birth and the first reproductive season); and Sa (the average adult survivorship).Table 1Summary data for a cohort of scrub lizards captured and followed for 4 consecutive years. Hatchling lizards (age 0) do not produce eggs during the summer they are born. Average clutch size for all other females is proportional to body size according to the function y = 0.21*(SVL)-7.5, where y is the clutch size and SVL is the snout-to-vent length in mm.Year Age Total NumberLivingNumber of LivingFemalesAvg. Female Size(mm)1 0 972 495 30.32 1 180 92 45.83 2 20 11 55.84 3 2 2 56.0Task 3: It has been conjectured that the parameters Fa , Sj , and Sa , are related to the size and amount of open sandy area of a scrub patch. Utilize the data provided in Table 2 to develop functions that estimate Fa, Sj , and Sa for different patches. In addition, develop a function that estimates C, the carrying capacity of scrub lizards for a given patch.Table 2Summary data for 8 scrub patches including vital rate data for scrub lizards. Annual female fecundity (Fa), juvenile survivorship (Sj), and adult survivorship (Sa) are presented for each patch along with patch size and the amount of open sandy habitat.Patch Patch Size (ha) Sandy Habitat (ha) Fa Sj Sa Density (lizards/ha)a 11.31 4.80 5.6 0.12 0.06 58b 35.54 11.31 6.6 0.16 0.10 60c 141.76 51.55 9.5 0.17 0.13 75d 14.65 7.55 4.8 0.15 0.09 55e 63.24 20.12 9.7 0.17 0.11 80f 132.35 54.14 9.9 0.18 0.14 82g 8.46 1.67 5.5 0.11 0.05 40h 278.26 84.32 11.0 0.19 0.15 115Task 4: There are many animal studies that indicate that food, space, shelter, or even reproductive partners may be limited within a habitat patch causing individuals to migrate between patches. There is no conclusive evidence on why scrub lizards migrate. However, about 10 percent of juvenile lizards do migrate between patches and this immigration can influence the size of the population within a patch. Adult lizards apparently do not migrate. Utilizing the data provided in the histogram below estimate the probability of lizards surviving the migration between any two patches i and patch j.Table 3HistogramMigration data for juvenile lizards marked, released, and recaptured up to 6 months later. Surveys for recapture were conducted up to 750m from release sites.Task 5: Develop a model to estimate the overall population size of scrub lizards for the landscape given in Table 3. Also, determine which patches are suitable for occupation by scrub lizards and which patches would not support a viable population.Patch size and amount of open sandy habitat for a landscape of 29 patches located on the Avon Park Air Force Range. See:/undergraduate/contests/icm/2002problem/map.jpg for a map of the landscape.Patch Identification Patch Size (ha) Sandy Habitat (ha)1 13.66 5.382 32.74 11.913 1.39 0.234 2.28 0.765 7.03 3.626 14.47 4.387 2.52 1.998 5.87 2.499 22.27 8.44。
全美数学建模往年试题2002
2002 Mathematical Contest in ModelingThe ProblemsProblem AAuthors: Tjalling YpmaTitle: Wind and 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 and 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 falls 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 BAuthors: Bill Fox and Rich WestTitle: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend 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 reservations 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 bumped 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 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 revenue received by an airline company 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 company's revenue is maximized. Insure that your model reflects the issues above, and consider alternatives for handling "bumped" passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings andanalysis.。
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历年美国大学生数学建模赛题目录MCM85问题-A 动物群体的管理 (3)MCM85问题-B 战购物资储备的管理 (3)MCM86问题-A 水道测量数据 (4)MCM86问题-B 应急设施的位置 (4)MCM87问题-A 盐的存贮 (4)MCM87问题-B 停车场 (5)MCM88问题-A 确定毒品走私船的位置 (5)MCM88问题-B 两辆铁路平板车的装货问题 (5)MCM89问题-A 蠓的分类 (5)MCM89问题-B 飞机排队 (6)MCM90-A 药物在脑内的分布 (6)MCM90问题-B 扫雪问题 (6)MCM91问题-B 通讯网络的极小生成树 (6)MCM 91问题-A 估计水塔的水流量 (7)MCM92问题-A 空中交通控制雷达的功率问题 (7)MCM 92问题-B 应急电力修复系统的修复计划 (7)MCM93问题-A 加速餐厅剩菜堆肥的生成 (7)MCM93问题-B 倒煤台的操作方案 (8)MCM94问题-A 住宅的保温 (8)MCM 94问题-B 计算机网络的最短传输时间 (9)MCM-95问题-A 单一螺旋线 (9)MCM95题-B A1uacha Balaclava学院 (10)MCM96问题-A 噪音场中潜艇的探测 (10)MCM96问题-B 竞赛评判问题 (10)MCM97问题-A Velociraptor(疾走龙属)问题 (11)MCM97问题-B为取得富有成果的讨论怎样搭配与会成员 (11)MCM98问题-A 磁共振成像扫描仪 (12)MCM98问题-B 成绩给分的通胀 (13)MCM99问题-A 大碰撞 (13)MCM99问题-B “非法”聚会 (13)MCM2000问题-A空间交通管制 (13)MCM2000问题-B: 无线电信道分配 (14)MCM2001问题- A: 选择自行车车轮 (14)MCM2001问题-B 逃避飓风怒吼（一场恶风...） .. (15)MCM2001问题-C我们的水系-不确定的前景 (15)MCM2002问题-A风和喷水池 (15)MCM2002问题-B航空公司超员订票 (16)MCM2002问题-C (16)MCM2003问题-A: 特技演员 (17)MCM2003问题-B: Gamma刀治疗方案 (18)MCM2003问题-C航空行李的扫描对策 (18)MCM2004问题-A：指纹是独一无二的吗？ (18)MCM2004问题-B：更快的快通系统 (18)MCM2004问题-C安全与否？ (19)MCM2005问题A.水灾计划 (19)MCM2005B.Tollbooths (19)MCM2005问题C：不可再生的资源 (20)MCM2006问题A: 用于灌溉的自动洒水器的安置和移动调度 (20)MCM2006问题B: 通过机场的轮椅 (20)MCM2006问题C : 抗击艾滋病的协调 (21)MCM2008问题A:给大陆洗个澡 (23)MCM2008问题B：建立数独拼图游戏 (23)MCM85问题-A 动物群体的管理在一个资源有限，即有限的食物、空间、水等等的环境里发现天然存在的动物群体。
试选择一种鱼类或哺乳动物(例如北美矮种马、鹿、免、鲑鱼、带条纹的欧洲鲈鱼)以及一个你能获得适当数据的环境，并形成一个对该动物群体的捕获量的最佳方针。
MCM85问题-B 战购物资储备的管理钴对许多工业是必不可少的(1979年仅国防需要就占了全世界钴生产量的17％)，但是钴不产生在美国。
大部分钴来自政治上不稳定的构F地区。
见图85B-1，85B-2，85B-3。
1946年制订的战略和稀有作战物资存贮法令要求钴的储存量应保证美国能渡过三年战争时期。
50年代政府按要求存贮了，并在70年代卖掉了大部分贮量，而在70年代后期决定重新贮存，贮存的指标是8540万磅，到1982年获得了贮量的一半。
试建立一个战略金属钴的储存管理数学模型。
你需要考虑诸如以下的问题；贮量应多大?应以多大的比率来获得贮量?买这些金属的合理价格应该是多少?还要求你考虑诸如以下的问题，贮量达到多大时应开始减少贮存量?应以多大的比率来减少？卖出这些金属的合理价格应该是多少?应该怎样分配(附页中有关于钴的资源、价格、需求及再循环等方面的信息)关于钴有用信息：1985年政府计划需要2500万磅钴。
进行周而复始的生产经营，从而每年可生产600万磅钴。
1980年占总消耗量70银的120万磅钴再循环了，得到了重新处理。
MCM86问题-A 水道测量数据表86A-1给出了在以码为单位的直角坐标为X，Y的水面一点处以英尺计的水Z．水深数据是在低潮时测得的。
船的吃水深度为5英尺。
在矩形区域(75，200)×(-50，150)里的哪些地方船要避免进入。
本题是由加州海军研究生院数学系的Richard Franke提供的，可阅他的论文Scattered Data Interpolation，Math，Comput.，38(1982)，18l-200。
MCM86问题-B 应急设施的位置“里奥兰翘镇”迄今还没有自己的应急设施。
1986年该镇得到了建立两个应急设施的安全拨款。
每个设施都把救护站、消防队和警察所合在一起。
图86B-1指出了1985年每个长方街区应急事件的次数。
在北边的上形状的区域是一个障碍，而在南边的长方形区域是一个有浅水池塘的公园。
应急车辆驶过一条南北向的街道平均要花15秒，而通过一条东西向的街道平均要花20秒。
你的任务就是确定这两个应急设施的位置，使得总的响应时间最少。
①假定需求集中在每个街道的中心而应急设施位于街角处；②假定需求是沿包围每个街区的街道上平均分布的，而应急设施可位于街道任何地方。
本题是由马里兰州沙里斯勃莱州立学院地理学与地区规划系的J.C. McGrew提供的。
MCM87问题-A 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9英寸。
本题是由印第安纳大学的M.Thompon 提供的，是从出现在1986年11月的Indianapolis Star的一个实际问题改造而成的。
MCM87问题-B 停车场在新英格兰地区一个镇上位于街角处的一个停车场的场主雇你来设计该停车场的安排，即设计“在地上的线应怎样划法”。
你一定认识到要把尽可能多的车塞进停车场会导致以直角停靠的方式一辆挨一辆地排成行。
但是缺乏经验的司机对于这种停靠方式是有困难的，这可能引起昂贵的保险费要求。
为了减少停靠车辆时可能造成的损坏，场主就要启用一些熟练的汽车司机作为“专职停靠司机”。
另一方面，如果汽车从通道进来有一个足够大的“转弯半径”的话，那么大多数司机看来都不会有很大的困难一次就停靠到该停靠的位置上去。
当然通道愈宽能容纳的车辆就愈少，这就会导致停车场场主收入的减少。
本题是由加州海军研究生院的M．D.Weir提供的。
MCM88问题-A 确定毒品走私船的位置相距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①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。
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。
MCM88问题-B 两辆铁路平板车的装货问题有七种规格的仪装箱要装到两辆铁路平板车上去。
包装箱的宽利高是一样的，但厚度(t，以厘米计)及重量(w，以公斤计)是不同的。
表88B-1给出了每种包装箱的厚度、重量以及数量。
图88B-1中每辆平板车有10.2米长的地方可用来装包装箱（象面包片那样)，载重为40吨。
由于当地货运的限制，对C5，C6，C7类的包装箱的总数有-个特别的限制；这类箱子所占的空间(厚度)不能超过302.7厘米。
试把包装箱装到干板车上去使得浪费的空间最小。
本题是由佐治亚理工学院的J．Bartholdi提供的。
这是出现在福特汽车公司的一个尚未解决的问题的修正与简化。
J．Bartholdi还写了一篇评论性文章The Outstanding Railroad Flatcar Papers，The UMAP Journal，v.9(1988)，no.4，399-103．MCM89问题-A 蠓的分类两种蠓Af和Apf己由生物学家W.L.Grongan和W.W.Wirth(1981年)根据它们的触角长度和翼长加以区分(见图89A-1)，9只Af蠓用标记，6只Apf鲸用“.”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?MCM89问题-B 飞机排队机场通常都是用“先来后到”的原则来分配飞机跑道，即当飞机准备好离开登机口时，驾驶员电告地面控制中心，加入等候跑道的队伍。
假设控制塔可以从快速联机数据库中得到每架飞机的如下信息：①预定离开登机口的时间；②实际离开登机口的时间；②机上乘客人数；④预定在下一站转机的人数和转机的时间；⑤到达下一站的预定时问。
又设共有七种飞机，载客量从100人起以50人递增，载客最多的一种是400人。
试开发和分析一种能使乘客和航空公司双方满意的数学模型。
本题是由纽约市立大学约克学院的Joseph Malkevitch 提供的。