2016年美国大学生数学建模竞赛题目

Contents1.Introduction (1)1.1 Background (1)1.2 Foundation & ROI (1)2 Task (1)3 Fundamental assumptions (2)4 Definitions and Notations (2)5 Models (3)5.1 Filter data (3)5.2 Object Selection Model (Grey Relational Analysis) (4)5.2.1 Model analysis (4)5.2.2 Model solution (4)5.3 ROI Model (Principal Component Analysis) (5)5.3.1 Model analysis (5)5.3.2 Model solution (6)5.4 Verify the possibility (9)5.4.1 Comparison (9)5.4.2 External factor (10)5.5 Investment Forecast Model (11)5.5.1 Linear Regression Forecasting Model (11)5.5.2 School potential Prediction (TOPSIS) (12)5.5.3 Final investment (TOPSIS) (13)6 Conclusions (16)7 Strengths and Weaknesses (18)7.1 Strengths (19)7.2 Weaknesses (20)8 Letter to Mr. Alpha Chiang (21)9 References (22)Team # 44952 Page 1 of 221 Introduction1.1 BackgroundThe Goodgrant Foundation is a charitable organization that wants to help improve educational performance of undergraduates attending colleges and universities in the United States. To do this, the foundation intends to donate a total of $100,000,000 (US100 million) to an appropriate group of schools per year, for five years, starting July 2016. In doing so, they do not want to duplicate the investments and focus of other large grant organizations such as the Gates Foundation and Lumina Foundation.Our team has been asked by the Goodgrant Foundation to develop a model to determine an optimal investment strategy that identifies the schools, the investment amount per school, the return on that investment, and the time duration that the organi zation’s money should be provided to have the highest likelihood of producing a strong positive effect on student performance. This strategy should contain a 1 to N optimized and prioritized candidate list of schools you are recommending for investment bas ed on each candidate school’s demonstrated potential for effective use of private funding, and an estimated return on investment (ROI) defined in a manner appropriate for a charitable organization such as the Goodgrant Foundation.1.2 Foundation & ROIFoundation (charitable foundation) refers to the nonprofit legal person who uses the property of the natural persons, legal persons or other organizations to engage in public welfare undertakings. In terms of its nature, foundation is a kind of folk non-profit organizations.ROI is a performance measure used to evaluate the efficiency of an investment or to compare the efficiency of a number of different investments. ROI measures the amount of return on an investment relative to the investment’s cost. To calculate ROI, the benefit (or return) of an investment is divided by the cost of the investment, and the result is expressed as a percentage or a ratio.2 Task●One-page summary for our MCM submission●Using our models to achieve the candidate list of schools●Calculate the time durati on that the organization’s money should be provided to have thehighest likelihood of producing a strong positive effect on student performance●Calculate the investment amount Goodgrant Foundation would pay for each school●Calculate the ROI of the Goodgrant Foundation●Forecast the development of this kind of investment mode●Write a letter to the CFO of the Goodgrant Foundation, Mr. Alpha Chiang, that describesthe optimal investment strategy。

近几年美国大学生数学建模竞赛（USMCM）的题目包括：
2019年：建立一个模型来模拟东海和黄海的湍流。

2018年：预测联合国安理会和联合国大会决策结果及党派之间的关系。

2017年：建立一个模型来识别投资者风险偏好并帮助他们优化投资组合。

2016年：建立一个模型来识别用户a浏览网页时的行为特征，以便更好地理解和预测用户的行为。

2015年：建立一个模型，根据通信终端的传输速率，识别用户的实时视听传输需求。

2014年：建立一个模型来模拟社会文化传播的影响。

2013年：建立一个模型，根据用户的行为来预测新闻传播的趋势，并建议相关策略。

2012年：建立一个模型来优化公共汽车系统，以满足不同地区乘客的旅行需求。

2011年：建立一个模型，根据居民就医环境的不同，构建卫生保健系统的合理结构。

2010年：建立一个模型，预测印度洋及其邻近海域的风暴强度，以及其对当地的影响。

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。

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。

B
太空垃圾
在地球轨道上的小碎片数量已成为日益严重的关注焦点。

据估计，超过500000块的空间碎片，也被称为轨道碎片，目前被锁定为对空间飞行器构成潜在危险的目标。

在2009年二月十号俄罗斯卫星kosmos-2251和美国卫星iridium-33相撞后，在新闻媒体上这个问题受到了更广泛的讨论。

已经有人提出消除碎片的一些方法。

这些方法包括用小型空间运载的喷水装置和高能激光瞄准碎片的特定位置，还有设计大卫星来清扫这些小碎片。

太空垃圾的范围包括从涂料碎片到废弃卫星，尺寸和质量也有很大变化范围。

碎片的高速度轨道使得捕捉变得困难。

开发一个随时间变化的模型，以确定一个私营企业可以采纳的最佳方案选择或组合方案，该企业将把解决空间碎片问题作为商业机会。

你的模型应该包括对成本，风险，收益，以及其他重要因素定量或定性的估计。

你的模型应该能够评估单独的解决方案，以及解决方案的组合，并能够探讨各种重要的可能情景。

使用你的模型，确定商机是否存在。

如果存在一个可行的商业机会作为可选的解决方案，请提供其与其他备选解决方案的比较，并且给出移除碎片的具体建议。

如果商业机会不存在，提出避免撞击的创新性解决方案。

除了要求的一页MCM提交摘要，你的报告必须包括二页的执行总结，介绍考虑的方案和主要的建模结果，并提供建议，是单独行动，联合行动，或不采取行动，依据你的工作作出选择。

执行总结
是为不具备技术背景的高层决策者和新闻媒体分析人士写的。

2016AMC12BProblem1What is the value of when?当时，的值是多少？Problem2The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of and is closest to which integer?两个数的调和平均值可以由它们乘积的2倍除以它们的和得到。

那么1和2016的调和平均值最接近下面哪个整数？Problem3Let.What is the value of?令，则的值是多少？The ratio of the measures of two acute angles is,and the complement of one of these twoangles is twice as large as the complement of the other.What is the sum of the degree measures of the two angles?两个锐角的度数比值是，且其中一个角的补角是另一个角的补角的2倍，那么这2个角的度数之和是多少?Problem5The War of started with a declaration of war on Thursday,June,.The peace treaty toend the war was signed days later,on December,.On what day of the week was thetreaty signed?1812英美之战以1812年6月18日星期四的宣战开始，结束战争的和平协议的签订是在919天后的1814年12月24日，问签订协议的那天是星期几？（A）Friday｜周五（B）Saturday｜周六（C）Sunday｜周日（D）Monday｜周一（E）Tuesday｜周二Problem6All three vertices of lie on the parabola defined by,with at the originand parallel to the-axis.The area of the triangle is.What is the length of?的3个顶点位于抛物线上，其中点A在原点，BC和x轴平行.，三角形的面积为64，问BC的长度为多少?Josh writes the numbers.He marks out ,skips the next number,marksout ,and continues skipping and marking out the next number to the end of the list.Then he goesback to the start of his list,marks out the first remaining number,skips the next number,marks out ,skips ,marks out ,and so on to the end.Josh continues in this manner until only one number remains.What is that number?Josh 写下一列数字1，2，3，…，99，100，他划掉1，跳过2，划掉3，并继续跳过和划掉接下来的数字，直到这列数字的末尾。

For office use only T1T2T3T4T eam Control Number42939Problem ChosenCFor office use onlyF1F2F3F42016Mathematical Contest in Modeling(MCM)Summary Sheet (Attach a copy of this page to each copy of your solution paper.)SummaryIn order to determine the optimal donation strategy,this paper proposes a data-motivated model based on an original definition of return on investment(ROI) appropriate for charitable organizations.First,after addressing missing data,we develop a composite index,called the performance index,to quantify students’educational performance.The perfor-mance index is a linear composition of several commonly used performance indi-cators,like graduation rate and graduates’earnings.And their weights are deter-mined by principal component analysis.Next,to deal with problems caused by high-dimensional data,we employ a lin-ear model and a selection method called post-LASSO to select variables that statis-tically significantly affect the performance index and determine their effects(coef-ficients).We call them performance contributing variables.In this case,5variables are selected.Among them,tuition&fees in2010and Carnegie High-Research-Activity classification are insusceptible to donation amount.Thus we only con-sider percentage of students who receive a Pell Grant,share of students who are part-time and student-to-faculty ratio.Then,a generalized adaptive model is adopted to estimate the relation between these3variables and donation amount.Wefit the relation across all institutions and get afitted function from donation amount to values of performance contributing variables.Then we divide the impact of donation amount into2parts:homogenous and heterogenous one.The homogenous influence is modeled as the change infit-ted values of performance contributing variables over increase in donation amount, which can be predicted from thefitted curve.The heterogenous one is modeled as a tuning parameter which adjusts the homogenous influence based on deviation from thefitted curve.And their product is increase in true values of performance over increase in donation amount.Finally,we calculate ROI,defined as increase in performance index over in-crease in donation amount.This ROI is institution-specific and dependent on in-crease in donation amount.By adopting a two-step ROI maximization algorithm, we determine the optimal investment strategy.Also,we propose an extended model to handle problems caused by time dura-tion and geographical distribution of donations.A Letter to the CFO of the Goodgrant FoundationDear Chiang,Our team has proposed a performance index quantifying the students’educational per-formance of each institution and defined the return of investment(ROI)appropriately for a charitable organization like Goodgrant Foundation.A mathematical model is built to help predict the return of investment after identifying the mechanism through which the donation generates its impact on the performance.The optimal investment strategy is determined by maximizing the estimated return of investment.More specifically,the composite performance index is developed after taking all the pos-sible performance indicators into consideration,like graduation rate and graduates’earnings. The performance index is constructed to represents the performance of the school as well as the positive effect that a college brings to students and the community.From this point of view, our definition manages to capture social benefits of donation.And then we adopt a variable selection method tofind out performance contributing vari-ables,which are variables that strongly affect the performance index.Among all the perfor-mance contributing variables we select,three variables which can be directly affected by your generous donation are kept to predict ROI:percentage of students who receive a Pell Grant, share of students who are part-time and student-to-faculty ratio.Wefitted a relation between these three variables and the donation amount to predict change in value of each performance contributing variable over your donation amount.And we calculate ROI,defined as increase in the performance index over your donation amount, by multiplying change in value of each performance contributing variable over your donation amount and each performance contributing variable’s effect on performance index,and then summing up the products of all performance contributing variables.The optimal investment strategy is decided after maximizing the return of investment according to an algorithm for selection.In conclusion,our model successfully produced an investment strategy including a list of target institutions and investment amount for each institution.(The list of year1is attached at the end of the letter).The time duration for the investment could also be determined based on our model.Since the model as well as the evaluation approach is fully data-motivated with no arbitrary criterion included,it is rather adaptable for solving future philanthropic educational investment problems.We have a strong belief that our model can effectively enhance the efficiency of philan-thropic educational investment and provides an appropriate as well as feasible way to best improve the educational performance of students.UNITID names ROI donation 197027United States Merchant Marine Academy21.85%2500000 102711AVTEC-Alaska’s Institute of Technology21.26%7500000 187745Institute of American Indian and Alaska Native Culture20.99%2000000 262129New College of Florida20.69%6500000 216296Thaddeus Stevens College of Technology20.66%3000000 229832Western Texas College20.26%10000000 196158SUNY at Fredonia20.24%5500000 234155Virginia State University20.04%10000000 196200SUNY College at Potsdam19.75%5000000 178615Truman State University19.60%3000000 199120University of North Carolina at Chapel Hill19.51%3000000 101648Marion Military Institute19.48%2500000187912New Mexico Military Institute19.31%500000 227386Panola College19.28%10000000 434584Ilisagvik College19.19%4500000 199184University of North Carolina School of the Arts19.15%500000 413802East San Gabriel Valley Regional Occupational Program19.09%6000000 174251University of Minnesota-Morris19.09%8000000 159391Louisiana State University and Agricultural&Mechanical Col-19.07%8500000lege403487Wabash Valley College19.05%1500000 Yours Sincerely,Team#42939An Optimal Strategy of Donation for Educational PurposeControl Number:#42939February,2016Contents1Introduction51.1Statement of the Problem (5)1.2Baseline Model (5)1.3Detailed Definitions&Assumptions (8)1.3.1Detailed Definitions: (8)1.3.2Assumptions: (9)1.4The Advantages of Our Model (9)2Addressing the Missing Values93Determining the Performance Index103.1Performance Indicators (10)3.2Performance Index via Principal-Component Factors (10)4Identifying Performance Contributing Variables via post-LASSO115Determining Investment Strategy based on ROI135.1Fitted Curve between Performance Contributing Variables and Donation Amount145.2ROI(Return on Investment) (15)5.2.1Model of Fitted ROIs of Performance Contributing Variables fROI i (15)5.2.2Model of the tuning parameter P i (16)5.2.3Calculation of ROI (17)5.3School Selection&Investment Strategy (18)6Extended Model186.1Time Duration (18)6.2Geographical Distribution (22)7Conclusions and Discussion22 8Reference23 9Appendix241Introduction1.1Statement of the ProblemThere exists no doubt in the significance of postsecondary education to the development of society,especially with the ascending need for skilled employees capable of complex work. Nevertheless,U.S.ranks only11th in the higher education attachment worldwide,which makes thefinancial support from large charitable organizations necessary.As it’s essential for charitable organizations to maximize the effectiveness of donations,an objective and systematic assessment model is in demand to develop appropriate investment strategies.To achieve this goal,several large foundations like Gates Foundation and Lumina Foundation have developed different evaluation approaches,where they mainly focus on spe-cific indexes like attendance and graduation rate.In other empirical literature,a Forbes ap-proach(Shifrin and Chen,2015)proposes a new indicator called the Grateful Graduates Index, using the median amount of private donations per student over a10-year period to measure the return on investment.Also,performance funding indicators(Burke,2002,Cave,1997,Ser-ban and Burke,1998,Banta et al,1996),which include but are not limited to external indicators like graduates’employment rate and internal indicators like teaching quality,are one of the most prevailing methods to evaluate effectiveness of educational donations.However,those methods also arise with widely acknowledged concerns(Burke,1998).Most of them require subjective choice of indexes and are rather arbitrary than data-based.And they perform badly in a data environment where there is miscellaneous cross-section data but scarce time-series data.Besides,they lack quantified analysis in precisely predicting or measuring the social benefits and the positive effect that the investment can generate,which serves as one of the targets for the Goodgrant Foundation.In accordance with Goodgrant Foundation’s request,this paper provides a prudent def-inition of return on investment(ROI)for charitable organizations,and develops an original data-motivated model,which is feasible even faced with tangled cross-section data and absent time-series data,to determine the optimal strategy for funding.The strategy contains selection of institutions and distribution of investment across institutions,time and regions.1.2Baseline ModelOur definition of ROI is similar to its usual meaning,which is the increase in students’educational performance over the amount Goodgrant Foundation donates(assuming other donationsfixed,it’s also the increase in total donation amount).First we cope with data missingness.Then,to quantify students’educational performance, we develop an index called performance index,which is a linear composition of commonly used performance indicators.Our major task is to build a model to predict the change of this index given a distribution of Goodgrant Foundation$100m donation.However,donation does not directly affect the performance index and we would encounter endogeneity problem or neglect effects of other variables if we solely focus on the relation between performance index and donation amount. Instead,we select several variables that are pivotal in predicting the performance index from many potential candidates,and determine their coefficients/effects on the performance index. We call these variables performance contributing variables.Due to absence of time-series data,it becomes difficult tofigure out how performance con-tributing variables are affected by donation amount for each institution respectively.Instead, wefit the relation between performance contributing variables and donation amount across all institutions and get afitted function from donation amount to values of performance contribut-ing variables.Then we divide the impact of donation amount into2parts:homogenous and heteroge-nous one.The homogenous influence is modeled as the change infitted values of performance contributing variables over increase in donation amount(We call these quotientsfitted ROI of performance contributing variable).The heterogenous one is modeled as a tuning parameter, which adjusts the homogenous influence based on deviation from thefitted function.And their product is the institution-specific increase in true values of performance contributing variables over increase in donation amount(We call these values ROI of performance contributing vari-able).The next step is to calculate the ROI of the performance index by adding the products of ROIs of performance contributing variables and their coefficients on the performance index. This ROI is institution-specific and dependent on increase in donation amount.By adopting a two-step ROI maximization algorithm,we determine the optimal investment strategy.Also,we propose an extended model to handle problems caused by time duration and geographical distribution of donations.Note:we only use data from the provided excel table and that mentioned in the pdffile.Table1:Data SourceVariable DatasetPerformance index Excel tablePerformance contributing variables Excel table and pdffileDonation amount PdffileTheflow chart of the whole model is presented below in Fig1:Figure1:Flow Chart Demonstration of the Model1.3Detailed Definitions&Assumptions 1.3.1Detailed Definitions:1.3.2Assumptions:A1.Stability.We assume data of any institution should be stable without the impact from outside.To be specific,the key factors like the donation amount and the performance index should remain unchanged if the college does not receive new donations.A2.Goodgrant Foundation’s donation(Increase in donation amount)is discrete rather than continuous.This is reasonable because each donation is usually an integer multiple of a minimum amount,like$1m.After referring to the data of other foundations like Lumina Foundation,we recommend donation amount should be one value in the set below:{500000,1000000,1500000, (10000000)A3.The performance index is a linear composition of all given performance indicators.A4.Performance contributing variables linearly affect the performance index.A5.Increase in donation amount affects the performance index through performance con-tributing variables.A6.The impact of increase in donation amount on performance contributing variables con-tains2parts:homogenous one and heterogenous one.The homogenous influence is repre-sented by a smooth function from donation amount to performance contributing variables.And the heterogenous one is represented by deviation from the function.1.4The Advantages of Our ModelOur model exhibits many advantages in application:•The evaluation model is fully data based with few subjective or arbitrary decision rules.•Our model successfully identifies the underlying mechanism instead of merely focusing on the relation between donation amount and the performance index.•Our model takes both homogeneity and heterogeneity into consideration.•Our model makes full use of the cross-section data and does not need time-series data to produce reasonable outcomes.2Addressing the Missing ValuesThe provided datasets suffer from severe data missing,which could undermine the reliabil-ity and interpretability of any results.To cope with this problem,we adopt several different methods for data with varied missing rate.For data with missing rate over50%,any current prevailing method would fall victim to under-or over-randomization.As a result,we omit this kind of data for simplicity’s sake.For variables with missing rate between10%-50%,we use imputation techniques(Little and Rubin,2014)where a missing value was imputed from a randomly selected similar record,and model-based analysis where missing values are substituted with distribution diagrams.For variables with missing rate under10%,we address missingness by simply replace miss-ing value with mean of existing values.3Determining the Performance IndexIn this section,we derive a composite index,called the performance index,to evaluate the educational performance of students at every institution.3.1Performance IndicatorsFirst,we need to determine which variables from various institutional performance data are direct indicators of Goodgrant Foundation’s major concern–to enhance students’educational performance.In practice,other charitable foundations such as Gates Foundation place their focus on core indexes like attendance and graduation rate.Logically,we select performance indicators on the basis of its correlation with these core indexes.With this method,miscellaneous performance data from the excel table boils down to4crucial variables.C150_4_P OOLED_SUP P and C200_L4_P OOLED_SUP P,as completion rates for different types of institutions,are directly correlated with graduation rate.We combine them into one variable.Md_earn_wne_p10and gt_25k_p6,as different measures of graduates’earnings,are proved in empirical studies(Ehren-berg,2004)to be highly dependent on educational performance.And RP Y_3Y R_RT_SUP P, as repayment rate,is also considered valid in the same sense.Let them be Y1,Y2,Y3and Y4.For easy calculation and interpretation of the performance index,we apply uniformization to all4variables,as to make sure they’re on the same scale(from0to100).3.2Performance Index via Principal-Component FactorsAs the model assumes the performance index is a linear composition of all performance indicators,all we need to do is determine the weights of these variables.Here we apply the method of Customer Satisfaction Index model(Rogg et al,2001),where principal-component factors(pcf)are employed to determine weights of all aspects.The pcf procedure uses an orthogonal transformation to convert a set of observations of pos-sibly correlated variables into a set of values of linearly uncorrelated variables called principal-component factors,each of which carries part of the total variance.If the cumulative proportion of the variance exceeds80%,it’s viable to use corresponding pcfs(usually thefirst two pcfs)to determine weights of original variables.In this case,we’ll get4pcfs(named P CF1,P CF2,P CF3and P CF4).First,the procedure provides the linear coefficients of Y m in the expression of P CF1and P CF2.We getP CF1=a11Y1+a12Y2+a13Y3+a14Y4P CF2=a21Y1+a22Y2+a23Y3+a24Y4(a km calculated as corresponding factor loadings over square root of factor k’s eigenvalue) Then,we calculate the rough weights c m for Y m.Let the variance proportions P CF1and P CF2 represent be N1and N2.We get c m=(a1m N1+a2m N2)/(N1+N2)(This formulation is justifiedbecause the variance proportions can be viewed as the significance of pcfs).If we let perfor-mance index=(P CF 1N 1+P CF 2N 2)/(N 1+N 2),c m is indeed the rough weight of Y m in terms of variance)Next,we get the weights by adjusting the sum of rough weights to 1:c m =c m /(c 1+c 2+c 3+c 4)Finally,we get the performance index,which is the weighted sum of the 4performance indicator.Performance index= m (c m Y m )Table 2presents the 10institutions with largest values of the performance index.This rank-ing is highly consistent with widely acknowledged rankings,like QS ranking,which indicates the validity of the performance index.Table 2:The Top 10Institutions in Terms of Performance IndexInstitutionPerformance index Los Angeles County College of Nursing and Allied Health79.60372162Massachusetts Institute of Technology79.06066895University of Pennsylvania79.05044556Babson College78.99269867Georgetown University78.90468597Stanford University78.70586395Duke University78.27719116University of Notre Dame78.15843964Weill Cornell Medical College 78.143341064Identifying Performance Contributing Variables via post-LASSO The next step of our model requires identifying the factors that may exert an influence on the students’educational performance from a variety of variables mentioned in the excel table and the pdf file (108in total,some of which are dummy variables converted from categorical variables).To achieve this purpose,we used a model called LASSO.A linear model is adopted to describe the relationship between the endogenous variable –performance index –and all variables that are potentially influential to it.We assign appropriate coefficient to each variable to minimize the square error between our model prediction and the actual value when fitting the data.min β1J J j =1(y j −x T j β)2where J =2881,x j =(1,x 1j ,x 2j ,...,x pj )THowever,as the amount of the variables included in the model is increasing,the cost func-tion will naturally decrease.So the problem of over fitting the data will arise,which make the model we come up with hard to predict the future performance of the students.Also,since there are hundreds of potential variables as candidates.We need a method to identify the variables that truly matter and have a strong effect on the performance index.Here we take the advantage of a method named post-LASSO (Tibshirani,1996).LASSO,also known as the least absolute shrinkage and selection operator,is a method used for variableselection and shrinkage in medium-or high-dimensional environment.And post-LASSO is to apply ordinary least squares(OLS)to the model selected byfirst-step LASSO procedure.In LASSO procedure,instead of using the cost function that merely focusing on the square error between the prediction and the actual value,a penalty term is also included into the objective function.We wish to minimize:min β1JJj=1(y j−x T jβ)2+λ||β||1whereλ||β||1is the penalty term.The penalty term takes the number of variables into con-sideration by penalizing on the absolute value of the coefficients and forcing the coefficients of many variables shrink to zero if this variable is of less importance.The penalty coefficient lambda determines the degree of penalty for including variables into the model.After min-imizing the cost function plus the penalty term,we couldfigure out the variables of larger essence to include in the model.We utilize the LARS algorithm to implement the LASSO procedure and cross-validation MSE minimization(Usai et al,2009)to determine the optimal penalty coefficient(represented by shrinkage factor in LARS algorithm).And then OLS is employed to complete the post-LASSO method.Figure2:LASSO path-coefficients as a function of shrinkage factor sFigure3:Cross-validated MSEFig2.displays the results of LASSO procedure and Fig3displays the cross-validated MSE for different shrinkage factors.As specified above,the cross-validated MSE reaches minimum with shrinkage factor between0.4-0.8.We choose0.6andfind in Fig2that6variables have nonzero coefficients via the LASSO procedure,thus being selected as the performance con-tributing variables.Table3is a demonstration of these6variables and corresponding post-LASSO results.Table3:Post-LASSO resultsDependent variable:performance_indexPCTPELL−26.453∗∗∗(0.872)PPTUG_EF−14.819∗∗∗(0.781)StudentToFaculty_ratio−0.231∗∗∗(0.025)Tuition&Fees20100.0003∗∗∗(0.00002)Carnegie_HighResearchActivity 5.667∗∗∗(0.775)Constant61.326∗∗∗(0.783)Observations2,880R20.610Adjusted R20.609Note:PCTPELL is percentage of students who receive aPell Grant;PPTUG_EF is share of students who are part-time;Carnegie_HighResearchActivity is Carnegie classifica-tion basic:High Research ActivityThe results presented in Table3are consistent with common sense.For instance,the pos-itive coefficient of High Research Activity Carnegie classification implies that active research activity helps student’s educational performance;and the negative coefficient of Student-to-Faculty ratio suggests that decrease in faculty quantity undermines students’educational per-formance.Along with the large R square value and small p-value for each coefficient,the post-LASSO procedure proves to select a valid set of performance contributing variables and describe well their contribution to the performance index.5Determining Investment Strategy based on ROIWe’ve identified5performance contributing variables via post-LASSO.Among them,tu-ition&fees in2010and Carnegie High-Research-Activity classification are quite insusceptible to donation amount.So we only consider the effects of increase in donation amount on per-centage of students who receive a Pell Grant,share of students who are part-time and student-to-faculty ratio.We denote them with F1,F2and F3,their post-LASSO coefficients withβ1,β2andβ3.In this section,wefirst introduce the procedure used tofit the relation between performance contributing variables and donation amount.Then we provide the model employed to calcu-latefitted ROIs of performance contributing variables(the homogenous influence of increase in donation amount)and the tuning parameter(the heterogenous influence of increase in dona-tion amount).Next,we introduce how to determine stly,we show how the maximiza-tion determines the investment strategy,including selection of institutions and distribution of investments.5.1Fitted Curve between Performance Contributing Variables and Donation AmountSince we have already approximated the linear relation between the performance index with the3performance contributing variables,we want to know how increase in donation changes them.In this paper,we use Generalized Adaptive Model(GAM)to smoothlyfit the relations. Generalized Adaptive Model is a generalized linear model in which the dependent variable depends linearly on unknown smooth functions of independent variables.Thefitted curve of percentage of students who receive a Pell Grant is depicted below in Fig4(see the other two fitted curves in Appendix):Figure4:GAM ApproximationA Pell Grant is money the U.S.federal government provides directly for students who needit to pay for college.Intuitively,if the amount of donation an institution receives from other sources such as private donation increases,the institution is likely to use these donations to alleviate students’financial stress,resulting in percentage of students who receive a Pell Grant. Thus it is reasonable to see afitted curve downward sloping at most part.Also,in commonsense,an increase in donation amount would lead to increase in the performance index.This downward sloping curve is consistent with the negative post-LASSO coefficient of percentage of students who receive a Pell Grant(as two negatives make a positive).5.2ROI(Return on Investment)5.2.1Model of Fitted ROIs of Performance Contributing Variables fROI iFigure5:Demonstration of fROI1Again,we usefitted curve of percentage of students who receive a Pell Grant as an example. We modeled the bluefitted curve to represent the homogeneous relation between percentage of students who receive a Pell Grant and donation amount.Recallfitted ROI of percentage of students who receive a Pell Grant(fROI1)is change in fitted values(∆f)over increase in donation amount(∆X).SofROI1=∆f/∆XAccording to assumption A2,the amount of each Goodgrant Foundation’s donation falls into a pre-specified set,namely,{500000,1000000,1500000,...,10000000}.So we get a set of possible fitted ROI of percentage of students who receive a Pell Grant(fROI1).Clearly,fROI1is de-pendent on both donation amount(X)and increase in donation amount(∆X).Calculation of fitted ROIs of other performance contributing variables is similar.5.2.2Model of the tuning parameter P iAlthough we’ve identified the homogenous influence of increase in donation amount,we shall not neglect the fact that institutions utilize donations differently.A proportion of do-nations might be appropriated by the university’s administration and different institutions allocate the donation differently.For example,university with a more convenient and well-maintained system of identifying students who needfinancial aid might be willing to use a larger portion of donations to directly aid students,resulting in a lower percentage of under-graduate students receiving Pell grant.Also,university facing lower cost of identifying and hiring suitable faculty members might be inclined to use a larger portion of donations in this direction,resulting in a lower student-to-faculty ratio.These above mentioned reasons make institutions deviate from the homogenousfitted func-tion and presents heterogeneous influence of increase in donation amount.Thus,while the homogenous influence only depends on donation amount and increase in donation amount, the heterogeneous influence is institution-specific.To account for this heterogeneous influence,we utilize a tuning parameter P i to adjust the homogenous influence.By multiplying the tuning parameter,fitted ROIs of performance con-tributing variables(fitted value changes)convert into ROI of performance contributing variable (true value changes).ROI i=fROI i·P iWe then argue that P i can be summarized by a function of deviation from thefitted curve (∆h),and the function has the shape shown in Fig6.The value of P i ranges from0to2,because P i can be viewed as an amplification or shrinkage of the homogenous influence.For example,P i=2means that the homogeneous influence is amplified greatly.P i=0means that this homogeneous influence would be entirely wiped out. The shape of the function is as shown in Fig6because of the following reasons.Intuitively,if one institution locates above thefitted line,when deviation is small,the larger it is,the larger P i is.This is because the institution might be more inclined to utilize donations to change that factor.However,when deviation becomes even larger,the institution grows less willing to invest on this factor.This is because marginal utility decreases.The discussion is similar if one institution initially lies under thefitted line.Thus,we assume the function mapping deviation to P i is similar to Fig6.deviation is on the x-axis while P i is on the y-axis.Figure6:Function from Deviation to P iIn order to simplify calculation and without loss of generality,we approximate the function。

衡量社会信息网络的演进与影响信息传播迅速在今天的技术连接的通信网络，有时它是由于信息本身的固有价值，以及其他时间，它是由于信息找到它的方式通过社交媒体加速传播的影响或中心网络节点。

虽然内容在19世纪不同，新闻更多的是当地的活动（如婚礼、风暴、死亡）而不是病毒猫咪视频或艺人的社会生活，当时的前提是，这种文化特征分享信息（这两个严肃的和微不足道的）一直在那里。

然而，信息流从来没有像今天这样容易或广泛，允许各种程度的新闻传播的重要性在我们的技术世界中快速穿越地球。

以历史的视角看信息与信息的内在价值，传媒学院（ICM）寻求了解社会网络的方法、目的和功能的演变。

具体来说，你的团队，作为ICM的信息分析部门的一部分，已分配分析基于信息的速度/流量与信息固有价值的关系5个阶段的思考：在19世纪70年代，当报纸被火车和故事发表了.在20世纪20年代，当收音机成为一种更常见的家居用品时，在20世纪70年代，当电视在大多数家庭中，在20世纪90年代，当家庭开始连接到早期互联网；在2010年，当我们可以连接到世界上我们的手机。

你的主管提醒你一定要报告你做的假设和你使用的数据建立你的模型。

您的具体任务是：（一）开发一个或多个模型（甲），让你可以探索信息流和过滤器的流程或者找到什么是新闻。

（乙）使用数据从过去的数据和预测能力验证您的模型的可靠性您的模型来预测信息通信的情况，今天和比较今天的现实。

（丙）利用你的模型预测通信网络的关系和能力2050年。

（4）利用信息对网络的影响的理论和概念来模拟公众在今天的互联世界中，可以通过信息网络改变兴趣和意见。

（一）确定信息价值，人们最初的意见和偏见，形式的信息或它的来源，以及在一个地区，国家或地区的信息网络的拓扑结构或强度。

世界范围内可用于传播信息和影响公众舆论。

可能的数据来源：当你开发你的模型并准备测试它，你需要收集数据的集合。

下面只是一些你在这个项目中发现有用的数据类型的例子。

根据您的确切模型中，某些类型的数据可能是非常重要的，其他的可能是完全无关的。

Contents1.Introduction (1)1.1 Background (1)1.2 Foundation & ROI (1)2 Task (1)3 Fundamental assumptions (2)4 Definitions and Notations (2)5 Models (3)5.1 Filter data (3)5.2 Object Selection Model (Grey Relational Analysis) (4)5.2.1 Model analysis (4)5.2.2 Model solution (4)5.3 ROI Model (Principal Component Analysis) (5)5.3.1 Model analysis (5)5.3.2 Model solution (6)5.4 Verify the possibility (9)5.4.1 Comparison (9)5.4.2 External factor (10)5.5 Investment Forecast Model (11)5.5.1 Linear Regression Forecasting Model (11)5.5.2 School potential Prediction (TOPSIS) (12)5.5.3 Final investment (TOPSIS) (13)6 Conclusions (16)7 Strengths and Weaknesses (18)7.1 Strengths (19)7.2 Weaknesses (20)8 Letter to Mr. Alpha Chiang (21)9 References (22)1 Introduction1.1 BackgroundThe Goodgrant Foundation is a charitable organization that wants to help improve educational performance of undergraduates attending colleges and universities in the United States. To do this, the foundation intends to donate a total of $100,000,000 (US100 million) to an appropriate group of schools per year, for five years, starting July 2016. In doing so, they do not want to duplicate the investments and focus of other large grant organizations such as the Gates Foundation and Lumina Foundation.Our team has been asked by the Goodgrant Foundation to develop a model to determine an optimal investment strategy that identifies the schools, the investment amount per school, the return on that investment, and the time duration that the organi zation’s money should be provided to have the highest likelihood of producing a strong positive effect on student performance. This strategy should contain a 1 to N optimized and prioritized candidate list of schools you are recommending for investment bas ed on each candidate school’s demonstrated potential for effective use of private funding, and an estimated return on investment (ROI) defined in a manner appropriate for a charitable organization such as the Goodgrant Foundation.1.2 Foundation & ROIFoundation (charitable foundation) refers to the nonprofit legal person who uses the property of the natural persons, legal persons or other organizations to engage in public welfare undertakings. In terms of its nature, foundation is a kind of folk non-profit organizations.ROI is a performance measure used to evaluate the efficiency of an investment or to compare the efficiency of a number of different investments. ROI measures the amount of return on an investment relative to the investment’s cost. To calculate ROI, the benefit (or return) of an investment is divided by the cost of the investment, and the result is expressed as a percentage or a ratio.2 Task●One-page summary for our MCM submission●Using our models to achieve the candidate list of schools●Calculate the time durati on that the organization’s money should be provided to have thehighest likelihood of producing a strong positive effect on student performance●Calculate the investment amount Goodgrant Foundation would pay for each school●Calculate the ROI of the Goodgrant Foundation●Forecast the development of this kind of investment mode●Write a letter to the CFO of the Goodgrant Foundation, Mr. Alpha Chiang, that describesthe optimal investment strategy3 Fundamental assumptions1) The indexes of GRA (such as ACT 、SA T 、Pell Grant 、Graduation Rate 、Retention Rate 、Graduates income)are the most influential factor that affect the use potential of school funds, what’s more, the indexes have the same weight2) For four-year universities, their C200_L4_POOLED_SUPP 、RET_FTL4、RET_PTL4 arezero; For two-year colleges, their C150_4_POOLED_SUPP 、RET_FT4、RET_PT4 are zero3) For public institutions ，their NPT4_PRIV 、NPT41_PRIV 、NPT42_PRIV 、NPT43_PRIV 、NPT44_PRIV 、NPT45_PRIV are zero; For private for-profit and nonprofit institutions, their NPT4_PUB 、NPT41_PUB 、NPT42_PUB 、NPT43_PUB 、NPT44_PUB 、NPT45_PUB are zero4) We define “NULL” appears in the data except appears in 3) as the average of that series5) Ignore the influence of degree-conferring situation, race, religion, region6) Schools’ data of SA T and ACT d evelop in a linear trend4 Definitions and NotationsTable A: The Excel which contain the IPEDS UID for Potential Candidate SchoolsTable B: The Excel which contain the Most Recent Cohorts Data (Scorecard Elements) : The weight of the first k index)(k iξ: Grey relational coefficient r i : Grey weight relation: Standardized index value: Index value: Sample average: Sample standard deviation:Standardized index vector:The correlation coefficient y i: Main components : Rate of contribution:The cumulative contribution rate : The comprehensive scorek w ~a ij a ij j μj s ~x jr ijb j pαZ5 Models5.1 Filter dataBecause the topic has a large number of additional data, we should classify the data based on the College Scorecard Data Dictionary which we can find from the official website .And then, according to the flow diagram5.1.1 as follow, we can filtering data. By using that flow diagram, we set up limits to filter the data circularly. We use SPSS to achieve above purpose, after that we get the valid data of 2936 potential candidate schools.InputTable A,TableBBased on the ID in TableA, merge Table A and BTable A has theID which TableB don’tDelete that IDin Table AOutputTable AOutputTableAFigure 5.1.1 Flow Diagram5.2 Object Selection Model (Grey Relational Analysis)5.2.1 Model analysisNot only there is a large amounts of missing in the original data, but also even after a preliminary screening, the data volume is still very large. We find there are more than 50 factors which affect us optimizing the school , what ’s more, the link between each factor we can't find accurately. So the normal model has no use to predict and evaluate the data .But Grey correlation analysis method is both suitable for irregular data and normal data. The quantitative results are consistent with qualitative analysis perfectly. Therefore, we choose that method to further narrowing the scope of the data.Grey correlation analysis is based on the similarity degree of various factors ’ changing curve , it can judge the correlation degree of each index .Through the quantitative analysis of the dynamic process, we can get the geometrical relationship of statistical data in the system and the grey correlation degree between the reference sequence and compare sequence. The greater the comparative sequence ’s correlation degree is , the closer the relationship between the reference sequence and compare sequence will be.The basic idea is to standardized the original observation and calculate the correlation coefficient, correlation degree. And then rank the indexes according to the size of the correlation. The application of GRA involves many fields, especially in the field of social economy, such as the ROI of the national economy departments , analyzing regional economic advantages , industrial structure adjustment, GRA has a good application effect.5.2.2 Model solution1) We have 2936 objects （potential candidate schools ）and 7 evaluation indexes （ACT 、SAT 、Pell Grant 、Graduation Rate 、Retention Rate 、Graduates income ），the reference sequence is}7,...,2,1|)({00==k k x x , the compare sequence is 2936,...,2,1},7,...,2,1|)({===i k k x i i x ；2) The weight of every index is ],...,[71w w w =，)7,...,2,1(=k w k means the first k index ’sweight , in that model we assume every index ’s weight is equal.3) |)()(|max max |)()(||)()(|max max |)()(|min min )(0000t x t x t x k x t x t x t x t x k s t s i s t s s t s i-+--+-=ρρξ 4) )(71k i k i i w r ξ∑==5) By using MATLAB, we can finish above data calculation process ,and then we can makethe figure of each evaluation objects’ grey weight relatio n as follow:Figure 5.2.1 Distribution Diagram of Grey Weight Relation According to Figure 5.2.1,we find that 90% objects’grey weight relation is under 0.60.We use MA TLAB to rank the evaluation objects, because the greater the grey weight relation is, the better the evaluation result is. We choose the first 300 objects as the new potential candidate schools.But during the process of GRA, the weight of evaluation indexes has deviation, we can’t get the accurate solution, so we need build another model to analyze in detail.5.3ROI Model (Principal Component Analysis)5.3.1 Model analysisAccording to the topic’s requirements, we should choose the potential candidate schools by their money using ability, but there are so many indexes influence the resul t, we can’t get reliable evaluation only by one factor.In the study of that practical problem, in order to comprehensively and systematically analyze problems, we must consider many factors. These involved factors generally referred to the index, also known as a variable in the multivariate statistical analysis. Because each variable reflects some information of the research question and all of them has a certain correlation , the information one index reflects may overlap another.PCA is to use less variables to explain most variables of the original data, it transforms many high correlation variables into uncorrelated variables. Usually the number of new variables is smaller than the original variables, we call it principal component and use it to explain the comprehensive index.This method simplify the problem, at the same time make the result more scientific and effectiveTherefore ,it seems easier to use PCA solving the Optimization problem. By reducing dimension, we transform many indexes(such as Net Price, Repayment of Debt, Repayment Rate, Graduates Income) into a few principal components.5.3.2 Model solution1) Standardized the original data,15,...,2,1,300,...,2,1,~==-=j i s j j ij ij a a μ;15,...,2,1,)(13001,3001300123001=--==∑∑==j a a i j ij j i ij j s μμ 15,...,2,1,~=-=j s x x j j jjμ2) Calculate the correlation coefficient matrix,15*15)(r ij R =，15,...,2,1,,1300*~3001~=-=∑=j i a a r kj k ki ij3) Calculate the eigenvalue and the feature vectors,By using SPSS, we calculate the eigenvalue of R ,0...1521≥≥≥≥λλλ，And the standardized feature vector ,,...,,1521μμμby using the feature vector, we get 15 newindexes ，,......,,...,...~151515~2215~111515~15152~222~1122~15151~221~1111x x x x x x x x x y yyμμμμμμμμμ+++=+++=+++= 4) Choose P （15≤P ）principal component ，calculate the comprehensive score①Calculate the contribution rate of )15,...,2,1(=j j λ and cumulative contribution rate∑∑====151151,j j p k k j j b b αλλ By using SPSS, we get the contribution rate of 15 eigenvalue as follow:Table 5.3.1 Contribution RateAnalyzing Table 5.3.1，we find the cumulative contribution rate of the first three eigenvalue is more than 90%, the model has a well result.②Choose the first three principal component for a comprehensive evaluation,y b j p j j Z ∑==1By using SPSS, we get the feature vectors of the first three characteristic root,shows in Table 5.3.2Table 5.3.2 The Feature VectorsWe make the first three principal components ’ contribution rate as weight, build up the principal component comprehensive evaluation model,y y y Z 3211031.01153.06892.0++= Put every optimized school’s three principal components into above equation ，we get the comprehensive evaluation result of 300 new potential candidate schools, shows in figure5.3.3 ，Fac_1 on behalf ofy 1，Fac_2 on behalf of y 2,Fac_3 on behalf of y 3， Total onbehalf of Z ，Figure 5.3.3 Comprehensive evaluation result of 300 schoolsWe choose the first 40 schools as the final potential candidates list ，the comprehensive evaluation result shows in figure 5.3.4，Figure 5.3.4 Comprehensive evaluation result of 40 final potential schoolsBecause the indexes (such as Net Price, Repayment of Debt, Repayment Rate, Graduates Income) of this model are closely related to ROI, we use the comprehensive score to evaluate schools’ ROI. We ranked it in Table 5.3.3,Table 5.3.3 Rank5.4 Verify the possibility5.4.1 ComparisonThrough comparison we found that many famous universities such as Harvard University does not appear in our preferred list, this can’t help but let us create confusion, what is the reason causes this kind of phenomenon ?Considering that phenomenon, we analyze from the model itself. The topic asks us to optimize the school list which is based on the potential of fund using .We've learned from the related literature , there is a positive correlation between the income of graduate individual and donation. As their income level become higher, the possibility and amount of donation is greater. According to statistics, every 1% increase in income, the possibility to donation increase 0.35% ~ 0.5%; when the personal income increased $10000, donation amount can increased 2%; per $10000 increase in household income, donation amount can increased 9%.So at the beginning, we choose Net Price, Repayment of Debt, Repayment Rate, Graduates Income as evaluation indexes.We standardized the indexes, and then compare the data of final potential schools with Harvard University, the result shows in Figure 5.4.1Figure 5.4.1 ComparisonAccording to Figure 5.4.1, we find the average net price and the monthly repayment of the debt are significantly less than final potential schools. The bigger the average net price and the monthly repayment of the debt are, the higher the students’ needs for money are, so the school has high potential to use the fund.Like Harvard University, there are many famous foundations invest it, its students have enough funds ,so there is no doubt that it has low average net price and monthly repayment of the debt. That is the reason which makes Harvard University get low comprehensive score in our model.5.4.2 External factorBecause the Goodgrant Foundation do not want to duplicate the investments and focus of other large grant organizations, we check out the Gates foundation's donation list of schools and compare it with the potential candidates list which we get in Figure 5.3.4, what’s more, we eliminate repetitive schools to get the final candidates list, shows in Table 5.4.2(the red name means that school should remove from the potential candidates list),Table 5.4.2 Real rank5.5 Investment Forecast Model5.5.1 Linear Regression Forecasting Model1)We find previous years’ (09-13years) Reference data which is closely related to students’performance(such as ACT，SAT, Enrollment of undergraduate degree-seeking students) from “https:///ipeds/datacenter/Default.aspx”.According to Table 5.4.2,we use that 36 schools’ data to do the linear regression prediction.2)There are many schools’ data lost in the table,so how to deal with those “NULL”?We choose other years data of that school, doing the linear regression prediction, and then getthe data of that year. If other years data have also lost , we take the average data of other schools as the lost data.3)Because the data we can use is limited(no more than 5 years),we can’t doing theComplex forecast. We choose Linear Regression prediction, by analyzing the variation trend of 36 schools’ data and forecast 2016~2020 years’ ACT and SCT . Based on forecast data, We make the diagram of 2016 ~ 2020 data variation trend which shows as Figure 5.5.1 and Figure 5.5.2 ,analyze the development of student performance primarily.Figure 5.5.1 Figure 5.5.25.5.2 School potential Prediction (TOPSIS)1)Based on forecast data which we get from Linear Regression Forecasting Model, we useTOPSIS method to evaluate the effect which The Goodgrant Foundation’s investment can takes in the future 5years.2)We take five factors (such as ACT and SAT mark、the growing trend of the mark and thegrowing trend of undergraduate degree-seeking students’ number) as indexes, evaluate the potential of its annual school development . We calculate the comprehensive evaluation score, and then sorting every candidate school by score.3)According to the influence the five factors have, we define the weight of every factor asfollow: ACT mark--0.25, SAT mark--0.25,both the growing trend of ACT and SAT are0.2, the growing trend of undergraduate degree-seeking students’ number--0.1.4)We use MA TLAB to calculate the comprehensive evaluation score and show the result inFigure 5.4.3,analyze the Figure, we find between 2016 and 2020 every school’s score changes little, it means every school’s development capacity is stable. What’s more, the score of the first 20 schools which get the higher score in 2016 are always higher than other school from 2017 to 2020,so we choose those 20 schools as the reliable potentialcandidate list, we think those school both has the better ROI and development potential.Figure 5.5.3 ScoreThe score of those 30 schools from2016 to 2020 is showed in Table 5.5.1ID Z[SP]_2016Z[SP]_2017Z[SP]_2018Z[SP]_2019Z[SP]_2020 1213090.581 0.653 0.591 0.595 0.600 1226120.574 0.609 0.575 0.576 0.577 1229310.684 0.656 0.683 0.682 0.681 1311590.543 0.527 0.526 0.518 0.511 1523180.543 0.490 0.616 0.610 0.604 1630460.628 0.490 0.512 0.509 0.506 1649880.520 0.571 0.561 0.555 0.550 1656620.574 0.725 0.601 0.598 0.596 1666560.607 0.497 0.639 0.645 0.652 1791590.548 0.551 0.534 0.528 0.523 1868670.734 0.734 0.737 0.739 0.740 1912410.611 0.549 0.602 0.598 0.594 1939000.694 0.585 0.683 0.678 0.673 1948240.556 0.524 0.549 0.547 0.544 2024800.584 0.594 0.583 0.582 0.582 2112910.624 0.526 0.613 0.607 0.602 2114400.740 0.654 0.737 0.735 0.732 2165970.601 0.488 0.587 0.580 0.573 2174930.546 0.483 0.535 0.530 0.525 2232320.611 0.595 0.607 0.605 0.603Table 5.5.1 Z[SP]5.5.3 Final investment (TOPSIS)5.5.3.1 Model AnalysisAccording to 5.5.2，we get the final 20 candidate school list, after that ,we will undertake the key process--investment allocation.There are many factors affect investment, but through the step-by-step modeling process, those factors finally can be summed up as follow : The comprehensive score of ROI model(Z[ROI]) ;The comprehensive score of School potential Prediction(Z[SP]) ;Percent of all federal undergraduate students receiving a federal student loan(PCTFLOAN);Median debtof the student (MD);The number of undergraduates(NG).Because TOPSIS method allows us to analyze the scheme by our own ideas which give us enough free space, what’s more,the result of TOPSIS method is clear, it has well operational flexibility, so we choose that to solve our problem.Weight distribution: all of the factors in 1) have great effects on investment, but both PCTFLOAN and MD belong to debt ,those share the weight of debt, so we can distribute the weight as follow:Z[ROI]--0.25,Z[SP]--0.25,NG--0.25,PCTFLOAN--0.125,MD--0.1255.5.3.2Model SolutionNG is changing with time ,we show it in Table 5.5.2,id NG_2016NG_2017NG_2018NG_2019NG_20201213092597263326692705274112261270777304753077577983122931551755795642570457671311597159725073427434752615231824102497258526722759163046414142004260431943791649881657016587166041662216639165662393740104083415642291666565866655172377923860817915985708711885389949136186867294430433142324133401912418505857886518724879719390022951231832341623648238811948245177514051045067503020248083688528868888489009211291348934853481347734732114405977601460506087612321659768916891689168916891217493203620482061207320862232321394914218144861475415023Table 5.5.2 NGWe can also get Z[ROI],PCTFLOAN,MD from previous data,ID Z[ROI]PCTFLOAN MD1213090.73034350.641249511226120.802319610.5115250001229310.765134390.3726205001311590.786649810.460323500152318 1.004578520.5776270001630460.845330140.5407270001649880.854815810.4235270001656620.924753020.567823312166656 1.112655080.783250001791590.808487460.348250001868670.785503970.6589270001912410.811966360.5467250001939000.982760460.4131232501948240.916736310.586627835.52024800.747331260.6791268632112910.767629720.4098270002114400.854345660.3964250002165970.771433880.435627000217493 1.154260170.390726024.52232320.825762390.491625281Table 5.5.3By using MATLAB, we calculate the final comprehensive score (Z [FN]) of the 20 candidate schools. According to the score, we decide how much money The Goodgrant Foundation should invest for each school. But Z[SP] is changing every year，so the investment to each school is changing. The final comprehensive score and investment from 2016 to 2020 is showed as follow table:ID Z[FN]_2016Z[FN]_2017Z[FN]_2018Z[FN]_2019Z[FN]_2020 1213090.252 0.339 0.268 0.274 0.280 1226120.280 0.343 0.292 0.298 0.303 1229310.329 0.301 0.334 0.332 0.331 1311590.211 0.225 0.206 0.204 0.203 1523180.373 0.370 0.430 0.426 0.423 1630460.366 0.288 0.281 0.281 0.282 1649880.426 0.489 0.457 0.453 0.449 1656620.304 0.460 0.337 0.336 0.336 1666560.514 0.472 0.560 0.575 0.589 1791590.264 0.293 0.262 0.261 0.261 1868670.451 0.445 0.456 0.457 0.457 1912410.351 0.317 0.347 0.344 0.342 1939000.674 0.605 0.670 0.667 0.663 1948240.367 0.371 0.366 0.365 0.364 2024800.352 0.381 0.358 0.360 0.362 2112910.311 0.250 0.304 0.299 0.295 2114400.465 0.382 0.464 0.461 0.457 2165970.321 0.269 0.312 0.307 0.302 2174930.423 0.416 0.420 0.419 0.419 2232320.437 0.446 0.442 0.444 0.446Table 5.5.4 Z[FN]ID INV_2016INV_2017INV_2018INV_2019INV_2020 121309$3,377,595$4,545,294$3,535,471$3,623,230$3,704,440 122612$3,753,266$4,594,187$3,863,051$3,934,467$4,003,692 122931$4,406,572$4,033,955$4,406,992$4,394,449$4,376,854 131159$2,817,760$3,009,942$2,716,954$2,696,351$2,682,999 152318$4,988,693$4,951,707$5,679,022$5,635,768$5,596,019 163046$4,893,351$3,860,524$3,716,334$3,720,512$3,726,668 164988$5,703,381$6,549,593$6,041,218$5,985,171$5,932,065 165662$4,069,234$6,165,503$4,455,247$4,447,100$4,439,571 166656$6,875,631$6,324,965$7,406,247$7,598,343$7,789,734 179159$3,533,653$3,922,658$3,462,914$3,456,195$3,453,114 186867$6,033,507$5,963,478$6,031,848$6,038,227$6,038,962 191241$4,695,197$4,249,163$4,587,144$4,551,653$4,517,746 193900$9,026,607$8,102,416$8,859,888$8,814,570$8,766,089 194824$4,917,651$4,973,672$4,833,172$4,824,265$4,817,097 202480$4,711,203$5,106,241$4,727,869$4,760,523$4,791,968 211291$4,167,850$3,355,796$4,021,230$3,959,898$3,900,586 211440$6,218,948$5,124,833$6,134,486$6,091,854$6,041,917 216597$4,301,641$3,607,390$4,124,471$4,055,891$3,990,970 217493$5,664,556$5,577,679$5,554,513$5,543,635$5,536,138 223232$5,843,702$5,981,004$5,841,930$5,867,899$5,893,373Table 5.5.5 INV6. ConclusionsThe candidate list of schools:Table 6.1During the evaluation of ROI Model, we get those40 schools as the first optimization list; When we compare the first optimization list with other Big Foundations’ list, the red names are deleted;During the evaluation of Investment Forecast Model, the blue names are deleted; In theend, the final candidate schools are the last 20 black names.Considering the distribution of that 40 optimized school, we can find that half of them are located in large city, the more developed the city is, the more the optimized schools locate in.Figure 6.1 LocaleThe investment list is showed in Table 6.2,We transform the data in Table 6.2 into a line chart which is showed in Figure 6.2.According to Figure 6.2 we can find , in different years The Goodgrant Foundation should undertakes different Investment strategy.Figure 6.2 INV7.Strengths and WeaknessesFor this model, we have used Grey Correlation Analysis, Principal Component Analysis, Comprehensive Evaluation Method(TOPSIS),Linear Regression Model. There are nearly 3000 schools' data with 50 index. In addition to the given data, we can't able to make a comprehensive evaluation for all university’ indexes.When we choose university at first time , the principle that according to is the ability to school’s cultivate, we can set aside most of the university due to the size of the numerical . The powerful of the university ability could be determined according to the size of the numerical. In the given index, we chose seven indicators related to the students' ability to predict the ability of university. However, these indicators without a typical distribution, so we can not use the exact formula or model to accurately judge the powerful of each university training students' ability.7.1StrengthsGrey Correlation Analysis Method, as a kind of Comprehensive Evaluation Method, it has not a exactly requirement for size of sample, also do not need the typical distribution, and the relatively small amount of calculation, the result will be the same as those of qualitative analysis, the advantages of simple and reliable. The most important thing is that it can build a relational sequence which can obtain correlation quantificationally. In this model, we can application the grey correlation method to get correlation, the size of the correlation willrepresent the strength of the school training students' ability. If size of the correlation more bigger, the evaluation results will be better.After the primary screening. we used Principal Component Analysis (PCA). Principal Component Analysis (PCA) can eliminate the mutual influence between the indexes, it also can reduce the workload of index selection, we can use a handful of composite indicator to replace the original indicators for evaluation. In this screening, we apply the indicator of return on investment regard as the university ability, we should select multiple indicators for principal component analysis, in order to determine the comprehensive variables. The return on investment required to our ability can be used a general comprehensive index to in place , according to the size of the index, we can select the list of schools that we need again. This method overcomes the defect of identified weighting, and having a standard calculation. Using the software can be implemented on the computer, the most important is that the comprehensive index value is objective and reasonable.Finally, using linear regression to forecast, then use TOPSIS to decide the last list. The reason is that we need five years list, so it must be make predictions. Linear regression method deal with multi-factor model is more simple and convenient, and the regression analysis applicable is easy to be affected by many factors. When Our model need predict index, simple easy to use regression analysis. At the end of the forecast, we using TOPSIS method to judge the development potential of the school by predict the index and the given index as influ ence factors. TOPSIS can join the evaluator’ like, it can analyze base on the director of the preference, and it can be carried out in accordance with the investors' willingness to change; The calculation results is more clear, and high maneuverability. The value that acquire by TOPSIS, can represent the development potential of the university. we can according to the size of the values to determine the amount of distribution, in order to distribution rationalization, and have a maximum functionality.7.2WeaknessesWhen we use grey correlation analysis, we should assume the value by ourselves. So it must have exist error, and the accuracy is not high.Principal component analysis model have disadvantage too. The Explain of meaning of the principal component have fuzziness, clear and exact compare with the original data.When predict date, there have a lot of missing, it must influential to forecast results.。

2016年美国大学生数学建模竞赛题目第５卷第２期２０１６年６月、・．．．．・‘．¨．．．‘－．．．．．’．．．Ⅲ’¨．．．．‘．．．．．．‘．．．¨．！数学建模及其应用ＭａｔｈｅｍａｔｉｃａＩＭＯｄｅ¨ｎｇａｎｄｌｔｓＡｐｐＩｉＣａｔｉＯｎｓＶＯＩ．５Ｎｏ．２Ｊｕｎ．２０１６｛竞赛论坛｝＾¨Ｉ・。

・－．．哪・．．．岫・．．．嘶・．．－‘・‘・．．．Ⅵ・‘‘“・・Ｉ２０１６年美国大学生数学建模竞赛题目韩中庚译（解放军信息工程大学四院，河南郑州４５０００１）问题Ａ：热水澡人们经常会通过用一个水龙头将浴缸注满热水，然后坐在浴缸里清洗和放松。

这个浴缸不是带有二次加热系统和循环喷流的温泉式浴缸，而是一个简单的水容器。

过一会儿，洗澡水就会明显变凉，所以洗澡的人需要不停地从水龙头注入热水，以加热洗浴的水。

该浴缸的设计是这样一种方式，即当浴缸里的水达到容量极限时，多余的水就会通过溢水口流出。

考虑空间和时间等因素，建立一个浴缸的水温控制模型，以确定最佳策略，使浴缸里的人可以利用这个策略让整个浴缸中的水保持或尽可能接近初始的温度，而且不浪费太多的水。

利用你们的模型来确定这个策略对浴缸的形状和体积，以及对浴缸中人的形状、体积、温度和活动等因素的依赖程度。

如果这个人一开始用了一种泡泡浴剂加入浴缸中以助清洗，这会对你们的模型结果有怎样的影响？除了要求提交１页的ＭＣＭ摘要之外，你们的报告必须包括１页为浴缸用户准备的非技术性的说明书，来阐述你们的策略，同时解释为什么保持洗澡水的恒温如此之难。

问题Ｂ：太空垃圾地球轨道上的小碎片数量已引起人们越来越多的关注。

据估计，目前有超过５０００００块的空间碎片，也被称为轨道碎片，由于被认为对空间飞行器是潜在的威胁而正在被跟踪。

２００９年２月１０日，俄罗斯卫星Ｋｏｓｍｏｓ一２２５１和美国卫星Ｉｒｉｄｉｕｍ一３３相撞之后，该问题受到了新闻媒体更广泛的讨论。

2016年数学建模美赛题目原文及翻译-A [个人思路] PROBLEM A:A Hot BathA 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.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.Use your model to determine the extent to which your strategy depends upon the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, and the motions made by the person in the bathtub. If the person used a bubble bath additive while initially filling the bathtub to assist in cleansing, how would this affect your model’s results?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 bathtubthat describes your strategy while explaining why it is so difficult to get an evenly maintained temperature throughout the bath water.A题一个热水澡一个人从一个单一的水龙头充满热水浴缸和落户到浴缸清洗和放松。

2016 Interdisciplinary Contest in Modeling®Press Release—April 8, 2016COMAP is pleased to announce the results of the 18th annual Interdisciplinary Contest in Modeling (ICM). This year 5025 teams representing institutions from eight countries participated in the contest. Fourteen teams were designated as OUTSTANDING WINNERS representing the following schools:∙Brown University, USA∙Chongqing University, China∙Communication University of China, China∙Huazhong University of Science and Technology, China, (INFORMS winner)∙NC School of Science and Mathematics, USA,(Vilfredo Pareto Award)∙NC School of Science and Mathematics, USA,(INFORMS winner)∙Northwestern Polytechnical University, China∙Renmin University of China, China∙Rensselaer Polytechnic Institute, USA,(Leonhard Euler Award)∙Shandong University, China∙Sun Yat-Sen University, China, (INFORMS winner)∙United States Military Academy, USA,(Rachel Carson Award)∙University of Colorado Denver, USA∙Xiamen University, ChinaThis year’s conte st ran from Thursday, January 28 to Monday, February 1, 2016. During that time, teams of three students researched, modeled, and communicated a solution to an open-ended interdisciplinary modeling problem. The 2016 ICM was primarily an online contest, where teams registered and obtained contest materials through COMAP’s ICM W ebsite.ICM teams chose one of the following three problems: The D Problem involved measuring the evolution of so ciety’s information networks. By taking a historical perspective of flow of information relative to value of information, the modelers sought to understand the methodology, purpose, andfuncti onality of society’s networks. The E Problem focused on the theme of environmental science. The teams needed to understand the drivers of water scarcity to create intervention strategies for a region to mitigate the water crisis. The new F Problem introduced policy modeling to the ICM. The problem for this first-time topic of policy modeling focused on the Middle East-Europe refugee migration issues. Consistent with other ICM problems, the policy problem challenged teams to utilize a diverse set of disciplinary skills including science, mathematics, politics, government operations, data science, and analysis in their modeling and problem solving. In all three cases, teams searched for pertinent data and grappled with how phenomena internal and external to the system under study needed to be considered. The student teams came up with creative and relevant solutions.These problems also had the ever-present ICM requirements to use thorough data analysis, creative modeling, and scientific methodology, along with effective writing and visualization to communicate their teams' results in a 20-page report. A selection from the Outstanding solution papers will be featured in The UMAP Journal, along with commentaries from the problem authors and judges. This year’s judges remarked that due to the multi-disciplinary nature of the problems, teams were able to solve these problems using a variety of methods and tools. This allowed teams to showcase their strengths in many diverse areas including history, information science, networks, ecology, environmental sciences, health sciences, public policy, dynamical systems, and geo-spatial techniques.2016 ICM Statistics∙5025 Teams participated∙864 Problem D submissions∙3209 Problem E submissions∙952 Problem F submissions∙91 US Teams (2%)∙4934 Foreign Teams (98%) from Australia, Canada, China, Hong Kong SAR, Indonesia, Singapore andUnited Kingdom∙14 Outstanding Winners (1%)∙15 Finalist Winners (1%)∙935 Meritorious Winners (18%)∙2287 Honorable Mentions (45%)∙1649 Successful Participants (33%)∙125 Unsuccessful Participants (2%)ICM is associated with COMAP’s Mathematical Contest in Modeling (MCM), which was held during the same weekend. ICM is designed to develop and advance interdisciplinary problem-solving skills in science, technology, engineering, mathematics (STEM) and the humanities, as well as competence in data science and written communication. Over the years the ICM problems have included topics in environmental science, biology, chemistry, resource management, operations research, information science, public health, and network science. Each team is expected to include advisors and team members who represent a range of disciplinary and interdisciplinary interests in applied problem solving and modeling. To obtain additional information about the ICM and to obtain a complete listing of all the team designations, please visit the ICM Website at: .Major start-up funding for the ICM was provided by a grant from the National Science Foundation (through Project INTERMATH) and COMAP. Additional support is provided by The Institute for Operations Research and the Management Sciences (INFORMS). COMAP's Mathematical Contest in Modeling and Interdisciplinary Contest in Modeling are unique among modeling competitions in that they are the only international contests in which students work in teams to find a solution. Centering its educational philosophy on mathematical modeling, COMAP uses mathematical tools to explore real-world problems. It serves the educational community as well as the world of work by preparing students to become better informed—and prepared—citizens, consumers, workers, and community leaders.Administered byThe Consortium for Mathematicsand Its ApplicationsContest DirectorsChris Arney, United States Military Academy, NY Tina Hartley, United States Military Academy, NY Executive DirectorSolomon A. Garfunkel, COMAP, Inc., MA。

2016 F题With hundreds of thousands of refugees moving across Europe and more arriving each day, considerable attention has been given to refugee integration policies and practices in many countries and regions.History has shown us that mass fleeing of populations occur as a result of major political and social unrest and warfare. These crises bring a set of unique challenges that must be managed carefully through effective policies. Events in the Middle East have caused a massive surge of refugees emigrating from the Middle East into safe haven countries in Europe and parts of Asia, often moving through the Mediterranean and into countries such as Turkey, Hungary, Germany, France, and UK. By the end of October 2015, European countries had received over 715,000 asylum applications from refugees.Hungary topped the charts with nearly 1,450 applications per 100,000 inhabitants, but with only a small percentage of those requests granted (32% in 2014), leaving close to a thousand refugees homeless per every 100K residents of the country. Europe has established a quota system where each country has agreed to take in a particular number of refugees, with the majority of the resettlement burden lying with France and Germany.每天有成千上万的难民移居欧洲，每天都更加到达，许多国家和地区对难民融入政策和做法给予了相当大的关注。

2016年AMC12真题及答案2016 AMC12 AProblem 1What is the value of ?SolutionProblem 2For what value of does ?SolutionProblem 3The remainder can be defined for all realnumbers and with by where denotes the greatest integer less than or equal to . What is the value of ?SolutionProblem 4The mean, median, and mode of the data values are all equal to . What is the value of ?SolutionProblem 5Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, ). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?SolutionProblem 6A triangular array of coins has coin in the first row, coins in the second row, coins in the third row, and so on up to coins in the th row. What is the sum of the digits of ?SolutionProblem 7Which of these describes the graph of ?SolutionProblem 8What is the area of the shaded region of the given rectangle?SolutionProblem 9The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of thevertices of the other four small squares as shown. The common side length is , where and are positive integers. What is ?SolutionProblem 10Five friends sat in a movie theater in a row containing seats, numbered to from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?SolutionProblem 11Each of the students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are students who cannot sing, students who cannot dance, and students who cannot act. How many students have two of these talents?SolutionProblem 12In , , , and . Point lies on ,and bisects . Point lies on , and bisects . The bisectors intersect at . What is the ratio : ?SolutionProblem 13Let be a positive multiple of . One red ball and green balls are arranged in a line in random order. Let be the probability that at least of the green balls are on the same side of the red ball. Observe that andthat approaches as grows large. What is the sum of the digits of the least value of such that ?SolutionProblem 14Each vertex of a cube is to be labeled with an integer from through , with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?SolutionProblem 15Circles with centers and , having radii and , respectively, lie on the same side of line and are tangent to at and , respectively,with between and . The circle with center is externally tangent to each of the other two circles. What is the area of triangle ?SolutionProblem 16The graphs of and are plotted on the same set of axes. How many points in the plane with positive -coordinates lie on two or more of the graphs?SolutionProblem 17Let be a square. Let and be the centers, respectively, of equilateral triangles with bases and each exterior to the square. What is the ratio of the area of square to the area of square ?SolutionProblem 18For some positive integer the number has positive integer divisors, including and the number How many positive integer divisors does the number have?SolutionProblem 19Jerry starts at on the real number line. He tosses a fair coin times. When he gets heads, he moves unit in the positive direction; when he gets tails, he moves unit in the negative direction. The probability that he reaches at some time during thisprocess is where and are relatively prime positive integers. What is (For example, he succeeds if his sequence of tosses is )SolutionProblem 20A binary operation has the properties that andthat for all nonzero real numbers and (Here the dot represents the usual multiplication operation.) The solution to theequation can be written as where and are relatively prime positive integers. What isSolutionProblem 21A quadrilateral is inscribed in a circle of radius Three of the sides of this quadrilateral have length What is the length of its fourth side?SolutionProblem 22How many ordered triples of positive integerssatisfy and ?SolutionProblem 23Three numbers in the interval are chosen independently and at random. What isthe probability that the chosen numbers are the side lengths of a triangle with positive area?SolutionProblem 24There is a smallest positive real number such that there exists a positive real number such that all the roots of the polynomial are real. In fact, for this value of the value of is unique. What is the value ofSolutionProblem 25Let be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with digits. Every time Bernardo writes a number, Silvia erases the last digits of it. Bernardo then writes the next perfect square, Silvia erases the last digits of it, and this process continues until the last two numbers that remain on the board differ by atleast 2. Let be the smallest positive integer not written on the board. For example,if , then the numbers that Bernardo writes are , and the numbers showing on the board after Silvia erases are and , andthus . What is the sum of the digits of ?2016 AMC 12A Answer Key1 B2 C3 B4 D5 E6 D7 D8 D9 E10 B11 E12 C13 A14 C15 D16 D17 B18 D19 B20 A21 E22 A23 C24 B25 E。