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美赛论文

注：LEO 低地球轨道MEO中地球轨道GeO 同步卫星轨道risk-profit 风险利润率fixed-profit rate 固定利润率提出一个合理的商业计划，可以使我们抓住商业机会，我们建立四个模型来分析三个替代方案（水射流，激光，卫星）和组合，然后确定是否存在一个经济上有吸引力的机会，从而设计了四种模型分析空间碎片的风险、成本、利润和预测。

首先，我们建立了利润模型基于净现值（NPV）模型，并确定三个最佳组合的替代品与定性分析：1）考虑了三个备选方案的组合时，碎片的量是巨大的；2）考虑了水射流和激光的结合，认为碎片的大小不太大；3）把卫星和激光的结合当尺寸的这些碎片足够大。

其次，建立风险定性分析模型，对影响因素进行分析在每一种替代的风险，并得出一个结论，风险将逐渐下降直到达到一个稳定的数字。

在定量分析技术投入和对设备的影响投资中，我们建立了双重技术的学习曲线模型,找到成本的变化规律与时间的变化。

然后，我们开发的差分方程预测模型预测的量在未来的四年内每年发射的飞机。

结合结果我们从预测中，我们可以确定最佳的去除选择。

最后，分析了模型的灵敏度，讨论了模型的优势和我们的模型的弱点，目前的非技术性的信，指出了未来工作。

目录1，简介1.1问题的背景1.2可行方案1.3一般的假设1.4我们的思想的轮廓2，我们的模型2.1 时间---利润模型2.1.1 模型的符号2.1.2 模型建立2.1.3 结果与分析2.2 . 差分方程的预测模型2.2.1 模型建立2.2.2 结果分析2.3 双因子技术-学习曲线模型2.3.1 模型背景知识2.3.2 模型的符号2.3.3 模型建立2.3.4 结果分析2.4风险定性分析模型2.4.1 模型背景2.4.2 模型建立2.4.3 结果与分析3．在我们模型的灵敏度分析3.1 差分方程的预测模型。

3.1.1 稳定性分析3.1.2 敏感性分析3.2 双因子技术学习曲线模型3.2.1 稳定性分析3.2.2 敏感性分析4 优点和缺点查分方程预测模型优点缺点双因子技术学习曲线模型优点缺点时间---利润模型优点缺点5..结论6..未来的工作7．参考双赢模式：拯救地球，抓住机遇1..简介问题的背景空间曾经很干净整洁。

建模美赛获奖范文

建模美赛获奖范文全文共四篇示例，供读者参考第一篇示例：近日，我校数学建模团队在全国大学生数学建模竞赛中荣获一等奖的喜讯传来，这是我校首次在该比赛中获得如此优异的成绩。

本文将从建模过程、团队合作、参赛经验等方面进行详细介绍，希望能为更多热爱数学建模的同学提供一些借鉴和参考。

让我们来了解一下比赛的背景和要求。

全国大学生数学建模竞赛是由中国工程院主办，旨在促进大学生对数学建模的兴趣和掌握数学建模的基本方法和技巧。

比赛通常会设置一些实际问题，参赛队伍需要在规定时间内通过建立数学模型、分析问题、提出解决方案等步骤来完成任务。

最终评选出的优胜队伍将获得一等奖、二等奖等不同级别的奖项。

在本次比赛中，我们团队选择了一道关于城市交通拥堵研究的题目，并从交通流理论、路网优化等角度进行建模和分析。

通过对城市交通流量、拥堵原因、路段限制等方面的研究，我们提出了一种基于智能交通系统的解决方案，有效缓解了城市交通拥堵问题。

在展示环节，我们通过图表、数据分析等方式清晰地呈现了我们的建模过程和成果，最终赢得了评委的认可。

在整个建模过程中，团队合作起着至关重要的作用。

每个成员都发挥了自己的专长和优势，在分析问题、建模求解、撰写报告等方面各司其职。

团队内部的沟通和协作非常顺畅，大家都能积极提出自己的想法和看法，达成共识后再进行实际操作。

通过团队合作，我们不仅完成了比赛的任务，也培养了团队精神和合作能力，这对我们日后的学习和工作都具有重要意义。

参加数学建模竞赛是一次非常宝贵的经历，不仅能提升自己的数学建模能力，也能锻炼自己的解决问题的能力和团队协作能力。

在比赛的过程中，我们学会了如何快速建立数学模型、如何分析和解决实际问题、如何展示自己的成果等，这些能力对我们未来的学习和工作都将大有裨益。

在未来，我们将继续努力，在数学建模领域不断学习和提升自己的能力，为更多的实际问题提供有效的数学解决方案。

我们也希望通过自己的经验和教训，为更多热爱数学建模的同学提供一些指导和帮助，共同进步，共同成长。

美赛论文(最终版)

For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number 46639Problem ChosenCFor office use onlyF1________________F2________________F3________________F4________________2016 MCM/ICM Summary SheetAn Optimal Investment Strategy ModelSummaryWe develop an optimal investment strategy model that appears to hold promise for providing insight into not only how to sort the schools according to investment priority, but also identify optimal investment amount of a specific school. This model considers a large number of parameters thought to be important to investment in the given College Scorecard Data Set.In order to develop the required model, two sub-models are constructed as follows: 1.For Analytic Hierarchy Process (AHP) Model, we identify the prioritizedcandidate list of schools by synthesizing the elements which have an influence on investment. First we define the specific value of any two elements’ effect on investment. And then the weight of each element’s influence on investment can be identified. Ultimately, we take the relevant parameters into the calculated weight, and then we get any school’s recommended value of investment.2.For Return On Investment M odel, it’s constructed on the basis of AHP Model.Let us suppose that all the investment is used to help the students to pay tuition fee.Then we can see optimal investment as that we help more students to the universities of higher return rate. However, because of dropout rate, there will be an optimization investment amount in each university. Therefore, we can change the problem into a nonlinear programming problem. We identify the optimal investment amount by maximizing return-on-investment.Specific attention is given to the stability and error analysis of our model. The influence of the model is discussed when several fundamental parameters vary. We attempt to use our model to prioritize the schools and identify investment amount of the candidate schools, and then an optimal investment strategy is generated. Ultimately, to demonstrate how our model works, we apply it to the given College Scorecard Data Set. For various situations, we propose an optimal solution. And we also analyze the strengths and weaknesses of our model. We believe that we can make our model more precise if more information are provided.Contents1.Introduction 21.1Restatement of the Problem (2)1.2Our Approach (2)2.Assumptions 23.Notations 34.The Optimal Investment Model 44.1Analytic Hierarchy Process Model (4)4.1.1Constructing the Hierarchy (4)4.1.2Constructing the Judgement Matrix (5)4.1.3Hierarchical Ranking (7)4.2Return On Investment Model (8)4.2.1Overview of the investment strategy (8)4.2.2Analysis of net income and investment cost (9)4.2.3Calculate Return On Investment (11)4.2.4Maximize the Total Net Income (11)5.Test the Model125.1Error Analysis (12)5.2Stability Analysis (13)6.Results136.1Results of Analytic Hierarchy Process (13)6.2Results of Return On Investment Model (14)7.Strengths and Weaknesses157.1Strengths (15)7.2Weaknesses (16)References16 Appendix A Letter to the Chief Financial Officer, Mr. Alpha Chiang.171.Introduction1.1Restatement of the ProblemIn order to help improve educational performance of undergraduates attending colleges and universities in the US, the Goodgrant Foundation intends to donate a total of $100,000,000 to an appropriate group of schools per year, for five years, starting July 2016. We are to develop a model to determine an optimal investment strategy that identifies the school, the investment amount per school, the return on that investment, and the time duration that the organization’s money should be provided to have the highest likelihood of producing a strong positive effect on student performance. Considering that they don’t want to duplicate the investments and focus of other large grant organizations, we interpret optimal investment as a strategy that maximizes the ROI on the premise that we help more students attend better colleges. So the problems to be solved are as follows:1.How to prioritize the schools by optimization level.2.How to measure ROI of a school.3.How to measure investment amount of a specific school.1.2Our ApproachWe offer a model of optimal investment which takes a great many factors in the College Scorecard Data Set into account. To begin with, we make a 1 to N optimized and prioritized candidate list of school we are recommending for investment by the AHP model. For the sake that we invest more students to better school, several factors are considered in the AHP model, such as SAT score, ACT score, etc. And then, we set investment amount of each university in the order of the list according to the standard of maximized ROI. The implement details of the model will be described in section 4.2.AssumptionsWe make the following basic assumptions in order to simplify the problem. And each of our assumptions is justified.1.Investment amount is mainly used for tuition and fees. Considering that theincome of an undergraduate is usually much higher than a high school students, we believe that it’s necessary to help more poor students have a chance to go to college.2.Bank rates will not change during the investment period. The variation ofthe bank rates have a little influence on the income we consider. So we make this assumption just to simplify the model.3.The employment rates and dropout rates will not change, and they aredifferent for different schools4.For return on investment, we only consider monetary income, regardlessof the intangible income.3.NotationsWe use a list of symbols for simplification of expression.4.The Optimal Investment ModelIn this section, we first prioritize schools by the AHP model (Section 4.1), and then calculate ROI value of the schools (Section 4.2). Ultimately, we identify investment amount of every candidate schools according to ROI (Section 4.3).4.1Analytic Hierarchy Process ModelIn order to prioritize schools, we must consider each necessary factor in the College Scorecard Data Set. For each factor, we calculate its weight value. And then, we can identify the investment necessity of each school. So, the model can be developed in 3 steps as follows:4.1.1Constructing the HierarchyWe consider 19 elements to measure priority of candidate schools, which can be seen in Fig 1. The hierarchy could be diagrammed as follows:Fig.1AHP for the investment decisionThe goal is red, the criteria are green and the alternatives are blue. All the alternatives are shown below the lowest level of each criterion. Later in the process, each alternatives will be rated with respect to the criterion directly above it.As they build their hierarchy, we should investigate the values or measurements of the different elements that make it up. If there are published fiscal policy, for example, or school policy, they should be gathered as part of the process. This information will be needed later, when the criteria and alternatives are evaluated.Note that the structure of the investment hierarchy might be different for other foundations. It would definitely be different for a foundation who doesn't care how much his score is, knows he will never dropout, and is intensely interested in math, history, and the numerous aspects of study[1].4.1.2Constructing the Judgement MatrixHierarchy reflects the relationship among elements to consider, but elements in the Criteria Layer don’t always weigh equal during aim measure. In deciders’ mind, each element accounts for a particular proportion.To incorporate their judgments about the various elements in the hierarchy, decision makers compare the elements “two by two”. The fundamental scale for pairwise comparison are shown in Fig 2.Fig 2Right now, let's see which items are compared. Our example will begin with the six criteria in the second row of the hierarchy in Fig 1, though we could begin elsewhere if we want. The criteria will be compared as to how important they are to the decisionmakers, with respect to the goal. Each pair of items in this row will be compared.Fig 3 Investment Judgement MatrixIn the next row, there is a group of 19 alternatives under the criterion. In the subgroup, each pair of alternatives will be compared regarding their importance with respect to the criterion. (As always, their importance is judged by the decision makers.) In the subgroup, there is only one pair of alternatives. They are compared as to how important they are with respect to the criterion.Things change a bit when we get to the alternatives row. Here, the factor in each group of alternatives are compared pair-by-pair with respect to the covering criterion of the group, which is the node directly above them in the hierarchy. What we are doing here is evaluating the models under consideration with respect to score, then with respect to Income, then expenditure, dropout rate, debt and graduation rate.The foundation can evaluate alternatives against their covering criteria in any order they choose. In this case, they choose the order of decreasing priority of the covering criteria.Fig 4 Score Judgement MatrixFig 5 Expenditure Judgement MatrixFig 6 Income Judgement MatrixFig 7 Dropout Judgement MatrixFig 8 Debt Judgement MatrixFig 9 Graduation Matrix4.1.3 Hierarchical RankingWhen the pairwise comparisons are as numerous as those in our example, specialized AHP software can help in making them quickly and efficiently. We will assume that the foundation has access to such software, and that it allows the opinions of various foundations to be combined into an overall opinion for the group.The AHP software uses mathematical calculations to convert these judgments to priorities for each of the six criteria. The details of the calculations are beyond the scope of this article, but are readily available elsewhere[2][3][4][5]. The software also calculates a consistency ratio that expresses the internal consistency of the judgments that have been entered. In this case the judgments showed acceptable consistency, and the software used the foundation’s inputs to assign these new priorities to the criteria:Fig 10.AHP hierarchy for the foundation investing decision.In the end, the AHP software arranges and totals the global priorities for each of the alternatives. Their grand total is 1.000, which is identical to the priority of the goal. Each alternative has a global priority corresponding to its "fit" to all the foundation's judgments about all those aspects of factor. Here is a summary of the global priorities of the alternatives:Fig 114.2 ROI Model4.2.1 Overview of the investment strategyConsider a foundation making investment on a set of N geographically dispersed colleges and university in the United States, D = {1, 2, 3……N }. Then we can select top N schools from the candidate list which has been sorted through analytic hierarchy process. The total investment amount is M per year which is donated by the Goodgrant Foundation. The investment amount is j m for each school j D ∈, satisfying the following balance constraint:j j D mM ∈=∑ (1)W e can’t invest too much or too little money to one school because we want to help more students go to college, and the student should have more choices. Then the investment amount for each school must have a lower limit lu and upper limit bu as follows:j lu m bu ≤≤ (2)The tuition and fees is j p , and the time duration is {1,2,3,4}j t ∈. To simplify ourmodel, we assume that our investment amount is only used for freshmen every year. Because a freshmen oriented investment can get more benefits compared with others. For each school j D ∈, the number of the undergraduate students who will be invested is j n , which can be calculated by the following formula ：,jj j j m n j D p t =∈⨯ (3)Figure12The foundation can use the ROI model to identify j m and j t so that it canmaximize the total net income. Figure1 has shown the overview of our investment model. We will then illustrate the principle and solution of this model by a kind of nonlinear programming method.4.2.2 Analysis of net income and investment costIn our return on investment model, we first focus on analysis of net income and investment cost. Obviously, the future earnings of undergraduate students are not only due to the investment itself. There are many meaning factors such as the effort, the money from their parents, the training from their companies. In order to simplify the model, we assume that the investment cost is the most important element and we don’t consider other possible influence factors. Then we can conclude that the total cost of the investment is j m for each school j D ∈.Figure 13For a single student, the meaning of the investment benefits is the expected earnings in the future. Assuming that the student is not going to college or university after graduating from high school and is directly going to work. Then his wage base is 0b as a high school graduate. If he works as a college graduate, then his wage base is 0a . Then we can give the future proceeds of life which is represented symbolically by T and we use r to represent the bank rates which will change over time. We assume that the bank rates will not change during the investment period. Here, we use bank rates in 2016 to represent the r . The future proceeds of life of a single undergraduate student will be different due to individual differences such as age, physical condition environment, etc. If we consider these differences, the calculation process will be complicated. For simplicity’s sake, we uniform the future proceeds of life T for 20 years. Then we will give two economics formulas to calculate the total expected income in the next T years for graduates and high school graduates:40(1)Tk k a u r +==+∑(4) 40(1)T kk b h r +==+∑(5) The total expected income of a graduate is u , and the total expected income of a highschool graduate is h .Then, we continue to analyze the net income. The net income can be calculated by the following formula:os NetIncome TotalIncome C t =- (6) For each school j D ∈, the net income is j P , the total income is j Q , and the cost is j m . Then we will get the following equation through formula (6):j j j P Q m =- (7)Therefore, the key of the problem is how to calculate j Q . In order to calculate j Q, weneed to estimate the number of future employment j ne . The total number of the invested is j n , which has been calculated above. Considering the dropout rates j α and the employment rates j β for each school j , we can calculate the number of future employment j ne through the following formula:(4)(1)jt j j j j n e n βα-=⨯⨯- (8)That way, we can calculate j Q by the following formula:()j j Q ne u h =⨯- (9)Finally, we take Eq. (2) (3) (4) (7) (8) into Eq. (6), and we will obtain Eq. (9) as follows:4(4)00400(1)()(1)(1)j TT t j j j j j k kk k j jm a b P m p t r r βα+-+===⨯⨯-⨯--⨯++∑∑ (10) We next reformulate the above equation of j P for concise presentation:(4)(1)j t j jj j j jc m P m t λα-⨯⨯=⨯-- (11)where jj j p βλ= and 400400(1)(1)TT k kk k a b c r r ++===-++∑∑ .4.2.3 Calculate Return On InvestmentROI is short of return on investment which can be determined by net income andinvestment cost [7]. It conveys the meaning of the financial assessment. For each schoolj D ∈ , the net income is j P , and the investment cost equals to j m . Then the j ROIcan be calculated by the following formula:100%j j jP ROI m =⨯ (12)We substitute Eq. (10) into Eq. (11), and we will get a new formula as follows:(4)((1)1)100%j t j j j jc ROI t λα-⨯=⨯--⨯ (13)4.2.4 Maximize the Total Net IncomeGiven the net income of each school, we formulate the portfolio problem that maximize the total net income, S=Max(4)((1))j t j jj j j j Dj Djc m P m t λα-∈∈⨯⨯=⨯--∑∑ (14)S. T.jj DmM ∈=∑,{1,2,3,4}t = ,j lu m bu ≤≤ ,Considering the constraint jj DmM ∈=∑, we can further simplify the model,S is equivalent to S’=Max(4)((1))j t j jj j j Dj Djc m P t λα-∈∈⨯⨯=⨯-∑∑ (15)S. T.jj DmM ∈=∑,{1,2,3,4t = ,j l u m b u ≤≤. By solving the nonlinear programming problem S’, we can get the sameanswer as problem S.5. Testing the Model 5.1 Error AnalysisSince the advent of analytic hierarchy process, people pay more attention to it due to the specific applicability, convenience, practicability and systematization of the method. Analytic hierarchy process has not reached the ideal situation whether in theory or application level because the results depend largely on the preference and subjective judgment. In this part, we will analyze the human error problem in analytic hierarchy process.Human error is mainly caused by human factors. The human error mainly reflects on the structure of the judgment matrix. The causes of the error are the following points:1. The number of times that human judge the factors’ importance is excessive.2. The calibration method is not perfect.Then we will give some methods to reduce errors:1. Reduce times of human judgment. One person repeatedly gave the samejudgment between two factors. Or many persons gave the same judgment between two factors one time. Finally, we take the average as result.2. Break the original calibration method. If we have defined the ranking vector111121(,...)n a a a a =between the factor 1A with others. Then we can get all theother ranking vector. For example : 12122111(,1...)na a a a a =.5.2 Stability AnalysisIt is necessary to analyze the stability of ranking result [6], because the strong subjectivefactors. If the ranking result changed a little while the judgment changed a lot, we can conclude that the method is effective and the result is acceptable, and vice versa. We assume that the weight of other factors will change if the weight of one factor changed from i ξ to i η:[8](1)(,1,2...,)(1)i j j i i j n i j ηξηξ-⨯==≠- (16)And it is simple to verify the equation:11nii η==∑ (17)And the new ranking vector ω will be:A ωη=⨯ (18)By this method, the Relative importance between other factors remain the same while one of the factor has changed.6. Results6.1 Results of Analytic Hierarchy ProcessWe can ranking colleges through the analytic hierarchy process, and we can get the top N = 20 schools as follows6.2 Results of Return On Investment ModelBased on the results above, we next use ROI model to distribute investment amountj m and time duration j t for each school j D ∈ by solving the following problem:Max (4)((1))j t j jj j j Dj Djc m P t λα-∈∈⨯⨯=⨯-∑∑S. T.jj DmM ∈=∑,{1,2,3,4t = , j l u m b u≤≤ . In order to solve the problem above, we collected the data from different sources. Inthe end, we solve the model with Lingo software. The program code is as follows:model: sets:roi/1..20/:a,b,p,m,t;endsets data:a = 0.9642 0.9250 0.9484 0.9422 0.9402 0.9498 0.90490.9263 0.9769 0.9553 0.9351 0.9123 0.9410 0.98610.9790 0.9640 0.8644 0.9598 0.9659 0.9720;b = 0.8024 0.7339 0.8737 0.8308 0.8681 0.7998 0.74920.6050 0.8342 0.8217 0.8940 0.8873 0.8495 0.87520.8333 0.8604 0.8176 0.8916 0.7527 0.8659;p = 3.3484 3.7971 3.3070 3.3386 3.3371 3.4956 3.22204.0306 2.8544 3.1503 3.2986 3.3087 3.3419 2.78452.9597 2.92713.3742 2.7801 2.5667 2.8058;c = 49.5528;enddatamax=@sum(roi(I):m(I)/t(I)/p(I)*((1-b(I))^4)*c*(1-a(I)+0.05)^(4-t(I)));@for(roi:@gin(t));@for(roi(I):@bnd(1,t(I),4));@for(roi(I):@bnd(0,m(I),100));@sum(roi(I):m(I))=1000;ENDFinally, we can get the investment amount and time duration distribution as follows:7.Strengths and Weaknesses7.1Strengths1.Fixing the bank rates during the investment period may run out, but it will haveonly marginal influences.2.For return on investment, we only consider monetary income, regardless of the3.intangible income. But the quantization of these intangible income is very importantand difficult. It needs to do too much complicated technical analysis and to quantify 4.too many variables. Considering that the investment persists for a short time, thiskind of random error is acceptable.5.Due to our investment which is freshmen oriented, other students may feel unfair.It is likely to produce adverse reaction to our investment strategy.6.The cost estimation is not impeccable. We only consider the investment amount andignore other non-monetary investment.5. AHP needs higher requirements for personnel quality.7.2Weaknesses1.Our investment strategy is distinct and clear, and it is convenient to implement.2.Our model not only identifies the investment amount for each school, but alsoidentifies the time duration that the organization’s money should be provide d.3.Data processing is convenient, because the most data we use is constant, average ormedian.4.Data sources are reliable. Our investment strategy is based on some meaningful anddefendable subset of two data sets.5.AHP is more simple, effective and universal.References[1] Saaty, Thomas L. (2008). Decision Making for Leaders: The Analytic Hierarchy Process for Decisions in a Complex World. Pittsburgh, Pennsylvania: RWS Publications. ISBN 0-9620317-8-X.[2] Bhushan, Navneet, Kanwal Rai (January 2004). Strategic Decision Making: Applying the Analytic Hierarchy Process. London: Springer-Verlag. ISBN 1-8523375-6-7.[3] Saaty, Thomas L. (2001). Fundamentals of Decision Making and Priority Theory. Pittsburgh, Pennsylvania: RWS Publications. ISBN 0-9620317-6-3.[4] Trick, Michael A. (1996-11-23). "Analytic Hierarchy Process". Class Notes. Carnegie Mellon University Tepper School of Business. Retrieved 2008-03-02.[5] Meixner, Oliver; Reiner Haas (2002). Computergestützte Entscheidungs-findung: Expert Choice und AHP – innovative Werkzeuge zur Lösung komplexer Probleme (in German). Frankfurt/Wien: Redline Wirtschaft bei Ueberreuter. ISBN 3-8323-0909-8.[6] Hazelkorn, E. The Impact of League Tables and Ranking System on Higher Education Decision Making [J]. Higher Education Management and Policy, 2007, 19(2), 87-110.[7] Leslie: Trainer Assessment: A Guide to Measuring the Performance of Trainers and Facilitors, Second Edition, Gower Publishing Limited, 2002.[8] Aguaron J, Moreno-Jimenea J M. Local stability intervals in the analytic hierarchy process. European Journal of Operational Research. 2000Letter to the Chief Financial Officer, Mr. Alpha Chiang. February 1th, 2016.I am writing this letter to introduce our optimal investment strategy. Before I describe our model, I want to discuss our proposed concept of a return-on-investment (ROI). And then I will describe the optimal investment model by construct two sub-model, namely AHP model and ROI model. Finally, the major results of the model simulation will be showed up to you.Considering that the Goodgrant Foundation aims to help improve educational performance of undergraduates attending colleges and universities in the US, we interpret return-on-investment as the increased income of undergraduates. Because the income of an undergraduate is generally much higher than a high school graduate, we suggest all the investment be used to pay for the tuition and fees. In that case, if we take both the income of undergraduates’ income and dropout rate into account, we can get the return-in-investment value.Our model begins with the production of an optimized and prioritized candidate list of schools you are recommending for investment. This sorted list of school is constructed through the use of specification that you would be fully qualified to provided, such as the score of school, the income of graduate student, the dropout rate, etc. With this information, a precise investment list of schools will be produced for donation select.Furthermore, we developed the second sub-model, ROI model, which identifies the investment amount of each school per year. If we invest more money in a school, more students will have a chance to go to college. However, there is an optimal investment amount of specific school because of the existence of dropout. So, we can identify every candidate school’s in vestment amount by solve a nonlinear programming problem. Ultimately, the result of the model simulation show that Washington University, New York University and Boston College are three schools that worth investing most. And detailed simulation can be seen in our MCM Contest article.We hope that this model is sufficient in meeting your needs in any further donation and future philanthropic educational investments within the United States.。

美赛论文模版

摘要：第一段：写论文解决什么问题1．问题的重述a. 介绍重点词开头：例1：“Hand move” irrigation, a cheap but labor-intensive system used on small farms, consists of a movable pipe with sprinkler on top that can be attached to a stationary main.例2:……is a real-life common phenomenon with many complexi t ies.例3：An (effective plan) is crucial to………b. 直接指出问题：例1：We find the optimal number of tollbooths in a highway toll-plaza for a given number of highway lanes: the number of tollbooths that minimizes average delay experienced by cars.例2：A brand-new university needs to balance the cost of information technology security measures wi t h the potential cost of attacks on its systems.例3：We determine the number of sprinklers to use by analyzing the energy and motion of water in the pipe and examining the engineering parameters of sprinklers available in the market.例4: After mathematically analyzing the …… problem, our modeling group would like to present our conclusions, strategies, (and recommendations )to the …….例5：Our goal is... that (mini mizes the time )……….2．解决这个问题的伟大意义反面说明。

美赛金奖论文

1
Team # 14604
Catalogue
Abstracts ........................................................................................................................................... 1 Contents ............................................................................................................................................ 3 1. Introduction ................................................................................................................................... 3 1.1 Restatement of the Problem ................................................................................................ 3 1.2 Survey of the Previous Research......................................................................................... 3 2. Assumptions .................................................................................................................................. 4 3. Parameters ..................................................................................................................................... 4 4. Model A ----------Package model .................................................................................................. 6 4.1 Motivation ........................................................................................................................... 6 4.2 Development ....................................................................................................................... 6 4.2.1 Module 1: Introduce of model A .............................................................................. 6 4.2.2 Module 2: Solution of model A .............................................................................. 10 4.3 Conclusion ........................................................................................................................ 11 5. Model B----------Optional model ................................................................................................ 12 5.1 Motivation ......................................................................................................................... 12 5.2 Development ..................................................................................................................... 12 5.2.1 Module B: Choose oar- powered rubber rafts or motorized boats either ............... 12 5.2.2 Module 2: Choose mix of oar- powered rubber rafts and motorized boats ............ 14 5.3 Initial arrangement ............................................................................................................ 17 5.4. Deepened model B ........................................................................................................... 18 5.4.1 Choose the campsites allodium .............................................................................. 18 5.4.2 Choose the oar- powered rubber rafts or motorized boats allodium ...................... 19 5.5 An example of reasonable arrangement ............................................................................ 19 5.6 The strengths and weakness .............................................................................................. 20 6. Extensions ................................................................................................................................... 21 7. Memo .......................................................................................................................................... 25 8. References ................................................................................................................................... 26 9. Appendices .................................................................................................................................. 27 9.1 Appendix I .................................................................................................. 27 9.2 Appendix II ....................................................................................................................... 29

美国数学建模竞赛优秀论文阅读报告

2.优秀论文一具体要求：1月28日上午汇报1）论文主要内容、具体模型和求解算法（针对摘要和全文进行概括）；In the part1, we will design a schedule with fixed trip dates and types and also routes. In the part2, we design a schedule with fixed trip dates and types but unrestrained routes.In the part3, we design a schedule with fixed trip dates but unrestrained types and routes.In part 1, passengers have to travel along the rigid route set by river agency, so the problem should be to come up with the schedule to arrange for the maximum number of trips without occurrence of two different trips occupying the same campsite on the same day.In part 2, passengers have the freedom to choose which campsites to stop at, therefore the mathematical description of their actions inevitably involve randomness and probability, and we actually use a probability model. The next campsite passengers choose at a current given campsite is subject to a certain distribution, and we describe events of two trips occupying the same campsite y probability. Note in probability model it is no longer appropriate to say that two trips do not meet at a campsite with certainty; instead, we regard events as impossible if their probabilities are below an adequately small number. Then we try to find the optimal schedule.In part 3, passengers have the freedom to choose both the type and route of the trip; therefore a probability model is also necessary. We continue to adopt the probability description as in part 2 and then try to find the optimal schedule.In part 1, we find the schedule of trips with fixed dates, types (propulsion and duration) and routes (which campsites the trip stops at), and to achieve this we use a rather novel method. The key idea is to divide campsites into different “orbits”that only allows some certain trip types to travel in, therefore the problem turns into several separate small problem to allocate fewer trip types, and the discussion of orbits allowing one, two, three trip types lead to general result which can deal with any value of Y. Particularly, we let Y=150, a rather realistic number of campsites, to demonstrate a concrete schedule and the carrying capacity of the river is 2340 trips.In part 2, we find the schedule of trips with fixed dates, types but unrestrained routes. To better describe the behavior of tourists, we need to use a stochastic model（随机模型）. We assume a classical probability model and also use the upper limit value of small probability to define an event as not happening. Then we use Greedy algorithm to choose the trips added and recursive algorithm together with Jordan Formula to calculate the probability of two trips simultaneously occupying the same campsites. The carrying capacity of the river by this method is 500 trips. This method can easily find theoptimal schedule with X given trips, no matter these X trips are with fixed routes or not. In part 3, we find the optimal schedule of trips with fixed dates and unrestrained types and routes. This is based on the probability model developed in part 2 and we assign the choice of trip types of the tourists with a uniform distribution to describe their freedom to choose and obtain the results similar to part 2. The carrying capacity of the river by this method is 493 trips. Also this method can easily find the optimal schedule with X given trips, no matter these X trips are with fixed routes or not.2）论文结构概述（列出提纲，分析优缺点，自己安排的结构）；1 Introduction2 Definitions3 Specific formulation of problem4 Assumptions5 Part 1 Best schedule of trips with fixed dates, types and also routes.5.1 Method5.1.1 Motivation and justification5.1.2 Key ideas5.2 Development of the model5.2.1Every campsite set for every single trip type5.2.2 Every campsite set for every multiple trip types5.2.3One campsite set for all trip types6 Part 2 Best schedule of trips with fixed dates and types, but unrestrained routes.6.1 Method6.1.1 Motivation and justification6.1.2 Key ideas6.2 Development of the model6.2.1 Calculation of p(T,x,t)6.2.2 Best schedule using Greedy algorithm6.2.3 Application to situation where X trips are given7 Part 3 Best schedule of trips with fixed dates, but unrestrained types and routes.7.1 Method7.1.1 Motivation and justification7.1.2 Key ideas7.2 Development of the model8 Testing of the model----Sensitivity analysis8.1Stability with varying trip types chosen in 68.2The sensitivity analysis of the assumption 4④8.3 The sensitivity analysis of the assumption 4⑥9 Evaluation of the model9.1 Strengths and weaknesses9.1.1 Strengths9.1.2 Weakness9.2 Further discussion10 Conclusions11 References12 Letter to the river managers3）论文中出现的好词好句（做好记录）；用于问题的转化We regard the carrying capacity of the river as the maximum total number of trips available each year, hence turning the task of the river managers into looking for the best schedule itself.表明我们在文中所做的工作We have examined many policies for different river…..问题的分解We mainly divide the problem into three parts and come up with three different….对我们工作的要求：Given the above considerations, we want to find the optimal。

美赛一等奖论文-中文翻译版

目录问题回顾 (3)问题分析： (4)模型假设： (6)符号定义 (7)4.1---------- (8)4.2 有热水输入的温度变化模型 (17)4.2.1模型假设与定义 (17)4.2.2 模型的建立The establishment of the model (18)4.2.3 模型求解 (19)4.3 有人存在的温度变化模型Temperature model of human presence (21)4.3.1 模型影响因素的讨论Discussion influencing factors of the model (21)4.3.2模型的建立 (25)4.3.3 Solving model (29)5.1 优化目标的确定 (29)5.2 约束条件的确定 (31)5.3模型的求解 (32)5.4 泡泡剂的影响 (35)5.5 灵敏度的分析 (35)8 non-technical explanation of the bathtub (37)Sｕmｍary人们经常在充满热水的浴缸里得到清洁和放松。

本文针对只有一个简单的热水龙头的浴缸，建立一个多目标优化模型，通过调整水龙头流量大小和流入水的温度来使整个泡澡过程浴缸内水温维持基本恒定且不会浪费太多水。

首先分析浴缸中水温度变化的具体情况。

根据能量转移的特点将浴缸中的热量损失分为两类情况:沿浴缸四壁和底面向空气中丧失的热量根据傅里叶导热定律求出；沿水面丧失的热量根据水由液态变为气态的焓变求出。

因涉及的参数过多,将系数进行回归分析的得到一个一元二次函数。

结合两类热量建立了温度关于时间的微分方程。

加入阻滞因子考虑环境温湿度升高对水温的影响,最后得到水温度随时间的变化规律(见图＊*)。

优化模型考虑保持水龙头匀速流入热水的情况。

将过程分为浴缸未加满和浴缸加满而水从排水口溢出的两种情况，根据能量守恒定律优化上述微分方程,建立一个有热源的情况下水的温度随时间变化的分段模型，（见图*＊)接下来考虑人在浴缸中对水温的影响。

数学建模美赛一等奖优秀专业论文

For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number52888Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________Mathematical Contest in Modeling (MCM/ICM) Summary SheetSummaryIt’s pleasant t o go home to take a bath with the evenly maintained temperature of hot water throughout the bathtub. This beautiful idea, however, can not be always realized by the constantly falling water temperature. Therefore, people should continually add hot water to keep the temperature even and as close as possible to the initial temperature without wasting too much water. This paper proposes a partial differential equation of the heat conduction of the bath water temperature, and an object programming model. Based on the Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), this paper illustrates the best strategy the person in the bathtub can adopt to satisfy his desires. First, a spatiotemporal partial differential equation model of the heat conduction of the temperature of the bath water is built. According to the priority, an object programming model is established, which takes the deviation of temperature throughout the bathtub, the deviation of temperature with the initial condition, water consumption, and the times of switching faucet as the four objectives. To ensure the top priority objective—homogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysis are conducted. The simulation and analytical results all imply that the top priority strategy is: The proper motions of the person making the temperature well-distributed throughout the bathtub. Therefore, the Partial Differential Equation model (PDE) can be simplified to the ordinary differential equation model.Second, the weights for the remaining three objectives are determined based on the tolerance of temperature and the hobby of the person by applying Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Therefore, the evaluation model of the synthesis score of the strategy is proposed to determine the best one the person in the bathtub can adopt. For example, keeping the temperature as close as the initial condition results in the fewer number of switching faucet while attention to water consumption gives rise to the more number. Third, the paper conducts the analysis of the diverse parameters in the model to determine the best strategy, respectively, by controlling the other parameters constantly, and adjusting the parameters of the volume, shape of the bathtub and the shape, volume, temperature and the motions and other parameters of the person in turns. All results indicate that the differential model and the evaluation model developed in this paper depends upon the parameters therein. When considering the usage of a bubble bath additive, it is equal to be the obstruction between water and air. Our results show that this strategy can reduce the dropping rate of the temperatureeffectively, and require fewer number of switching.The surface area and heat transfer coefficient can be increased because of the motions of the person in the bathtub. Therefore, the deterministic model can be improved as a stochastic one. With the above evaluation model, this paper present the stochastic optimization model to determine the best strategy. Taking the disparity from the initial temperature as the suboptimum objectives, the result of the model reveals that it is very difficult to keep the temperature constant even wasting plentiful hot water in reality.Finally, the paper performs sensitivity analysis of parameters. The result shows that the shape and the volume of the tub, different hobbies of people will influence the strategies significantly. Meanwhile, combine with the conclusion of the paper, we provide a one-page non-technical explanation for users of the bathtub.Fall in love with your bathtubAbstractIt’s pleasant t o go home to take a bath with the evenly maintained temperature of hot water throughout the bathtub. This beautiful idea, however, can not be always realized by the constantly falling water temperature. Therefore, people should continually add hot water to keep the temperature even and as close as possible to the initial temperature without wasting too much water. This paper proposes a partial differential equation of the heat conduction of the bath water temperature, and an object programming model. Based on the Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), this paper illustrates the best strategy the person in the bathtub can adopt to satisfy his desires. First, a spatiotemporal partial differential equation model of the heat conduction of the temperature of the bath water is built. According to the priority, an object programming model is established, which takes the deviation of temperature throughout the bathtub, the deviation of temperature with the initial condition, water consumption, and the times of switching faucet as the four objectives. To ensure the top priority objective—homogenization of temperature, the discretization method of the Partial Differential Equation model (PDE) and the analytical analysis are conducted. The simulation and analytical results all imply that the top priority strategy is: The proper motions of the person making the temperature well-distributed throughout the bathtub. Therefore, the Partial Differential Equation model (PDE) can be simplified to the ordinary differential equation model.Second, the weights for the remaining three objectives are determined based on the tolerance of temperature and the hobby of the person by applying Analytic Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Therefore, the evaluation model of the synthesis score of the strategy is proposed to determine the best one the person in the bathtub can adopt. For example, keeping the temperature as close as the initial condition results in the fewer number of switching faucet while attention to water consumption gives rise to the more number. Third, the paper conducts the analysis of the diverse parameters in the model to determine the best strategy, respectively, by controlling the other parameters constantly, and adjusting the parameters of the volume, shape of the bathtub and the shape, volume, temperature and the motions and other parameters of the person in turns. All results indicate that the differential model and the evaluation model developed in this paper depends upon the parameters therein. When considering the usage of a bubble bath additive, it is equal to be the obstruction between water and air. Our results show that this strategy can reduce the dropping rate of the temperature effectively, and require fewer number of switching.The surface area and heat transfer coefficient can be increased because of the motions of the person in the bathtub. Therefore, the deterministic model can be improved as a stochastic one. With the above evaluation model, this paper present the stochastic optimization model to determine the best strategy. Taking the disparity from the initial temperature as the suboptimum objectives, the result of the model reveals that it is very difficult to keep the temperature constant even wasting plentiful hotwater in reality.Finally, the paper performs sensitivity analysis of parameters. The result shows that the shape and the volume of the tub, different hobbies of people will influence the strategies significantly. Meanwhile, combine with the conclusion of the paper, we provide a one-page non-technical explanation for users of the bathtub.Keywords:Heat conduction equation; Partial Differential Equation model (PDE Model); Objective programming; Strategy; Analytical Hierarchy Process (AHP) Problem StatementA person fills a bathtub with hot water and settles into the bathtub to clean and relax. However, the bathtub is not a spa-style tub with a secondary hearing system, as time goes by, the temperature of water will drop. In that conditions,we need to solve several problems:(1) Develop a spatiotemporal model of the temperature of the bathtub water to determine the best strategy to keep the temperature even throughout the bathtub and as close as possible to the initial temperature without wasting too much water;(2) Determine the extent to which your strategy depends on 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.(3)The influence of using b ubble to model’s results.(4)Give a one-page non-technical explanation for users that describes your strategyGeneral Assumptions1.Considering the safety factors as far as possible to save water, the upper temperature limit is set to 45 ℃;2.Considering the pleasant of taking a bath, the lower temperature limit is set to 33℃;3.The initial temperature of the bathtub is 40℃.Table 1Model Inputs and SymbolsSymbols Definition UnitT Initial temperature of the Bath water ℃℃T∞Outer circumstance temperatureT Water temperature of the bathtub at the every moment ℃t Time hx X coordinates of an arbitrary point my Y coordinates of an arbitrary point mz Z coordinates of an arbitrary point mαTotal heat transfer coefficient of the system 2（）⋅/W m K1SThe surrounding-surface area of the bathtub 2m 2S The above-surface area of water2m 1H Bathtub’s thermal conductivity/W m K ⋅（） D The thickness of the bathtub wallm 2H Convection coefficient of water2/W m K ⋅（） a Length of the bathtubm b Width of the bathtubm h Height of the bathtubm V The volume of the bathtub water3m c Specific heat capacity of water/()J kg ⋅℃ ρ Density of water3/kg m ()v t Flooding rate of hot water3/m s r TThe temperature of hot water ℃Temperature ModelBasic ModelA spatio-temporal temperature model of the bathtub water is proposed in this paper. It is a four dimensional partial differential equation with the generation and loss of heat. Therefore the model can be described as the Thermal Equation.The three-dimension coordinate system is established on a corner of the bottom of the bathtub as the original point. The length of the tub is set as the positive direction along the x axis, the width is set as the positive direction along the y axis, while the height is set as the positive direction along the z axis, as shown in figure 1.Figure 1. The three-dimension coordinate systemTemperature variation of each point in space includes three aspects: one is the natural heat dissipation of each point in space; the second is the addition of exogenous thermal energy; and the third is the loss of thermal energy . In this way , we build the Partial Differential Equation model as follows:22212222(,,,)(,,,)()f x y z t f x y z t T T T T t x y z c Vαρ-∂∂∂∂=+++∂∂∂∂ (1) Where● t refers to time;● T is the temperature of any point in the space;● 1f is the addition of exogenous thermal energy;● 2f is the loss of thermal energy.According to the requirements of the subject, as well as the preferences of people, the article proposes these following optimization objective functions. A precedence level exists among these objectives, while keeping the temperature even throughout the bathtub must be ensured.Objective 1(.1O ): keep the temperature even throughout the bathtub;22100min (,,,)(,,,)t t V V F t T x y z t dxdydz dt t T x y z t dxdydz dt ⎡⎤⎡⎤⎛⎫=-⎢⎥ ⎪⎢⎥⎢⎥⎣⎦⎝⎭⎣⎦⎰⎰⎰⎰⎰⎰⎰⎰ (2) Objective 2(.2O ): keep the temperature as close as possible to the initial temperature;[]2200min (,,,)tV F T x y z t T dxdydz dt ⎛⎫=- ⎪⎝⎭⎰⎰⎰⎰ (3) Objective 3(.3O ): do not waste too much water;()30min tF v t dt =⋅⎰ (4) Objective 4(.4O ): fewer times of switching.4min F n = (5)Since the .1O is the most crucial, we should give priority to this objective. Therefore, the highest priority strategy is given here, which is homogenization of temperature.Strategy 0 – Homogenization of T emperatureThe following three reasons are provided to prove the importance of this strategy. Reason 1-SimulationIn this case, we use grid algorithm to make discretization of the formula (1), and simulate the distribution of water temperature.(1) Without manual intervention, the distribution of water temperature as shown infigure 2. And the variance of the temperature is 0.4962. 00.20.40.60.8100.51 1.5200.5Length WidthH e i g h t 4242.54343.54444.54545.5Distribution of temperature at the length=1Distribution of temperatureat the width=1Hot water Cool waterFigure 2. Temperature profiles in three-dimension space without manual intervention(2) Adding manual intervention, the distribution of water temperature as shown infigure 3. And the variance of the temperature is 0.005. 00.5100.51 1.5200.5 Length WidthH e i g h t 44.744.7544.844.8544.944.9545Distribution of temperatureat the length=1Distribution of temperature at the width=1Hot water Cool waterFigure 3. Temperature profiles in three-dimension space with manual interventionComparing figure 2 with figure 3, it is significant that the temperature of water will be homogeneous if we add some manual intervention. Therefore, we can assumed that222222()0T T T x y zα∂∂∂++≠∂∂∂ in formula (1). Reason 2-EstimationIf the temperature of any point in the space is different, then222222()0T T T x y zα∂∂∂++≠∂∂∂ Thus, we find two points 1111(,,,)x y z t and 2222(,,,)x y z t with:11112222(,,,)(,,,)T x y z t T x y z t ≠Therefore, the objective function 1F could be estimated as follows:[]2200200001111(,,,)(,,,)(,,,)(,,,)0t t V V t T x y z t dxdydz dt t T x y z t dxdydz dt T x y z t T x y z t ⎡⎤⎡⎤⎛⎫-⎢⎥ ⎪⎢⎥⎢⎥⎣⎦⎝⎭⎣⎦≥->⎰⎰⎰⎰⎰⎰⎰⎰ (6) The formula (6) implies that some motion should be taken to make sure that the temperature can be homogeneous quickly in general and 10F =. So we can assumed that: 222222()0T T T x y zα∂∂∂++≠∂∂∂. Reason 3-Analytical analysisIt is supposed that the temperature varies only on x axis but not on the y-z plane. Then a simplified model is proposed as follows:()()()()()()()2sin 000,0,,00,000t xx x T a T A x l t l T t T l t t T x x l π⎧=+≤≤≤⎪⎪⎪==≤⎨⎪⎪=≤≤⎪⎩ (7)Then we use two ways, Fourier transformation and Laplace transformation, in solving one-dimensional heat equation [Qiming Jin 2012]. Accordingly, we get the solution:()()2222/22,1sin a t l Al x T x t e a l πππ-=- (8) Where ()0,2x ∈, 0t >, ()01|x T f t ==(assumed as a constant), 00|t T T ==.Without general assumptions, we choose three specific value of t , and gain a picture containing distribution change of temperature in one-dimension space at different time.00.20.40.60.811.2 1.4 1.6 1.8200.511.522.533.54Length T e m p e r a t u r e time=3time=5time=8Figure 4. Distribution change of temperature in one-dimension space at different timeT able 2.V ariance of temperature at different timet3 5 8 variance0.4640 0.8821 1.3541It is noticeable in Figure 4 that temperature varies sharply in one-dimensional space. Furthermore, it seems that temperature will vary more sharply in three-dimension space. Thus it is so difficult to keep temperature throughout the bathtub that we have to take some strategies.Based on the above discussion, we simplify the four dimensional partial differential equation to an ordinary differential equation. Thus, we take the first strategy that make some motion to meet the requirement of homogenization of temperature, that is 10F =.ResultsTherefore, in order to meet the objective function, water temperature at any point in the bathtub needs to be same as far as possible. We can resort to some strategies to make the temperature of bathtub water homogenized, which is (,,)x y z ∀∈∀. That is,()(),,,T x y z t T t =Given these conditions, we improve the basic model as temperature does not change with space.112213312()()()()/()p r H S dT H S T T H S T T c v T T c V V dt D μρρ∞⎡⎤=++-+-+--⎢⎥⎣⎦(9) Where● 1μis the intensity of people’s movement ;● 3H is convection between water and people;● 3S is contact area between water and people;● p T is body surface temperature;● 1V is the volume of the bathtub;● 2V is the volume of people.Where the μ refers to the intensity of people ’s movement. It is a constant. However , it is a random variable in reality, which will be taken into consideration in the following.Model T estingWe use the oval-shaped bathtub to test our model. According to the actual situation, we give initial values as follows:0.19λ=,0.03D =,20.54H =,25T ∞=,040T =00.20.40.60.8125303540Time T e m p e r a t u r eFigure 5. Basic modelThe Figure 5 shows that the temperature decreases monotonously with time. And some signs of a slowing down in the rate of decrease are evident in the picture. Reaching about two hours, the water temperature does not change basically and be closely to the room temperature. Obviously , it is in line with the actual situation, indicating the rationality of this model.ConclusionOur model is robust under reasonable conditions, as can be seen from the testing above. In order to keep the temperature even throughout the bathtub, we should take some strategies like stirring constantly while adding hot water to the tub. Most important of all, this is the necessary premise of the following question.Strategy 1 – Fully adapted to the hot water in the tubInfluence of body surface temperatureWe select a set of parameters to simulate two kinds of situation separately.The first situation is that do not involve the factor of human1122()()/H S dT H S T T cV dt D ρ∞⎡⎤=+-⎢⎥⎣⎦(10) The second situation is that involves the factor of human112213312()()()/()p H S dT H S T T H S T T c V V dt D μρ∞⎡⎤=++-+--⎢⎥⎣⎦(11) According to the actual situation, we give specific values as follows, and draw agraph of temperature of two functions.33p T =,040T =204060801001201401601803838.53939.540TimeT e m p e r a t u r eWith body Without bodyFigure 6a. Influence of body surface temperature50010001500200025003000350025303540TimeT e m p e r a t u r eWith body Without bodyCoincident pointFigure 6b. Influence of body surface temperatureThe figure 6 shows the difference between two kinds of situation in the early time (before the coincident point ), while the figure 7 implies that the influence of body surface temperature reduces as time goes by . Combing with the degree of comfort ofbath and the factor of health, we propose the second optimization strategy: Fully adapted to the hot water after getting into the bathtub.Strategy 2 –Adding water intermittentlyInfluence of adding methods of waterThere are two kinds of adding methods of water. One is the continuous; the other is the intermittent. We can use both different methods to add hot water.1122112()()()/()r H S dT H S T T c v T T c V V dt D μρρ∞⎡⎤=++-+--⎢⎥⎣⎦(12) Where r T is the temperature of the hot water.To meet .3O , we calculated the minimum water consumption by changing the flow rate of hot water. And we compared the minimum water consumptions of the continuous with the intermittent to determine which method is better.A . Adding water continuouslyAccording to the actual situation, we give specific values as follows and draw a picture of the change of temperature.040T =, 37d T =, 45r T =5001000150020002500300035003737.53838.53939.54040.5TimeT e m p e r a t u r eadd hot waterFigure 7. Adding water continuouslyIn most cases, people are used to have a bath in an hour. Thus we consumed that deadline of the bath: 3600final t =. Then we can find the best strategy in Figure 5 which is listed in Table 2.T able 3Strategy of adding water continuouslystart t final tt ∆ vr T varianceWater flow 4 min 1 hour56 min537.410m s -⨯45℃31.8410⨯0.2455 3mB . Adding water intermittentlyMaintain the values of 0T ,d T ,r T ,v , we change the form of adding water, and get another graph.5001000150020002500300035003737.53838.53939.540TimeT e m p e r a t u r et1=283(turn on)t3=2107(turn on)t2=1828(turn off)Figure 8. Adding water intermittentlyT able 4.Strategy of adding water intermittently()1t on ()2t off 3()t on vr T varianceWater flow 5 min 30 min35min537.410m s -⨯45℃33.610⨯0.2248 3mConclusionDifferent methods of adding water can influence the variance, water flow and the times of switching. Therefore, we give heights to evaluate comprehensively the methods of adding hot water on the basis of different hobbies of people. Then we build the following model:()()()2213600210213i i n t t i F T t T dtF v t dtF n -=⎧=-⎪⎪⎪=⎨⎪⎪=⎪⎩⎰∑⎰ (13) ()112233min F w F w F w F =++ (14)12123min ..510mini i t s t t t +>⎧⎨≤-≤⎩Evaluation on StrategiesFor example: Given a set of parameters, we choose different values of v and d T , and gain the results as follows.Method 1- AHPStep 1:Establish hierarchy modelFigure 9. Establish hierarchy modelStep 2: Structure judgment matrix153113511133A ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦Step 3: Assign weight1w 2w3w 0.650.220.13Method 2-TopsisStep1 ：Create an evaluation matrix consisting of m alternatives and n criteria, with the intersection of each alternative and criteria given as ij x we therefore have a matrixStep2：The matrix ij m n x ⨯（）is then normalised to form the matrix ij m n R r ⨯=（）, using thenormalisation method21r ,1,2,,;1,2,ijij mij i x i n j m x====∑…………，Step3：Calculate the weighted normalised decision matrix()(),1,2,,ij j ij m n m nT t w r i m ⨯⨯===⋅⋅⋅where 1,1,2,,nj j jj w W Wj n ===⋅⋅⋅∑so that11njj w==∑, and j w is the original weight given to the indicator,1,2,,j v j n =⋅⋅⋅.Step 4： Determine the worst alternative ()w A and the best alternative ()b A()(){}{}()(){}{}max 1,2,,,min 1,2,,1,2,,n ,min 1,2,,,max 1,2,,1,2,,n ,w ij ij wjbijij bjA t i m j J t i m j J t j A t i m j J t i m j J tj -+-+==∈=∈====∈=∈==where, {}1,2,,J j n j +==⋅⋅⋅ associated with the criteria having a positive impact, and {}1,2,,J j n j -==⋅⋅⋅associated with the criteria having a negative impact. Step 5： Calculate the L2-distance between the target alternative i and the worst condition w A()21,1,2,,m niw ij wj j d tt i ==-=⋅⋅⋅∑and the distance between the alternative i and the best condition b A()21,1,2,,m nib ij bj j d t t i ==-=⋅⋅⋅∑where iw d and ib d are L2-norm distances from the target alternative i to the worst and best conditions, respectively .Step 6 ：Calculate the similarity to the worst condition Step 7 ： Rank the alternatives according to ()1,2,,iw s i m =⋅⋅⋅ Step 8 : Assign weight1w2w 3w 0.55 0.170.23ConclusionAHP gives height subjectively while TOPSIS gives height objectively. And the heights are decided by the hobbies of people. However, different people has different hobbies, we choose AHP to solve the following situations.Impact of parametersDifferent customers have their own hobbies. Some customers prefer enjoying in the bath, so the .2O is more important . While other customers prefer saving water, the .3O is more important. Therefore, we can solve the problem on basis of APH . 1. Customers who prefer enjoying: 20.83w =,30.17w =According to the actual situation, we give initial values as follows:13S =,11V =,2 1.4631S =,20.05V =,33p T =,110μ=Ensure other parameters unchanged, then change the values of these parameters including 1S ,1V ,2S ,2V ,d T ,1μ. So we can obtain the optimal strategies under different conditions in Table 4.T able 5.Optimal strategies under different conditions2.Customers who prefer saving: 20.17w =,30.83w =Just as the former, we give the initial values of these parameters including1S ,1V ,2S ,2V ,d T ,1μ, then change these values in turn with other parameters unchanged. So we can obtain the optimal strategies as well in these conditions.T able 6.Optimal strategies under different conditionsInfluence of bubbleUsing the bubble bath additives is equivalent to forming a barrier between the bath water and air, thereby slowing the falling velocity of water temperature. According to the reality, we give the values of some parameters and gain the results as follows:5001000150020002500300035003334353637383940TimeT e m p e r a t u r eWithour bubbleWith bubbleFigure 10. Influence of bubbleT able 7.Strategies (influence of bubble)Situation Dropping rate of temperature (the larger the number, the slower)Disparity to theinitial temperatureWater flow Times of switchingWithout bubble 802 1.4419 0.1477 4 With bubble 34499.85530.01122The Figure 10 and the Table 7 indicates that adding bubble can slow down the dropping rate of temperature effectively . It can decrease the disparity to the initial temperature and times of switching, as well as the water flow.Improved ModelIn reality , human ’s motivation in the bathtub is flexible, which means that the parameter 1μis a changeable measure. Therefore, the parameter can be regarded as a random variable, written as ()[]110,50t random μ=. Meanwhile, the surface of water will come into being ripples when people moves in the tub, which will influence the parameters like 1S and 2S . So, combining with reality , we give the range of values as follows:()[]()[]111222,1.1,1.1S t random S S S t random S S ⎧=⎪⎨=⎪⎩Combined with the above model, the improved model is given here:()[]()[]()[]11221121111222()()()/()10,50,1.1,1.1a H S dT H S T T c v T T c V V dt D t random S t random S S S t random S S μρρμ∞⎧⎡⎤=++-+--⎪⎢⎥⎣⎦⎨⎪===⎩(15)Given the values, we can get simulation diagram:050010001500200025003000350039.954040.0540.140.15TimeT e m p e r a t u r eFigure 11. Improved modelThe figure shows that the variance is small while the water flow is large, especially the variance do not equals to zero. This indicates that keeping the temperature of water is difficult though we regard .2O as the secondary objective.Sensitivity AnalysisSome parameters have a fixed value throughout our work. By varying their values, we can see their impacts.Impact of the shape of the tub0.70.80.91 1.1 1.2 1.3 1.433.23.43.63.84Superficial areaT h e t i m e sFigure 12a. Times of switching0.70.80.91 1.11.21.31.43890390039103920393039403950Superficial areaV a r i a n c eFigure 12b. V ariance of temperature0.70.80.91 1.1 1.2 1.3 1.40.190.1950.20.2050.21Superficial areaW a t e r f l o wFigure 12c. Water flowBy varying the value of some parameters, we can get the relationships between the shape of tub and the times of switching, variance of temperature, and water flow et. It is significant that the three indexes will change as the shape of the tub changes. Therefore the shape of the tub makes an obvious effect on the strategies. It is a sensitive parameter.Impact of the volume of the tub0.70.80.91 1.1 1.2 1.3 1.4 1.533.544.55VolumeT h e t i m e sFigure 13a. Times of switching。

美赛数模论文

MCM 2015 Summary Sheet for Team 35565For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number35565Problem ChosenBFor office use onlyF1________________F2________________F3________________F4________________ SummaryThe lost MH370 urges us to build a universal search plan to assist searchers to locate the lost plane effi-ciently and optimize the arrangement of search plans.For the location of the search area, we divided it into two stages, respectively, to locate the splash point and the wreckage‟s sunk point. In the first stage, we consider the types of crashed aircraft, its motion and different position out of contact. We also consider the Earth‟s rotation, and other factors. Taking all these into account, we establish a model to locate the splash point. Then we apply this model to MH370. we can get the splash point in the open water is 6.813°N 103.49°E and the falling time is 52.4s. In the second stage, considering resistances of the wreckage in different shapes and its distribution affected by ocean currents, we establish a wreckage sunk point model to calculate the horizontal displacement and the angle deviation affected by the ocean currents. The result is 1517m and 0.11°respectively. Next, we extract a satellite map of submarine topography and use MATLAB to depict seabed topography map, determining the settlement of the wreckage by using dichotomy algorithm under different terrains. Finally, we build a Bayesian model and calculate the weight of corresponding area, sending aircrafts to obtain new evidence and refresh suspected wreckage area.For the assignment of the search planes, we divide it into two stages, respectively, to determine the num-ber of the aircraft and the assignment scheme of the search aircraft. In the first stage, we consider the search ability of each plane and other factors. And then we establish global optimization model. Next we use Dinkelbach algorithm to select the best n search aircrafts from all search aircrafts. In the second stage, we divide the assignment into two cases whether there are search aircrafts in the target area. If there is no search aircraft, we take the search area as an arbitrary polygon and establish the subdivision model. Considering the searching ability of each plane, we divide n small polygons into 2n sub-polygons by using NonconvexDivide algorithm, which assigns specific anchor points to these 2n sub-polygons re-spectively. If there exist search aircrafts, we divide the search area into several polygons with the search aircrafts being at the boundary of the small polygons. To improve search efficiency, we introduce” ma x-imize the minimum angle strategy” to maximize right-angle subdivision so that we can reduce the turning times of search aircraft. When we changed the speed of the crashed plane about 36m/s, the latitude of the splash point changes about 1°.When a wreck landing at 5.888m out from the initial zone, it will divorce from suspected searching area, which means our models are fairly robust to the changes in parameters. Our model is able to efficiently deal with existing data and modify some parameters basing the practical situation. The model has better versatility and stability. The weakness of our model is neglect of human factors, the search time and other uncontrollable factors that could lead to deviation compared to practical data. Therefore, we make some in-depth discussions about the model, modifying assumptions establish-Searching For a Lost PlaneControl#35565February 10, 2014Team # 35565 Page 3 of 47 Contents1 Introduction (5)1.1 Restatement of the Problem (5)1.2 Literature Review (6)2 Assumptions and Justifications (7)3 Notations (7)4 Model Overview (10)5 Modeling For Locating the Lost Plane (10)5.1 Modeling For Locating the Splash Poin t (11)5.1.1 Types of Planes (11)5.1.2 Preparation of the Model—Earth Rotation (12)5.1.3 Modeling (13)5.1.4 Solution of The Model (14)5.2 Modeling For Locating Wreckage (15)5.2.1 Assumptions of the Model (16)5.2.2 Preparation of the Model (16)5.2.3 Modeling (21)5.2.4 Solution of the Model (25)5.3 Verification of the Model (26)5.3.1 Verification of the Splash Point (26)5.3.2 Verification of the binary search algorithm (27)6 Modeling For Optimization of Search Plan (29)6.1 The Global Optimization Model (29)6.1.1 Preparation of the Model (29)6.1.2 Modeling (31)6.1.3 Solution of the Model (31)6.2 The Area Partition Algorithm (33)6.2.1 Preparation of the Model (33)6.2.2 Modeling (34)6.2.3 Solution of the Model (35)6.2.4 Improvement of the Model (36)7 Sensitivity Analysis (38)8 Further Discussions (39)9 Strengths and Weaknesses (41)9.1 Strengths (41)9.2 Weaknesses (42)10 Non-technical Paper (42)1 IntroductionAn airplane (informally plane) is a powered, fixed-wing aircraft that is propelled for-ward by thrust from a jet engine or propeller. Its main feature is fast and safe. Typi-cally, air travel is approximately 10 times safer than travel by car, rail or bus. Howev-er, when using the deaths per journey statistic, air travel is significantly more danger-ous than car, rail, or bus travel. In an aircraft crash, almost no one could survive [1]. Furthermore, the wreckage of the lost plane is difficult to find due to the crash site may be in the open ocean or other rough terrain.Thus, it will be exhilarating if we can design a model that can find the lost plane quickly. In this paper, we establish several models to find the lost plane in seawater and develop an op-timal scheme to assign search planes to model to locate the wreckage of the lost plane.1.1 Restatement of the ProblemWe are required to build a mathematical model to find the lost plane crashed in open water. We decompose the problem into three sub-problems:●Work out the position and distributions of the plane‟s wreckage●Arrange a mathematical scheme to schedule searching planesIn the first step, we seek to build a model with the inputs of altitude and other factors to locate the splash point on the sea-level. Most importantly, the model should reflect the process of the given plane. Then we can change the inputs to do some simulations. Also we can change the mechanism to apply other plane crash to our model. Finally, we can obtain the outputs of our model.In the second step, we seek to extend our model to simulate distribution of the plane wreckage and position the final point of the lost plane in the sea. We will consider more realistic factors such as ocean currents, characteristics of plane.We will design some rules to dispatch search planes to confirm the wreckage and de-cide which rule is the best.Then we attempt to adjust our model and apply it to lost planes like MH370. We also consider some further discussion of our model.1.2 Literature ReviewA model for searching the lost plane is inevitable to study the crashed point of the plane and develop a best scheme to assign search planes.According to Newton's second law, the simple types of projectile motion model can work out the splash point on the seafloor. We will analyze the motion state ofthe plane when it arrives at the seafloor considering the effect of the earth's rotation,After the types of projectile motion model was established, several scientists were devoted to finding a method to simulate the movement of wreckage. The main diffi-culty was to combine natural factors with the movement. Juan Santos-Echeandía introduced a differential equation model to simplify the difficulty [2]. Moreover,A. Boultif and D. Louër introduced a dichotomy iteration algorithm to circular compu-ting which can be borrowed to combine the motion of wreckage with underwater ter-rain [3]. Several conditions have to be fulfilled before simulating the movement: (1) Seawater density keeps unchanged despite the seawater depth. (2) The velocity of the wreck stay the same compared with velocity of the plane before it crashes into pieces.(3) Marine life will not affect our simulation. (4) Acting forceof seawater is a function of the speed of ocean currents.However the conclusion above cannot describe the wreckage zone accurately. This inaccuracy results from simplified conditions and ignoring the probability distribution of wreckage. In 1989, Stone et.al introduced a Bayesian search approach for searching problems and found the efficient search plans that maximize the probability of finding the target given a fixed time limit by maintaining an accurate target location probabil-ity density function, and by explicitly modeling the target‟s process model [4].To come up with a concrete dispatch plan. Xing Shenwei first simulated the model with different kinds of algorithm. [5] In his model, different searching planes are as-sessed by several key factors. Then based on the model established before, he use the global optimization model and an area partition algorithm to propose the number of aircrafts. He also arranged quantitative searching recourses according to the maxi-mum speed and other factors. The result shows that search operations can be ensured and effective.Further studies are carried out based on the comparison between model andreality.Some article illustrate the random error caused by assumptions.2 Assumptions and JustificationsTo simplify the problem, we make the following basic assumptions, each ofwhich is properly justified.●Utilized data is accuracy. A common modeling assumption.●We ignore the change of the gravitational acceleration. The altitude of anaircraft is less than 30 km [6]. The average radius of the earth is 6731.004km, which is much more than the altitude of an aircraft. The gravitational accele-ration changes weakly.●We assume that aeroengine do not work when a plane is out of contact.Most air crash resulted from engine failure caused by aircraft fault, bad weather, etc.●In our model, the angle of attack do not change in an air crash and thefuselage don’t wag from side to side. We neglect the impact of natural and human factors●We treat plane as a material point the moment it hit the sea-level. Thecrashing plane moves fast with a short time-frame to get into the water. The shape and volume will be negligible.●We assume that coefficient of air friction is a constant. This impact is neg-ligible compared with that of the gravity.●Planes will crash into wreckage instantly when falling to sea surface.Typically planes travel at highly speed and may happen explosion accident with water. So we ignore the short time.3 NotationsAll the variables and constants used in this paper are listed in Table 1 and Table 2.Table 1 Symbol Table–ConstantsSymbol DefinitionωRotational angular velocity of the earthg Gravitational accelerationr The average radius of the earthC D Coefficient of resistance decided by the angle of attack ρAtmospheric densityφLatitude of the lost contact pointμCoefficient of viscosityS0Area of the initial wrecking zoneS Area of the wrecking zoneS T Area of the searching zoneK Correction factorTable 2 Symbol Table-VariablesSymbol DefinitionF r Air frictionF g Inertial centrifugal forceF k Coriolis forceW Angular velocity of the crash planev r Relative velocity of the crash planev x Initial velocity of the surface layer of ocean currentsk Coefficient of fluid frictionF f Buoyancy of the wreckagef i Churning resistance of the wreckage from ocean currents f Fluid resistance opposite to the direction of motionG Gravity of the wreckageV Volume of the wreckageh Decent height of the wreckageH Marine depthS x Displacement of the wreckageS y Horizontal distance of S xα Deviation angle of factually final position of the wreckage s Horizontal distance between final point and splash point p Probability of a wreck in a given pointN The number of the searching planeTS ' The area of sea to be searched a i V ˆ The maximum speed of each planeai D The initial distance from sea to search planeai A The search ability of each plane is),(h T L i The maximum battery life of each plane isi L The mobilized times of each plane in the whole search )1(N Q Q a a ≤≤ The maximum number of search plane in the searching zone T(h) The time the whole action takes4 Model OverviewMost research for searching the lost plane can be classified as academic and practical. As practical methods are difficult to apply to our problem, we approach theproblem with academic techniques. Our study into the searching of the lost plane takes several approaches.Our basic model allows us to obtain the splash point of the lost plane. We focus on the force analysis of the plane. Then we We turn to simple types of projectile motion model. This model gives us critical data about the movement and serves as a stepping stone to our later study.The extended model views the problem based on the conclusion above. We run diffe-rential equation method and Bayesian search model to simulate the movement of wreckage. The essence of the model is the way to combine the effect of natural factors with distribution of the wreckage. Moreover, using distributing conditions, we treat size of the lost plane as “initial wreckage zone” so as to approximately describe the distribution. Thus, after considering the natural factors, we name the distribution of wreckage a “wreck zone” to minimize searching zone. While we name all the space needed to search “searching zone”.Our conclusive model containing several kinds of algorithm attempts to tackle a more realistic and more challenging problem. We add the global optimization model and an area partition algorithm to improve the efficiency of search aircrafts according to the area of search zone. An assessment of search planes consisting of search capabili-ties and other factors are also added. The Dinkelbach and NonConvexDivide algo-rithm for the solutions of the results are also added.We use the extended and conclusive model as a standard model to analyze the problem and all results have this two model at their cores.5 Modeling For Locating the Lost PlaneWe will start with the idea of the basic model. Then we present the Bayesian search model to get the position of the sinking point.5.1 Modeling For Locating the Splash PointThe basic model is a academic approach. A typical types of projectile behavior con-sists of horizontal and vertical motion. We also add another dimension consider-ing the effect of the earth's rotation. Among these actions, the force analysis is the most crucial part during descent from the point out of contact to the sea-level. Types of plane might impact trajectory of the crashing plane.5.1.1 Types of PlanesWe classify the planes into six groups [7]:●Helicopters: A helicopter is one of the most timesaving ways to transfer be-tween the city and airport, alternatively an easy way to reach remote destina-tions.●Twins Pistons: An economical aircraft range suitable for short distance flights.Aircraft seating capacity ranging from 3 to 8 passengers.●Turboprops: A wide range of aircraft suitable for short and medium distanceflights with a duration of up to 2-4 hours. Aircraft seating capacity ranging from 4 to 70 passengers.●Executive Jets:An Executive Jet is suitable for medium or long distanceflights. Aircraft seating capacity ranging from 4 to 16 passengers●Airliners:Large jet aircraft suitable for all kinds of flights. Aircraft seatingcapacity ranging from 50 to 400 passengers.●Cargo Aircrafts:Any type of cargo. Ranging from short notice flights carry-ing vital spare parts up to large cargo aircraft that can transport any volumin-ous goods.The lost plane may be one of these group. Then we extract the characteristics of planes into three essential factors: mass, maximum flying speed, volume. We use these three factors to abstract a variety of planes:●Mass: Planes of different product models have their own mass.●Maximum flying speed: Different planes are provided with kinds of me-chanical configuration, which will decide their properties such as flying speed.●Volume: Planes of distinct product models have different sizes and configura-tion, so the volume is definitive .5.1.2 Preparation of the Model —Earth RotationWhen considering the earth rotation, we should know that earth is a non-inertial run-ning system. Thus, mobile on the earth suffers two other non-inertial forces except air friction F r . They are inertial centrifugal force F g and Coriolis force F k . According to Newton ‟s second law of motion, the law of object relative motion to the earth is:Rotational angular velocity of the earth is very small, about .For a big mobile v r , it suffers far less inertial centrifugal force than Coriolis force, so we can ignore it. Thus, the equation can be approximated as follows:Now we establish a coordinate system: x axis z axis pointing to the east and south re-spectively, y axis vertical upward, then v r , ω and F r in the projection coordinate system are as follows:⎪⎪⎩⎪⎪⎨⎧++=⋅⋅-⋅⋅=++=kdt dz j dt dy i dt dx m v k j w kF j F i F F r rz ry rx r φωφωcos sinφis the latitude of the lost contact point of the lost plane. Put equation 1-3 and equa-tion 1-2 together, then the component of projectile movement in differential equation is:ma FF F k g r=++srad ⋅⨯=-5103.7ωmamv F r r =+ω2⎪⎪⎪⎩⎪⎪⎪⎨⎧+⋅=+⋅=+⎪⎭⎫ ⎝⎛+⋅-=m F dt dx w dt z d m F dt dx w dt y d m F dt dz dt dy w dtx d rz ry rx φφφφsin 2cos 2sin cos 22222225.1.3 ModelingConsidering the effect caused by earth rotation and air draught to plane when crashing to sea level, we analyze the force on the X axis by using Newton ‟s second law, the differential equation on x y and axis, we can conclude:In conclusion, we establish the earth rotation and types of projectile second order dif-ferential model:()⎪⎩⎪⎨⎧+-⋅'⋅⋅=''-⋅'+⋅'⋅⋅-=''-⋅'⋅⋅=''m gf y w m z m f z x w m y m f y w m x m obj 321cos 2cos sin 2sin 2.φφφφAccording to Coriolis theorem, we analyze the force of the plane on different direc-tions. By using the Newton ‟s laws of motion, we can work out the resultant accelera-tion on all directions:⎪⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎪⎨⎧'+'+'⋅'⋅⋅⨯+='+'+'⋅'⋅⋅⨯+='+'+'⋅'⋅⋅⨯+=⋅⋅-⋅⋅=⋅⨯=⋅'''⋅===-2222222225)()()(21)()()(21)()()(21cos sin 103.704.022z y x z c F f z y x y c F f z y x x c F f k j w s rad S y x F c D rz D ryD rx D ρρρφωφωωμφC D is the angle of attack of a plane flew in the best state, w is the angular speed of a moving object, vector j and k are the unit vector on y and z direction respectively,μisrx F y w m x m -⋅'⋅⋅⨯=''φsin 2()ry F z x w m y m -'+⋅'⋅⨯-=''φφcos sin 2mg F y w m z m rz +-⋅'⋅⋅⋅=''φcos 2the coefficient of viscosity of the object.5.1.4 Solution of the ModelWhen air flows through an object, only the air close to layer on the surface of the ob-ject in the laminar airflow is larger, whose air viscosity performance is more noticea-ble while the outer region has negligible viscous force [8]. Typically, to simplify cal-culation, we ignore the viscous force produced by plane surface caused by air resis-tance.Step 1: the examination of dimension in modelTo verify the validity of the model based on Newton ‟s second theorem, first, we standardize them respectively, turn them into the standardization of dimensionless data to diminish the influence of dimensional data. The standard equation is:Step 2: the confirmation of initial conditionsIn a space coordinate origin based on plane, we assume the earth's rotation direc-tion for the x axis, the plane's flight heading as y axis, the vertical downward di-rection for z axis. Space coordinate system are as follows:Figure 1 Space coordinate systemStep 3: the simplification and solutionAfter twice integrations of the model, ignoring some of the dimensionless in thesxx y i -=integral process, we can simplify the model and get the following:⎪⎪⎪⎩⎪⎪⎪⎨⎧+'⋅⋅⋅-⋅'⋅⨯='''-⋅⋅⋅-⋅'⋅⨯-=''⋅'⋅⨯=''g z m s c y w z y v m s c z w y y w x D D 220)(2cos 2)(2cos 2sin 2ρφρφφWe can calculate the corresponding xyz by putting in specific data to get the in-formation about the point of losing contact.Step 4: the solution of the coordinateThe distance of every latitude on the same longitude is 111km and the distance ofevery longitude on the same latitude is 111*cos (the latitude of this point) (km). Moreover, the latitude distance of two points on the same longitude is r ×cos(a ×pi/180) and the longitude distance of two points on the same latitude is: r ×sin(a ×pi/180)[9].We assume a as the clockwise angle starting with the due north direction and r as the distance between two points; X 、Y are the latitude and longitude coordinates of the known point P respectively; Lon , Lat are the latitude and longitude coordi-nates of the unknown point B respectively.Therefore, the longitude and latitude coordinates of the unknown point Q is:⎪⎪⎩⎪⎪⎨⎧⨯⨯+=⨯⨯⨯⨯+=111)180/cos()180/cos(111)180/sin(pi a r Y Lat pi Y pi a r X LonThus, we can get coordinates of the point of splash by putting in specific data.5.2 Modeling For Locating WreckageIn order to understand how the wreckage distributes in the sea, we have to understand the whole process beginning from the plane crashing into water to reaching the seaf-loor. One intuition for modeling the problem is to think of the ocean currents as astochastic process decided by water velocity. Therefore, we use a differential equation method to simulate the impact on wreckage from ocean currents.A Bayesian Searching model is a continuous model that computing a probability dis-tribution on the location of the wreckage (search object) in the presence of uncertain-ties and conflicting information that require the use of subjective probabilities. The model requires an initial searching zone and a set of the posterior distribution given failure of the search to plan the next increment of search. As the search proceeds, the subjective estimates of the detection will be more reliable.5.2.1 Assumptions of the ModelThe following general assumptions are made based on common sense and weuse them throughout our model.●Seawater density keeps unchanged despite the seawater depth.Seawater density is determined by water temperature, pressure, salinity etc.These factors are decided by or affected by the seawater density. Considering the falling height, the density changes slightly. To simplify the calculation, we consider it as a constant.●The velocity of the wreck stay the same compared with velocity of theplane before it crashes into pieces. The whole process will end quickly witha little loss of energy. Thus, we simplify the calculation.●Marine life will not affect our simulation.Most open coast habitats arefound in the deep ocean beyond the edge of the continental shelf, while the falling height of the plane cannot hit.●Acting force of seawater is a function of the speed and direction of oceancurrents. Ocean currents is a complicated element affected by temperature, wide direction, weather pattern etc. we focus on a short term of open sea.Acting force of seawater will not take this factors into consideration.5.2.2 Preparation of the Model●The resistance of objects of different shapes is different. Due to the continuityof the movement of the water, when faced with the surface of different shapes, the water will be diverted, resulting in the loss of partial energy. Thus the pressure of the surface of objects is changed. Based on this, we first consider the general object, and then revise the corresponding coefficients.●Ocean currents and influencing factorsOcean currents, also called sea currents, are large-scale seawater movements which have relatively stable speed and direction. Only in the land along the coast, due to tides, terrain, the injection of river water, and other factors, the speed and direction of ocean currents changes.Figure 2Distribution of world ocean currentsIt can be known from Figure 2 that warm and cold currents exist in the area where aircraft incidences happened. Considering the fact that the speed of ocean currents slows down as the increase of the depth of ocean, the velocity with depth sea surface currents gradually slowed down, v x is set as the initial speed of ocean currents in subsequent calculations.●Turbulent layerTurbulent flow is one kind of state of the fluid. When the flow rate is very low, the fluid is separated into different layers, called laminar flow, which do not mix with each other. As the flow speed increases, the flow line of the fluid begins to appear wavy swing. And the swing frequency and amplitude in-creases as the flow rate increases. This kind of stream regimen is called tran-sition flow. When the flow rate becomes great, the flow line is no longer clear and many small whirlpools, called turbulence, appeared in the flow field.Under the influence of ocean currents, the flow speed of the fluid changes as the water depth changes gradually, the speed and direction of the fluid is un-certain, and the density of the fluid density changes, resulting in uneven flow distribution. This indirectly causes the change of drag coefficient, and the re-sistance of the fluid is calculated as follows:2fkvGLCM texture of submarine topographyIn order to describe the impact of submarine topography, we choose a rectan-gular region from 33°33…W, 5°01…N to 31°42‟W , 3°37‟N. As texture is formed by repetitive distribution of gray in the spatial position, there is a cer-tain gray relation between two pixels which are separated by a certain dis-tance, which is space correlation character of gray in images. GLCM is a common way to describe the texture by studying the space correlation cha-racter of gray. We use correlation function of GLCM texture in MATLAB:I=imread ('map.jpg'); imshow(I);We arbitrarily select a seabed images and import seabed images to get the coordinate of highlights as follows：Table 1Coordinate of highlightsNO. x/km y/km NO. x/km y/km NO. x/km y/km1 154.59 1.365 13 91.2 22.71 25 331.42 16.632 151.25 8.19 14 40.04 18.12 26 235.77 13.93 174.6 14.02 15 117.89 14.89 27 240.22 17.754 172.38 19.23 16 74.51 12.29 28 331.42 24.455 165.71 24.82 17 45.6 8.56 29 102.32 19.486 215.75 26.31 18 103.43 5.58 30 229.1 18.247 262.46 22.96 19 48.934 3.51 31 176.83 9.188 331.42 22.34 20 212.42 2.85 32 123.45 3.239 320.29 27.55 21 272.47 2.48 33 32.252 11.7910 272.47 27.55 22 325.85 6.45 34 31.14 27.811 107.88 28.79 23 230.21 7.32 35 226.88 16.0112 25.579 27.05 24 280.26 9.93 36 291.38 5.46Then we use HDVM algorithm to get the 3D image of submarine topography, which can be simulated by MATLAB.Figure 3 3D image of submarine topographyObjects force analysis under the condition of currentsf is the resistance, f i is the disturbance resistance, F f is the buoyancy, G isgravity of object.Figure 4Force analysis of object under the conditions of currentsConsidering the impact of currents on the sinking process of objects, wheninterfered with currents, objects will sheer because of uneven force. There-。

2013美赛A题一等奖论文

3. What is the distribution of heat across the outer edge of a pan?
3.1 Model establishment 3.1.1 Model Ⅰ: Micro-point model We build a micro-point model to show the different distribution of heat in different position for different shapes of brownie pan. As is mentioned in the problem, when baking in a rectangular pan heat is concentrated in the 4 corners and the product gets overcooked at the corners (and to a lesser extent at the edges).
Figure 1 A kind of brownie pan used in daily life
2. General assumption for all models
The oven is rectangular The heat distribute evenly in the oven, that is the temperature is constant everywhere The baking pan is heated evenly The thickness of the pan is constant Without consideration of the influence caused by various material of the pan

美赛论文模板(超实用)

TitileSummaryDuring cell division, mitotic spindles are assembled by microtubule-based motor proteins1, 2. The bipolar organization of spindles is essential for proper segregation of chromosomes, and requires plus-end-directed homotetrameric motor proteins of the widely conserved kinesin-5 (BimC) family3. Hypotheses for bipolar spindle formation include the 'push−pull mitotic muscle' model, in which kinesin-5 and opposing motor proteins act between overlapping microtubules2, 4, 5. However, the precise roles of kinesin-5 during this process are unknown. Here we show that the vertebrate kinesin-5 Eg5 drives the sliding of microtubules depending on their relative orientation. We found in controlled in vitro assays that Eg5 has the remarkable capability of simultaneously moving at 20 nm s-1 towards the plus-ends of each of the two microtubules it crosslinks. For anti-parallel microtubules, this results in relative sliding at 40 nm s-1, comparable to spindle pole separation rates in vivo6. Furthermore, we found that Eg5 can tether microtubule plus-ends, suggesting an additional microtubule-binding mode for Eg5. Our results demonstrate how members of the kinesin-5 family are likely to function in mitosis, pushing apart interpolar microtubules as well as recruiting microtubules into bundles that are subsequently polarized by relative sliding. We anticipate our assay to be a starting point for more sophisticated in vitro models of mitotic spindles. For example, the individual and combined action of multiple mitotic motors could be tested, including minus-end-directed motors opposing Eg5 motility. Furthermore, Eg5 inhibition is a major target of anti-cancer drug development, and a well-defined and quantitative assay for motor function will be relevant for such developmentsContentTitile (1)Summary (1)1Introduction (1)1.1Restatement of the Problem (1)1.2Background (1)1.1.1Common Solving Technique (1)1.1.2Previous Works (1)1.3Example (1)2Analysis of the Problem (1)2.1Outline of the Approach (1)2.2Basic Assumptions (2)2.3Definitions and Key Terms (2)3Calculating and Simplifying the Model (2)4The Model Results (3)5Validating the Model (3)6Strengths and Weaknesses (3)6.1Strengths (3)6.2Weaknesses (3)7Food for Thought (3)8Conclusion (3)References (4)Appendices (4)Appendix A Source Code (4)Appendix B (4)1Introduction1.1Restatement of the Problem …1.2Background…1.1.1Common Solving Technique…1.1.2Previous Works…1.3Example…2Analysis of the Problem …2.1Outline of the Approach…2.2Basic Assumptions●●●●●2.3Definitions and Key Terms●●●●Table 1.…Symbol Meaning Unit3Calculating and Simplifying the Model …4The Model Results……5Validating the Model…6Strengths and Weaknesses6.1S trengths●●●●6.2W eaknesses●●●●7Food for Thought…8Conclusion….References…AppendicesAppendix A Source CodeHere are the simulation programmes we used in our model as follow. Input matlab source:……….Appendix B…….Input C++ source：…………..…………..。

美国大学生数学建模大赛优秀论文一等奖摘要

SummaryChina is the biggest developing country. Whether water is sufficient or not will have a direct impact on the economic development of our country. China's water resources are unevenly distributed. Water resource will critically restrict the sustainable development of China if it can not be properly solved.First, we consider a greater number of Chinese cities so that China is divided into 6 areas. The first model is to predict through division and classification. We predict the total amount of available water resources and actual water usage for each area. And we conclude that risk of water shortage will exist in North China, Northwest China, East China, Northeast China, whereas Southwest China, South China region will be abundant in water resources in 2025.Secondly, we take four measures to solve water scarcity: cross-regional water transfer, desalination, storage, and recycling. The second model mainly uses the multi-objective planning strategy. For inter-regional water strategy, we have made reference to the the strategy of South-to-North Water Transfer[5]and other related strategies, and estimate that the lowest cost of laying the pipeline is about 33.14 billion yuan. The program can transport about 69.723 billion cubic meters water to the North China from the Southwest China region per year. South China to East China water transfer is about 31 billion cubic meters. In addition, we can also build desalination mechanism program in East China and Northeast China, and the program cost about 700 million and can provide 10 billion cubic meters a year.Finally, we enumerate the east China as an example to show model to improve. Other area also can use the same method for water resources management, and deployment. So all regions in the whole China can realize the water resources allocation.In a word, the strong theoretical basis and suitable assumption make our model estimable for further study of China's water resources. Combining this model with more information from the China Statistical Yearbook will maximize the accuracy of our model.。

美赛论文模板

T eam Control NumberFor office use only0000For office use onlyT1 F1T2 F2T3 Problem Chosen F3T4 A F42014 Mathematical Contest in Modeling (MCM) Summary Sheet(Attach a copy of this page to each copy of your solution paper.)Repeaters Coordination And DistributionFebruary 6,2015AbstractIn this paper, it aims to computing problem on Relay Strategy (repeaters coordination and distribution). According to advanced radio cellular coverage technology, usage of frequency attenuation and geometric mapping methods, Hata model， cellular coverage solution and FDM (Frequency Division Multiplexing) model were established. The algorithms used MATLAB to simulate, with the final modeling results of sensitivity analysis and improvement & promotion on models.Question one : For a circular flat area of radius 40 miles radius, determine the minimum number of repeaters necessary to accommodate 1,000 simultaneous users. Assume that the spectrum available is 145 to 148 MHz, the transmitter frequency in a repeater is either 600 kHz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available.Answer:1. Based on Frequency attenuation expression and calculation with MATLAB, it figuredout the eligible coverage radiuses, which are 30km for BS (base station), and 14.9km for repeater.2. Assuming the users in a given area under uniform distribution, using advancedcellular coverage solution, we can calculate that minimum number of required repeater is 36 under cellular features.3. Based on the US VHF spectrum allocation standard, the minimum spacing for adjacentchannels is 30kHz. And with up to 54 different PL tones, maximum 4320 channels can be allocated to provide 1000 simultaneous users to use at the same time. Conclusion:The minimum number of repeaters necessary to accommodate 1,000 simultaneous users is 36.Question Two : How does your solution change if there are 10,000 users?Answer:1. Since the given spectrum is in a fixed range, even if 54 different PL tones can not be allocated enough channels for 10,000 simultaneous users. So the number of repeaters will be increased, meanwhile, the given area will be divided into different parts.2. On the assumption that uniform distribution of the population in the given area, it will be divided into 3 sub-regions equally by analyzing the binding domain, frequency spectrum and PL tones three independent factors. And then the number of repeaters within each sub-region will be classified discussion.3. The FDM (Frequency Division Multiplexing) model is established here to improve channel efficiency to accommodate up to 10,000 simultaneous users Conclusion:The minimum number of repeaters necessary to accommodate 10,000 simultaneous users is 126.Question Three : Discuss the case where there might be defects in line-of-sight propagation caused by mountainous areas. Answer:Basically, under the same condition for question 1&2, the mountainous area will be analyzed as following:1. The function for relationship between radio attenuation x caused by obstacles and the eligible coverage radius d for repeater is 2249.354371.4110x d -=, which is to analyze the impact on the number of repeaters under full signal coverage. 2. For the mountain barrier, based on the different situation of mountains, the addition of repeaters on the suitable location will be discussed to achieve full coverage. This paper describes model established by using of cellular coverage technology and frequency attenuation expression, to achieve simple, fast, accurate algorithm. And also illustrated the effect takes the entire article. In the end, the sensitivity analysis and error calculation are applied for modeling, making the model practically.Key words: Cellular Coverage technology, frequency attenuation expression, channel allocation, MatlabRepeaters coordination and distributionContent1 Restatement of the Problem (1)1.1 Introduction (1)1.2 The Problem (1)2 Simplifying Assumption (1)3 Phrase explain (1)4 Model (2)4.1 Model I (2)4.1.1 Analysis of the Problem (2)4.1.2 Model Design (2)5 Sensitivity analysis (2)6 Model extension (2)7 Evaluating our model (2)7.1 The strengths of model (2)7.2 The weaknesses of model (2)References (3)1 Restatement of the Problem1.1 IntroductionThe VHF radio spectrum involves line-of-sight transmission and reception. This limitation can be overcome by “repeaters,” which pick up weak signals, amplify them, and retransmit them on a different frequency. Thus, using a repeater, low-power users (such as mobile stations) can communicate with one another in situations where direct user-to-user contact would not be possible. However, repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.1.2 The ProblemYour job is to:◆Design a scheme that determines the minimum number of repeaters necessaryto accommodate 1,000 simultaneous users in a circular flat area of radius40 miles radius.And assume that the spectrum available is 145 to 148 MHz,the transmitter frequency in a repeater is either 600 kHz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available.◆Change your scheme to accommodate 1,0000 simultaneous users base on yourmodel.◆Discuss the case where there might be defects in line-of-sight propagationcaused by mountainous areas.2 Simplifying Assumption3 Phrase explain4 Model4.1 Model I4.1.1 Analysis of the Problem4.1.2 Model Design5 Sensitivity analysisSymbol◆N: the number of total repeaters in the circle area ◆Q: the number of the users in the circle area◆k: the number of the red circle in figure 2最前面最好有一个Symbol List6 Model extension7 Evaluating our model7.1 The strengths of model7.2 The weaknesses of modelReferences参考文献不要引用非常差的期刊的论文，要引用比较厉害的英文期刊，证明你有足够的阅读文献量。

美国大学生数学建模大赛模拟1论文

又实际中影响生猪存栏量和猪肉价格的因素复杂多变因此我们的预测模型只能进行短期预成艳聂锐灰色预测模型gm11在江苏省能源消费量预测中的应用中国矿业大学管理学院江苏徐州221116我国生猪生产的波动性研究辽宁农业职业技术学院辽宁营口1150093盛建华程功鹏金喜新任广智高腾云李晓峰河南省猪肉价格变化规律及其预测研究河南农业大学学报1315生猪年末存栏量及猪肉价格周期性波动研究附录
生猪年末存栏量及猪肉价格周期性波动研究
摘要
本文是关于时间序列数据的预测探究的，我们利用灰色系统理论和自己建立的模拟函数对生猪的年末存栏量和 36 个大中城市的猪肉价格进行了模拟和预测。在对第一问生猪年末存栏量的预测中，由于 1997 年以前的统计数据不真实，从而导致可用信息贫乏，据此特点我们采用了灰色系统理论中的 GM(1,1)模型。通过对已知数据和生猪养殖业的分析我们发现生猪养殖随供求关系的变化呈周期性的波动，波动周期为 3 至 4 年，又鉴于我国经济社会和人民生活水平的不断发展，生猪养殖业总体上呈现上升趋势。GM(1,1)并无对周期性变化数据的预测能力，但对单调趋势的小信息量数据有较好的预测能力，据此我们剔除数据中的波动成分，即选取原始序列时每隔 3 年取一点，得出 2010 的生猪存栏量为 44770.6 （万头）。最后对预测结果的检验发现其与原数据的偏差不会超过 3.03%，从而保证了预测结果的可靠性。利用生猪年末存栏量随时间增长但呈现周期性波动的特性，我们还建立了与之适应的模拟函数模型，模拟函数包括由一次函数表征的增长部分、由正弦函数表征的波动部分和由正态函数表征的冲击部分（如 03 年和 06 年生猪养殖业遭到了重大变故，反应在存栏量上有很大波动）。以最小二乘法确定此模拟函数的各项参数，进而预测出 2010 年末的生猪存栏量为 46813 万头。由于模拟函数是在机理分析的基础上建立的，所以很好的吻合了已知数据，从而对未来数据的预测也就有了保证。两种预测模型预测结果还是有较大差异的，但我们认为机理分析在预测中是更有效的方式，所以更倾向于把 46813 万头作为最终预测结果。第二问要求对 36 个大中城市的猪肉价格做出预测。通过对猪肉市场和已有数据的分析研究，我们发现猪肉价格在总体上仍然呈现 3 到 4 年为一个周期的波动，而在每一年中，猪肉价格受节假日、经济波动和病疫等诸多因素的影响，呈现小幅度短周期的波动情况。鉴于此变化规律，我们构造的模拟函数包括表示初始价格的常数、表示总体波动的周期为 4 年的正弦函数和表示众多繁杂因素影响的傅里叶级数（出于计算考虑取级数前 50 项）。通过最小二乘法确定各参数，得出的模拟函数对已知数据完全贴合。在模型检验中，我们只用部分已知数据来确定模拟函数，发现模拟结果与未使用的数据也有很好的吻合程度。为了进一步检验模型我们查阅了新的猪肉价格信息发现其与预测价格有相近的走势。这些都验证了模拟方法的正确性和模拟函数的有效性。

美赛论文模板(中文版)

For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number 26282Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________2014 Mathematical Contest in Modeling (MCM) Summary Sheet (Attach a copy of this page to your solution paper.)1.Introduction近年来，世界上的交通拥堵问题越来越严重，严重的交通拥堵问题引发了人们的对现行交通规则的思考。

在汽车驾驶规则是右侧的国家多车道高速公路经常遵循除非超车否则靠右行驶的交通规则，那么这个交通规则是否能够对交通拥堵起着什么作用呢？在汽车驾驶规则是右侧的国家多车道高速公路经常遵循以下原则：司机必须在最右侧驾驶，除非他们正在超车，超车时必须先移到左侧车道在超车后再返回。

根据这个规则，在美国单向的3车道高速公路上，最左侧的车道是超车道，这条车道的目的就是超车。

现在我们提出了4个问题：1、什么是低负荷和高负荷，如何界定他们？2、这条规则在提升车流量的方面是否有效？3、这条规则在安全问题上所起的作用？4、这条规则对速度的限制？1.1 Survey of Previous Research1.2 Restatement of the problem本题需要我们建立一个数学模型对这个规则进行评价。

我们需要解决的问题如下：●什么是低负荷和高负荷，如何界定他们？●这条规则在提升车流量的方面是否有效？●这条规则在安全问题上所起的作用？●这条规则对速度的限制？●对于靠左行的规则，该模型能否可以使用？？（待定）●如果交通运输完全在智能系统的控制下，会怎样影响建立的模型？针对以上问题，我们的解题思路和方法如下所示：◆我们根据交通密度对低负荷和高负荷进行界定，交通密度是指：在某时刻，每单位道路长度内一条道路的车辆数。

2018美赛A题论文1

Team #88255Page 1of 20Team Control Number 88255Problem Chosen A2018MCM/ICM Summary SheetThe traditional communication methods of high frequency (HF)radio are still widely used worldwide,which has longer propagation distance than other kinds of radio communication systems,and can cover mountainous,deserts and oceans.At present,the studyof high frequency radio multi-hop propagation mode has been relatively systematic,which provide various methods to simulate its trajectory and attenuation in the air,ionosphere and earth surface.In this paper,we mainly focus on the reflection properties on the ocean surface,including calm and turbulent conditions.What’s more,the overall propagation process and practical application of HF radio are also involved.For part one,on the basis of calm surface reflection model proved in the relative literature,we can calculate the reflection coefficient and the attenuation L cal (dB )by using relative permittivity εr and electrical conductivity σ,then we estimate the first reflection strength off the calm ocean is from 0.5-5.5mW .Next,a P-M spectrum was used to simulate the distribution of sea wave,which is related to wind speed.Based on it,we use finite element analysis to calculate the correction factor ρof tough surface.An alternative way is using the international general standard published by CCIR to calculate ρin the condition of different wind speed,and the attenuation L tur (dB ).The estimation of the first reflection strength off the turbulent ocean is 0.2-2.5mW .Afterwards,according to the analysis of the air and ionosphere,we can get the Path Transmission Loss L b (dB )and the attenuation inthe ionosphere L ion (dB ).The total loss L t (dB )of the overall propagation process is the sum of L b ,L ion ,L cal or L tur .Given the source power is 100-watt and the SNR threshold is 10dB ,by estimating the external noises of receiver,we calculated that the maximal attenuation in the propagation path have to be lower than 151dB .For f =3MHz ,our model shows that the signal will take 6to 7hops before the total loss L t reach 151dB .With the increase of frequency,the number of hops will decrease gradually.For part two,the difference between land and oceans including different εr and σ.In addition,the amplitude of fluctuation of land is much larger than the sea surface,which is relatively stable and not related to wind speed.By putting the different parameters to the turbulent surface reflection model,we can calculate the reflection attenuation and estimate the first reflected power is respectively 0.4-4.8mW and 0.2-2.4mW for rugged terrain and smooth terrain.The estimation of number of hops is 3reflected by the rugged terrain,and 5to 6for smooth terrain.Generally,the attenuation of ground is much larger than ocean.For part three,for a moving or shaky object,generally we take the circular polarization ways to maintain the stability of signal,which means that there must be a Mismatch Factor v as a part of loss.For a sailing ship,we calculated the coverage area of each hop of signal when the frequency is 24MHz ,between 1900-2676km (1st hop)、3894-5535km (2nd hop)、5768-8302km (3rd hop )、8682-11070km (4th hop),theship could receive effective signal.The time that can receive the signal could be calculated if the direction and speed of the ship is known.For office use onlyT1T2T3T4For office use only F1F2F3F4Team#88255Page2of20 Key words:high frequency radio,ocean surface reflection,multi-hop propagationContents1Introductions (3)2General Assumptions and Variable Description (3)2.1General Assumptions (3)2.2Variable Description (4)3Model Establishment and Analysis (4)3.1Ocean Surface Reflection Attenuation (4)3.1.1Reflection Properties of Seawater (4)3.1.2Calm Oceans Reflection (5)3.1.3Turbulent Oceans Reflection (6)3.1.3.1The Sea Surface Simulation By P-M Spectrum (6)3.1.3.2Finite Element Analysis on Rough Surface (6)3.1.3.3International Standard of Sea Surface Roughness (8)3.1.4The Reflection Attenuation Difference Between Calm andTurbulent Sea Surface (8)3.2The Maximum Number of Hops (10)3.2.1Free Space Transmission Loss and Path Transmission Loss (10)3.2.2The Properties of Ionosphere Reflection (11)3.2.2.1Physical Properties of Ionosphere (11)3.2.2.2The Absorption of Ionosphere (12)3.2.3The Calculation of Maximum Hops (13)3.3The Reflection Loss of rugged and smooth terrain (14)3.4Model Improvement About Shipboard Receiver (15)3.4.1The Properties of Moving Receiver (15)3.4.2The range of the communication area of the same multi-hop path (15)4Stability and Sensitivity Analysis (16)5Strengths and Weakness (18)5.1Strengths (18)5.2Weakness (18)6Short note in IEEE Communication magazine (19)7Reference (20)1IntroductionsTo high frequency radio range from3MHz~30MHz,the ionosphere can reflect the transmission electromagnetic back to the earth or oceans,and reflect back to the sky again and again,thus the remote distance communication could be achieved,figure1displays the progress. The hop distance could reach several thousand kilometers,so that it could be widely used in international short wave communication business[1].The propagation mode is also called sky wave.Figure1:Propagation path of sky waveThe refraction of ionosphere is depend on its electron density,which was generated by solar radiation.Generally,when the frequencies exceed the maximum usable frequencies(MUF),the radio will penetrate the ionosphere and won’t be back to the earth.Similarly,if the radio frequencies are lower than lowest usable frequency(LUF),the signal-to-noise ratio(SNR)would be lower than a usable threshold.The reflection point on the ground could be possible for both ground or oceans,which have the different reflection properties.They have something in common that the reflection ismainly depends on the electromagnetic properties of medium and the surface shape.There have been a lot of previous researches and international general standard about the reflection of tough sea surface.After multi-hops,the signal will be received by a receiver,to acquire the highest SNR signal, what we have to do is to reduce the attenuation and lost in the process of transmission. Generally,the most frequently used frequency nearest0.85MUF[1]2General Assumptions and Variable Description2.1General Assumptions•The effects of fading,multipath delay,dispersion in the propagation of radio in the atmosphere are not considered.That is to say,even though the signal is not below SNR threshold when received,it's still possible to be distorted.•The electrical parameter of seawater in a partial area is constant and doesn’t change by the influence of ocean wave or other reasons.•Suppose that the background noises are the same in each receiving point on the earth surface, and the internal noises of the signal receiver is not considered.2.2Variable Description R h Reflection Coefficient Of Horizontal Polarized Wave~S(ω)Pierson-Moscowtc SpectrumρCorrection FactorσElectrical ConductivityL cal The Attenuation of Clam OceanL tur The Attenuation Factor of Turbulent Surfacef FrequencyL bf Free Space Transmission LossL b Path Transmission Lossεr Relative PermittivityA ion The Attenuation Factor of IonosphereL tTotal LossL shipConstant Attenuation 3Model Establishment and Analysis 3.1Ocean Surface Reflection AttenuationTo estimate the reflection loss off calm oceans and turbulent oceans,we have to figure out the reflection properties of seawater as well as the difference between calm surface and tough surface.3.1.1Reflection Properties of SeawaterThe electromagnetic properties of ocean can affect the refection of radio,it mainly depends on the temperature and salinity of seawater,as well as the radio frequency.The parameter to define the electromagnetic properties of ocean surface is complex permittivity,which consists ofrelative permittivity εr ，electrical conductivity σ,and wavelength λ,the expression is:ε=εr +i 60λσ(1)To εr and σ,there are already international standard published by International Radio Consultative Committee in the real applications.On the frequency of 3~30MHz ,the approximate estimation is:εr =7,σ=5.[2]According to Snell's law,we can get the Fresnel reflection coefficient of horizontal polarized wave and vertically polarized wave [3]:R h =R v =(2)According to the sea surface reflection,the loss of horizontal polarized wave is smaller than vertically polarized wave,so we use horizontal polarization wave as transmitting signal,the horizontal reflection coefficient R h will be used in calculation.θε-cos 2θsin θ+ε-cos 2θεsin θ-ε-cos 2θεsin θ+ε-cos 2θεn =-⎰θ3.1.2Calm Oceans ReflectionIn the ideal condition,we regard the surface of calm oceans as a totally flat plane,on which the radio has the reflection properties in accordance with Snell's law [3].Meanwhile,we adopt the horizontal polarization to decrease the reflection loss as far as possible,the attenuation factor can be calculated by:θmaxL cal (dB )10lg min R h d θ(3)To define θmax and θmin ,due to the radio was reflected by the ionosphere,we have to analyze from the refraction properties of ionosphere.R is the radius of the earth,Z is the height of ionosphere,N n is the electron density of the n threflection point.According to the law of refraction:n 0sin θ0=n 1sin θ1=n 2sin θ2==n n sin θn (4)θ1θ2θn Z θ0θmaxRFigure 2:The reflection process in the ionosphereThe boundary conditions that radio can reflect out from the ionosphere is:n 0=1,θn =90(5)Putting the boundary conditions into formula (4):sin θ0==(6)θmax =sin -1Due to the curvature of the earth,θ0can’t reach 90°.The radio reach the maximum incident angle θmin when the projection of it is horizontal.θmin =sin -1R (7)1-80.8N n f 21-80.8N n f 2g or ocean current,the surface of ocean must be rippled.We can regard the rippled surface as a slightly turbulent surface of ocean which will be considered in the next paragraph.3.1.3Turbulent Oceans Reflection3.1.3.1The Sea Surface Simulation By P-M SpectrumThe turbulent ocean can be regarded as the combination of infinite harmonic waves that have different amplitudes,frequencies,directions and phases,the contribution of the harmonic waves consist of the sea spectrum.The sea spectrum is the statistical property of a random process,contains each harmonic component about the distributions of frequency and directions.There are a lot of previous researches about sea spectrum,among which Pierson-Moscowtc spectrum ,JONSWAP spectrum and Elfouhaily spectrum were widely used.Now we are using the first one as our ocean reflection surface.Moscowitz [4]evaluated the spectrum of wind waves in the North Atlantic Ocean by averaging the observed 54spectrums then got the Pierson-Moscowtc spectrum (P-Mspectrum):2S (ω)=αexp[-β(ω2U g )-4]ω(8)19.5α=8.1×10-3,β=0.74,U 19.5is the wind speed of 19.5m above the sea.Figure 3display the wave simulation distributed by P-M spectrum ,it's easy to find that the peak and roughness of waves increased following the increase of wind speed.Figure 3:the ocean wave simulated by P-M spectrum (wind speed=5,10,15and 20m/s )3.1.3.2Finite Element Analysis on Rough SurfaceThe differences between turbulent oceans and calm oceans include wave heights,shapes and frequencies,on which the electromagnetic wave can be reflected to all directions.Figure 4shows the reflectionproperties:h h h hh Figure 4:the reflection forms on the turbulent surfaceWe can simulate the reflection model by Finite Element Analysis ,then calculate the reflection coefficient of a relatively large ocean surface.By dividing the ocean surface into finite microplane,the combination of each reflections about their own normal vectors determine the properties of diffuse reflection.The model consist of two parts:one is the distribution expression of the surface,the other is the reflection model to describe the reflection properties of each microplane,which has already discussed in 3.1.1.Blinn (1977)gave an exponential decay model of normal vectors distributions of microplane [5].In this model,the microplane which have the highest probability density is horizontal,so the criterion normal vector is vertical.The reflection wave shall decay along with the decrease of the angle of normal vector until to be horizontal.To a smooth surface,the decay is fast,while to rough surface,the decay is progressive.The distribution function of Blinn’s model is proportional to the dot product of halfway vector ωh and criterion normal factor n .D (ω)∝(ω⋅n )e (9)To guarantee the physical effects,the distribution of microplane shall be standardized.That is to say,there must be a height field that has the distributions of D(ωn ).the total projected area of every microplanes in the height field equals to 1.⎰H 2D (ω)cos θd ω=1(n )(10)So,the normalized Blinn’s microplane distribution is:3.1.3.3International Standard of Sea Surface RoughnessTo the rough sea surface reflection,the International Radio Consultative Committee(CCIR) gave the expression of correction factorρ[6]:3.1.4The Reflection Attenuation Difference Between Calm andTurbulent Sea SurfaceBy importing the data to formulas(14)and(15),take the wind speed U as20m/s,figure5 shows the relationship between the angle if incidenceθand the attenuation of different sea surface L cal and L tur when the frequency f=3MHz.figure5shows the relationship between frequencies and attenuation.Figure5:the relationship between L cal or L tur andθon calm surfaceversus tough surface,f=3MHzFrom figure5we can find that on the calm surface,no matter what angle of incidence,the attenuation is very small that can be ignored,while on a turbulent surface,the attenuation increase along with the increase ofθ,which can reach10.578dB whenθapproach90°.Figure6：The relationship between L cal or L tur and fon calm surface versus tough surfaceFrom Figure6we can find that no matter what frequency it is,the attenuation on calm surface are almost invariable and very small,which below1dB.While the attenuation on the turbulent surface increased along with the increase of frequency,which can reach4dB when the frequency approach30MHz.Figure7:the comparison of reflected power between calm and turbulent ocean The signal is reflected by the calm and turbulent sea surface after the first reflection of ionosphere.The loss in the air and ionosphere will be mentioned in the next passage3.2.As shown in the figure7,the signal strength decreases as the frequency increases.To calm ocean, the range of the reflected signal strength is0.5to5.5mW.To turbulent ocean,the range is0.2to 2.5mW.At the most commonly used frequency of0.85MUF,the signal power is about0.5062mW.3.2The Maximum Number of HopsTo calculate the maximum number of hops,it’s necessary to estimate the signal attenuation in different processes,including path transmission loss,ionosphere reflection attenuation and sea surface reflection attenuation.3.2.1Free Space Transmission Loss and Path Transmission LossFree space is the indefinite space that is filled with homogeneous and dissipationless medium,which has the character of isotropy,electrical conductivityσ=0,relative permittivity εr=1,and relative permeabilityμr=1.Suppose that two ideal point signal source antenna(Gain:G t=G r=1)are respectively transmitting and receiving antenna,P t is the input(transmitting)power,P r is the output (receiving)power,th e free space transmission loss L bf can be defined as:So,when double the radio frequency or propagation distance,the free space transmission loss will be increased by6dB.Figure8:The relationship between L bf(dB)and distance(km),f=3,12,21,30MHzbf different frequencies.The higher frequency,the higher free space transmission loss there will be.In the atmosphere environment,except for free space transmission loss,electromagnetic wave may also suffer such effects like attenuation,fading,depolarization,time and frequency domain distortion,which can cause a complex change.Suppose that an antenna is set in a free space,the field strength in maximal direction E 0can be defined as:|E 0|=(V /m )(20)The field strength of receiving point is:|E |=|E 0|A air =60P t (W )G t A air (V /m )(21)A air is the attenuation factor in the atmosphere,which is related to frequency,distance,electrical parameter of medium,propagation mode and so on.We can also convert A air to dB form:Apparently,L b described the power propagation situation in the medium,so L b is called path transmission loss or basic transmission loss.So,we can use L b to define the transmission loss in the atmosphere (not include the ionosphere),which is related with propagation distance d ,wavelength λand attenuation factor A air ,L b .3.2.2The Properties of Ionosphere Reflection3.2.2.1Physical Properties of IonosphereThe ionosphere is the ionized part of Earth's upper atmosphere,from about 60km to 1,000km altitude [7],which contains electrons and electrically charged atoms and molecules that surrounds the monly,we use electron density (electron amounts/m 3)to describe its ionization degree.The ionosphere is a kind of random,dispersive and anisotropic semiconductor medium,the parameters (thickness,electron density and distribution)change at random.60P t G tFigure 9:The layers and electron density of ionosphere [8]Within ionosphere,there are also four parts [8]:D layer :60～90km above the surface of the Earth.The thickness of D layer decrease gradually as night comes.In the dark night,D layer almost disappear.E layer :90～150km,relatively stable at an altitude of 110km.Electron density decrease at night.F1layer :170～220km,always appear in the summer daytime and disappear at night and winter.F2layer :225～450km,Electron density in the daytime is higher than night,which is also higher in winter than summer.3.2.2.2The Absorption of IonosphereThe absorption of ionosphere includes deviative absorption and non-deviative absorption.The non-deviative region is the region whose refractive index n approach 1[9],the radio almost travels in straight lines.The non-deviative absorption is mainly in D layer,which contains large amounts of neutral molecule and ion so that the collision frequency v is very high.E and the inferior region of F layer are also included but can be ignored compared with D layer.To calculate the non-deviative absorption of D layer,according to electromagnetic theory,known εr and σof the dissipationless medium,the attenuation factor A ion is:A ion =ω(24)ε0is vacuum permittivity,the approximate value is 8.854187817×10-12F/m.the vacuum permeability μ0is 4π×10-7H/m.To high frequency wave propagation,the condition σ/(ωσ)is met,then:The deviative region is the region whose refractive index n is very small,the radio will change the propagation direction with a curve and back to the earth.Generally there is εr ≈1[10],the total attenuation in the ionosphere L ion can be calculated by e -⎰A 2dl ,l is the route of radio inthe ionosphere.Generally the absorption is quite small and usually under 10dB [11].μ0ε0[2ε+2r (60λσ)-ε]2r 13.2.3The Calculation of Maximum HopsThe main reasons to affect SNR include internal noises and external noises,the internal noises come from receiving system itself,which can be ignored in this situation.The external noises origin from the universe,the atmosphere and the earth surface,among which the cosmic background radiation[12]is the most important part,which can be representedby:Pnoise=kTB≈8.0⨯10-15W(26) k is Boltzmann constant,the value is1.3806505×10-23J/K,T is the background temperature, the general value is290K,B is the bandwidth of the signal which is depends on the quality of the signal,we take2MHz to calculate.The SNR threshold is10dB,which meansSNR=10lg PsignalPnoise=10dB(27)Psignal_min=8⨯10-14WThe given power of signal source is100W,so the maximal attenuation that allowed in the transmission process is:Lmax=-10lg P100=150.969dB(28)signal_minBy calculating the total loss L t,compared with Lmax,we can estimate the number of hops that the signal can take before its strength falls below the SNR threshold.According to the conclusions in3.1,3.2.1and3.2.2,the radio suffered several kinds of loss and attenuation in the process of transmission,including the path transmission loss L b,the ionosphere attenuation L ion,the sea surface attenuation L cal or L tur.The total loss can be expressed by:L t =Lb+Lion+Lcal/tur(29)Figure10:The relationship between distance(km)and the total loss L t(dB) n:number of hops,f=3MHz,reflected by calm oceansth th hops,the attenuation of a part of signal has already exceed151dB,and almost all of the8th hophave been attenuated to the threshold.So in the frequency of3MHz,the hops of signal could reach6to7times,along with the transmission distance reach15,000km.What’s more,with the increase of frequency,the total number of hops will decrease gradually and finally drop to only one hop.3.3The Reflection Loss of rugged and smooth terrainCompared with sea surface,the surface of the ground have totally different relative permittivityεr and electrical conductivityσ,the table below shows the differentεr andσin different geologies.Table1:Differentεr andσin different geologies.[13]GeologyεrσRange Average Range AverageSeawater70700.66~6.65Fresh Water808010-3~2.4×10-210-3Wet Soil10~30203×10-3~3×10-210-2Dry Soil2~64 1.1×10-3~2×10-310-3In addition,the amplitude of fluctuation is much larger than the sea surface.The radio would also scatter to all directions.Take hilly topography(amplitude under500m,sparse hills)as an example to correct the model in3.1.3.2.We can use the reflection coefficient formula(1)(2)in 3.1.1,replaceεr andσto calculate the reflection coefficient of the ground R g.It is notable that the reflection off mountainous or rugged terrain is not related with wind speed,which is different from sea surface.If we have known the terrain distribution of the ground, it’s not hard to calculate R g,so that the attenuation of ground can also be calculated by formula (3)given in3.1.2.Figure11:The comparison of reflected power between smooth and rugged terrainsmooth terrain,the range of the reflected signal strength is0.4to4.8mW.To rugged terrain,the range is0.2to2.4mW.Figure12:The relationship between distance(km)and the total loss L t(dB) n:number of hops,f=3MHz,reflected by smooth(left)and rugged(right)terrainFigure12shows that when the signal was reflected by smooth terrain,f=3MHz,there will be5to6hops of signal could be received,while reflected by rugged terrain,there are only3 hops of signal.The difference reveals that when the fluctuation of terrain could affect the transmission distance and hops by change the direction of reflected radio.Compared with the reflection off calm oceans,the land reflection has more attenuation and less reflecting times,it’s also correspond to the smallerεr andσof soil than seawater.3.4Model Improvement About Shipboard Receiver3.4.1The Properties of Moving ReceiverIn the process of sailing,to the receiver on the ship,we should not only consider SNR of receiving signal,but also the stability of receiving ually,in a shaky or moving subject, using the circular polarization antenna can keep the signal as stable as possible.There are two ways of circular polarization,one is linear polarization transmitting and circular polarization receiver,another is on the contrary.No matter what kind of polarization way we take,the Mismatch Factor v is always1/2.So our model need to add a constant attenuation:L ship=-10lg0.5=3.01dBL=L+L+L+L(30)t b ion cal/tur ship3.4.2The Range of The Communication Area of The Same Multi-HopPathFigure13displays the attenuation degree of the signal reflected by the turbulent ocean and received by a shipboard receiver,it’s not hard to find the signal will take3to4hops before attenuated to151dB.On this basis,we can also calculate the distance range among which the ship can receive the effective signal.The result was displayed in figure14.ΔdFigure13:The multi-hop signal received by the shipboard receivern:number of hops,f=3MHz,reflected by turbulent oceand ship-signal019002767389455355786The ship8302868211070d r/kmFigure14:the range of effective signald r is the relative distance,suppose that the earth is a regular sphere,every reflected pointon the earth must not in the same plane,so we use d r to describe the curve distance that the ship move on the earth.In figure14,the red lines are the ranges that could receive the signal,that is to say within the red ranges,the SNR of signals>10dB.The specific ranges are between1900-2676km(1st hop)、3894-5535km(2nd hop)、5768-8302km(3rd hop)、8682-11070km(4th hop).Between each range,there are areas like“exclusion zone”where cannot receive the effective signal.Known the ranges,as long as we know the direction and speed of the ship,it’s quite easy to calculate the time that could receive the signal.4Stability and Sensitivity AnalysisIn the reflection model we established in3.1,the changes of height,shape of sea surface are mainly caused by wind speed.What’s more,in different sea areas,the temperature and salinity of seawater couldn't be the same,which could changeεr andσof seawater.So,we mainly analyze the influence of wind speed,εr andσto the stability and sensitivity of our model.We considered the wind speed near the sea surface from5m/s to20m/s,and compare them:Figure15:The change of L tur when wind speed=5,10,15and20m/s From figure15we can find that wind speed has a relatively larger influence on the reflection attenuation of sea surface,the attenuation increased following the increase of wind speed,as well as the worse communication effect.Consider the20%rise and fall ofεr andσ,the results are:Figure16:the change of the change of L tur when there are small changes ofεr andσThe four curves of figure16are almost overlapped.Therefore,the small change within20 percent ofεr andσhas very little influence on the attenuation on sea surface.We can almost ignore the differences.Figure17:the change of L tur when there are large changes ofεr andσHowever,when there are large changes of reflecting medium,such as the seawater versus the ground.The attenuation of ground is obviously much more than seawater.Figure17shows the difference.5Strengths and weakness5.1Strengths•There have been systematic previous researches about sky wave.About the sea surface reflection model,most of them were calculated by the empirical model about frequency and incident angle.Our model used P-M model to simulate the wave of sea surface,Finite Element Analysis is also used to calculate the reflection coefficient,which is more accurate and universal.•To the receiving problem of shipboard receiver,we know the particularity of the moving and shaky objects,in the condition of circular polarization only half of the signal could be received.To calculate the effective range of reflection,we can display the useable receiving area and attenuation.5.2Weakness•In the model of ground reflection,we only change the relative parameters of the sea surface reflection model,there may be some inapplicable places and some errors.•To partⅢ,the weaknesses are due to the complexity of ionosphere,the calculation is very difficult,it’s not very accurate to represent the transmission path in the ionosphere by the function about incident angle,which may affect the result.。

2020年美赛C题论文

2020年美赛C题论文引言在2020年的美赛C题中，我们将研究某城市的停车问题。

停车问题在现代城市中非常普遍，而且经常引起交通拥堵和资源浪费。

因此，寻找一种合理的停车方案对于城市的可持续发展至关重要。

本文将介绍我们对该停车问题的建模过程、假设和模型结果。

问题描述该城市位于一个山区，拥有许多旅游景点，吸引了大量游客。

然而，停车场的数量有限，传统的交通管理方式导致了拥堵和停车困难。

因此，我们需要提出一种新的停车方案，以改善交通状况和资源利用。

我们需要回答以下问题：1.如何确定合理的停车位价格以确保公平性和减少拥堵？2.如何确定合理的停车位数量以满足游客的需求？3.如何指导游客选择合适的停车场？数据处理和建模为了解决上述问题，我们从该城市收集了大量的交通数据和停车场信息。

首先，我们对数据进行处理，包括数据清洗、整理和校验。

然后，我们使用Python编程语言对数据进行分析和建模。

下面是我们的建模过程：1.确定停车需求模型：我们将游客的停车需求建模为一个随机变量，可以以概率密度函数的形式表示。

为了准确地估计需求模型，我们使用了大量的历史停车数据和游客统计数据。

2.确定停车位定价模型：我们考虑了停车位价格对停车需求的影响，并建立了一个停车位定价模型。

该模型将考虑停车位的成本、游客的支付意愿和其他相关因素。

3.确定停车场选择模型：我们使用了多属性决策分析方法来确定游客选择停车场的因素和权重。

通过评估每个停车场的特点和游客的偏好，我们可以为游客提供选择停车场的指导。

假设为了简化问题和建立数学模型，我们做出了以下假设：1.停车需求是服从某种概率分布的随机变量。

2.停车位定价的主要目标是确保公平性和减少拥堵。

3.游客的停车选择主要受停车位价格和距离的影响。

4.停车场之间没有容量限制。

这些假设可以帮助我们建立合理的模型和解决方案，但也需要在实际应用中考虑其他可能的因素。

模型结果基于我们的建模过程和假设，我们得到了以下模型结果：1.停车需求模型：通过对历史停车数据和游客统计数据的分析，我们得到了停车需求的概率密度函数模型。