2013年全美大学生数学建模大赛水源调配18569

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2013年美国数学建模比赛提交论文-二等奖

Print This Page Close This WindowFor office use only T1________________T2________________T3________________T4________________Team Control Number22940Problem ChosenAFor office use only F1________________F2________________F3________________F4________________2013Mathematical Contest in Modeling (MCM)Summary Sheet(Attach a copy of this page to your solution paper.)Type a summary of your results on this page.Do not include the name of your school,advisor,or team members on this page.SummaryThis paper aims to design the optimal shape of Brownie pan on the given conditions.The influencing factors include the shape of pan,the ratio of width and the length of the oven,and the weight be conferred to the quantity of Brownie pans that can be put in a oven and evenness of heat distribution.We have a scientific study about the three typical shapes,namely rectangular circular and oval,and we get the following conclusion that the time required by circular pans is the shortest,while rectangular pan can maximize the use of space.To solve the problem,we simplify the influencing factors and assume that the area of being effective baking inside the oven can be exactly fulfilled by rectangular pans.So that we can easily solve the condition 1,and be aware of that the rectangular pans is the optimal shape;and the same to the condition 2,we believe that the temperature distribution on the outer edge of circular pans is completely uniform,so the optimal solution is circular.When it comes to the third condition,it’s a typical linear programming problem.We use the idea of normalizing to construct design objectivefunction:(1)s A nZ p p N A=+−.In the formula above,the calculation of area being effective heated adopt the view of discrete,which is very innovative (formula (k)).By using the above objective function,whose results is roughly the same with the results of MATLAB,This can be used to confirm each other.We obtain the desired optimal solution to a maximum of 1.215,which is larger than other shapes’maximum by 21.5%.Its advantages are obvious,so this further illustrates the correctness of the ideal optimal shape.So we can say the pan we design is the Ultimate Brownie Pan!ContentsI.Introduction (4)II.Assumptions (4)III.Analysis&Models (5)IV.Solutions (12)4.1rectangular Brownie Pans (12)4.2circular Brownie Pans (14)4.3oval Brownie Pans (15)V.Optimization of the model (12)5.1rectangular Brownie Pans (16)5.2circular Brownie Pans (16)5.3oval Brownie Pans (17)References (18)Advertising sheet (19)Appendix (20)List of Figures1.Region of discrete (8)2.Volumetric controlled by internal node (8)3.The simulation isotherms of rectangular pan (10)4.The simulation isotherms of circular pan (10)5.The simulation isotherms of oval pan (10)6.Temperature curve of the rectangular pan center (11)7.Temperature curve of the circular pan center (11)8.Temperature curve of the oval pan center (11)9.How the rectangular pans are placed in the oven (14)10.The placement of round Brownie Pan in the oven (14)11.The ideal shape of the Brownie Pan (15)12.The distribution of the pans in the oven (15)13.The placement of rectangular Brownie Pan in the oven (16)14.The zoning figure of the extended the oven (17)15.The arrangement of Brownie Pans in theoretical optimal shape in the oven (18)16.Distribution of temperature of rectangular pan (20)17.Distribution of temperature of circular pan (20)18.Distribution of temperature of oval pan (20)1Introduction:When baking in a rectangular pan heat is concentrated in the4corners and the product gets overcooked at the corners(and to a lesser extent at the edges).In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges.However,since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven.Since both have their own advantages and can complement each other,we consider that a combination of the two will be a better choice.The purpose of this article is to identify the pan in best shape to satisfy various needs.2Assumption:1.The heat inside the oven is Changeless,After a Brownie Pan have been put in the oven,the heat is conducted to the center from the edge.2.The initial temperature of the food is25C o.3.The coefficient of thermal conductivity of the food is0.1.4.To make full use of the area of a rectangular oven,We assume that the area of being effective baking inside the oven can just be fulfilled by rectangular Brownie Pans.Table 1NotationParameter MeaningFThe heat of per unit time through a given area λThermal conductivity TTemperaturetThe temperature field in the Cartesian coordinate system A Cross-sectional areaΔx,ΔyThe amount of change of the x-direction,y-direction τ∆Time variation amount ∆o F The number of grid FourierT ij The temperature of the arbitrary point (i,j)T oThe temperature of the center of the pant j i T ,The temperature of the arbitrary point (i,j)at time t 0div ∇The gradient value at a certain point3Analysis &ModelsWe solved this problem using numerical solution of heat conduction.The numerical solution is based on the basic knowledge of heat transfer,and a method of using partial differential equations solving a heat conduction problem.Heat conduction partial differential equations:⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂=∇=∂∂conditions Initial conditions dary B z T y T x T T t T oun 2222222ααThe basic idea:putting the problem of the temperature within the object continuously changing with time and space transformed into the problem of the temperature value in a finite number of discrete points within the area of the field of time and space.Further,using the temperature value of these discrete units to approximate the continuous temperature distribution.Models:1.The establishment of the physical modelAccording to the meaning of the questions,we construct three models,which are rectangular pan,circular pan and oval pan.2.The establishment of the mathematical modelOne of the main purpose of the study heat conduction problem is obtained under certain boundary conditions the state of objects within the temperature distribution with spatial location and time.⑴.Equations of heat conductionThe equations of the temperature distribution in the description of thermal conductivity inside the object is called the partial differential equation of heat conduction,which also known as the heat diffusion w of conservation of energy and the Fourier law are its fundamental basis.Once solving the temperature distribution,the heat conduction rate of inside the object or surface at any point on can be obtained by Fourier law.That is the heat flux density.Unsteady heat conduction problem for the temperature field varies with time,by obtaining the temperature distribution inside the object at different times,can determine the thermal stress and thermal deformation of the various parts.The general form of the equation of heat conduction in the Cartesian coordinate system:v ztz y t y x t x t cφλλλτρ+∂∂∂∂+∂∂∂∂+∂∂∂∂=∂∂)(((................................(a)This formula reflects the relationship between the thermal conductivity of the object within the total energy conservation.In the calculation,the thermal conductivity is seen as a constant,so the above equation reduces to:c zty t x t t v ρφατ+∂∂+∂∂+∂∂=∂∂)(222222...............................................................(b)It is composed by three physical properties of a physical parameter,called thermal diffusivity,can also be referred to the coefficient of thermal conductive,whose unit iss m /2.It indicates the temperature tends to be uniform capacity of the object in the heating or cooling process.This comprehensive parameters in the non-steady-state heat transfer process is a very important parameter.⑵.Single -Valued Property conditionsAll pure heat conduction problem can be described with the equation of heat conduction in the corresponding coordinates,including one-dimensional and multidimensional,steady-state and non-steady-state,constant material properties and Variable physical properties,the internal heat source and no internal heat source heat conduction problems.Differential equations (The mathematical said general solution)must contain the pending integration constant.In addition to the differential equation,for these constants to be determined uniquely determined must be attached to a certain characteristics of solving this particular heat conduction problem and limited of external environment or supplementary explanation.These additional instructions and restriction condition is Single -Valued Property conditions,and mathematics known as boundary conditions.Mathematical model of any one specific full heat conduction problems,in addition to the equation of heat conduction in the appropriately selected coordinates,must be given the corresponding Single -Valued Property conditions.In terms of the general problem of thermal conductivity,the Single -Valued property conditions consist of the following two parts:①.Time conditionsFor unsteady-state heat conduction,the temperature field of the object at the start time must be given,so the time conditions are known as initial conditions.In the three-dimensional Cartesian coordinate system,the initial conditions can generally be expressed as the following form:),,(0z y x f t ==τ....................................................................(c)Under normal circumstances,the object having a completely uniform temperature at the beginning of the process,so we can think that the distribution function (),,(z y x f )is a constant in the above formula.②.Boundary conditionsThe boundary conditions refers to the contact and interaction in the heat exchange of hot objects between the boundary surface and the external environment.For unsteady heat conduction,it is often the external driving force making process take place and develop.Here we consider the first category.Stipulate:The surface temperature is a room temperature.),(y x f t w =,The boundary temperature.Notes:tcons t w tan =3.Discrete the solution domainDiscretization is divided the substantially continuous object into a series of tiny unit,while the center of the unit is called a node.Divided the spatial domain x into n subparagraph,the step is Δx,get 0,l,2,...,i,...,n x node of the n +1;divided the spatial domain y into n subparagraph,while the step is Δy,get 0,l,2,...,i,...,n y node of the n +1.Intersection point (i,j)of the separation line represents the spatial domain position.The size of space and time step depends on the specific circumstances of the problem,and sometimes cannot bearbitrarily selected,needing to consider the node temperature equation solving stability problems which are as shown in the following figure:Figure 1Region of discreteTime can also be divided into many inter-cell.The physical significance of the zone discretization is that we can think a node focus the heat capacity around its tiny area.Thus the node temperature is the average temperature of this tiny area.Thus,the temperature of all the nodes together represents the distribution of the temperature within this continuous region.4.Establish the difference equation of nodes’temperatureUsing difference quotient instead of the derivative,and then on the basis of the relationship of the heat balance we can a establish differential equation.The same as the steady heat conduction,using the method of heat balance,you can establish the temperature difference equations of objects internal nodes and boundary nodes at unsteady heat conduction.As shown in the following figure:Figure 2V olumetric controlled by internal nodeFor the two-dimensional unsteady heat conduction problems of constant material properties and no internal heat source,the internal nodes (i,j)represents the thermal eq uilibrium of the control volume is expressed as:In unit time,the heat flow rate jQ λand iQ λcoming from the adjacent control volume ),1()1j i j i +−，，（and)1,()1,(+−j i j i ，is equal to the control volume thermodynamic energy increase dU .dU Q Q j i =+λλ.Therefore,for each node Differential established as follows:tj,i 2222j,i y T x T t T ⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂α=⎟⎠⎞⎜⎝⎛∂∂................................................(d)for the left side:tT T t T (t j,i t t j ,i j ,i ∆−=∂∂∆+.................................................................(e)for the right side:2,1,,1,222)(x T T T x Tt j i t j i t j i j i ∆+−=∂∂−+........................................................(f)21,,1,,222(y T T T y Tt j i t j i t j i j i ∆+−=∂∂−+...................................................................(g)Making y x ∆=∆，2xF o ∆∆=∆τα，∆o F :called grid Fourier number.(e),(f)and (g)substituted into the formula (d).After finishing,the following equations can be obtained:t j i o t j i t j i t j i t j i o t t j i T F T T T T F T ,1,1,,1,1,)41()(∆+−+−∆∆+−++++=......(h)This equation can draw two conclusions:1).Any node temperature of an internal node at a particular moment can be obtained directly from the temperature of the node and its collar node at the former time.Do not have the simultaneous solution of the equations.This is the advantages of the explicit difference scheme.So we can start from the initial temperature sequentially obtainsτ∆,τ∆2,...,τ∆k each time nodes temperature.2).In front of the t j i T ,coefficient in the formula (h)can not be negative.So041≥−∆o F ,which is equivalent to 41≤∆o F ................................（i ）This is the stability conditions of two-dimensional non-steady-state thermal conductivity inside the object node temperature equation explicit difference scheme.Physical meaning:The temperature of any internal node at time t t ∆+all dependson the temperature of this node and its surrounding nodes at time t .When the temperature of surrounding nodes to are known,the higher the temperature of the node is,the higher the temperature of the node will be at time t t ∆+.Therefore,the coefficient in front of the formula can not be negative,which must meet the formula (i)that represents the stability condition.5.Solve temperature algebra equation and obtain the temperature of all nodes Assuming the area of each Brownie Pan is 2600cm A =and it will be heated for ing Matlab programming solve the problem of rectangular,round and oval pan heat conduct to the center from all sides into the incubator.As shown in the following figures:Figure 6The simulation isothermsFigure 7The simulation isothermsof rectangular panof circular panFigure 8The simulation isothermsof oval panyy102030405060708090100yWe can know from the figures above:whatever the shape of the pan is,the isothermal lines are arcuate curve,and they are nearly circular in its center position.Therefore,in the design of the shape of baking pan,we should try to design it in arc-shape in order to make the baking pan to be evenly heated.Temperature curve of the pan centerFigure 9Temperature curve of Figure 10Temperature curve ofthe rectangular pan centerthe circular pan centerFigure 11Temperature curve of oval pan center6.Conclusion⑴.The circular baking pan temperature rises very quickly.When the food reaches a certain temperature,the time required is the shortest.However,the space utilization of the pan in the rectangular oven is low,not efficient of using energy;⑵.Rectangular baking pan to maximize the use of space,effective use of energy.However,the temperature difference between the center and the periphery will cause the food being heated unevenly;⑶.The oval pan space utilization and degree of heat evenly is between the circular baking pan and rectangular baking pan.T e m p e r a t u r e /T e m p e r a t u r e /T e m p e r a t u r e /4Solutions Solutions::For the condition 1:Based on the assumption 4,it’s obvious that if we use a rectangular Brownie Pan,The Pans can exactly fulfill the oven.That’s to say,the number of pans in rectangular shape used is maximum.For the condition 2:①.How to determine the area where temperature is uniformly distributedWe use temperature variance of all the discrete points as a threshold value to determine whether a point is in the area where temperature is uniformly ly,whether its temperature can satisfy the following inequality:00ij T T div −≤∇..............................(j)If the above equation can be satisfied,we believe that this point is at the place where temperature is evenly distributed.②.The area of the region where temperature is uniformly distributedWhen we calculating the area of the region where temperature is uniformly distributed,We apply the previous concept of discretization.Firstly,we use MATLAB to calculate the number of points(N)in a given range,then,we can know that the area occupied by each discrete point isyx A d d d ×=...........................................(k)so,the total area occupied by all the discrete point isyx a d d N A ××=............................................(l)According to the temperature distribution figure of various Brownie Pan in geometric shapes in the first question,we measure the uniformity of the temperature distribution with the temperature variance22211((,))n navg i j T i j T n σ==−=∑∑..............................(m)and it’s obvious that:2rectangle22circle σσσ>>oval And we can know that only when Brownie Pan is round,can the heat on outer edge ofthe Brownie Pan be completely evenly distributed.For the condition 3:Based on the above two considerations,We think the combination of the above two optimization conditions we can use the following normalized method and get the Objective function ：(1)sA nZ p p N A=+−...................................(n)In addition,there are other restrictions:1.Different kinds of pan share the same area;2.The temperature reached after a specific time(6mins);3.The coefficient of thermal conductivity of the food is similar.(0.1)4.1Firstly,for rectangular Brownie Pans whose heating time is 6mins,we know in the objective function above,n=N,while the area where temperature is uniformly distributed ,A s ,need to be worked out.In this paper we can use the MATLAB program programmed in the Question 1,and we can get the area where temperature is uniformly distributed by directly substituting into the length and width of a rectangle,and the objective function turns into:(1)s AZ p p A=+−........................(o)the equation shows objective function becomes a single-valued function of p,and ithas no relationship with WL,while 'L NL =By using the MATLAB program written in the Analysis &Models,we can get we can obtain the output of 12sets of data(in the table 2),Fitting the relationship between a and b,and we can get the following function:2234.49()469.05599.16s W WA L L=−+Notes:to obtain their solutions,we assume that the area of a rectangle is 600cm 2,And the objective function turns into600/)16.59905.469(49.234)(1(2+−−+=LWL W p p Z With the constrains:..(0,1)[0,1]s t p W L∈∈And the optimal solution is Z=1,if and only if 1&Wp L=it can be satisfied Table 2Scatter values calculated by the MATLABL W 0.800.760.630.600.580.55A s /cm 2374.78378.69397.52402.94406.01412.92LW 0.490.450.420.380.350.32A s /cm 2426.43436.38444.33455.59464.53473.89Figure 12How the rectangular pans are placed in the oven4.2Secondly,for circular Brownie Pan which have been heated in the oven for 6mins,We already know from the question1that heat is evenly distributed on the outer edge of the round pan.Therefore,we can think in round Brownie Pans A A=and the objective function turns into:1nZ p p N=+−........................................(p)Figure 13The placement of round Brownie Pan in the ovenTo simplify the calculation,We make'W L for the fixed value k,namely 'Wk L =,And we know that by calculating the horizontal distance between two round pan is(n −............................(q)And we can obtain constraint condition of circular as following:(1n L −+≤.....................(r)4.3thirdly,according to the above calculation results,we think it will be a good ideato combine the advantages of round Brownie Pan and rectangle Brownie Pan.And we can draw the following diagram of ideal shape of Brownie Pan:We will have a study on the ideal Brownie Pan.In order to ensure the area of the pan unchanged,we can see how Brownie Pan's width changes:224'''2r r L L W rπ−=+−.............(s)To ensure the pan will not exceed the scope of the oven,which is impossible,we can get a restriction on the oven:224''2r r n L NL W r π⎛⎞−+≤⎜⎟−⎝⎠..........(t)Figure 14The ideal shape of the Brownie Pan And the distribution of the pans in the oven as follows.Figure 15The distribution of the pans in ideal shape in ovenAnd the objective function turns into:(1)s A nZ p p N A=+−........................................................(u)By using the MATLAB as well,we know that the optimal solution isWhen p=0.75and 0.8WL=,Z can get the optimal solution,andZ=1.215Taking that the oven has two racks into account ,all of the above data required to be multiplied by a expansion of the coefficient α=2,which has no influence on the relative difference among different shape.5Optimization of the modelIn the optimized design above,we consider that the Brownie Pans can only be placed in a row in the oven.Next,we intend to expand the previously conclusion.And we consider that the Brownie Pans can be placed in N rows in the oven.5.1Firstly,for rectangular Brownie PansFigure 16The placement of rectangular Brownie Pan in the ovenAt this point,we know that:the specifications of Brownie pan is:L W N M×It’s obvious that the Brownie Pan optimization function does not change,which still is:**(1)(1)**ss A A M N Z p p p p M N A A =+−=+− (v)Namely max 1Z =.5.2Secondly,for round Brownie Pans,we divide the extended the oven into threeparts,which is as follows:Figure 17The zoning figure of the extended ovenThe reasons why we adopt the above area dividing method are based on the following two considerations:1.To Simplify the calculation;2.To take advantage of the inequality (r)above.And we can obtain constraint condition of circular are as follows:1.Region 1:the number of the circulars is1([1)([]1)W Ln d d=−×−...........................................(w)Notes:[]tan W Ws ds for the Gauss number ofd d2.Region 2:We assume the number of the circulars is n 2and the constraint condition is(21n L −≤..............................(x)Notes:L L =',d dWW W )1(['−−=3.Region 2:We assume the number of the circulars is n 3and the constraint condition is(31'n L −≤....................(y)Notes:d n L L 1''−=,d n W 1''=and it’s obvious that max 1Z <5.3Thirdly,for Brownie Pans in theoretical optimal shapeFigure18The arrangement of Brownie Pans in theoretical optimal shape in the oven It’s obvious that in the oven that can arrange multi-row Brownie Pans the optimal solution keeps unchanged.Namely,Z f=Z=1.215From all above,we can know that the Brownie Pan in the theoretical optimal shape can get greater optimal solution than either rectangular pans or round pans.So it’s a smart idea to choose a Brownie Pan in the theoretical optimal shape to cook your Brownie cakes!6References[1]Lars Mönch Robert Unbehaun You In Choung,Minimizing Earliness–Tardiness on a Single Burn-in Oven with a Common due Date and Maximum Allowable Tardiness Constraint, /static/pdf/374/art%253A10.1007%252Fs00291-005-0013-4.pdf?aut h66=1360982955_54758b58d2754d71008329400df3c11e&ext=.pdf,10Dec.2005[2]K.Venkateshmurthy&K.S.M.S.Raghavarao,Analysis of Modes of Heat Transfer in Baking Indian Rice Pan Cake(Dosa,)a Breakfast Food, /static/pdf/517/art%253A10.1007%252Fs13197-010-0204-0.pdf?aut h66=1360983288_e6bf23e753296afd9fcd43eea80dad4a&ext=.pdf,06Dec.2010[3]Jiang Qiyuan and Xie Jinxing,Mathematical model Mathematical programming model,3rd edn, Mathematical Modeling,Accessed Feb.2003[4]D.Pitts et al,Schaum's Outline of Theory and Problems of Heat Transfer,2nd edn,Science Press,2002[5]M.N.Aoqi Sigg,Heat Conduction,Higher Education Press,1984A dvertising sheet:AppendixFigure3Distribution of temperature of rectangular pan199.5198.5Figure4Distribution of temperature of circular panFigure5Distribution of temperature of oval pan。

2013年美国数学建模竞赛B题

水资源计划摘要本文是要设计一个有效的，可行的，低成本的用水计划，来满足某国2025年的用水需求。

我们选择中国为研究对象，根据中国各地区历年的水资源总量并求出其均值，参考各地区历年用水总量来预测2025年的用水总量，将两者相减得出差值，并以此为依据将中国各地区分为缺水地区，不缺水地区，水资源丰富地区三类。

经研究分析有两种可行性高的方案。

第一种，由水资源丰富地区向缺水地区提供水。

第二种，是由沿海缺水城市进行海水淡化并运往其他缺水城市。

我们主要考虑经济因素对两种方案进行分析研究，最终得出结论由水资源丰富地区铺设管道向缺水地区提供水为最优方案。

并以各省的省会作为核心城市，说明全省的需水和调水情况，并以省会城市或直辖市为顶点构成一个赋权图，即把问题转换为求水资源丰富地区到缺水地区的最短路问题，并用图论的知识来解决问题。

在此基础上考虑到此方案会改变就业，生产力，水资源利用等因素，从而对经济，物理，环境产生不同程度的影响，并用层次分析加以研究，最终以报告的方式向政府反映。

关键词：回归分析最小生成树层次分析法一、问题重述淡水是世界大部分地区的发展限制。

试建立一个数学模型，用来确定一个有效的、可行的和低成本的水资源战略，以满足2025年预计的用水需求，特别是，您的数学模型必须解决存储和输送，去盐碱化和环境保护等问题。

如果可能的话，用你的模型探讨此战略在经济，物理和环境等方面的影响。

试提供一个非技术性的文件，向政府相关部门介绍你的方法以及其可行性和成本，并说明为什么它是“最好的水战略”。

二、符号说明ˆy：预测得出的2025年用水量；S：输水的造价；1S：海水淡化的造价；2d1: 输水工程的单位造价；d2：海水淡化的单位造价；2R：拟合度.三、模型假设1.从2013年到2025年各外部因素对水资源总量无影响，例如：雪灾、地震、洪水、战争等对环境的影响；2.各地区海水淡化单位费用相同；3.不同地区淡水转移的单位费用相同；4.人们的消费水平及劳动力费用不会随意外事故发生明显改变。

2013年美国大学生数学建模竞赛(MCMICM)参赛规则中英文对照

2013年美国大学生数学建模竞赛（MCMICM）参赛规则中英文对照2 ICM:The InterdisciplinaryContest in ModelingICM：交叉学科建模竞赛ContestRules, Registration and Instructions比赛规则，报名注册和指导(All rules and instructions apply to both ICM and MCM contests, except where otherwisenoted.)（所有MCM的说明和规则除特别说明以外都适用于ICM）To participate in a contest, each team must be sponsored by a faculty advisor fromits institution.参加MCM的每个队伍需有一名在职的高校老师负责指导。

TeamAdvisors: Please read these instructions carefully. It isyour responsibility to make sure that teams are correctly registered and thatall of the following steps required for participation in the contest arecompleted:Pleaseprint a copy of these contest instructions for reference before, during, andafter the contest. Click here for the printer friendly version.指导老师：请认真阅读这些说明事项，确保完成了所有相关的项。

每位指导教师的责任包括确保每个参赛队正确注册并正确完成参加MCM/ICM所要求的相关步骤。

2013年美赛数模A题答案

中国水资源战略摘要Summary为了确定中国最佳的水资源战略，将中国分为九大流域，首先借助MATLAB建立多项式拟合模型来预测出中国2013年到2025年每年各流域的供水量和需水量，接着在可持续发展的原则指导下建立区域水资源合理配置模型，对每一个流域，采用水资源综合短缺度最小为目标函数, 对地表水、地下水等多种水源统筹考虑, 用权重区别对待工业、农业、生活、生态环境等不同领域的用水需求, ,从而求出各个流域最小的缺水量。

再根据前面的两个模型所预测出来的各流域的缺水量，建立最佳的补水模型解决缺水问题：通过对实际问题的分析，可能的补水方案有两个：方案一是直接从珠江流域调水到缺水的流域，方案二是沿海流域采取海水淡化补水，内陆流域采取直接从珠江流域调水过去，经过分析、计算发现方案二是最佳的。

最后，我们统筹考虑我们所制定的水策略，发现其无论是对经济、社会还是生态环境都将产生重大影响。

In order to determine the best water resources strategy, we divided China into nine basins. Firstly, we established polynomial fitting model with the use of MATLAB to predict the water supply and the water demand of every basin from 2013 to 2025. Secondly, we established the regional water resources rational allocation model under the guidance of the principle of sustainable development. In this model, through taking the minimum comprehensive water shortage degree as objective , surface water , groundwater and other water are considered, and different weightings are used for industrial, agricultural, domestic and ecological water users in order to realize regional water resources rational allocation .In this way can we obtained the minimum amount of water scarcity in every basin. Thirdly, according to the data predicted based on the previous two models, we can establish the optimal replenishment model to solve the problem of water shortage. We identified two possible replenishment program based on the analysis of the actual problems. One is to transfer the water of the Pearl River to basins where lack of water resources, another is to transfer the water of the Pearl River to inland basins directly while we meet the water shortage of coastal basins by desalination. After analysis and calculation, we find second program is the best. Finally, we find the water strategy we developed has a significant impact on the economic, social and ecological environment after we considered the models we established.关键字：水策略多项式拟合模型区域水资源合理配置模型补水模型Keywords：Water strategythe Polynomial fitting modelThe Regional water resources rational allocation modelthe Replenishment model§1.问题重述Problem restatement水是生命之源, 是人类生存和发展不可替代的资源, 是经济、社会可持续发展的基础。

2013年数学建模美赛B题论文

2013建模美赛B题思路摘要水资源是极为重要生活资料，同时与政治经济文化的发展密切相关，北京市是世界上水资源严重缺乏的大都市之一。

本文以北京为例，针对影响水资源短缺的因素，通过查找权威数据建立数学模型揭示相关因素与水资源短缺的关系，评价水资源短缺风险并运用模型对水资源短缺问题进行有效调控。

首先，分析水资源量的组成得出影响因素。

主要从水资源总量（供水量）和总用水量（需水量）两方面进行讨论。

影响水资源总量的因素从地表水量，地下水量和污水处理量入手。

影响总用水量的因素从农业用水，工业用水，第三产业及生活用水量入手进行具体分析。

其次，利用查得得北京市2001-2008年水量数据，采用多元线性回归，建立水资源总量与地表水量，地下水量和污水处理量的线性回归方程yˆ=-4.732+2.138x1+0.498x2+0.274x3根据各个因数前的系数的大小，得到风险因子的显著性为r x1>r x2>r x3(x1, x2，x3分别为地表水、地下水、污水处理量)。

再次，利用灰色关联确定农业用水、工业用水、第三产业及生活用水量与总用水量的关联程度r a=0.369852，r b= 0.369167，r c=0.260981。

从而确定其风险显著性为r a>r b>r c。

再再次，由数据利用曲线拟合得到农业、工业及第三产业及生活用水量与年份之间的函数关系，a=0.0019（t-1994）3-0.0383（t-1994）2-0.4332（t-1994）+20.2598;b=0.014（t-1994）2-0.8261t+14.1337;c=0.0383（t-1994）2-0.097（t-1994）+11.2116;D=a+b+c；预测出2009-2012年用水总量。

最后，通过定义缺水程度S=（D-y）/D=1-y/D，计算出1994-2008的缺水程度，绘制出柱状图，划分风险等级。

我们取多年数据进行比较，推测未来四年地表水量和地下水量维持在前八年的平均水平，污水处理量为近三年的平均水平，得出2009-2012年的预测值，并利用回归方程yˆ=-4.732+2.138x1+0.4982x2+0.274x3计算出对应的水资源总量。

2013年美赛数学建模b题的分析

Байду номын сангаас
美国2025年可提供的实际淡水量怎样预测？
（可以先预测不同地区不同领域供水量，再相加）
如果实际淡水量小于需求量，可通过那些方法提供不足的淡水？（例如：海水淡化、不同地区淡水转移）
这些方法那个更好？（比较的依据是什么？例如费用，环境）需要做出那些假设？
1. 从现在到2025年之前不会有战争，自然灾害等影响因素 2.同一个地区的居民年用水量相同 3.同一个地区的居民量的年增长率为常数 4.不同地区海水淡化的单位费用相同 5.不同地区淡水转移的单位费用相同
注：模型假设一般是在问题分析中根据需要提出的，所以可以先提出基本假设，以后再补充
Vx pre S
1.预测2025年供水量的模型（以一个州的降水量为例）
其中，表示州i 的可用降水量，表示该州的单位面积平均年降水量， Si 表示该州的淡水地表面积
Vx pre S
2.预测2025年需水量的模型（以一个州的居民用水量为例）
若 V大于0，则说明2025年美国的淡水量是足够的，然后再看每个州需要补充的淡水量，即 V V ，
si ri
若每个州需要补充的淡水量均大于0，则不需要采取任何措施，
否则，需要在不通州之间进行淡水调度。
若 V小于0，则说明2025年美国的淡水量是不足的，需要采取海水淡化等措施
注：为使表达更清晰，可以在论文中采用图、表等形式
其中，表示州i 在2025年人口数量，表示该州在2010年人口数量，表示该州的人口平均年增长率则该州在2025年的居民用水量为
i Vri w N 2 0 0 5
其中，w 表示该州居民的平均年用水量
3. 2025年美国需要补充的淡水量

2013年全国大学生数学建模竞赛A题

车道被占用对城市道路通行能力的影响摘要在城市道路常会发生交通异常事件，导致车道被占用，事发地段的通行能力也会因此受到影响。

当交通需求大于事发断剩余通行能力时，车辆排队，产生延误，行程时间增加，交通流量发生变化。

根据这些特点，我们以城市道路基本路段发生交通事故为例，主要分析了交通事故发生后道路的通行能力的变化，以及不同时间段事故点及其上下游路段交通流量的变化，用于以后进一步突发事件下交通流的预测。

针对问题一，根据道路通行能力的定义，考虑到车身大小不同，我们把所有车辆进行标准化。

运用统计估算模型对视频一的车辆进行分段统计，得出未发生事故前道路通行能力2555(辆/h )。

因为车辆所占车道未达到数学理论计算要求，所以我们利用修正过后城市干道通行能力的数学计算模型，计算出交通事故发生至撤离期间的理论通行能力为1356（辆/h ），进而与实际数据对比，得出相对误差。

针对问题二，我们基于问题一的模型，以及附件三数据分析所得，不同车道的通行流量比例不同，对视频二的车辆各项数据的分段统计分析，得到道路实际通行能力。

再根据修正的理论数学计算模型，得出理论通行能力。

得到的结果与问题一的结果相比较，得出结论：在同一横断面上的实际通行能力与交通事故所占车道的车流量呈负相关性。

针对问题三，我们运用了两种模型，一种结合层次分析与线性回归模型，得到理想化的函数关系式。

基于层次分析模型，我们将进行问题分解，把车辆长度作为目标层，其他三个量作为准则层。

通过查阅资料对各因素进行打分，计算出事故持续时间、车道通行能力、上游车流量对车辆排队长度的权重。

层次分析模型得到各个指标对目标层的影响关系的大小，然后我们用线性回归模型求出各指标与目标层的具体的函数关系式为130.0430.09263.623y x x =-+-。

第二，我们运用车流波动相关理论，得到理论模型，继而得出它们之间的关系。

针对问题四，我们首先考虑的是上游来车在红绿灯下的时间间断问题，所以把来车的情况作周期性分析，假设来车是间隔相同的时间连续的到来，求出一个周期能通过的最大车流量数。

2013美赛结果分析

2013年美国大学生数学建模竞赛结果发布COMAP非常高兴地宣布第29届大学生数学建模竞赛（MCM）结果。

今年共有5636支队伍参加了比赛，分别代表14个国家和地区。

以下11支队伍提交的论文被评定为优胜论文（OUTSTANDING WINNERS）：Beijing Univ. of Posts and Telecomm, China（北京邮电大学：郭众鑫、吴帆、王蓓丹；指导教师：贺祖国）Bethel University, Arden Hills, MNColorado College, Colorado Springs, COFudan University, China（复旦大学：王坤睿、许晶、曾溦；指导教师：杨翎）Nanjing University, China（南京大学：陈炜、刘威志、杨岑莹；指导教师：瞿慧）Peking University, China（北京大学：金冲、刘博闻、吴蒙; 指导教师：刘旭峰）Shandong University, China（山东大学：宋炎侃、徐珂、伊凡；指导教师：Hengxu Zhang）Shanghai Jiaotong University, China（上海交通大学:文理斌、吴婧元、王聪; 指导教师：Yuehui Zhang）Tsinghua University, China（清华大学: 高鹏飞、何博硕、邹天忻；指导教师：吴昊）University of Colorado Boulder, Boulder, CO (2)今年的竞赛时间是从2013年1月31日（星期四）到2013年2月4日（星期一）。

在这段时间里，由三名学生组成的本科生或高中生队伍从两个竞赛问题中选择一个，认真研究并建模，最终提交一份解决方案。

今年MCM的主要形式通过网络展开。

参赛队伍需要在规定的时间内通过COMAP的MCM网站注册、获得竞赛材料并下载题目和数据。

今年MCM的两个问题被公认为具有很大的挑战性。

2000-2013美国数学建模竞赛(MCM、ICM)历年试题汇总(1)

目录2000 年美国大学生数学建模竞赛MCM、ICM 试题 (3)2000 MCM A: Air Traffic Control (3)2000 MCM B: Radio Channel Assignments (3)2000 ICM: Elephants: When is Enough, Enough? (5)2001 年美国大学生数学建模竞赛MCM、ICM 试题 (7)2001 MCM A: Choosing a Bicycle Wheel (7)2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...). (8)2001 ICM: Our Waterways - An Uncertain Future (10)2002 年美国大学生数学建模竞赛MCM、ICM 试题 (14)2002 MCM A: Wind and Waterspray (14)2002 MCM B: Airline Overbooking (14)2002 ICM: Scrub Lizards (15)2003 年美国大学生数学建模竞赛MCM、ICM 试题 (19)2003 MCM A: The Stunt Person (19)2003 MCM B: Gamma Knife Treatment Planning (19)2003 ICM: Aviation Baggage Screening Strategies: To Screen or Not to Screen, that is the Question (20)2004 年美国大学生数学建模竞赛MCM、ICM 试题 (24)2004 MCM A: Are Fingerprints Unique? (24)2004 MCM B: A Faster QuickPass System (24)2004 ICM: To Be Secure or Not to Be? (24)2005 年美国大学生数学建模竞赛MCM、ICM 试题 (25)2005 MCM A: Flood Planning (25)2005 MCM B: Tollbooths (25)2005 ICM: Nonrenewable Resources (25)2006 年美国大学生数学建模竞赛MCM、ICM 试题 (27)2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation (27)2006 MCM B: Wheel Chair Access at Airports (27)2006 ICM: Trade-offs in the fight against HIV/AIDS (28)2007 年美国大学生数学建模竞赛MCM、ICM 试题 (32)2007 MCM A: Gerrymandering (32)2007 MCM B: The Airplane Seating Problem (32)2007 ICM: Organ Transplant: The Kidney Exchange Problem (33)2008 年美国大学生数学建模竞赛MCM、ICM 试题 (38)2008 MCM A: Take a Bath (38)2008 MCM B: Creating Sudoku Puzzles (38)2008 ICM: Finding the Good in Health Care Systems (38)2009 年美国大学生数学建模竞赛MCM、ICM 试题 (40)2009 MCM A: Designing a Traffic Circle (40)2009 MCM B: Energy and the Cell Phone (40)2009 ICM: Creating Food Systems: Re-Balancing Human-Influenced Ecosystems41 2010年美国大学生数学建模竞赛 MCM、ICM 试题 (42)2010 MCM A: The Sweet Spot (42)2010 MCM B: Criminology (43)2010 ICM: The Great Pacific Ocean Garbage Patch (44)2011年美国大学生数学建模竞赛 MCM、ICM 试题 (45)2011 MCM A: Snowboard Course (45)2011 MCM B: Repeater Coordination (45)2011 ICM: Environmentally and Economically Sound (46)2012年美国大学生数学建模竞赛 MCM、ICM 试题 (48)2012 MCM A: The Leaves of a Tree (48)2012 MCM B: Camping along the Big Long River (50)2012 ICM: Modeling for Crime Busting (51)2013年美国大学生数学建模竞赛 MCM、ICM 试题 (59)2013 MCM A: The Ultimate Brownie Pan (59)2013 MCM B: Water, Water, Everywhere (61)2013 ICM: NetworkModeling of Earth's Health (62)2000 年美国大学生数学建模竞赛MCM、ICM 试题2000 MCM A: Air Traffic ControlTo improve safety and reduce air traffic controller workload, the Federal Aviation Agency (FAA) is considering adding software to the air traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analysit at the FAA has posed the following problems.Requirement A: Given two airplanes flying in space, when should the air traffic controller consider the objects to be too close and to require intervention? Requirement B: And airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how do we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of aircraft simultaneously passing through that sector1.at any one instant?2.during any given interval of time?3.during a particular time of day?How does the number of potential conflicts arising during those periods affect complexity? Does the presence of additional software tools to automatically predict conflicts and alert the controller reduce or add to this complexity? In addition to the guidelines for your report, write a summary (no more than two pages) that the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your conclusions.2000 MCM B: Radio Channel AssignmentsWe seek to model the assignment of radio channels to a symmetric network of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grix (honeycomb-style), as shown in Figure 1, where a transmitter is located at the center of each hexagon.An interval of the frequency spectrum is to be alloted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1,2,3, … . Each transmitter wil be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided.Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assugn channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span.Let s be the length of a side of one of the hexagons. We concentrate on the case that there are two levels of interference.Requirement A: There are several contrainsts on the frequency assignments. First, no two transmitters within distance 4s of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these contraints, what can we say about the span in Figure 1?Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in all directions.Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance 2s differ by at least some given integer k, while those at distance at most 4s must still differ by at least one. What cna we say about the span and about efficient strategies for designing assignments, as a function of k?Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider?Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining your findings.2000 ICM: Elephants: When is Enough, Enough?“Ultimately, if a habitat is undesirably changed by elephants, then their removal should be considered -even by culling.”National Geographic (Earth Almanac) –December 1999 A large National Park in South Africa contains approximately 11,000 elephants. Management policy requires a healthy environment that can maintain a stable herf of 11,000 elephants. Each year park rangers count the elephant population. During the past 20 years whole herds have been removed to keep the population as close to 11,000 as possible. The process involved shooting (for the most part) and occasionally relocating approximately 600 to 800 elephants per year.Recently, there has been a public outcry against the shooting of these elephants. In addition, it is no longer feasible to relocate even a small population of elephants each year. A contraceptive dart, however, has been developed that can prevent a mature elephant cow from conceiving for a period of two years.Here is some information about eh elephants in the Park:∙There is very little emigration of immigration of elephants.∙The gender ratio is very close to 1:1 and control measures have endeavored to maintain parity.∙The gender ratio of newborn calves is also about 1:1. Twins are bornaverage, a calf every 3.5 years until they reach an age of about 60.Gestation is approximately 22 months.∙The contraceptive dart causes an elephant cow to come into oestrus every month (but not conceiving). Elephants usually have courtship only once in 3.5 years, so the monthly cycle can cause additional stress.∙ A cow can be darted every year without additional detrimental effects. A mature elephant cow will not be able to conceive for 2 years after thelast darting.∙Between 70% and 80% of newborn calves survive to age 1 year.Thereafter, the survival rate is uniform across all ages and is very high(over 95%), until about age 60; it is a good assumption that elephantsdie before reading age 70.There is no hunting and negligible poaching in the Park.The park management has a rough data file of the approximate ages and2 years. This data is available on website: icm2000data.xls. Unfortunately no data is available for the elephants that have been shot or remain in the Park.Your overall task is to develop and use models to investigate how the contraceptive dart might be used for population control. Specifically:Task 1: Develop and use a model to speculate about the likely survival rate for elephants aged 2 to 60. Also speculate about the current age structure of the elephant population.Task 2: Estimate how many cows would need to be darted each year to keep the population fixed at approximately 11,000 elephants. Show how the uncertainty in the data at your disposal affects your estimate. Comment on any changes in the age structure of the population and how this might affect tourists. (You may want to look ahead about 30-60 years.)Task 3: If it were feasible to relocate between 50 and 300 elephants per year, how would this reduce the number of elephants to be darted? Comment on the trade-off between darting and relocation.large number of elephants (due to disease or uncontrolled poaching), even if darting stopped immediately, the ability of the population to grow again would be seriously impeded. Investigate and respond to this concer.Task 5: The management in the Park is skeptical about modeling. In particular, they argue that a lack of complete data makes a mockery of any attempt to use models to guide their decision. In addition to your technical report, include a carefully crafted report (3-page maximum) written explicitly for the park management that responds to their concerns and provides advice. Also suggest ways to increase the park managers confidence in your model and your conclusions.Task 6: If your model works, other elephant parks in Africa would be interested in using it. Prepare a darting plan for parks of various sizes (300-25,000 elephants), with slightly different survival rates and transportation possibilities.2001 年美国大学生数学建模竞赛MCM、ICM 试题2001 MCM A: Choosing a Bicycle WheelCyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike.Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course.Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the right.The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear.Task 1. Provide a table giving the wind speed at which the power required for a solid rear wheel is less than for a spoked rear wheel. The table should include the wind speeds for different road grades startingfrom zero percent to ten percent in one percent increments. (Roadgrade is defined to be the ratio of the total rise of a hill divided by thelength of the road. If the hill is viewed as a triangle, the grade is the sine of the angle at the bottom of the hill.) A rider starts at the bottom of the hill at a speed of 45 kph, and the deceleration of the rider is proportionalto the road grade. A rider will lose about 8 kph for a five percent grade over 100 meters.∙Task 2. Provide an example of how the table could be used for a specific time trial course.∙Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision.2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...)Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare.The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading inland from Myrtle Beach and Hilton Head is also planned.A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of the coastal strip. (More accurate data, if sought, are widely available.)The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three. Columbia, another metro area of around 500,000 people, does not have sufficient hotel space to accommodate the evacuees (including some coming from farther north by other routes), so some traffic continues outbound on I-26 towards Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. In 1999, traffic leaving Columbia going northwest was moving only very slowly. Construct a model for the problem to investigate what strategies may reduce the congestion observed in 1999. Here are the questions that need to be addressed:1.Under what conditions does the plan for turning the two coastal-boundlanes of I-26 into two lanes of Columbia-bound traffic, essentiallyturning the entire I-26 into one-way traffic, significantly improveevacuation traffic flow?2.In 1999, the simultaneous evacuation of the state's entire coastal regionwas ordered. Would the evacuation traffic flow improve under analternative strategy that staggers the evacuation, perhapscounty-by-county over some time period consistent with the pattern of how hurricanes affect the coast?3.Several smaller highways besides I-26 extend inland from the coast.Under what conditions would it improve evacuation flow to turn around traffic on these?4.What effect would it have on evacuation flow to establish moretemporary shelters in Columbia, to reduce the traffic leaving Columbia?5.In 1999, many families leaving the coast brought along their boats,campers, and motor homes. Many drove all of their cars. Under whatconditions should there be restrictions on vehicle types or numbers ofvehicles brought in order to guarantee timely evacuation?6.It has been suggested that in 1999 some of the coastal residents ofGeorgia and Florida, who were fleeing the earlier predicted landfalls ofHurricane Floyd to the south, came up I-95 and compounded the traffic problems. How big an impact can they have on the evacuation trafficflow? Clearly identify what measures of performance are used tocompare strategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.Clearly identify what measures of performance are used to compare strategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.2001 ICM: Our Waterways - An Uncertain FutureZebra mussels, Dreissena polymorpha, are small, fingernail-sized, freshwater mollusks unintentionally introduced to North America via ballast water from a transoceanic vessel. Since their introduction in the mid 1980s, they have spread through all of the Great Lakes and to an increasing number of inland waterways in the United States and Canada. Zebra mussels colonize on various surfaces,such as docks, boat hulls, commercial fishing nets, water intake pipes and valves, native mollusks and other zebra mussels. Their only known predators, some diving ducks, freshwater drum, carp, and sturgeon, are not numerous enough to have a significant effect on them. Zebra mussels have significantly impacted the Great Lakes ecosystem and economy. Many communities are trying to control or eliminate these aquatic pests. SOURCE: Great Lakes Sea Grant Network /.Researchers are attempting to identify the environmental variables related to the zebra mussel infestation in North American waterways. The relevant factors that may limit or prevent the spread of the zebra mussel are uncertain. You will have access to some reference data to include listings of several chemicals and substances in the water system that may affect the spread of the zebra mussel throughout waterways. Additionally, you can assume individual zebra mussels grow at a rate of 15 millimeters per year with a life span between 4 - 6 years. The typical mussel can filter 1 liter of water each day.Requirement A: Discuss environmental factors that could influence the spread of zebra mussels.Requirement B: Utilizing the chemical data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAChem1.xls, and the mussel population data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAPopulation 1.xls model the population growth of zebra mussels in Lake A. Be sure to review the Information about the collection of the zebra mussel data. Requirement C: Utilizing additional data on Lake A from another scientist provided at :ap/undergraduate/contests/icm/imagesdata/LakeAChem2.xls and additional mussel population data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAPopulation 2.xls corroborate the reasonableness of your model from Requirement B. As a result of this additional data, adjust your earlier model. Analyze the performance of your model. Discuss the sensitivity of your model. Requirement D: Utilizing the Chemical data from two lakes (Lake B and Lake C) in the United States provided atap/undergraduate/contests/icm/imagesdata/LakeB.xls and ap/undergraduate/contests/icm/imagesdata/LakeC.xls determine if these lakes are vulnerable to the spread of zebra mussels. Discuss your prediction.Requirement E: The community in the vicinity of Lake B (in requirement D) is considering specific policies for the de-icing of roadways near the lake duringthe winter season. Provide guidance to the local government officials regarding a policy on “de-icing agents.”In your guidance include predictions on the long-term impact of de-icing on the zebra mussel population. Requirement F: It has been recommended by a local community in the United States to introduce round goby fish. Zebra mussels are not often eaten by native fish species so they represent a dead end ecologically. However, round gobies greater than 100 mm feed almost exclusively on zebra mussels. Ironically, because of habitat destruction, the goby is endangered in its native habitat of the Black and Caspian Seas in Russia. In addition to your technical report, include a carefully crafted report (3-page maximum) written explicitly for the local community leaders that responds to their recommendation to introduce the round goby. Also suggest ways to help reduce the growth of the mussel within and among waterways.Information about the collection of the zebra mussel dataThe developmental state of the Zebra mussel is categorized by three stages: veligers (larvae), settling juveniles, and adults. Veligers (microscopic zebra mussel larvae) are free-swimming, suspended in the water for one to three weeks, after which they begin searching for a hard surface to attach to and begin their adult life. Looking for zebra mussel veligers is difficult because they are not easily visible by the naked eye. Settled juvenile zebra mussels can be felt on smooth surfaces like boats and motors. An advanced zebra mussel infestation can cover a surface, even forming thick mats sometimes reaching very high densities. The density of juveniles was determined along the lake using three 15×15 cm settling plates. The top plate remained in the water for the entire sampling season (S - seasonal) to estimate seasonal accumulation. The middle and bottom plates are collected after specific periods (A –alternating ) of time denoted by “Lake Days”in the data files.The settling plates are placed under the microscope and all juveniles on the undersides of the plate are counted and densities are reported as juveniles/m^2.2002 年美国大学生数学建模竞赛MCM、ICM 试题2002 MCM A: Wind and WatersprayAn ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area. Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.2002 MCM B: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat.Historically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook-that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations. Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.Consider the overbooking issue in light of the current situation: Less flights by airlines from point A to point B Heightened security at and around airports Passengers' fear Loss of billions of dollars in revenue by airlines to dateBuild a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the company's revenue is maximized. Insure that your model reflects the issues above, andconsider alternatives for handling “bumped”passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.2002 ICM: Scrub LizardsThe Florida scrub lizard is a small, gray or gray-brown lizard that lives throughout upland sandy areas in the Central and Atlantic coast regions of Florida. The Florida Committee on Rare and Endangered Plants classified the scrub lizard as endangered.You will find a fact sheet on the Florida Scrub Lizard at/undergraduate/contests/mcm/contests/2002/problem s/icm2002data/scrublizard.pdfThe long-term survival of the Florida scrub lizard is dependent upon preservation of the proper spatial configuration and size of scrub habitat patches.Task 1: Discuss factors that may contribute to the loss of appropriate habitat for scrub lizards in Florida. What recommendations would you make to the state of Florida to preserve these habitats and discuss obstacles to the implementation of your recommendations?average fecundity of adult lizards); Sj (the survivorship of juvenile lizards- between birth and the first reproductive season); and Sa (the average adult survivorship).Table 1Summary data for a cohort of scrub lizards captured and followed for 4 consecutive years. Hatchling lizards (age 0) do not produce eggs during the summer they are born. Average clutch size for all other females is proportional to body size according to the function y = 0.21*(SVL)-7.5, where y is the clutch size and SVL is the snout-to-vent length in mm.Year Age Total NumberLivingNumber of LivingFemalesAvg. Female Size(mm)1 0 972 495 30.32 1 180 92 45.83 2 20 11 55.84 3 2 2 56.0Task 3: It has been conjectured that the parameters Fa , Sj , and Sa , are related to the size and amount of open sandy area of a scrub patch. Utilize the data provided in Table 2 to develop functions that estimate Fa, Sj , and Sa for different patches. In addition, develop a function that estimates C, the carrying capacity of scrub lizards for a given patch.Table 2Summary data for 8 scrub patches including vital rate data for scrub lizards. Annual female fecundity (Fa), juvenile survivorship (Sj), and adult survivorship (Sa) are presented for each patch along with patch size and the amount of open sandy habitat.Patch Patch Size (ha) Sandy Habitat (ha) Fa Sj Sa Density (lizards/ha)a 11.31 4.80 5.6 0.12 0.06 58b 35.54 11.31 6.6 0.16 0.10 60c 141.76 51.55 9.5 0.17 0.13 75d 14.65 7.55 4.8 0.15 0.09 55e 63.24 20.12 9.7 0.17 0.11 80f 132.35 54.14 9.9 0.18 0.14 82g 8.46 1.67 5.5 0.11 0.05 40h 278.26 84.32 11.0 0.19 0.15 115Task 4: There are many animal studies that indicate that food, space, shelter, or even reproductive partners may be limited within a habitat patch causing individuals to migrate between patches. There is no conclusive evidence on why scrub lizards migrate. However, about 10 percent of juvenile lizards do migrate between patches and this immigration can influence the size of the population within a patch. Adult lizards apparently do not migrate. Utilizing the data provided in the histogram below estimate the probability of lizards surviving the migration between any two patches i and patch j.Table 3HistogramMigration data for juvenile lizards marked, released, and recaptured up to 6 months later. Surveys for recapture were conducted up to 750m from release sites.Task 5: Develop a model to estimate the overall population size of scrub lizards for the landscape given in Table 3. Also, determine which patches are suitable for occupation by scrub lizards and which patches would not support a viable population.Patch size and amount of open sandy habitat for a landscape of 29 patches located on the Avon Park Air Force Range. See:/undergraduate/contests/icm/2002problem/map.jpg for a map of the landscape.Patch Identification Patch Size (ha) Sandy Habitat (ha)1 13.66 5.382 32.74 11.913 1.39 0.234 2.28 0.765 7.03 3.626 14.47 4.387 2.52 1.998 5.87 2.499 22.27 8.44。

2013年美国数学建模B题

PROBLEM B: Water, Water, EverywhereFresh water is the limiting constraint for development in much of the world. Build a mathematical model for determining an effective, feasible, and cost-efficient water strategy for 2013 to meet the projected water needs of [pick one country from the list below] in 2025, and identify the best water strategy. In particular, your mathematical model must address storage and movement; de-salinization; and conservation. If possible, use your model to discuss the economic, physical, and environmental implications of your strategy. Provide a non-technical position paper to governmental leadership outlining your approach, its feasibility andcosts, and why it is the “best water strategy choice.”Countries: United States, China, Russia, Egypt, or Saudi Arabia水, 水, 无处不在(美国竞赛2013年B题)淡水资源逐渐成为这个世界大多数国家发展的极限约束。

2013高教社杯全国大学生数学建模竞赛A题长江水质的评价和预测

2013高教社杯全国大学生数学建模竞赛承诺书我们仔细阅读了《全国大学生数学建模竞赛章程》和《全国大学生数学建模竞赛参赛规则》（以下简称为“竞赛章程和参赛规则”，可从全国大学生数学建模竞赛网站下载）。

我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。

我们知道，抄袭别人的成果是违反竞赛章程和参赛规则的，如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。

我们郑重承诺，严格遵守竞赛章程和参赛规则，以保证竞赛的公正、公平性。

如有违反竞赛章程和参赛规则的行为，我们将受到严肃处理。

我们授权全国大学生数学建模竞赛组委会，可将我们的论文以任何形式进行公开展示（包括进行网上公示，在书籍、期刊和其他媒体进行正式或非正式发表等）。

我们参赛选择的题号是（从A/B/C/D中选择一项填写）：我们的参赛报名号为（如果赛区设置报名号的话）：所属学校（请填写完整的全名）：参赛队员(打印并签名) ：1.2.3.指导教师或指导教师组负责人(打印并签名)：（论文纸质版与电子版中的以上信息必须一致，只是电子版中无需签名。

以上内容请仔细核对，提交后将不再允许做任何修改。

如填写错误，论文可能被取消评奖资格。

）日期：年月日赛区评阅编号（由赛区组委会评阅前进行编号）：2013高教社杯全国大学生数学建模竞赛编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）：长江水质的评价和预测摘要本文在所给资料的基础上通过建立数学模型和各种分析的方法对长江水质问题做出了水质评测，定位分析，还对未来十年长江污水治理问题做出了预测。

针对问题一：我们根据长江各流域各月份的污染物指标统计出各地区的污染物指标状况，然后通过各指标的污染比例权重建立模型计算各地段总的污染情况。

2013全国大学生数学建模竞赛C题参考答案

2013全国大学生数学建模竞赛C题参考答案第一篇：2013全国大学生数学建模竞赛C题参考答案2013高教社杯全国大学生数学建模竞赛C题评阅要点[说明]本要点仅供参考，各赛区评阅组应根据对题目的理解及学生的解答，自主地进行评阅。

问题1(1)补充1986年和1996年缺失的数据（第13层第5点），可用外推法或几何方法补充数据。

(2)因各层基本处于同一平面内，可先拟合出各层所在平面，将各测量点投影到拟合平面内，然后再用均匀物体的重心公式计算中心坐标。

注：(1)对1986年和1996年第13层，不补充数据，直接用7个点的数据计算中心坐标是错误的。

(2)用各层测量点坐标的平均值作为中心点坐标，不是一种好方法。

问题2(1)倾斜程度：对中心点作线性拟合，中轴线与水平面法向的夹角可作为倾斜程度的度量。

(2)弯曲程度：对中心点作三次样条拟合，三次样条曲线各点曲率的平均值可作为弯曲程度的度量。

也可用离散方法：连接各层的对应点，折线各顶点角度的平均值可作为弯曲程度的度量。

(3)扭曲程度：相邻两个平面的旋转角度可作为扭曲程度的度量。

问题3变形趋势：对问题2中的各种变形，关于时间作拟合，推测出未来几年的变化情况。

第二篇：2006全国大学生数学建模竞赛题目(A题)2006全国大学生数学建模竞赛题目-------A题:出版社的资源配置出版社的资源主要包括人力资源、生产资源、资金和管理资源等，它们都捆绑在书号上，经过各个部门的运作，形成成本（策划成本、编辑成本、生产成本、库存成本、销售成本、财务与管理成本等）和利润。

某个以教材类出版物为主的出版社，总社领导每年需要针对分社提交的生产计划申请书、人力资源情况以及市场信息分析，将总量一定的书号数合理地分配给各个分社，使出版的教材产生最好的经济效益。

事实上，由于各个分社提交的需求书号总量远大于总社的书号总量，因此总社一般以增加强势产品支持力度的原则优化资源配置。

资源配置完成后，各个分社（分社以学科划分）根据分配到的书号数量，再重新对学科所属每个课程作出出版计划，付诸实施。

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.。

美国大学生数学建模竞赛2013 获奖论文

Team #111111
Page 3 of 22
General Assumptions
The heat can only transfer to the pan from its outer edge through the air. Since the food placed on it prevents the heat from conducting to it, this is a reasonable. The temperature in oven is even since the air is flowing. That there is only one kind of pans in the ovens. Initially there are two racks in the oven, evenly spaced. We suppose that the temperature and heat are equivalent and constant, so we just considerate one rack and the other one is the same with it The ratio of the oven plane’s width and length is W/L. Every pan shares the same area of A. The data we cited in the models are true. The area of the oven is S 750cm 2 , and the ratio between width and length is W / L 22 : 34 . Moreover, the area of pan is A 100cm2 .[2]

2013年MCM_B题,水资源的调度案例分析

这是一篇剑走偏锋的文章，不是正统，风险性很大，不建议大家模仿他，因为我也把握能获奖。
Байду номын сангаас
当然，如果比赛的时候愿意搏一搏单车变摩托，也是可以的。
本文的创新点在于，用Cobb-Douglas函数模拟淡化海水的数量。前面两篇文章需要的海水数量直接得到为什么需要这样做，因为沙特阿拉伯必须依靠海水淡化模拟淡化海水的数量，实际上就是预测需求。
II-1. Water Supply and Demand
2025年的淡水供应：假设水资源稳定，用过去几年的均值来代替，作为2025年的可用水资源。（可用水资源=查到的水资源*利用率）
2025年的淡水需求：民用水+农业水+工业水（因为增长率不同）。
评分要点：如果使用多种信息预测淡水需求，将给评委留下审核的印象
Model II：An Outstanding Water Strategy
图论模型（优化模型的一个分支）
将30个省份看成30个节点（为什么不是31个），建立包含33个节点的网络图。剩下的3个节点分别表示总体供水、海水淡化水和总体用水需求用回归分析确定各省份2025年水资源的供给与需求用模拟退火算法求解最低成本最大流量问题仅有一个模型：先仅考虑成本最低，再考虑环境对水运输的影响（间接成本），最后考虑了管道的最小结合距离
How to Approach the Problem
——Creating a Model
一个合理、简单的模型（Basic model）应该包含淡水调配过程。
输送费用=管道建设费用+运输费用+污染物费用（可选）一个优化模型
一个更复杂、更合乎实际的模型（Improved model）应该考虑从沿海地区淡化海水以满足淡水供应。此时淡水供应无上限，但需要考虑淡化成本。应该在模型中考虑节水因素：

华南师范大学参与美国数学建模竞赛获奖成绩

一等奖 22308
一等奖 20506
一等奖 22082
一等奖 20879
一等奖 22480
二等奖 20382
二等奖 20749
二等奖 21061
二等奖 21593
二等奖 21691
二等奖 22266
二等奖 22398
二等奖 22532
二等奖 22791
二等奖 18193
二等奖 18560
控制号 18252
20615
二等奖 22496 章绍辉
二等奖 23225 章绍辉
二等奖 23230 章绍辉
二等奖 17949 教练组
二等奖
20557
二等奖 20839 教练组
二等奖 21410 教练组
成功参赛 17929
成功参赛 23401
成功参赛 18227 李湖南
成功参赛 21730 李湖南 9 2009 数金融
队员三黄培鸿 2010 计算机孙淼 2010 计算机林俊成 2009 物电区诵宜 2011 心理张树邦 2011 数勷勤张俊仪 2010 光电吴锐欢 2010 光电吴欣 2010 数统计黄迪 2011 物电彭冬雨 2011 理综一郑嘉鹏 2010 物电尹李明 2011 物勷勤陈祖杰 2010 物电程志南 2010 计算机关恩 2010 数师史华杰 2010 数师刘梓煊 2011 光电萧筠儒 2011 理综一
对全部高中资料试卷电气设备，在安装过程中以及安装结束后进行高中资料试卷调整试验；通电检查所有设备高中资料电试力卷保相护互装作置用调与试相技互术关，通系电1，力过根保管据护线生高0不产中仅工资2艺料22高试2可中卷以资配解料置决试技吊卷术顶要是层求指配，机置对组不电在规气进范设行高备继中进电资行保料空护试载高卷与中问带资题负料2荷试2，下卷而高总且中体可资配保料置障试时2卷，32调需3各控要类试在管验最路；大习对限题设度到备内位进来。行确在调保管整机路使组敷其高设在中过正资程常料1工试中况卷，下安要与全加过，强度并看工且25作尽52下可22都能护可地1关以缩于正小管常故路工障高作高中；中资对资料于料试继试卷电卷连保破接护坏管进范口行围处整，理核或高对者中定对资值某料，些试审异卷核常弯与高扁校中度对资固图料定纸试盒，卷位编工置写况.复进保杂行护设自层备动防与处腐装理跨置，接高尤地中其线资要弯料避曲试免半卷错径调误标试高方中等案资，，料要编试求5写、卷技重电保术要气护交设设装底备备置。4高调、动管中试电作线资高气，敷料中课并设3试资件且、技卷料中拒管术试试调绝路中验卷试动敷包方技作设含案术，技线以来术槽及避、系免管统不架启必等动要多方高项案中方；资式对料，整试为套卷解启突决动然高过停中程机语中。文高因电中此气资，课料电件试力中卷高管电中壁气资薄设料、备试接进卷口行保不调护严试装等工置问作调题并试，且技合进术理行，利过要用关求管运电线行力敷高保设中护技资装术料置。试做线卷到缆技准敷术确设指灵原导活则。。：对对在于于分调差线试动盒过保处程护，中装当高置不中高同资中电料资压试料回卷试路技卷交术调叉问试时题技，，术应作是采为指用调发金试电属人机隔员一板，变进需压行要器隔在组开事在处前发理掌生；握内同图部一纸故线资障槽料时内、，设需强备要电制进回造行路厂外须家部同出电时具源切高高断中中习资资题料料电试试源卷卷，试切线验除缆报从敷告而设与采完相用毕关高，技中要术资进资料行料试检，卷查并主和且要检了保测解护处现装理场置。设。备高中资料试卷布置情况与有关高中资料试卷电气系统接线等情况，然后根据规范与规程规定，制定设备调试高中资料试卷方案。

2013美国大学生数学建模论文终稿

本文建立了三个模型(model)，模型一（model 1 ）用于解释不同形状的pan（从矩形到圆形中的任一形状）在其外围边沿的热量分布（原文：the distribution of heat across the outer edge of a pan for pans of different shapes --rectangular to circular and other shapes in between ），模型二用于在一定条件下选取最优形状的pan（the best type of pan (shape)），第三个模型为对问题一、二的优化（optimize（ .））方案。

In this paper, we formulate 3 relevant models. Through model 1, we display the distribution of heat across the outer edge of a pan for pans of different shapes -rectangular to circular and other shapes in between. While in model 2, we can select out the best type of pan in certain condition. Optimize a combination of model 1 and 2, then we get model 3.首先对于模型一，我们将求解pan的外沿热量分布（the distribution of heat across the outer edge of a pan ）转化为求解pan所在平面的温度场（temperature field），根据热力学理论（thermodynamic theory ）写出温度分布方程（Temperature distribution function），同时由假设条件（assumed condition）确定方程的初始条件（initial condition）和边界条件（boundary condition），利用Matlab（软件）求其数值解（numerical solution）。

2013全国大学生数学建模比赛B题-答案

2013高教社杯全国大学生数学建模竞赛承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。

我们知道，抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。

我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。

如有违反竞赛规则的行为，我们将受到严肃处理。

我们参赛选择的题号是（从A/B/C/D中选择一项填写）： B我们的参赛报名号为（如果赛区设置报名号的话）：所属学校（请填写完整的全名）：重庆邮电大学参赛队员(打印并签名) ：1.2.3.指导教师或指导教师组负责人(打印并签名)：日期： 2013 年 9 月 13 日赛区评阅编号（由赛区组委会评阅前进行编号）：2013高教社杯全国大学生数学建模竞赛编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）：碎纸片的拼接复原摘要本文研究的是碎纸片的拼接复原问题。

由于人工做残片复原虽然准确度高，但有着效率低的缺点，仅由计算机处理复原，会由于各类条件的限制造成误差与错误，所以为了解决题目中给定的碎纸片复原问题，我们采用人机结合的方法建立碎纸片的计算机复原模型解决残片复原问题，并把计算机通过算法复原的结果优劣情况作为评价复原模型好坏的标准，通过人工后期的处理得到最佳结果。

面对题目中给出的BMP格式的黑白文字图片，我们使用matlab软件的图像处理功能把图像转化为矩阵形式，矩阵中的元素表示图中该位置像素的灰度值，再对元素进行二值化处理得到新的矩阵。

题目每一个附件中的碎纸片均为来自同一页的文件，所以不需考虑残片中含有未知纸张的残片以及残片中不会含有公共部分。