2016年数学建模美赛A题题面及翻译
美赛数学建模A题翻译版论文
美赛数学建模A题翻译版论文The document was finally revised on 2021数学建模竞赛(MCM / ICM)汇总表基于细胞的高速公路交通模型自动机和蒙特卡罗方法总结基于元胞自动机和蒙特卡罗方法,我们建立一个模型来讨论“靠右行”规则的影响。
首先,我们打破汽车的运动过程和建立相应的子模型car-generation的流入模型,对于匀速行驶车辆，我们建立一个跟随模型,和超车模型。
然后我们设计规则来模拟车辆的运动模型。
我们进一步讨论我们的模型规则适应靠右的情况和,不受限制的情况, 和交通情况由智能控制系统的情况。
我们也设计一个道路的危险指数评价公式。
我们模拟双车道高速公路上交通(每个方向两个车道,一共四条车道),高速公路双向三车道（总共6车道)。
通过计算机和分析数据。
我们记录的平均速度,超车取代率、道路密度和危险指数和通过与不受规则限制的比较评估靠右行的性能。
我们利用不同的速度限制分析模型的敏感性和看到不同的限速的影响。
左手交通也进行了讨论。
根据我们的分析,我们提出一个新规则结合两个现有的规则(靠右的规则和无限制的规则)的智能系统来实现更好的的性能。
1介绍术语假设2模型设计的元胞自动机流入模型跟随模型超车模型超车概率超车条件危险指数两套规则CA模型靠右行无限制行驶规则3补充分析模型加速和减速概率分布的设计设计来避免碰撞4模型实现与计算机5数据分析和模型验证平均速度快车的平均速度密度超车几率危险指数6在不同速度限制下敏感性评价模型7驾驶在左边8交通智能系统智能系统的新规则模型的适应度智能系统结果9结论10优点和缺点优势弱点引用附录。
1 Introduction今天,大约65%的世界人口生活在右手交通的国家和35%在左手交通的国家交通流量。
[worldstandards。
欧盟,2013] 右手交通的国家,比如美国和中国,法规要求驾驶在靠路的右边行走。
多车道高速公路在这些国家经常使用一个规则,要求司机在最右边开车除非他们超过另一辆车,在这种情况下,他们移动到左边的车道、通过，返回到原来的车道。
2016年美赛COMAP50481作答
For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number50481Problem ChosenBFor office use onlyF1________________F2________________F3________________F4________________2016 Mathematical Contest in Modeling (MCM)Space JunkSummaryWith the rapid pace of peaceful exploitation and utilization of outer space resources and increasing frequency of activities of space launch, environment of space debris is worse and worse and amount of space debris is ever-increasing. Space debris poses great hazard to the safety of spacecraft, which arouses widespread concern, especially when the Russian satellite Kosmos-2251 and the USA satellite Iridium-33 collided on 10 February, 2009.In order to deal with the issue in a better way and search for possible business opportunity, we take into consideration of four sub problems in our paper.In the first model, we find out a program of curves concerning temporal and spatial distribution of space debris. To further understand their intrinsic link and trend, we choose the best fitting function and get distribution rule of space debris in time and space.In the second model, we consider to divide the space into several ball layers. We set probability of collision in space superposition of every ball layer in certain proportion. Then adding probability of unsuccessful launch, we can estimate the risk probability in the whole process is.In the third model, firstly we divide space debris into three categories, concerning each of which carry out revenue analysis quantitatively respectively. We can get revenue gained from three equipment disposing of debris, of which lasers satellites and water jets can achieveand $.In the fourth model, considering that investment of a firm is limited, we establish model of optimization, in which it enables the firm to achieve maximum benefit in every single day. Assume that finance of a firm is 200 billion dollars. We can program in Lingo and get the solution that the amount of each invested equipment is one and benefit of every day is dollars.Content1. Introduction (1)1.1 Background—Space debris and its urgency (1)1.2 Development of research and study (2)1.3 Qur Work (3)2. Model Analysis (3)2.1 Spatial and Temporal Distribution of Space Debris (4)2.1.1 Change of spatial density of space debris with latitude (4)2.1.2 Change of number of space debris with year (7)2.2 Estimate of risk probability (9)2.2.1 Symbol description (9)2.2.2 Assumption (10)2.2.3 Model building (10)2.2.4 Model solving (12)2.3 Number of space debris mitigation and revenue (14)2.3.1 Symbol description (14)2.3.2 Assumption (15)2.3.3 Model building (15)2.3.4 Model solving (16)2.4 Investment strategy of the private firm (18)2.4.1 Symbol description (18)2.4.2 Assumption (19)2.4.3 Model building (19)2.4.4 Model solving (19)3. Innovative Alternative (20)3.1 Ultraviolet Rays Focusing Apparatus (20)3.2 Shield of magnetic field (20)4. Assessment (21)5. Acknowledgements (21)6. Executive Summary (21)Reference (23)1. Introduction1.1Background—Space debris and its urgencySpace junk,also called orbital debris, coupled with activities of space launch, is referred collectively to artificial objects and debris with no function whatsoever. Ever since the Soviet Union launched the first artificial satellite of the world in 1957, 4300 activities of space launch have been carried out and over 5500 spacecraft have been launched into orbits by man. Space debris mainly derive from invalid spacecraft, the end of rocket body, debris left by astronauts and debris from collapse of spacecraft[1]. It’s estimated that there is over 500000 space debris. With the rapid pace of peaceful exploitation and utilization of outer space resources and increasing frequency of activities of space launch, environment of space debris is worse and worse[1]. The collision between space debris and spacecraft will pose hazards to astronautic system in many ways, many of which is fatal. The direct influence on astronautic activities made by space debris mainly aims at spacecraft. Different sizes of space debris will pose damages of different degrees to different parts of spacecraft[2]. The issue sparked widespread attention all over the world, particularly since the Russian satellite Kosmos-2251 and the USA satellite Iridium-33 collided on 10 February, 2009. How to mitigate the space debris effectively has become a problem to be solved.Figure 1.Artists impression of space debris1.2Development of research and studyMicrometeoroids and space debris (orbital debris) are collectively called as M/OD environment[3]. In 1981, American Institute Aeronautics and Astronautics(AIAA) officially proposed report about space debris for the first time. In the same year, National Aeronautics and Space Administration (NASA) started to carry out the 10-year research program of surveillance, modelling and control of M/OD environment. Besides, in 1985, ESA convened a symposium about the reentry and falling of space debris and M/OD research group was set up in the next year. In1993, launched by NASA, ESA, Russia, Japan, IADC was founded and intended to coordinate astronautic countries to act in concert, promote cooperation and communication in the field of space debris one another and address problem of space debris hand-by-hand. Chinese National Space Administration (CNSA) joined IADC officially in 1995. Under the support of national finance, Chinese National Defense and Science and Industry Council has started and carried out the special research work of Action Plan of Space Debris[1].In recent years, through ongoing research of M/OD by many astronautic countries all over the world, remarkable development has been achieved in the technical field of space debris and many approaches have been offered to mitigate space junk.It’s reported by England network section of the m agazine New Scientist that a nanoscale satellite named Cubesail uses solar sail by the aid of solar energy as a propulsion system. In addition to that, another unique function equipped by the sail is that it can help the debris to deorbit and fall into the atmosphere as a rail brake. Vauls Lappas, in charge of the project, said that if Cubesail can work well as expected, an analogous sail also can be fixed onto the future satellite and when it finishes its tasks in space and becomes space debris, the sail can bring them back and burn them down automatically. Or, Cubesails can be launched in swarm to the low earth orbit to trap those floating space debris and mitigate it in a mutually destructive way[4].What’s more, America invented a windmill type equipment for mitigation of space debris, which involves that when debris collides with blades, minute debris will be embedded into the metal fans, which achieves the goal thereby. Even if a little big debris punches through the fan, due to great decrease of the speed by collision and without adequate speed to maintain movement around the earth, it will fall into the atmosphere gradually and be burnt down[4].Another assumption is that when the satellite with a collection net linked by light-duty electronic ropes, reaches the specified location, it will loosen the rope automatically. After the collection net is loaded a certain amount of debris, they will fall into the atmosphere together under the influence of Earth’s magnetic field[4].Actually, NASA set about the tests of Laser Broom since 2000, planning to assist to mitigate the space debris in the trajectory of ISS’s moving with a diameter from 1cm to 10cm. Once certain space debris is targeted, the Laser Broom will send out a laser beam to the side of the space debris back to the earth and gasify it. And thenwith the counterforce of the gas the space debris will be forced to move to the earth and be burned down eventually. In addition, method of suicide satellite and space debris perishing together is also adopted[4].Despite floods of ways to mitigate the space debris, none has been found out to be efficient both technically and economically in the field of astronautics. When it comes to combat with space debris, it’s indispensable to cooperate with all parties, with combination of multiple means policies of prevention and control, which has hope to tackle both the cause and effect of the problem[4].1.3Qur WorkThe problem requires us to do severer jobs to search for business opportunity in the process of mitigating space debris. In order to solve it, we divided this problem into four sub-problems:1).Spatial and Temporal Distribution of Space Debris2).Risk Probability3).Mitigation of Space Debris and Revenue4).Investment Strategy of the Private FirmIn the first model, we can explicitly know about the trend of space debris with the change of time and space. Through fitting the exported data, functions of space debris in time and space can be obtained respectively, which lays solid foundation for our following work.In the second model, we take into consideration of probability of unsuccessful launch and collision with space debris. So we use risk probability to examine and weigh the whole problem and try to make it quantitative. Finally, risk probability can be obtained.In the third model, the ultimate goal of the question is to search for a business opportunity. So we need to analyze the relationship between number of mitigation of debris and revenue. Then, iscretization of the time and determination of daily revenue can get a result.In the last model, to maximize the benefit of a firm, we try to use integral optimization to finish the reasonable and economical distribution, with the assumption of limited investment.2. Model AnalysisIn order to solve it, we divided this problem into four sub-problems:1).Spatial and Temporal Distribution of Space Debris2).Risk Probability3).Mitigation of Space Debris and Revenue4).Investment Strategy of the Private Firm2.1 Spatial and Temporal Distribution of Space Debris2.1.1 Change of spatial density of space debris with latitudeAccording to the information[1], we can get a graph of change of spatial density of space debris with change of latitude. Then through Digitize of Origin software, we export the relevant detailed data shown in the Table 1.Then we fit the data through Origin and the fitting effect is shown in the below Figure 2.Figure 2. The original fitting of latitude and spatial densityIn Origin software, means significant analysis of mathematical statistics, which is an index that verifies whether the assumption is reasonable. Due to, we think the fitting effect is not pretty good. To get better effect, we modify some data (from 785.99738 to 963.50839 ) of abnormal fluctuations. The modified results are shown in the below Table 2.0.00E+0001.00E+0082.00E+0083.00E+0084.00E+0085.00E+0086.00E+008S p a c i a l D e n s i t y (n u /k m ^3)Latitude(km)After modification, we fit the data again and the fitting effect is shown as follows.Figure 3. The modified fitting of latitude and spatial densityNow, , so we can get the function of spatial density ( ) and latitude ( ) of space debris:0.00E+0001.00E+0082.00E+0083.00E+0084.00E+0085.00E+0086.00E+008S p a c i a l D e n s i t y (n u /k m ^3)Latitude(km)2.1.2 Change of number of space debris with yearBesides, we also find graph concerning change of space debris in quantity with year. Similarly, we export the relevant data through Digitize of Origin software shown in the below Table 3.We make the graph by linear fitting the data and get the following fitting effect.Figure 4. The original fitting of number of objects and yearThe fitting effect is not ideal, either, with .Figure 5. Historical growth of space debris through 2010-2000020004000600080001000012000140001600018000N u m b e r o f O b j e c t sYearAccording to the information, some severe collisions and explosions occurred from 2007, such us Fengyun-1c, Iridium-33 and so on and the number of debris soured. So, to avoid abnormal data affecting the results, we decide to analyze it piecewise. Namely, to fit the data and get the piecewise function from 1961 to 2006 and from 2007 to 2013 respectively. And the results are as below.Figure 6. Number of objects from 1961 to 2006 Figure 7. Number of objects from 2007 to 2013From Figure.6 we can get and the function ofnumber of objects with year from 1961-2006 isFrom Figure.7 we can get , and similarly the function of number of objects with year from 2007 to 2013 is2.2 Estimate of risk probability2.2.1Symbol descriptionN u m b er o f O b i e c t sYear N u m b e r o f O b j e c t sYear2.2.2Assumption●Assume that probability of successful launch of required equipment into space by the private firm mentioned in the question is unchangeable.●Assume that orbital altitude of space debris required to be mitigated has an upper bound and a lower bound.●Assume that range of possible collision with spacecraft is a ball layer and spacing between two contiguous layers is unchangeable.●Assume that space debris in every ball layer is distributed evenly.●Assume that both probability of mitigating space debris in every ball layer and probability of collision relate to spatial density.2.2.3Model buildingWhy can we assume that spacing between two contiguous ball layers is?We can consider the fact that when people wait for train pulling into the station, they are forced to st and inside the safe line so that they won’t be trapped in by airflow of running train.Based on the fact above, we can set a spacing. With it, we can describe that when the distance between spacecraft and space debris is beyond the spacing, collision will scarcely happen. So, we divide the range between low earth orbit and the upper bound of the space debris into several ranges with spacing. The schematic diagram is as below.Figure 8. Schematic diagram of space debris ball layers From the assumptions, we know that probability of collision in every ball layer is proportional to the spatial density in the ball layer. To make an estimate, we use the density of the middle of two contiguous layers. Probability of collision happening in the th ball layer isNote: in which is a very small scale coefficient.Based on conditional probability, probability of collision in the th ball layer is The risk probability of working in space born by the firm isBecause the spacing of every two ball layers is unchangeable and equals, we can identify the number of layers by dividing the spatial range. SoThe model of risk probability can be established as follows.According to the reference[7], we can get the function of probability of collision between single space debris and equipment isin which is radius of compound body; is distance on the short axis; is distance on the long axis; and is distance projection of intersection distance on the intersection plane; and is corresponding standard deviation.Figure9. Schematic diagram of parameters of intersection plane2.2.4Model solvingTo solve the model, values of some necessary parameters will be shown as below. And according to data, upper bound of orbital altitude is 2000km, lower bound orbital altitude is 100km and replaced by probability of successful launch of American satellite is 0.87.Considering that double integral is needed for probability of collision between single space debris and spacecraft and double integral is complicated, we hope to get the relationship between probability of collision and distance.Fortunately, noted in some documents[7], function of maximum probability of collision and intersection distance can bein which means intersection distance.To be more intuitive, we plot the function with MATLAB software, shown as below.Figure 10. Relationship of intersection distance and probability From the figure above, we can see that the slope of the curve is ever-decreasing with the increase of intersection distance, which means as intersection increases, the probability of collision becomes smaller.Combining the information in the figure and speed magnitude in factual orbit, we set spacing between layersProbability of collision of single space debris and equipment isDue to pretty small probability of collision, according to function of density, we setThen we substitute the data above in the expression and simplify it. Through MATLAB program we can calculate risk probability and probability of collision, as shown in the below Table 5.According to the results, probability is pretty small. But once collision occurs, it will cause enormous loss and more new debris. Without measures implemented, space in the future will be sieged by space debris.2.3 Number of space debris mitigation and revenue2.3.1Symbol description2.3.2Assumption● Assume that equipment won’t break down for no reason.● Assume that other objects apart from space debris are not taken into account.● Assume that all of the energy of the equipment used in space comes from solar energy and operating it every time may cause wastage of equipment.●Assume that maintenance cost of the equipment used and staff cost is proportional to revenue.●Assume that every equipment has a certain life span and will scrap directly till the time comes2.3.3Model buildingFrom the information[8], we can get the overview of space debris shown in the below Table 6.The amount of debris of cm dominates, about 99.67% and the mass of it only occupies 0.035%. This type of debris poses minute hazard to spacecraft and can be defensed. It’s followed by cm debris with0.031% amount and 0.035% mass , which can destroy spacecraft and is the greatest offender. The amount of debris with over 10 cm in size is the smallest with 99.93% mass however. This kind of debris poses enormous hazard and yet it can be avoided in a negative way.Clearly，total revenue from mitigating space debris in the th way in the required time isAnd total time isFrom model2, we know that the probability of equipment against debris launched successfully isAnd the probability of collision with debris in space is.The probability of unsuccessful launch of the th equipment is, and the corresponding revenue isThe probability of collision with the space debris on the first working day is, and the corresponding revenue isThe probability of collision with the space debris on the second working day is, and the corresponding revenue isBy mathematical induction, the probability of collision with the space debris on the th working day is, and the corresponding revenue isThe revenue gained in the first and second occasion is the same, so it can be induced in the first day. Detailed formula are listed in the following table.According to meaning of expectation, we can deduce that revenue gained every day is2.3.4Model solvingAssume that revenue gained from disposing of three types of debris can be shown as follows.Table8. Revenue of disposing of three types of debrisCalculation of average revenue of space-based laserWhen, it means space-based laser is the equipment of disposing of debris. According to material[9,10], we know about some parameters of the laser shown as follows.Due to lack of data, based on some characteristics, such as laser its major target for debris with 1-10cm in size, we can assume some data shown as below.By programming in MATLAB software, average revenue is. Calculation of average revenue of large satelliteWhen, it means large satellite is the equipment of disposing of debris. According to material[11], we know about some parameters of the satellite shown as follows.Due to lack of data, based on some characteristics, we can assume some data shown as below.Similarly with MATLAB, average revenue is.Calculation of average revenue of space-based water jetsWhen, it means space-based water jets is the equipment of disposing of debris. According to material[12], we know about some parameters of the satellite shown as follows.Due to lack of data, based on some characteristics, we can assume some data shown as below.Also with MATLAB, average revenue is.2.4 Investment strategy of the private firm2.4.1Symbol2.4.2Assumption●Assume that a firm sets earning profit as main goal.2.4.3Model buildingConstraint of finance isObjective function, benefit gained in time isBut the objective function has two variations, to be more convenient, to maximize the benefit, we consider to set benefit gained every day maximal.The final model of optimization is2.4.4Model solvingAssume that a firm has 200 billion dollars. According to model 3, benefit gained by lasers, large satellites, water jets in every single day equals to ratio of total benefit to days, the results of which are shown as follows.From the table above, it's apparent to see that the highest revenue gained is the second equipment----satellites, of which the benefit is 654670$, followed by which is the first method----lasers; the lowest one is water jets, 448854$.So the model can be established as follows:This is a problem of integer programming and the results are shown as below via Lingo.According to the table, ultimate investment strategy of a firm is shown that the firm spend 200 billion$ in investing water jets. Through Internet, the price of water jets is found cheap relatively and the efficiency is favorable, so the redults are credible comp aratively.3. Innovative AlternativeIf no commercial opportunity is possible, we will provide two innovative alternatives for avoiding collisions.3.1 Ultraviolet Rays Focusing ApparatusTo avoid collision in space, we take into consideration of the technology of ultraviolet rays, which are abundant in space and pose enormous harm to living beings on earth. So if we human can take fully advantage of ultraviolet and mitigate the ultraviolet rays in space in a degree, it will kill two birds with one stone.We surmise that a device with high temperature resistance, radiation resistance and heat resistance, can be fixed on the spacecraft and probes and absorbs ultraviolet rays at the time of operating in space. The device can transform the ultraviolet rays to enormous energy. Once encountering approaching space debris, the spacecraft will recognize it and release high energy to the target. The space debris will absorb energy and accelerate and transform the orbit, to avoid collision.3.2 Shield of magnetic fieldWith geomagnetic field as a protective umbrella of the earth, our home won’t be intruded upon by many kinds of radiation in outer space. Can we consider such a circumstance that a magnetic field can be artificially established inside the spacecraft according to the factual size of the spacecraft, of which applied force tends to be in a reasonable range. Neither does it have an effect on components of the spacecraft, nordoes it exert counteraction to debris outside to make it far away from the spacecraft and avoid collision accordingly. If spacecraft can absorb energy of sunlight used by created magnetic field, cost can be reduced effectively.4. AssessmentWe establish four models above and find out some advantages and disadvantages through some analysis.AdvantagesModel1：Fully fit temporal and spatial distribution of space debris.Model2:●Divide space into several ball layers to increase accuracy of probability calculation.● Use fitting functions to simplify calculating of probability of collision. Model 3: Take full consideration of space debris of different types.WeaknessModel1: Leave out individual data and decrease accuracy of trend.Model2: Ignore other factors in space and bring about errors with scale coefficient. Model3: Lack data and decrease accuracy of the final results.5. AcknowledgementsSo far, we have basically finished the paper. Recollecting the process of setting up the model, we have encountered couples of obstacles, but via constant absorption and updating pool of information and knowledge, we gradually sorted out our thoughts and tackled the problem in a better way finally. We hereby express great gratitude to the relevant authors and websites providing document and web data. It’s you that make our results more credible and rich the contents of our paper. Thank you very much.6. Executive SummaryBe a member of mitigating space debrisAs we all know, the amount of space debris in orbit around earth has been a heated issue recently. It is estimated that over 500,000 pieces of space debris, also called orbital debris, are currently being tracked as potential hazards to spacecraft. The issue itself became more widely discussed in the news media when the Russian satellite Kosmos-2251 and the USA satellite Iridium-33 collided on 10 February, 2009.Generally, space debris can be divided into three types. The first type is spacecraft debris caused by explosion with or without intention. Most of it comes from the so-called preview of space war between Union Soviet and America. The second type is inadvertent carelessness of astronauts, such as leaving bolts, spacers, pen and even life garbage. The last type is wreckage of rockets and satellites. Some invalid satellites are still floating in orbit, which will be like ghosts for possible decades of years.To address the big problem, what we should do is to be a member to devote to mitigating space debris. Saying it in a more serious way, we ought to act in concert with astronautic countries, promote cooperation and communication in the field of space debris one another and address problem of space debris hand-by-hand.Concerning the severity of the issue, we want to propose our concept to devote to dealing with space debris, which will help the private firm to gain certain profit. The detailed project is as follows.According to the results by reasonable analysis and accurate calculation, we consider that two hundred small, space-based water jets and zero high energy laser used to target specific pieces of debris and zero large satellite designed to sweep up the debris should be produced and dispatched to carry out clearing work, which will obtain over $ per day.A lot of elements can account for our idea. Some technical reasons will be left out and the following might be the critical ones.Firstly, in term of their working mechanism, a water jet loaded by a rocket into space creates a special liquid wall, which will reduce the speed of debris once it bumps into the wall. A laser can directly gasify the debris with high temperature or decelerate and deorbit the debris. Large satellite can capture the running debris and return it to the earth atmosphere.Secondly, in term of their best targeted debris in different sizes, water jets are target for space debris with below 10cm in size; Lasers are mainly tools for space debris with 1-10cm in size; Satellites are good at dealing with space debris with over 1cm in size. So, clear to see that every type of equipment possesses respective specialty and applying medicine according to indications is of great importance and necessity to proceed the severe task.Thirdly, in term of their advantages and disadvantages, water jets won’t bring about any extra pollution as other method does, but its feasibility is not favorable. Lasers will not be susceptible by atmospheric scattering and is able to deal with debris at any location with high efficiency. However, it’s too expensive to afford many of it。
美赛a题思路
美赛a题思路
美赛A题思路
美国大学生数学建模竞赛（USAMO）A题是一道关于四边形的问题，要求给出一个正方形ABCD，以及在正方形边上的4个点P，Q，R，S。
要求在这4个点上分别划出4个三角形，使得这4个三角形的外接圆的半径最小。
该题的解法应用了几何学中的相关定理，如三角形外接圆的半径和内接圆的半径之间的关系。
此外，还要考虑正方形ABCD的对称性，这样可以将原问题简化。
首先，我们可以将正方形ABCD平分成四个三角形，即ACD，ADB，BCS，CSA。
根据这4个三角形的特点可知，它们的外接圆必须通过正方形ABCD的4个顶点。
同时，由于三角形外接圆的半径和内接圆的半径之间的关系，我们可以确定每个三角形的外接圆的半径。
接下来，我们可以利用正方形ABCD的对称性，将原问题简化为另一个问题，即在正方形ABCD的四条边上，找出两个点，使得它们分别在四个三角形的外接圆上，且外接圆的半径最小。
在此基础上，我们可以利用斜率的概念，判断某一点在哪个三角形的外接圆上。
具体来说，我们可以计算正方形ABCD的4条边上的各点的斜率，与之前求出的每个三角
形外接圆的斜率进行比较，从而判断某一点在哪个三角形的外接圆上。
最后，我们可以通过对四条边上的点的排列组合，找出最优解，即得到4个三角形的外接圆的半径最小的情况。
以上就是美赛A题的思路。
解决该题，需要综合运用几何学、代数学中的相关定理，以及斜率的概念，最终求出最优解。
【免费阅读】2016全国大学生数学建模竞赛A题题目及参考答案
2011高教社杯全国大学生数学建模竞赛题目（请先阅读“全国大学生数学建模竞赛论文格式规范”）A 题 城市表层土壤重金属污染分析随着城市经济的快速发展和城市人口的不断增加，人类活动对城市环境质量的影响日显突出。
对城市土壤地质环境异常的查证，以及如何应用查证获得的海量数据资料开展城市环境质量评价，研究人类活动影响下城市地质环境的演变模式，日益成为人们关注的焦点。
按照功能划分，城区一般可分为生活区、工业区、山区、主干道路区及公园绿地区等，分别记为1类区、2类区、……、5类区，不同的区域环境受人类活动影响的程度不同。
现对某城市城区土壤地质环境进行调查。
为此，将所考察的城区划分为间距1公里左右的网格子区域，按照每平方公里1个采样点对表层土（0~10 厘米深度）进行取样、编号，并用GPS 记录采样点的位置。
应用专门仪器测试分析，获得了每个样本所含的多种化学元素的浓度数据。
另一方面，按照2公里的间距在那些远离人群及工业活动的自然区取样，将其作为该城区表层土壤中元素的背景值。
附件1列出了采样点的位置、海拔高度及其所属功能区等信息，附件2列出了8种主要重金属元素在采样点处的浓度，附件3列出了8种主要重金属元素的背景值。
现要求你们通过数学建模来完成以下任务：(1) 给出8种主要重金属元素在该城区的空间分布，并分析该城区内不同区域重金属的污染程度。
(2) 通过数据分析，说明重金属污染的主要原因。
(3) 分析重金属污染物的传播特征，由此建立模型，确定污染源的位置。
(4) 分析你所建立模型的优缺点，为更好地研究城市地质环境的演变模式，还应收集什么信息？有了这些信息，如何建立模型解决问题？分分工会舒服的规划法规f x c f 是否撒的发生的发生fx c f 是否撒的发生的发生f x c f 是否撒的发生的发生fx c f 是否撒的发生的发生fx c f 是否撒的发生的发生f x c f 是否撒的发生的阿斯顿发斯蒂芬斯蒂芬题 目 A 题 城市表层土壤重金属污染分析摘 要：本文研究的是某城区警车配置及巡逻方案的制定问题，建立了求解警车巡逻方案的模型，并在满足D1的条件下给出了巡逻效果最好的方案。
2016美国数模竞赛题目中文翻译
A: 一个人充满热水的浴缸从一个单一的水龙头，并落户到浴缸里清洁和放松。
不幸的是，浴缸不是温泉式浴盆与辅助加热系统和循环飞机，而是一个简单的水安全壳。
一段时间后，在浴变得明显冷却器，所以该人增加的热水恒定涓流从龙头再加热洗澡水。
所述浴缸的设计以这样的方式，当所述桶达到其容量，过量的水逸出通过溢流漏极。
开发浴缸水的空间和时间，以确定该人在浴缸可以采用，以保持温度甚至整个浴缸和尽可能接近到初始温度，而不会浪费过多的水的最佳策略的温度的模型。
使用模型确定哪个你的策略取决于形状和桶，该人在浴缸的形状/体积/温度的体积，并且由该人在浴缸所作的运动的程度。
如果对方使用了泡泡浴的添加剂，而最初填补了浴缸，以协助清洁，怎么会变成这样影响模型的结果吗？除了必需的单页摘要您的MCM提交，你的报告必须包括一份一页纸的非技术性解释浴缸的用户描述你的策略，同时解释了为什么它是如此难以得到整个均匀保持温度浴缸里的水。
B: 小碎片在轨道上绕地球金额已日益受到关注。
据估计，超过50万件的空间碎片，也被称为轨道碎片，目前都被跟踪的潜在危害航天器。
这个问题本身变得更广泛的讨论，在新闻媒体时，俄罗斯卫星的Kosmos-2251和美国铱卫星-33相撞2009年2月10日。
有许多方法以除去碎片已被提出。
这些方法包括小的，基于空间的水射流和用于针对特定的碎片高能激光器和设计，以清扫杂物，其中包括大型卫星。
从漆片的废弃卫星的大小和质量的碎片范围。
碎片“高速轨道进行采集困难。
开发时间依赖模型来确定的替代品，私人公司可以采取作为一个商业机会，以解决空间碎片问题的最佳替代品或组合。
您的模型应该包括成本，风险，效益的定量和/或定性的估计，以及其他重要的因素。
您的模型应该能够评估独立的替代品的替代品，以及组合和能够探索各种重要的“如果？”的情景。
使用模型，确定了经济上有吸引力的机会是否存在，或没有这样的机会是可能的。
如果一个可行的商业机会的存在作为一种替代解决方案，提供了不同的选择去除杂物进行比较，并包括具体建议，以碎片应该如何去除。
2016年数学建模竞赛A题优秀论文
将(1)式代入得：
dy mg T1 sin 1
dx
T1 cos1
(2)
对于锚链，m=σs ，其中 s 是 AB 锚链的长度，σ是锚链的线密度，即单位长
度锚链的质量[1]。代入(2)式得：
dy sg T1 sin 1
(3)
dx
T1 cos1
根据勾股定理可以得到弧长公式：
ds
1
dy dx
dp dx
T1
cos1
g
1 p2
然后对 x 和 p 分离变量并对两端进行积分得到：
dp
1 p2
T1
g cos 1
dx
即：sinh 1
p
g T1
x
C1
(4)
其中 C1 可以由 x=0,y=0 时的值确定，原点 A 处 p y ' tan 1 ，可得 C1 为：
当海面风速一定且海水静止时，钢桶和各节钢管的倾斜角度、锚链形状、浮 标的吃水深度和游动区域，与锚链线的方程、系泊系统各部分之间的受力平衡和 力矩平衡的约束密切相关。由于传输节点各部分相互影响，根据力学相关知识， 可以按照锚链→钢桶和重物球→钢管→浮标的顺序依次进行受力分析，从而得到 各部分受力平衡时的定量解析式，通过这些表达式可以确定钢桶和各节钢管的倾 斜角度、锚链形状。由于吃水深度与浮标受力直接相关，还可以确定浮标的吃水 深度。对于浮标的游动区域，可以由稳定后系泊系统各个部分在水平方向投影的 总长度来计算游动区域的最大半径。
多无人机协同任务规划-2016年全国研究生数学建模竞赛A题答案
2016年全国研究生数学建模竞赛A题多无人机协同任务规划无人机（Unmanned Aerial Vehicle，UAV）是一种具备自主飞行和独立执行任务能力的新型作战平台，不仅能够执行军事侦察、监视、搜索、目标指向等非攻击性任务，而且还能够执行对地攻击和目标轰炸等作战任务。
随着无人机技术的快速发展，越来越多的无人机将应用在未来战场。
某无人机作战部队现配属有P01~P07等7个无人机基地，各基地均配备一定数量的FY系列无人机（各基地具体坐标、配备的无人机类型及数量见附件1，位置示意图见附件2）。
其中FY-1型无人机主要担任目标侦察和目标指示，FY-2型无人机主要担任通信中继，FY-3型无人机用于对地攻击。
FY-1型无人机的巡航飞行速度为200km/h，最长巡航时间为10h，巡航飞行高度为1500m；FY-2型、FY-3型无人机的巡航飞行速度为300km/h，最长巡航时间为8h，巡航飞行高度为5000m。
受燃料限制，无人机在飞行过程中尽可能减少转弯、爬升、俯冲等机动动作，一般来说，机动时消耗的燃料是巡航的2~4倍。
最小转弯半径70m。
FY-1型无人机可加载S-1、S-2、S-3三种载荷。
其中载荷S-1系成像传感器，采用广域搜索模式对目标进行成像，传感器的成像带宽为2km（附件3对成像传感器工作原理提供了一个非常简洁的说明，对性能参数进行了一些限定，若干简化亦有助于本赛题的讨论）；载荷S-2系光学传感器，为达到一定的目标识别精度，对地面目标拍照时要求距目标的距离不超过7.5km，可瞬时完成拍照任务；载荷S-3系目标指示器，为制导炸弹提供目标指示时要求距被攻击目标的距离不超过15km。
由于各种技术条件的限制，该系列无人机每次只能加载S-1、S-2、S-3三种载荷中的一种。
为保证侦察效果，对每一个目标需安排S-1、S-2两种不同载荷各自至少侦察一次，两种不同载荷对同一目标的侦察间隔时间不超过4小时。
为保证执行侦察任务的无人机与地面控制中心的联系，需安排专门的FY-2型无人机担任通信中继任务，通信中继无人机与执行侦察任务的无人机的通信距离限定在50km范围内。
2016年美国大学生数学建模大赛A题获奖论文A Hot Bath
The first part has five sections: air’s heat radiation, bathtub wall’s heat radiation, person in, hot water in, bubble existed. We discuss some factors that affect water temperature, such as the shape and the volume of the bathtub and person, especially the motions made by the person in the bathtub because the temperature in the bathtub has a great connection with person. Finally, we get the water temperature variation and distribution model.
In this article, we establish two models. One is water temperature variation and distr one is finding best strategy model. We put forward some acceptable hypothesis to simplify the model. What’s more, we clear the meaning of the word “noticeably”.
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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.
一个人用一个水龙头让浴缸装满了热水，（然后？）睡进去来清洗和放松。
//那就先放到一定的程度，泡进去，然后边加水这样。
Unfortunately, the bathtub is not a spa-style tub with a secondary heating system and circulating jets, but rather a simple water containment vessel.
不幸的是，这个浴缸没有温泉热水模式，就是没有另外的加热系统和循环的喷嘴，而是个简单的水密闭容器。
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.
不一会儿，这个水池明显的变冷了，所以这个人打开水龙头，加入了持续的细细的水，来加热这个浴缸里面的水。
The bathtub is designed in such a way that when the tub reaches its capacity, excess water escapes through an overflow drain.
这个浴缸设计成一种形式，当这个池子到达了它的容量，多余的水会通过一个溢出排水系统排出。
Develop a model of the temperature of the bathtub water in space 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.
设计一个浴缸里面的水温度关于空间和时间上的模型，去决定最好的策略，让这个人在浴缸里能够在不浪费太多的水的前提下，去尽量的靠近初始的温度。
Use your model to determine the extent to which your strategy depends upon the shape and volume of the tub, the shapeolume/temperature of the person in the bathtub, and the motions made by the person in the bathtub.
用你的模型去决定你的策略对以下因素的依赖程度（依赖关系）。
因素为：缸的形状和容量，这个浴缸里面的人的形状，体积，温度，还有这个人在浴缸里面的动作。
If the person used a bubble bath additive while initially filling the bathtub to assist in cleansing, how would this affect you r model’s results?
如果这个人在一开始装满这个浴缸的时候，就加入了泡泡添加剂去帮助清洗，这个会如何影响你的模型的结果呢？
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.
除了已经要求的一页的总结，你的报告必须含有一页的对浴缸使用者的非技术性的解释，去描述你的策略，同时解释为什么如此难以做到保持整个洗澡水的水温是均匀的。