美國數(shù)學(xué)建模輔導(dǎo)課

1、 一、美國數(shù)學(xué)建模競賽介紹一、美國數(shù)學(xué)建模競賽介紹 MCM: Mathematical Contest in Modeling MCM: Mathematical Contest in Modeling ICM: Interdisciplinary Contest in ICM: Interdisciplinary Contest in ModelingModeling 主辦：美國運(yùn)籌學(xué)和管理科學(xué)協(xié)會、工業(yè)及應(yīng)用主辦：美國運(yùn)籌學(xué)和管理科學(xué)協(xié)會、工業(yè)及應(yīng)用數(shù)學(xué)協(xié)會、美國數(shù)學(xué)協(xié)會以及國家安全局贊助，數(shù)學(xué)協(xié)會、美國數(shù)學(xué)協(xié)會以及國家安全局贊助，美國數(shù)學(xué)及應(yīng)用協(xié)會承辦美國數(shù)學(xué)及應(yīng)用協(xié)會承辦 時間（美國東

2、部）：每年二月份的第二個周末，時間（美國東部）：每年二月份的第二個周末，中國時間：周四晚上中國時間：周四晚上8 8：0000開始，周一晚上開始，周一晚上8 8：0000結(jié)束結(jié)束 題目：題目： MCM競賽一般有兩個問題，都是根據(jù)各競賽一般有兩個問題，都是根據(jù)各 行各業(yè)的專家建議精選出來的實際問題。行各業(yè)的專家建議精選出來的實際問題。 問題的答案不是唯一的，完全是開放的，問題的答案不是唯一的，完全是開放的， 主要考察參賽者處理問題的巧妙性，靈主要考察參賽者處理問題的巧妙性，靈 活性，新穎性?；钚?，新穎性。 ICM只有一道題，涵蓋數(shù)學(xué)、環(huán)境科學(xué)、只有一道題，涵蓋數(shù)學(xué)、環(huán)境科學(xué)、 環(huán)境工程及資源管理等

3、概念的建模問題。環(huán)境工程及資源管理等概念的建模問題。評審標(biāo)準(zhǔn)及評審程序評審標(biāo)準(zhǔn)及評審程序 四個等級：四個等級： Outstanding (特等獎），特等獎）， Meritorious(一等獎一等獎)， Honorable Mention(二等獎二等獎)， Successful Participation(成功參賽獎成功參賽獎). 三輪審稿三輪審稿Initial round：至少兩人審稿：至少兩人審稿 重點看重點看summary：文字通順，語言精練、準(zhǔn)：文字通順，語言精練、準(zhǔn)確；包括解題的方法和結(jié)論；摘要和全文結(jié)確；包括解題的方法和結(jié)論；摘要和全文結(jié)論要一致。不合格者被淘汰。論要一致。不合格者被

4、淘汰。Screening round： 由第二組人審稿由第二組人審稿 重點檢查是否滿足進(jìn)入下一輪的最低要求：重點檢查是否滿足進(jìn)入下一輪的最低要求：文字通順，語法正確仍然是必須的；文章前文字通順，語法正確仍然是必須的；文章前后的一致性更加重要；清楚地說明哪些是自己做后的一致性更加重要；清楚地說明哪些是自己做的，哪些是引用的，引用資源的出處要明確。的，哪些是引用的，引用資源的出處要明確。Final rounds：由第三組人審稿，每篇文章多：由第三組人審稿，每篇文章多人閱讀，每次審閱時間延長；按照加分元素人閱讀，每次審閱時間延長；按照加分元素篩選進(jìn)入下一輪的參賽隊，被留下時間越長，篩選進(jìn)入下一輪的參

5、賽隊，被留下時間越長，得最高獎的可能性越高。得最高獎的可能性越高。 重點除了重點除了summary以外，全篇的一致性以外，全篇的一致性更加重要，每一解題部分都要經(jīng)過審查。更加重要，每一解題部分都要經(jīng)過審查?？梢约臃值脑乜梢约臃值脑?、文字優(yōu)美，流暢，語言豐富，表達(dá)清晰，、文字優(yōu)美，流暢，語言豐富，表達(dá)清晰，方程、標(biāo)點引用格式正確。方程、標(biāo)點引用格式正確。2、文摘的要求：包括問題的簡單概括，建模、文摘的要求：包括問題的簡單概括，建模使用的模型和方法，所得的結(jié)論。使用的模型和方法，所得的結(jié)論。3、合理的假設(shè)以及必要的解釋。、合理的假設(shè)以及必要的解釋。4、多個模型由淺入深地對問題求解、多個模型由

6、淺入深地對問題求解5、對模型使用的方法條件及結(jié)論進(jìn)行分析、對模型使用的方法條件及結(jié)論進(jìn)行分析6、模型的敏感度和穩(wěn)定度分析、模型的敏感度和穩(wěn)定度分析7、當(dāng)可引用的資源很多時，要對你引用的資、當(dāng)可引用的資源很多時，要對你引用的資源進(jìn)行分析源進(jìn)行分析8、理論分析和數(shù)字模擬相結(jié)合，對于理論結(jié)、理論分析和數(shù)字模擬相結(jié)合，對于理論結(jié)果要有數(shù)字模擬加以說明果要有數(shù)字模擬加以說明9、圖表的合理使用，讓閱卷人對圖表的解釋、圖表的合理使用，讓閱卷人對圖表的解釋和討論一目了然和討論一目了然10、模型的優(yōu)勢、弱勢分析、模型的優(yōu)勢、弱勢分析二、論文的組成部分二、論文的組成部分*Summary/Abstract1、簡單的

7、背景介紹、簡單的背景介紹2、建立和使用的模型由淺入深地介紹；求解、建立和使用的模型由淺入深地介紹；求解模型的方法，包括解析解、數(shù)值解或者圖模型的方法，包括解析解、數(shù)值解或者圖解。解。3、主要的結(jié)論和建議、主要的結(jié)論和建議*Overview/Analysis of the problem 主要是對問題的理解做一解釋，對問題主要是對問題的理解做一解釋，對問題的解答建立在哪些前提和背景下的解答建立在哪些前提和背景下*Assumptions 要對假設(shè)做些合理性的解釋要對假設(shè)做些合理性的解釋*Design of the models1、Effects of2、Determination of3、Equa

8、tions of *Sensitivity and stability analysis1、Sensitivity of the equation2、Stability of the model to the parameter*Conclusions 總結(jié)建立模型的條件、方法；使用模型的總結(jié)建立模型的條件、方法；使用模型的注意事項；模型推廣的可能性，模型存在的注意事項；模型推廣的可能性，模型存在的問題。問題。*Strengths and weaknesses1、參數(shù)不是任意給出的，而是用、參數(shù)不是任意給出的，而是用.方法得到方法得到的的2、模型運(yùn)算時間很短，具有使用性、模型運(yùn)算時間很短，具有

9、使用性*References 順序與文章中引用的順序要一致順序與文章中引用的順序要一致三、論文目錄舉例三、論文目錄舉例Example 1 Contents Introduction . 2 2. The Plan . 2 Objectives . 3 4. Defining the “Sweet Spot” . 4 5. Model A- Center of Percussion: Physics Sweet Spot. 5 5-b. Example I . 5 6. Model B: Brody Power Model . 6 6-a. Facts and Assumptions . 6 6

10、-b. General Form . 7 6-c. Example II. 9 6-d. Torque . 9 7. Discussion . 10 7-a. “Corking” . .10 7-b. Cork Model Augmentation . 11 7-c . Aluminum vs. Wood . 13 7-d. Aluminum Model Augmentation .13 8. Sensitivity Analysis .15 9. Conclusion . 19 10. References . 22 Example 2Contents Abstract . 2 2. Sta

11、tement of the problem and approach. 5 2.1 Survey of Previous Research : Environmental Criminology . 6 2.2 Assumptions . 6 2.3 Propositions and Foundation . 8 Methods . 11 3.1 Construction of the Map Geographical Method . 11 3.2 Static and Dynamic Risk Intensity Method . 18 Simulation Results and Dis

12、cussion . 30 4.1 Results of the Geographical Method. 31 4.2 Results of the Risk Intensity, “Static” and “Dynamic” Method . 36 4.3 Discussion . 37 4.3.1 Sensitivity and Robustness Testing . 37 4.3.2 Accuracy of the Prediction . 40 4.3.3 Combination of the Two Methods . 40 Strength and Weakness of the

13、 Model. 41 6. Conclusion and Recommendation . 41 Appendices . 43 A Bibliography . 43 B Data . 45 C Code . 46 Example 3Contents1 Introduction .12 The Models 3 2.1 A Simplified Model . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 An Intermediate Model . . . . . . . . . . . . . . . . . . . . . . 3

14、 2.3 A Congestion Model . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Extending the Model Using Computer Simulation . . . 4 2.4.1 Simulation Assumptions . . . . . . . . . . . . . . . . 5 2.4.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . 73 Analyzing the Models .8 3.1 The Simplest

15、 Model . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Intermediate Model . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Congestion Model . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 124 Conclusion. 15四、摘要舉例四、

16、摘要舉例Abstract 1 The advent of computer and technologicalprogress has introduced a new stage in the development of criminology. Investigatorscan now use computational techniques of geographic profiling in order to determine the patterns of movement of their suspects. We propose a model that aims to pr

17、edict areas with high probability of being the next on the criminals target list. We haveassumed that serial crimes are instrumental rather than expressive, thus ensuring that the criminal follows a predictable pattern of movement. We also assume that this pattern is characterized by a certain stabi

18、lity and continuity which facilitates a correlationwith the actions of other criminals in that area. Our model first uses an initial “geographical method”which reduces the areas under consideration based on parameters such as location coordinates, area, population and criminal rate, as well as the h

19、istory and psychological value This input is used to determine the shape of a Gaussian 2D function showing the distribution of the areas with the highest probability of becoming the location of a future crime. We improve these results by using the risk intensity method, a combination of two schemes,

20、 a“static” and a “dynamic” . The static method consists of first generating the risk intensities of different locations based on variables such as crimerates and distances from the anchor point,by using tools such as the distance-decay function. We then assign crime coefficients, which indicate the

21、extent to which the crime can becategorized as murder, rape, arson or robbery. In the dynamic model, we categorize the static parameters into homotypic, heterotypic and cumulative types by computing the mean and covariance matrix of these parameters. We apply different algorithms: logistic regressio

22、n, linear regression and nearest neighbor algorithm respectively to these types and then weigh them differently to obtain a parameter probability. This is then combined with the results of the static process to generate the probability of a crime at a certain location. We tested our model using exam

23、plesfrom different categories of serial crimes: robbery, murder, arson, which demonstrateddistinct criminal patterns. The surfaces generated using the geographic methodand the final predicted probabilities generallyagreed with our expectations of areas where the criminal will attack again. The test

24、for sensitivity suggested that parameters such as crime rate or populationdensity (area and location) are well taken into consideration by our model. Small changes in location, however, affected to a significant extent our results, probably because the differences in coordinates of the locations wer

25、e not large to begin with. Summary 2 Our goal was to design a model that couldaccount for the dynamics of vehicles in a traffic circle. We mainly focused on the rate of entry into the circle to determine the best way to regulate traffic. We assumed that vehicles circulate in a single lane and that o

26、nly incoming traffic can be regulated. For our model, the adjustable parameters were the rate of entry into the queue, the rate of entry into the circle (service rate), the maximum capacity of the traffic circle, and the rate of departure from the circle (departure rate). Vehicles first enter the qu

27、eue from the outside world, then enter the traffic circle from the queue, and lastly exit the traffic circle to the outside world. We modeled both the service rate and the departure rate asdependent on the number of vehicles inside the traffic circle. In addition, we ran computer simulations to have

28、 a visual representation of what happens in traffic circles during different situations. In many cases, we found that a fast service rate was the optimal way to maintain traffic flow. However, when the circle became more Heavily trafficked, a slower service rate better accommodated the traffic, indi

29、cating that a traffic light should be used. with variable timing depending on the expected amount of traffic. The main advantage of our approach was that the model was very simple and allowed usto clearly see the dynamics of the system. Also,the computer simulations that we ran provided more in-dept

30、h information abouttraffic flow under conditions that the model could not easily show visual observation of the traffic. Some disadvantages to our approachwere that we could not analyze the affects of multiple lanes nor stop lights that controlled the flow of traffic inside the circle. In addition,

31、we had no way of analyzing singularities in the situation, such as vehicles that drivefaster or slower than the rest of the traffic circle and pedestrians.Summary 3 Our basic model can be divided into two parts.The first part is to find a half-pipe shape which can maximize “Vertical Air”; in the sec

32、ond part, we adapt the shape to maximize the possible total angle of rotation. In the extended model, we analyze the snowboarders subjective influence on “Vertical Air” and the total degree of rotation. Finally, we discuss the feasibility and the trade-off of building a practical course. In the firs

33、t part of our basic model, we obtain a differential equation of Elost based on force analysis and Energy Conservation. Then we derive the representation of Elost by solving the equation, which is a functional. In the second part of the basic model, we derive the representation of the initial angular

34、 momentum before the fly, and discuss the factors influencing it. In our extended model, we calculate the “Vertical Air” when taking the snowboarders subjective influence into account. We use numerical method to solve the model and compare analytical results with reality and validate our method to b

35、e correct and robust. We analyzed the effects of factors such aswidth, height and gradient angle of half-pipe on “vertical Air” by controlling other factors. Using Gene Algorithm, we globally optimized the course shape to provide largest “vertical air” and total degree of rotation. Implementing a hy

36、brid scoring system as the objective function, we optimize the course shape to a “half-blood” shape.Yet, the model hasnt provided analytic solution of optimal course.五、寫作練習(xí)五、寫作練習(xí)1、本文是關(guān)于汽車公司產(chǎn)值總利潤最大化問、本文是關(guān)于汽車公司產(chǎn)值總利潤最大化問題，目的是在一定的生產(chǎn)條件和市場需求題，目的是在一定的生產(chǎn)條件和市場需求下創(chuàng)造最大利潤。首先求解出每一種車型下創(chuàng)造最大利潤。首先求解出每一種車型的生產(chǎn)成本和預(yù)計銷售利潤表，并通過分的生產(chǎn)成本和預(yù)計銷售利潤表，并通過分析得出每小時內(nèi)生產(chǎn)不同種配置車型所產(chǎn)析得出每小時內(nèi)生產(chǎn)不同種配置車型所產(chǎn)生的利潤，即小時利潤是不同的。以小時生的利潤，即小時利潤是不同的。以小時利潤為變量，導(dǎo)出總利潤的表達(dá)式，把問利潤為變量，導(dǎo)出總利潤的表達(dá)式，把問題化為以總利潤為目標(biāo)函數(shù)的線性規(guī)劃，題化為以總利潤為目標(biāo)函數(shù)的線性規(guī)劃，用用Excel 和和Matlab對此進(jìn)行求解。對此進(jìn)行求解。2、我們給出了一個腫瘤（我們給出了一個腫瘤（tumor）生長的簡）生長的簡單模型，此模型會得出腫瘤的實際形狀。單模型，此模型會得出腫瘤的實際形