2012年美国数学建模赛真题

2001年A题（一）Choosing a Bicycle Wheel选择自行车车轮有不同类型的车轮可以让自行车手们用在自己的自行车上。

两种基本的车轮类型是分别用金属辐条和实体圆盘组装而成（见图1）。

辐条车轮较轻，但实体车轮更符合空气动力学原理。

对于一场公路竞赛，实体车轮从来不会用作自行车的前轮但可以用作后轮。

职业自行车手们审视竞赛路线，并且请一位识文断字的人推断应该使用哪种车轮。

选择决定是根据沿途山丘的数量和陡度，天气，风速，竞赛本身以及其他考虑作出的。

你所喜爱的参赛队的教练希望准备妥当一个较好的系统，并且对于给定的竞赛路线已经向你的参赛队索取有助于确定宜用哪种车轮的信息。

这位教练需要明确的信息来帮助作出决定，而且已经要求你的参赛队完成下面列出的各项任务。

对于每项任务都假定，同样的辐条车轮将总是装在前面，而装在后面的车轮是可以选择的。

任务1. 提供一个给出风速的表格，在这种速度下实体后轮所需要的体能少于辐条后轮。

这个表格应当包括相应于从百分之零到百分之十增量为百分之一的不同公路陡度的风速。

（公路陡度定义为一座山丘的总升高除以公路长度。

如果把山丘看作一个三角形，它的陡度是指山脚处倾角的正弦。

）一位骑手以初始速度45kph从山脚出发，他的减速度与公路陡度成正比。

对于百分之五的陡度，骑上100米车速要下降8kph左右。

任务2. 提供一个例证，说明这个表格怎样用于一条时间试验路线。

任务3. 请判明这个表格是不是一件决定车轮配置的适当工具，并且关于如何作出这个决定提出其他建议。

MCM2001B题Escaping a Hurricane's Wrath逃避飓风怒吼(一场恶风…)1999年，在Floyd飓风预报登陆之前，撤离南卡罗来纳州沿海地区的行动导致一场永垂青史的交通拥塞。

车水马龙停滞在州际公路I-26上，那是内陆上从Charleston通往该州中心Columbia相对安全处所的主要干线。

正常时轻松的两个小时驱车路要用上18个小时才能开到头。

2012 AMC 12B ProblemsProblem 1Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?SolutionProblem 2A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?SolutionProblem 3For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?SolutionProblem 4Suppose that the euro is worth 1.30 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money?SolutionProblem 5Two integers have a sum of 26. when two more integers are added to the first two, the sum is 41. Finally, when two more integers are added to the sum of the previous 4 integers, the sum is 57. What is the minimum number of even integers among the 6 integers?SolutionProblem 6In order to estimate the value of where and are real numbers with , Xiaoli rounded up by a small amount, rounded down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?SolutionProblem 7Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the 3rd red light and the 21st red light?Note: 1 foot is equal to 12 inches.SolutionProblem 8A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?SolutionProblem 9It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How seconds would it take Clea to ride the escalator down when she is not walking?SolutionProblem 10What is the area of the polygon whose vertices are the points of intersection of the curves and ?SolutionProblem 11In the equation below, and are consecutive positive integers, and , , and represent number bases: What is ?SolutionProblem 12How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?SolutionProblem 13Two parabolas have equations and , where , , , and are integers, eachchosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have a least one point in common?SolutionProblem 14Bernardo and Silvia play the following game. An integer between 0 and 999 inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she addes 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000. Let N be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of N?SolutionProblem 15Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of 120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?SolutionProblem 16Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?SolutionProblem 17Square lies in the first quadrant. Points and lie on lines and , respectively. What is the sum of the coordinates of the center of the square ?SolutionProblem 18Let be a list of the first 10 positive integers such that for each either oror both appear somewhere before in the list. How many such lists are there?SolutionProblem 19A unit cube has vertices and . Vertices , , and are adjacent to , and forvertices and are opposite to each other. A regular octahedron has one vertex in each of the segments , , , , , and . What is the octahedron's side length?SolutionProblem 20A trapezoid has side lengths 3, 5, 7, and 11. The sums of all the possible areas of the trapezoid can be written in the form of , where , , and are rational numbers and and are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to ?SolutionProblem 21Square is inscribed in equiangular hexagon with on , on , and on . Suppose that , and . What is the side-length of the square?SolutionProblem 22A bug travels from to along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?SolutionProblem 23Consider all polynomials of a complex variable, , where and are integers, , and the polynomial has a zero with What is the sum of all values over all the polynomials with these properties?SolutionProblem 24\item Define the function on the positive integers by setting and if is the prime factorization of , then For every , let. For how many in the range is the sequenceunbounded?Note: A sequence of positive numbers is unbounded if for every integer , there is a member of the sequence greater than .Problem 25\item Let . Let be the set of all right triangles whose vertices are in . For every right triangle with vertices , , and in counter-clockwise order and right angle at , let . What is。

2012 MCM ProblemsPROBLEM A:The Leaves of a Tree"How much do the leaves on a tree weigh?" How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical mode l to describe and classify leaves. Consider and answer the following:• Why do leaves have the various shapes that they have?• Do the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the “volume” of the tree and its branches effect the shape?• Speaking of profiles, is leaf shape (general characteristics) related to tree profile/branching structure?• How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)?In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings.“多少钱树的叶子有多重？”怎么可能估计的叶子（或树为此事的任何其他部分）的实际重量？会如何分类的叶子吗？建立了一个数学模型来描述和分类的叶子。

2012年美国数学建模题目中文版第一篇：2012年美国数学建模题目解析2012年美国数学建模竞赛题目分为3个部分：A、B、C 部分，其中A、B两部分每个题目都设计成了开放式问题，而C部分则是两道严谨的数学证明题目。

A部分共有四个问题，分别为：1、搜索引擎的自动补充功能对于使用者的输入进行了什么样的预测和补全？如果这种功能可以被改变，在搜索引擎中进行必要的优化，会对搜索引擎的使用产生什么影响？2、在一个公共交通的网络中，如何合理地分配车辆保证所有的车辆在一定时间内都能够按时到达各自的终点站？3、如何在餐馆排队时，给不同的桌子和不同的人分配最佳位置，以便让顾客在餐厅等待的时间最短？4、针对特定的树木，如何编写算法来找到该树生长的变化，在叶片的数量和大小、气孔的数量和大小等方面的特征？对于这四个问题，考生需要通过分析问题，理清思路，构思模型，进行数据分析，最后得出自己的结论。

需要注意的是，每个问题都是非常开放式的，没有标准答案，最终得分并不会仅仅取决于观点是否正确，具体的解题过程、数据展示和准确度也是非常关键的。

B部分共有三个问题，分别为：1、如何通过旅游者在社交网络上的信息，帮助旅游者更好地定制旅游计划？2、如何在残缺不全的传媒报道中，找到事实并从中解读该事件？3、针对滑雪者在滑雪过程中的各种情况，如何预测他们的滑雪技巧以及未来的滑雪表现？对于B部分的三个问题，其实也都是很自由的问题，可以根据自己所擅长领域进行分析，构思自己的模型和算法，注重细节和数据展示。

C部分共有两个题目：1、已知一个最小二乘问题，其正则化后的解为稀疏的，试设计一个迭代算法在有效的处理机制下对其进行数值求解。

2、已知一个对象向一条线段上匀速运动，在线段的中途，运动的对象突然重力下落，如果目标是在最短的时间内捕捉该运动的对象，该怎样运动才是最优策略？对于C部分两个题目，需要在数学基础扎实的基础上进行思考，深入分析，构建出严谨的证明过程，注重逻辑和方法。

SummaryMany scholars conclude that leaf shape is highly related with the veins. Based on this theory, we assume the leaf growth in each direction satisfies a function. For the leaves in the same tree, the parameters are different; for those of separate trees, the function mode is different. Thus the shape of leaf differs from that of another. In the end of section 3, we simulate one growing period and depict the leaf shape.Through thousands of years of evolution, the leaves find various wa ys to make a full use of natural resources, including minimizing overlapping individual shadows. In order to find the main factors promoting the evolution of leaves, we analyze the distribution of adjacent leaves and the equilibrium point of photosynthesis and respiration. Besides, we also make a coronary hierarchical model and transmission model of the solar radiation to analyze the influence of the branches.As to the tree structure and the leaf shape, first we consider one species. Different tree shapes have different space which is built up by the branch quality and angle, effect light distribution, ventilation and humidity and concentration of CO2 in the tree crown. These are the factors which affect the leaf shape according to the model in section 1. Here we analyze three typical tree shapes: Small canopy shape, Open center shape and Freedom spindle shape, which can be described by BP network and fractional dimension model. We find that the factors mainly affect the function of Sthat affects the additional leaf area. Factors are assembled in different ways to create different leaf shapes. So that the relationship between leaf shape and tree profile/branching structure is proved.Finally we develop a model to calculate the leaf mass from the basic formula of . By adjusting the crown of a tree to a half ellipsoid, we first define thefunction of related factors,such as the leaf density and the effective ratio of leaf area. Then we develop the model using calculus. With this model, weapproximately evaluate the leaf mass of a middle-sized tree is 141kg.Dear editor,How much the leaves on a tree weigh is the focus of discussion all the time. Our team study on the theme following the current trend and we find something interesting in the process.The tree itself is component by many major elements. In our findings, we analyze the leaf mass with complicated ones, like leaf shape, tree structure and branch characteristics, which interlace with each other.With the theory that leaf shape is highly related with the veins, we assume theleaf growth in each direction satisfies a function . For the leaves in thesame tree, the parameters are different; for those of separate trees, the function mode is di fferent. That’s why no leaf shares the same shape. Also, we simulate one growing period and depict the leaf shape.In order to find the main factors promoting the evolution of leaves, we analyze the distribution of adjacent leaves and the equilibrium point of photosynthesis and respiration. Besides, we also make a coronary hierarchical model and transmission model of the solar radiation to analyze the influence of the branches.As to the tree structure and the leaf shape, different tree shapes have differe nt space which is built up by the branch quality and angle, effect light distribution, ventilation and humidity and concentration of CO2 in the tree crown that affect leaf shapes. Here we analyze three typical tree shapes which can be described by BP network and fractional dimension model. We find that the factors mainly affect the function of Sthat affects the additional leaf area. Factors are assembled in different ways to create different leaf shapes. So that the relationship between leaf shape and tree profile or branching structure is proved.Finally we develop the significant model to calculate the leaf mass from thebasic formula of . By adjusting the crown of a tree to a half ellipsoid,we first define the function of related factors and then we develop the model using calculus. With this model, we approximately evaluate the leaf mass of a middle-sized tree is 141kg.We are greatly appreciated that if you can take our findings into consideration. Thank you very much for your precious time for reading our letter.Yours sincerely,Team #14749Contents1. Introduction (4)2. Parameters (4)3. Leaves have their own shapes (5)3.1 Photosynthesis is important to plants (5)3.2 How leaves grow? (6)3.3 Build our model (7)3.4 A simulation of the model (10)4. Do the shapes maximize exposure? (14)4.1 The optimum solution of reducing overlapping shadows (14)4.1.1 The distribution of adjacent leaves (14)4.1.2 Equilibrium point of photosynthesis and respiration (15)4.2 The influence of the “volume” of a tree and its branches (17)4.2.1 The coronary hierarchical model (17)4.2.2 Spatial distribution model of canopy leaf area (18)4.2.3 Transmission model of the solar radiation (19)5. Is leaf shape related to tree structure? (20)5.1 The experiment for one species (21)5.2 Different tree shapes affect the leaf shapes (23)5.2.1 The light distribution in different shapes (23)5.2.2 Wind speed and humidity in the canopy (23)5.2.3 The concentration of carbon dioxide (24)5.3 Conclusion and promotion (25)6. Calculus model for leaf mass (26)6.1 How to estimate the leaf mass? (26)6.2 A simulation of the model (28)7. Strengths and Weakness (29)7.1 Strengths (29)7.2 Weaknesses (30)8. Reference (30)1. IntroductionHow much do the leaves on a tree weigh? Why do leaves have the various shapes that they have? How might one estimate the actual weight of the leaves? How might one classify leaves?We human-beings have never stopped our steps on exploring the natural world. But, as a matter of fact, the answer to those questions is still unresolved. Many scientists continue to study on this area. Recently , Dr. Benjamin Blonder (2010) achieved a new breakthrough on the venation networks and the origin of the leaf econo mics spectrum. They defined a standardized set of traits – density , distance and loopiness and developed a novel quantitative model that uses these venation traits to model leaf-level physiology .Now, it is commonly thought that there are four key leaf functional traits related to leaf economics: net carbon assimilation rate, life span, leaf mass per area ratio and nitrogen content.2. Parametersthe area a leaf grows decided by photosynthesisthe additional leaf area in one growing periodthe leaf growing obliquity Pthe total photosynthetic rate 0d R the dark respiration rate of leavesn P the net photosynthetic rateh the height of the canopyd the distance between two branchesdi the illumination intensity of scattered light from a given directionthe solar zenith angleh the truck highh the crown high13. Leaves have their own shapes3.1 Photosynthesis is important to plantsIt is widely accepted that two leaves are different, no matter where they are chosen from; even they are from the very tree. To understand how leaves grow is helpful to answer why leaves have the various shapes that they have.The canopy photosynthesis and respiration are the central parts of most biophysical crop and pasture simulation models. In most models, the acclamatory responses of protein and the environmental conditions, such as light, temperature and CO2 concentration, are concerned[1].In 1980, Farquhar et al developed a model named FvCB model to describe photosynthesis[2]:The FvCB model predicts the net assimilation rate by choosing the minimum between the Rubisco-limited net photosynthetic rate and the electron transport-limited net photosynthetic rate.Assume A n, A c, A j are the symbols for net assimilation rate, the Rubisco-limited net photosynthetic rate and the electron transport-limited net photosynthetic rate respectively, and the function can be described as:(1)(2)where and are the intercellular partial pressures of CO2 and O2,respectively, and are the Michaelis–Menten coefficients ofRubisco for CO2and O2, respectively, is the CO2compensation point inthe absence of (day respiration in andis the photosystem II electron transport rate that is used for CO2fixation and photorespiration[3][4].We apply the results of this model to build the relationship between the photosynthesis and the area a leaf grows during a period of time. It can be released as:(3)and are the area of the target leaf and the period of time it grows.is a function which can transfer the amount ofCO2into the area the leaf grows and the are parameters which affect S p. S p can be used as a constraint condition in our model.3.2 How leaves grow?As the collocation of computer hardware and software develops, people can refer to bridging biology, morphogenesis, applied mathematics and computer graphics to simulate living organisms[5], thus how to model leaves is of great challenge. In 2001, Dengler and Kang[6]brought up the thought that leaf shape is highly related to venation patterns. Recently, Runions[7] brought up a method to portray the leaf shape by analyzing venation patterns. Together with the Lindenmayer system (L-system), an advanced venation model can adjust the growth better that it solved the problem occurred in the previous model that the secondary veins are retarded.We knowleaves have various shapes.For example, leaves can be classified in to simple leaves which have an undivided blade and compound leaves whose blade is divided into two or more distinctleaflets such as the Fabaceae. As to the shape of a leaf, it may have marginal dentations of the leaf blades or not, and like a palm with various fingers or an elliptical cake. Judd et al defined a set of terms which describe the shape of leaves as follows [8]:We chose entire leaves to produce this model as a simplification. What’s more, they confirmed again that the growth of venations relates with that of the leaf.To disclose this relationship, Relative Elementary Rate of Growth (RERG) can be introduced to depict leaves growth [9]. RERG is defined as the growth rate per distance, in the definitive direction l at a point p of the growing object, yielding(4)Considered RERG , the growth patterns of leaves are also different. Roth-Nebelsick et al brought up four styles in their paper [10]:3.3 Build our modelWe chose marginal growth to build our model. Amid all above-mentioned studies, weFigure 3.1 Terms pertinent to the description of leaf shapes.Figure 3.2 A sample leaf (a) and the results of its: (b) marginal growth, (c) uniform isotropic (isogonic) growth, (d) uniform anisotropic growth, and (e) non-uniform anisotropic growth.Figure 3.3 The half of a leaf is settled in x-y plane like this with primary vein overlapping x-axis. The leaf grows in the direction of .assume that the leaf produce materials it needs to grow by photosynthesis to expand its leaf area from its border and this process is only affected by what we have discussed in the previous section about photosynthesis. The border can be infinitesimally divided into points. Set as the angle between the x-axis and thestraight line connecting the grid origin and one point on the curve,andas the growth distance in the direction of . To simplify the model, we assume that the leaf grows symmetrical. We put half of the leaf into the x-y plane and make the primary vein overlap x-axis.This is how we assume the leaf grows.In one circle of leaf growth, anything that photosynthesis provided transfers into theadditional leaf area, which can be described as:(5)while in the figure.In this case, we can simulate leaf growth thus define the leaf shape by using iterative operations the times N a leaf grow in its entire circle ①.① For instance, if the vegetative circle of a leaf is 20 weeks on average, the times of iterative operations N can beset as 20 when we calculate on a weekly basis.Figure 3.4 The curves of the adjacent growing period and their relationship.First, we pre-establish the border shape of a leaf in the x-y plane, yielding . where , the relevant satisfies:(6)(7) In the first growing period, assume , the growth distance in the direction of ,satisfies:(8)(9) It releases the relationship of the coordinates in the adjacent growing period. In this case, we can use eq.(9)to predict the new border of the leaf after one period of growth②:(10)And the average simple recursions are③:②That means, in the end of period 1.③As we both change the x coordinate and the y coordinate, in the new period, these two figures relate through those in the last period in the functions.(11)After simulate the leaf borders of the interactive periods, use definite integral ④ to settle parametersin the eq.(8) then can be calculated in each direction of , thus the exact shape of a leaf in the next period is visible.When the number of times N the leaf grows in its life circle applies above-mentioned recursions to iterate N times and the final leaf shape can be settled.By this model, we can draw conclusions about why leaves have different shapes. For the leaves on the same tree, they share the same method of expansion which can be described as the same type of function as Eq.(8). The reason why they are different, not only in a sense of big or small, is that in each growing period they acquire different amount of materials used to expand its own area. In a word, the parameters in the fixedly formed eq.(8) are different for any individual leaf on the same tree. For the leaves of different tree species, the corresponding forms of eq.(8) are dissimilar. Some are linear, some are logarithmic, some are exponential or mixtures of that, which settle the totally different expansion way of leaf, are related with the veins. On that condition, the characters can be divided by a more general concept such as entire or toothed.3.4 A simulation of the modelWe set the related parameters by ourselves to simulate the shape of a leaf and to express the model better.First we initialize the leaf shape by simulating the function of a leaf border at the④The relationship must meet eq.(5).Figure 3.5 and the recurrence relations.beginning of growing period 1 in the x-y plane. By observation, we assume that themovement of the initial leaf border satisfied:(12)Suppose the curve goes across the origin of coordinates, then the constraint conditionscan be:(13)Thus the solution to eq.(13) is:(14)By using Mathematica we calculatewhereandthe area of the half leaf is: (15)Wesettleaccording to the research by S. V . Archontoulisin et al [11] in eq.(3). On a weekly basis, theparameter. In eq.(15), we have .Figure 3.6When assume ,thus constraint condition eq.(5) becomes: (16)When eq.(8) is linear and after referring to Runions’s paper, we assume that Y -valuedecreases when X-value increases, which means the leaf grows faster at the end of theprimary vein. If the grow rate at the end of the primary vein is 0,as ,eq.(8) can be described as: (17)where b is decided by eq.(16).To settle the value of b , we calculate multiple sets of data by Excel then use a planecurve to trace them and get the approximation of b . In this method, we use grid toapproximate .Apparently,is monotone.When b is 2: The square of each square is 0.0139cm 2, and the total number of the squares in theadditional area is about 240. SoFigure 3.7When b is 2.5:Thesquare of each squareis 0.0240cm 2, and the total number of the squares in theadditional area is about 201. SoWhen b is 3:The square of each square is 0.0320cm 2, and the total number of the squares in the additional area is about 193. SoAfter comparing, we can draw a conclusion that fit eq.(16)best; accordingly, in period 1: (18)In each growing period, may be different for the amount of material produced isFigure 3.9Figure 3.8related with various factors, such as the change of relative location and CO 2 or O 2concentration, and other reasons. By using the same method, the leaf shape in period2 or other period can be generated on the basis of the previous growing period untilthe end of its life circle.What’s more, when eq.(8) is remodeled, the corresponding leaf shape can be changed.With , the following figure shows the transformationof a leaf shape in the firstgrowing period with the samesettled above and we can see that the shape will be dissected in the end.4. Do the shapes maximize exposure?4.1 The optimum solution of reducing overlapping shadows4.1.1 The distribution of adjacent leavesV ein is the foundation of the leaves. With the growing of veins, the leaves graduallyexpand around. The distribution of main and lateral veins plays an important decisiverole in the shapes of leaves. The scientists created a mathematical model which usesthree decisive factors - the relationship between the rate of photosynthesis, leaf life,carbon consumption or nitrogen consumption, to simulate the leaves’ shape. Becauseof carbon consumption is a constant for one tree, and we take the neighboring leavesin the same growth cycle to observe. So, we can only focus on one factor - the rate ofphotosynthesis.Figure 3.10The shape differs from figure 3.7-3.9, as the kind offunction of is different. In this case, it is a cubic model while a linear model in figure 3.7-3.9.Through the observation of dicotyledon, leaves on a branch will grow in a staggered way that can reduce the overlapping individual shadows of adjacent leaves and make them get more sunlight (Figure 4.1).Figure 4.1 The rotation distribution of leavesBase on the similar environment, we assume that adjacent leaves nearly have the same shape. From the perspective of looking down, the leaves grow from a point on the branch. So, we can simplify the vertical view of leaves as a circle of which the radius is the length of a vein which is represented with r. The width of the leaf is represented with w. The angle between two leaves is represented with β(Figure4.2).Figure 4.2 The vertical view of leavesThe leaves should use the space as much as possible, and for the leaf with one main vein, oval is the best choice. In general, βis between 15°and 90°. In this way, effectively reduce the direct overlapping area. According to the analysis of the first question, r and w are determined by the rate of photosynthesis and respiration. Besides, the width of leaf is becoming narrower when the main vein turns to be thinner.4.1.2 Equilibrium point of photosynthesis and respirationThe organism produced by photosynthesis firstly satisfies needs of leaf itself. Then the remaining organism delivered to the root to meet the growth needs of the tree. As we know, respiration needs to consume organism. If the light is not sufficient, organism produced by photosynthesis may no longer be able to afford the materials required for the growth of leaves. There should be an equilibrium point so as toprevent the leaf is behindhand in its circumstances.The formula [12] that describes the photosynthetic rate in response to light intensity with gradual exponential growth index can be expressed as:(19) Where P is total photosynthetic rate, m ax P is maximum photosynthetic rate ofleaves, a is initial solar energy utilization and I is photosynthetic photon quanta flux density .The respiration rate is affected by temperature ，using the formal of index to describe as follows:025102T d d R R -=⋅ (20)Where T is temperature and 0d R is the dark respiration rate of leaves, when the temperature is 25 degrees. As a model parameter, 0d R can be determined bynonlinear fitting. Therefore, the net photosynthetic rate can be expressed in the indexform as follows:(21)Where n P is the net photosynthetic rate, which does not include the concentration ofcarbon dioxide and other factors. The unit of n P is 21m ol m s μ--⋅⋅.We assume that the space for leaf growth is limited, the initial area of the leaf is 0S . At this point, the entire leaf happens to be capable of receiving sunlight. If the leaf continue to grow, some part of the leaf will be in the shadows and the area in the shadows is represented with x . Ignore the fluctuation cycle of photosynthesis and respiration, on average, the duration of photosynthesis is six hours per day . In the meanwhile, the respiration is ongoing all the time. When the area increased to S , we established an equation as follows:(22)where μ is the remaining organism created by the leaf per day . And from where 0μ=, which means the organism produced by photosynthesis has all been broken down completely in the respiration, we get the equilibrium point.4n n d S P x P R ⋅=+ (23) The proportion of the shaded area in the total area:4nn d P xS P R =+ (24)Cite an example of oak trees, we found the following data (Figure 4.3), which shows the relationship between net photosynthesis and dark respiration during 160 days. Thehabitat of the trees is affected by the semi-humid monsoon climate.Figure 4.3 The diurnal rate of net photosynthesis and dark respiration [13]In general, 217n P m ol m s μ--=⋅⋅，214d R m ol m s μ--=⋅⋅. Taking the given numbersinto the equation, we can get the result.0025xS =From the result, we can see that in the natural growth of leaves, with the weakening of photosynthesis, the leaves will naturally stop growing once they come across the blade between blocked. Therefore, the leaves can always keep overlapping individual shadows about 25%, so as to maximize exposure.4.2 The influence of the “volume” of a tree and its branches4.2.1 The coronary hierarchical modelMr. Bōken and Dr. J. Fischer (1987) found that in order to adequate lighting, leaves have different densities and the branches is distributed according to certain rules. They observed tropical plants in Miami, found that the ratio of main branch and two side branches is 1:0.94:0.87, and the angles between them are 24.4°and 36.9°. According to the computer simulation, the two angles can maximize the exposure of leaves.For simply, we use hemisphere to simulate the shape of the canopy. According to the light transmittance rate, we can divide the canopy into outer and inner two layers(Figure 4.3). The volume of the hemisphere depends on the size and distribution of branches.Figure 4.3 The coronary hierarchical modelThe angle between the branches will affect the depth of penetration of sun radiation and the distribution of leaves. Besides, thickness of the branches will affect the transfer of nutrients to the leaves.4.2.2 Spatial distribution model of canopy leaf areaAssume that the distribution of leaves is uniform in the section xz but not uniform in the y direction, as the figure 4.3. Then, we can get the formula of leaf area index (LAI) as follows [14]:/20/21(,)h d d dz x z dx LAI d -∂=⎰⎰(25)Where h is the height of the canopy , d is the distance between two branches, and (,)x z ∂ is the leaf area density function of the micro-body at the point (,)x z . x a is the function of leaf area which means the distribution of cross-section of the X direction, and Its value is a dimensionless, defined as follows: /2/21()d x d a x dx LAI d -=⎰ (26)()()x s a x d LAI C x =⋅⋅ (27)For the canopy which is not uniform in the horizontal direction, the distribution of its leaves is not entirely clear. We use s C to represent the distribution function of thedensity of leaves. s C can be expressed as a quadratic function or a Gaussiandistribution function.4.2.3 Transmission model of the solar radiationAssume that the attenuation of the solar radiation accords with the law of Beer-Lambert. The attenuation value at a point of canopy where the light arrives with a certain angle of incidence and azimuth is proportional to the length of the path, and the length can be calculated by Goudriaan function. Approximate function of the G function is shown as follows [15]：00(12)cos G G k G θ=+- (28)where k is a parameter decided by different plants.Take a micro unit in the canopy . Direct sunlight intercepted in this micro unit can be expressed in the following form:()(,)b b L dI I z G z dLAI =-Θ (29)Where b I means the direct sunlight, Θ is the solar zenith angle and L dL A I is the LAI on the path of light.Figure 4.4 Micro unit of the canopyThrough the integral and the chain rule, we can get the transmittance at the point (',')x z as follows:(,)[()]tan sin ()(',')exp()()h b b z G z a x z a z I x z I h ΘΘ∆Φ+=-⎰ (30)Average direct transmission rate:/2/21(')(',')'d b b d t z t x z dx d --=⎰(31)Different with the direct light, the scattered light in all directions is intercepted by theleaf surface from the upper hemisphere. The irradiance d dI of scattering at the point(',')x z can be expressed as follows:'cos d d b dI i t d ωθ=⋅⋅ (32) where d i is the illumination intensity of scattered light from a given direction.The transmission rate of the sun scattered light is as follows:2/2'00(',')1sin cos ()d b d I x z t d d I h ππθθθϕπ=⎰⎰ (33) Average scattering transmission rate:/2/21(')(',')'d d d d t z t x z dx d --=⎰(34)Global solar radiation reaching the canopy with a given depth z :(')()[(1)(')(')]d b d d I z I h k t z k t z ---=-+ (35)According to eq.(30) and eq.(33), the maximum depth the solar radiation can reach has a major link with h and θ. The shape of leaves have a relationship with photosynthesis. So, the h and θ of branches do have an influence on the leaves. Results showed that the light level and light utilization of high stem and open ce nter shape as well as small and sparse canopy shape were better than others ．Double canopy shape ，spindle shape and center shape took second place ，while big canopy shape had the lowest light distribution [16]．5. Is leaf shape related to tree structure?Due to internal and external factors, there are many kinds of tree shapes in the nature. For example, the apple's tree shape is semi-ellipsoidal, the willow's is hemispherical, the peach's likes a cup, the pine's likes cone and so on. The shape of their leaves varies. The leaf shape of apple is oval, pine's is needle, and the Indus's is palm. Is leaf shape related to tree shape? Even for the same species, there are many kinds of tree shapes. For instance, Small canopy shape, Open center shape, Freedom spindle shape, high stem and open center shape, Double canopy shape and so on. The sizes of leaves are different. Does the tree profile/branching structure effects the leaf shape?5.1 The experiment for one speciesTo solve this problem, we first consider the relationship of one tree species such as apple, which is semi-ellipsoidal. According to the second question, the hemispherical model is further extended to the semi-ellipsoidal model. Every tree individual shows irregularly because of natural and man-made factors, which reflects on the diversity of the crown in the canopy height direction of the changing relationship between the crown and canopy height. Figure 5.1 shows the relationship.Figure5.1 Schematic diagram of tree crown contourWhere 0h means the truck high, 1h means the crown high, 8710,,,,d d d d ⋅⋅⋅ meanscrown diameter. Tree shape with the scale change can be characterized in fractal dimension. Across the same scales, a fixed fractal dimension indicates the boundary shape's self-similarity; on different scales, the change of fractal dimension means that different processes or limiting factor has superiority. (Wiens, 1989)[17] According to the application of BP network and fractional dimension, we describe the tree shapes.[18]According to Li Guodong and Zhang Junke's work which detected and evaluated in different tree shapes of ‘Fuji’ apple. We choose three kinds of tree shapes to study; they are Small canopy shape, Open center shape and Freedom spindle shape as figure5.2 shows.。

题目：How Much Gas Should I Buy This Week?题目来源：2012年第十五届美国高中生数学建模竞赛（HiMCM）B题获奖等级：特等奖，并授予INFORMS奖论文作者：深圳中学2014届毕业生李依琛、王喆沛、林桂兴、李卓尔指导老师：深圳中学张文涛AbstractGasoline is the bleed that surges incessantly within the muscular ground of city; gasoline is the feast that lures the appetite of drivers. “To fill or not fill?” That is the question flustering thousands of car owners. This paper will guide you to predict the gasoline prices of the coming week with the currently available data with respect to swift changes of oil prices. Do you hold any interest in what pattern of filling up the gas tank can lead to a lower cost in total?By applying the Time series analysis method, this paper infers the price in the imminent week. Furthermore, we innovatively utilize the average prices of the continuous two weeks to predict the next two week’s average price; similarly, employ the four-week-long average prices to forecast the average price of four weeks later. By adopting the data obtained from 2011and the comparison in different aspects, we can obtain the gas price prediction model ：G t+1=0.0398+1.6002g t+−0.7842g t−1+0.1207g t−2+ 0.4147g t−0.5107g t−1+0.1703g t−2+ε .This predicted result of 2012 according to this model is fairly ideal. Based on the prediction model,We also establish the model for how to fill gasoline. With these models, we had calculated the lowest cost of filling up in 2012 when traveling 100 miles a week is 637.24 dollars with the help of MATLAB, while the lowest cost when traveling 200 miles a week is 1283.5 dollars. These two values are very close to the ideal value of cost on the basis of the historical figure, which are 635.24 dollars and 1253.5 dollars respectively. Also, we have come up with the scheme of gas fulfillment respectively. By analyzing the schemes of gas filling, we can discover that when you predict the future gasoline price going up, the best strategy is to fill the tank as soon as possible, in order to lower the gas fare. On the contrary, when the predicted price tends to decrease, it is wiser and more economic for people to postpone the filling, which encourages people to purchase a half tank of gasoline only if the tank is almost empty.For other different pattern for every week’s “mileage driven”, we calculate the changing point of strategies-changed is 133.33 miles.Eventually, we will apply the models -to the analysis of the New York City. The result of prediction is good enough to match the actual data approximately. However, the total gas cost of New York is a little higher than that of the average cost nationally, which might be related to the higher consumer price index in the city. Due to the limit of time, we are not able to investigate further the particular factors.Keywords: gasoline price Time series analysis forecast lowest cost MATLABAbstract ---------------------------------------------------------------------------------------1 Restatement --------------------------------------------------------------------------------------21. Assumption----------------------------------------------------------------------------------42. Definitions of Variables and Models-----------------------------------------------------4 2.1 Models for the prediction of gasoline price in the subsequent week------------4 2.2 The Model of oil price next two weeks and four weeks--------------------------5 2.3 Model for refuel decision-------------------------------------------------------------52.3.1 Decision Model for consumer who drives 100 miles per week-------------62.3.2 Decision Model for consumer who drives 200 miles per week-------------73. Train and Test Model by 2011 dataset---------------------------------------------------8 3.1 Determine the all the parameters in Equation ② from the 2011 dataset-------8 3.2 Test the Forecast Model of gasoline price by the dataset of gasoline price in2012-------------------------------------------------------------------------------------10 3.3 Calculating ε --------------------------------------------------------------------------12 3.4 Test Decision Models of buying gasoline by dataset of 2012-------------------143.4.1 100 miles per week---------------------------------------------------------------143.4.2 200 miles per week---------------------------------------------------------------143.4.3 Second Test for the Decision of buying gasoline-----------------------------154. The upper bound will change the Decision of buying gasoline---------------------155. An analysis of New York City-----------------------------------------------------------16 5.1 The main factor that will affect the gasoline price in New York City----------16 5.2 Test Models with New York data----------------------------------------------------185.3 The analysis of result------------------------------------------------------------------196. Summery& Advantage and disadvantage-----------------------------------------------197. Report----------------------------------------------------------------------------------------208. Appendix------------------------------------------------------------------------------------21 Appendix 1(main MATLAB programs) ------------------------------------------------21 Appendix 2(outcome and graph) --------------------------------------------------------34The world market is fluctuating swiftly now. As the most important limited energy, oil is much accounted of cars owners and dealer. We are required to make a gas-buying plan which relates to the price of gasoline, the volume of tank, the distance that consumer drives per week, the data from EIA and the influence of other events in order to help drivers to save money.We should use the data of 2011 to build up two models that discuss two situations: 100miles/week or 200miles/week and use the data of 2012 to test the models to prove the model is applicable. In the model, consumer only has three choices to purchase gas each week, including no gas, half a tank and full tank. At the end, we should not only build two models but also write a simple but educational report that can attract consumer to follow this model.1.Assumptiona)Assume the consumer always buy gasoline according to the rule of minimumcost.b)Ignore the difference of the gasoline weight.c)Ignore the oil wear on the way to gas stations.d)Assume the tank is empty at the beginning of the following models.e)Apply the past data of crude oil price to predict the future price ofgasoline.(The crude oil price can affect the gasoline price and we ignore thehysteresis effect on prices of crude oil towards prices of gasoline.)2.Definitions of Variables and Modelst stands for the sequence number of week in any time.(t stands for the current week. (t-1) stands for the last week. (t+1) stands for the next week.c t: Price of crude oil of the current week.g t: Price of gasoline of the t th week.P t: The volume of oil of the t th week.G t+1: Predicted price of gasoline of the (t+1)th week.α,β: The coefficient of the g t and c t in the model.d: The variable of decision of buying gasoline.(d=1/2 stands for buying a half tank gasoline)2.1 Model for the prediction of gasoline price in the subsequent weekWhether to buy half a tank oil or full tank oil depends on the short-term forecast about the gasoline prices. Time series analysis is a frequently-used method to expect the gasoline price trend. It can be expressed as:G t+1=α1g t+α2g t−1+α3g t−2+α4g t−3+…αn+1g t−n+ε ----Equation ①ε is a parameter that reflects the influence towards the trend of gasoline price in relation to several aspects such as weather data, economic data, world events and so on.Due to the prices of crude oil can influence the future prices of gasoline; we will adopt the past prices of crude oil into the model for gasoline price forecast.G t+1=(α1g t+α2g t−1+α3g t−2+α4g t−3+⋯αn+1g t−n)+(β1g t+β2g t−1+β3g t−2+β4g t−3+⋯βn+1g t−n)+ε----Equation ②We will use the 2011 data set to calculate the all coefficients and the best delay periods n.2.2 The Model of oil price next two weeks and four weeksWe mainly depend on the prediction of change of gasoline price in order to make decision that the consumer should buy half a tank or full tank gas. When consumer drives 100miles/week, he can drive whether 400miles most if he buys full tank gas or 200miles most if he buys half a tank gas. When consumer drives 200miles/week, full tank gas can be used two weeks most or half a tank can be used one week most. Thus, we should consider the gasoline price trend in four weeks in future.Equation ②can also be rewritten asG t+1=(α1g t+β1g t)+(α2g t−1+β2g t−1)+(α3g t−2+β3g t−2)+⋯+(αn+1g t−n+βn+1g t−n)+ε ----Equation ③If we define y t=α1g t+β1g t,y t−1=α2g t−1+β2g t−1, y t−2=α3g t−2+β3g t−2……, and so on.Equation ③can change toG t+1=y t+y t−1+y t−2+⋯+y t−n+ε ----Equation ④We use y(t−1,t)denote the average price from week (t-1) to week (t), which is.y(t−1,t)=y t−1+y t2Accordingly, the average price from week (t-3) to week (t) isy(t−3,t)=y t−3+y t−2+y t−1+y t.4Apply Time series analysis, we can get the average price from week (t+1) to week (t+2) by Equation ④,G(t+1,t+2)=y(t−1,t)+y(t−3,t−2)+y(t−5,t−4), ----Equation ⑤As well, the average price from week (t+1) to week (t+4) isG(t+1,t+4)=y(t−3,t)+y(t−7,t−4)+y(t−11,t−8). ----Equation ⑥2.3 Model for refuel decisionBy comparing the present gasoline price with the future price, we can decide whether to fill half or full tank.The process for decision can be shown through the following flow chart.Chart 1For the consumer, the best decision is to get gasoline with the lowest prices. Because a tank of gasoline can run 2 or 4 week, so we should choose a time point that the price is lowest by comparison of the gas prices at present, 2 weeks and 4 weeks later separately. The refuel decision also depends on how many free spaces in the tank because we can only choose half or full tank each time. If the free spaces are less than 1/2, we can refuel nothing even if we think the price is the lowest at that time.2.3.1 Decision Model for consumer who drives 100 miles per week.We assume the oil tank is empty at the beginning time(t=0). There are four cases for a consumer to choose a best refuel time when the tank is empty.i.g t>G t+4and g t>G t+2, which means the present gasoline price is higherthan that either two weeks or four weeks later. It is economic to fill halftank under such condition. ii. g t <Gt +4 and g t <G t +2, which means the present gasoline price is lower than that either two weeks or four weeks later. It is economic to fill fulltank under such condition. iii. Gt +4>g t >G t +2, which means the present gasoline price is higher than that two weeks later but lower than that four weeks later. It is economic to fillhalf tank under such condition. iv. Gt +4<g t <G t +2, which means the present gasoline price is higher than that four weeks later but lower than that two weeks later. It is economic to fillfull tank under such condition.If other time, we should consider both the gasoline price and the oil volume in the tank to pick up a best refuel time. In summary, the decision model for running 100 miles a week ist 2t 4t 2t 4t 2t 4t 2t 4t 11111411111ˆˆ(1)1((1)&max(,))24442011111ˆˆˆˆ1/2((1)&G G G (&))(0(1G G )&)4424411ˆˆˆ(1)0&(G 4G G (G &)t i t i t t t t i t i t t t t t t i t t d t or d t g d d t g or d t g d t g or ++++----+++-++<--<<--<>⎧⎪=<--<<<--<<<⎨⎪⎩--=><∑∑∑∑∑t 2G ˆ)t g +<----Equation ⑦d i is the decision variable, d i =1 means we fill full tank, d i =1/2 means we fill half tank. 11(1)4t i tdt ---∑represents the residual gasoline volume in the tank. The method of prices comparison was analyzed in the beginning part of 2.3.1.2.3.2 Decision Model for consumer who drives 200 miles per week.Because even full tank can run only two weeks, the consumer must refuel during every two weeks. There are two cases to decide whether to buy half or full tank when the tank is empty. This situation is much simpler than that of 100 miles a week. The process for decision can also be shown through the following flow chart.Chart 2The two cases for deciding buy half or full tank are: i. g t >Gt +1, which means the present gasoline price is higher than the next week. We will buy half tank because we can buy the cheaper gasoline inthe next week. ii. g t <Gt +1, which means the present gasoline price is lower than the next week. To buy full tank is economic under such situation.But we should consider both gasoline prices and free tank volume to decide our refueling plan. The Model is111t 11t 111(1)1220111ˆ1/20(1)((1)0&)22411ˆ(1&G )0G 2t i t t i t i t t t t t i t t d t d d t or d t g d t g ----++<--<⎧⎪=<--<--=>⎨⎪⎩--=<∑∑∑∑ ----Equation ⑧3. Train and Test Model by the 2011 datasetChart 33.1 Determine all the parameters in Equation ② from the 2011 dataset.Using the weekly gas data from the website and the weekly crude price data from , we can determine the best delay periods n and calculate all the parameters in Equation ②. For there are two crude oil price dataset (Weekly Cushing OK WTI Spot Price FOB and Weekly Europe Brent SpotPrice FOB), we use the average value as the crude oil price without loss of generality. We tried n =3, 4 and 5 respectively with 2011 dataset and received comparison graph of predicted value and actual value, including corresponding coefficient.（A ） n =3(the hysteretic period is 3)Graph 1 The fitted price and real price of gasoline in 2011(n=3)We find that the nearby effect coefficient of the price of crude oil and gasoline. This result is same as our anticipation.（B）n=4(the hysteretic period is 4)Graph 2 The fitted price and real price of gasoline in 2011（n=4）(C) n=5(the hysteretic period is 5)Graph 3 The fitted price and real price of gasoline in 2011（n=5）Via comparing the three figures above, we can easily found that the predictive validity of n=3(the hysteretic period is 3) is slightly better than that of n=4(the hysteretic period is 4) and n=5(the hysteretic period is 5) so we choose the model of n=3 to be the prediction model of gasoline price.G t+1=0.0398+1.6002g t+−0.7842g t−1+0.1207g t−2+ 0.4147g t−0.5107g t−1+0.1703g t−2+ε----Equation ⑨3.2 Test the Forecast Model of gasoline price by the dataset of gasoline price in 2012Next, we apply models in terms of different hysteretic periods(n=3,4,5 respectively), which are shown in Equation ②,to forecast the gasoline price which can be acquired currently in 2012 and get the graph of the forecast price and real price of gasoline:Graph 4 The real price and forecast price in 2012（n=3）Graph 5 The real price and forecast price in 2012（n=4）Graph 6 The real price and forecast price in 2012（n=5）Conserving the error of observation, predictive validity is best when n is 3, but the differences are not obvious when n=4 and n=5. However, a serious problem should be drawn to concerns: consumers determines how to fill the tank by using the trend of oil price. If the trend prediction is wrong (like predicting oil price will rise when it actually falls), consumers will lose. We use MATLAB software to calculate the amount of error time when we use the model of Equation ⑨to predict the price of gasoline in 2012. The graph below shows the result.It’s not difficult to find the prediction effect is the best when n is 3. Therefore, we determined to use Equation ⑨as the prediction model of oil price in 2012.G t+1=0.0398+1.6002g t+−0.7842g t−1+0.1207g t−2+ 0.4147g t−0.5107g t−1+0.1703g t−2+ε3.3 Calculating εSince political occurences, economic events and climatic changes can affect gasoline price, it is undeniable that a ε exists between predicted prices and real prices. We can use Equation ②to predict gasoline prices in 2011 and then compare them with real data. Through the difference between predicted data and real data, we can estimate the value of ε .The estimating process can be shown through the following flow chartChart 4We divide the international events into three types: extra serious event, major event and ordinary event according to the criteria of influence on gas prices. Then we evaluate the value: extra serious event is 3a, major event is 2a, and ordinary event is a. With inference to the comparison of the forecast price and real price in 2011, we find that large deviation of data exists at three time points: May 16,2011, Aug 08,2011 andOct 10,2011. After searching, we find that some important international events happened nearly at the three time points. We believe that these events which occurred by chance affect the international prices of gasoline so the predicted prices deviate from the actual prices. The table of events and the calculation of the value of a areTherefore, by generalizing several sets of particular data and events, we can estimate the value of a:a=26.84 ----Equation ⑩The calculating process is shown as the following graph.Since now we have obtained the approximate value of a, we can evaluate the future prices according to currently known gasoline prices and crude oil prices. To improve our model, we can look for factors resulting in some major turning point in the graph of gasoline prices. On the ground that the most influential factors on prices in 2012 are respectively graded, the difference between fact and prediction can be calculated.3.4 Test Decision Models of buying gasoline by the dataset of 2012First, we use Equation ⑨to calculate the gasoline price of next week and use Equation ⑤and Equation ⑥to calculate the gasoline price trend of next two to four weeks. On the basis above, we calculate the total cost, and thus receive schemes of buying gasoline of 100miles per week according to Equation ⑦and Equation ⑧. Using the same method, we can easily obtain the pattern when driving 200 miles per week. The result is presented below.We collect the important events which will affect the gasoline price in 2012 as well. Therefore, we calculate and adjust the predicted price of gasoline by Equation ⑩. We calculate the scheme of buying gasoline again. The result is below:3.4.1 100 miles per weekT2012 = 637.2400 （If the consumer drives 100 miles per week, the total cost inTable 53.4.2 200 miles per weekT2012 = 1283.5 （If the consumer drives 200 miles per week, the total cost in 2012 is 1283.5 USD）. The scheme calculated by software is below：Table 6According to the result of calculating the buying-gasoline scheme from the model, we can know: when the gasoline price goes up, we should fill up the tank first and fill up again immediately after using half of gasoline. It is economical to always keep the tank full and also to fill the tank in advance in order to spend least on gasoline fee. However, when gasoline price goes down, we have to use up gasoline first and then fill up the tank. In another words, we need to delay the time of filling the tank in order to pay for the lowest price. In retrospect to our model, it is very easy to discover that the situation is consistent with life experience. However, there is a difference. The result is based on the calculation from the model, while experience is just a kind of intuition.3.4.3 Second Test for the Decision of buying gasolineSince the data in 2012 is historical data now, we use artificial calculation to get the optimal value of buying gasoline. The minimum fee of driving 100 miles per week is 635.7440 USD. The result of calculating the model is 637.44 USD. The minimum fee of driving 200 miles per week is 1253.5 USD. The result of calculating the model is 1283.5 USD. The values we calculate is close to the result of the model we build. It means our model prediction effect is good. (we mention the decision people made every week and the gas price in the future is unknown. We can only predict. It’s normal to have deviation. The buying-gasoline fee which is based on predicted calculation must be higher than the minimum buying-gasoline fee which is calculated when all the gas price data are known.)We use MATLAB again to calculate the total buying-gasoline fee when n=4 and n=5. When n=4,the total fee of driving 100 miles per week is 639.4560 USD and the total fee of driving 200 miles per week is 1285 USD. When n=5, the total fee of driving 100 miles per week is 639.5840 USD and the total fee of driving 200 miles per week is 1285.9 USD. The total fee are all higher the fee when n=3. It means it is best for us to take the average prediction model of 3 phases.4. The upper bound will change the Decision of buying gasoline.Assume the consumer has a mileage driven of x1miles per week. Then, we can use 200to indicate the period of consumption, for half of a tank can supply 200-mile x1driving. Here are two situations:<1.5①200x1>1.5②200x1In situation①, the consumer is more likely to apply the decision of 200-mile consumer’s; otherwise, it is wiser to adopt the decision of 100-mile consumer’s. Therefore, x1is a critical value that changes the decision if200=1.5x1x1=133.3.Thus, the mileage driven of 133.3 miles per week changes the buying decision.Then, we consider the full-tank buyers likewise. The 100-mile consumer buys half a tank once in four weeks; the 200-mile consumer buys half a tank once in two weeks. The midpoint of buying period is 3 weeks.Assume the consumer has a mileage driven of x2miles per week. Then, we can to illustrate the buying period, since a full tank contains 400 gallons. There use 400x2are still two situations:<3③400x2>3④400x2In situation③, the consumer needs the decision of 200-mile consumer’s to prevent the gasoline from running out; in the latter situation, it is wiser to tend to the decision of 100-mile consumer’s. Therefore, x2is a critical value that changes the decision if400=3x2x2=133.3We can find that x2=x1=133.3.To wrap up, there exists an upper bound on “mileage driven”, that 133.3 miles per week is the value to switch the decision for buying weekly gasoline. The following picture simplifies the process.Chart 45. An analysis of New Y ork City5.1 The main factors that will affect the gasoline price in New York CityBased on the models above, we decide to estimate the price of gasoline according to the data collected and real circumstances in several cities. Specifically, we choose New York City as a representative one.New York City stands in the North East in the United States, with the largest population throughout the country as 8.2 million. The total area of New York City is around 1300 km2, with the land area as 785.6 km2(303.3 mi2). One of the largest trading centers in the world, New York City has a high level of resident’s consumption. As a result, the level of the price of gasoline in New York City is higher than the average regular oil price of the United States. The price level of gasoline and its fluctuation are the main factors of buying decision.Another reasonable factor we expect is the distribution of gas stations. According to the latest report, there are approximately 1670 gas stations in the city area (However, after the impact of hurricane Sandy, about 90 gas stations have been temporarily out of use because of the devastation of Sandy, and there is still around 1580 stations remaining). From the information above, we can calculate the density of gas stations thatD(gasoline station)= t e amount of gas stationstotal land area =1670 stations303.3 mi2=5.506 stations per mi2This is a respectively high value compared with several other cities the United States. It also indicates that the average distance between gas stations is relatively small. The fact that we can neglect the distance for the cars to get to the station highlights the role of the fluctuation of the price of gasoline in New York City.Also, there are approximately 1.8 million residents of New York City hold the driving license. Because the exact amount of cars in New York City is hard to determine, we choose to analyze the distribution of possible consumers. Thus, we can directly estimate the density of consumers in New York City in a similar way as that of gas stations:D(gasoline consumers)= t e amount of consumerstotal land area = 1.8 million consumers303.3 mi2=5817consumers per mi2Chart 5In addition, we expect that the fluctuation of the price of crude oil plays a critical role of the buying decision. The media in New York City is well developed, so it is convenient for citizens to look for the data of the instant price of crude oil, then to estimate the price of gasoline for the coming week if the result of our model conforms to the assumption. We will include all of these considerations in our modification of the model, which we will discuss in the next few steps.For the analysis of New York City, we apply two different models to estimate the price and help consumers make the decision.5.2 Test Models with New York dataAmong the cities in US, we pick up New York as an typical example. The gas price data is downloaded from the website () and is used in the model described in Section 2 and 3.The gas price curves between the observed data and prediction data are compared in next Figure.Figure 6The gas price between the observed data and predicted data of New York is very similar to Figure 3 in US case.Since there is little difference between the National case and New York case, the purchase strategy is same. Following the same procedure, we can compare the gas cost between the historical result and predicted result.For the case of 100 miles per week, the total cost of observed data from Feb to Oct of 2012 in New York is 636.26USD, while the total cost of predicted data in the same period is 638.78USD, which is very close. It proves that our prediction model is good. For the case of 200 miles per week, the total cost of observed data from Feb to Oct of 2012 in New York is 1271.2USD, while the total cost of predicted data in the same period is 1277.6USD, which is very close. It proves that our prediction model is good also.5.3 The analysis of resultBy comparing, though density of gas stations and density of consumers of New York is a little higher than other places but it can’t lower the total buying-gas fee. Inanother words, density of gas stations and density of consumers are not the actual factors of affecting buying-gas fee.On the other hand, we find the gas fee in New York is a bit higher than the average fee in US. We can only analyze preliminary it is because of the higher goods price in New York. We need to add price factor into prediction model. We can’t improve deeper because of the limited time. The average CPI table of New York City and USA is below:Datas Statistics website(/xg_shells/ro2xg01.htm)6. Summery& Advantage and disadvantageTo reach the solution, we make graphs of crude oil and gasoline respectively and find the similarity between them. Since the conditions are limited that consumers can only drive 100miles per week or 200miles per week, we separate the problem into two parts according to the limitation. we use Time series analysis Method to predict the gasoline price of a future period by the data of several periods in the past. Then we take the influence of international events, economic events and weather changes and so on into consideration by adding a parameter. We give each factor a weight consequently and find the rules of the solution of 100miles per week and 200miles per week. Then we discuss the upper bound and clarify the definition of upper bound to solve the problem.According to comparison from many different aspects, we confirm that the model expressed byEquation ⑨is the best. On the basis of historical data and the decision model of buying gasoline(Equation ⑦and Equation ⑧), we calculate that the actual least cost of buying gasoline is 635.7440 USD if the consumer drives 100 miles per week (the result of our model is 637.24 USD) and the actual least cost of buying gasoline is 1253.5 USD(the result of our model is 1283.5 USD) if the consumer drives 100 miles per week. The result we predicted is similar to the actual result so the predictive validity of our model is finer.Disadvantages:1.The events which we predicted are difficult to quantize accurately. The turningpoint is difficult for us to predict accurately as well.2.We only choose two kinds of train of thought to develop models so we cannotevaluate other methods that we did not discuss in this paper. Other models which are built up by other train of thought are possible to be the optimal solution.。

2004年A题题目是研究人的指纹相同导致确认身份时产生错误的可能性和因为DNA相同导致产生错误的可能性。

这道题与生物有关。

2004年，美国提出了“科技展望——生物盾国家纳米技术”，针对微分子进行研究，与人体相关的DNA也就成为焦点。

指纹是区分人的重要参数，它由DNA决定，由于现在自然环境改变，人体的DNA也有可能发生变异，这样引起相同指纹出现也是有可能的。

研究这种变异的可能性有利于刑事案件的侦破，提高政府的防御能力等。

本题目可以采用机理分析法来建模。

可以采用计算机模拟人体染色体中的基因排列进而检验、统计排列结果与指纹形成情况B题题目：现在的快通系统在收费站、娱乐公园和其他的地方，正在越来越频繁的使用，来减少人们排队等候的时间，现在我们考虑为一个娱乐公园所设计的快通系统，在一次测试中，这个公园在几个游客比较多的景点旁都设置的快通系统，这个系统的设计创意是对于那些比较热门的景点，可以到旁边的一个机器，将其门票插入后出来一张纸条，上面写着具体的你可以回来时间段，比如说你把你的门票在1：15插到机子里，系统就会告诉你，你可以在3:30-4:30回来，这个时候队伍就比较短，你可以凭你的纸条加入这个队伍，很快就可以进入景点，为了防止游客同时在几个景点使用这个系统。

系统的机器只允许你一次在一个景点排队等候。

现在你是几个被公园雇佣的互相竞争的一个，你的职责是改善快通系统的运行。

很多游客都在抱怨测试期间系统的异常现象，比如说有一次系统提供的回到景点的时间是4小时以后，但是才过一会儿，在相同景点系统提供的时间只有1小时。

在另外一些时候根据快通系统组织起来的游客的等候队伍，就和普通的队伍一样长一样慢。

现在的问题是要提出并且测试一个模型，这个模型能让快通系统的等候纸条的发放能增加人们在公园的乐趣的目的。

问题的一部分就是首先决定衡量不同模型的标准，在你提交的报告里还要附带一份技术性的总结，以便公园的领导在不同的顾问所提出的模型中选择。