2012美赛翻译

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美赛竞赛试题

1.2013 MCM A: The Ultimate Brownie Pan2013 MCM B: Water, Water, Everywhere2013 ICM: Network Modeling of Earth's Health2. 2012 MCM A: The Leaves of a Tree2012 MCM B: Camping along the Big Long River2012 ICM: Modeling for Crime Busting3. 2011 MCM A: Snowboard Course2011 MCM B: Repeater Coordination2011 ICM: How environmentally and economically sound are electric vehicles? Is their widespread use feasible and practical4. 2010 MCM A: The Sweet Spot2010 MCM A: The Sweet Spot2010 ICM: The Great Pacific Ocean Garbage Patch5. 2009 MCM A: Designing a Traffic Circle2009 MCM B: Energy and the Cell Phone2009 ICM: Creating Food Systems: Re-Balancing Human-Influenced Ecosystems6. 2008 MCM A: Take a Bath2008 MCM B: Creating Sudoku Puzzles2008 ICM: Finding the Good in Health Care Systems7. 2007 MCM A: Gerrymandering2007 MCM B: The Airplane Seating Problem2007 ICM: Organ Transplant: The Kidney Exchange Problem8. 2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation2006 MCM B: Wheel Chair Access at Airports2006 ICM: Trade-offs in the fight against HIV/AIDS9. 2005 MCM A: Flood Planning2005 MCM B: Tollbooths2005 ICM: Nonrenewable Resources10. 2004 MCM A: Are Fingerprints Unique?2004 MCM B: A Faster QuickPass System2004 ICM: To Be Secure or Not to Be?11. 2003 MCM A: The Stunt Person2003 MCM B: Gamma Knife Treatment Planning2003 ICM: Aviation Baggage Screening Strategies: To Screen or Not to Screen, that is the Question12. 2002 MCM A: Wind and Waterspray2002 MCM B: Airline Overbooking2002 ICM: Scrub Lizards13. 2001 MCM A: Choosing a Bicycle Wheel2001 MCM B: Escaping a Hurricane's Wrath (An III Wind...)14. 2000 MCM A: Air Traffic Control2000 MCM B: Radio Channel Assignments2000 ICM: Elephants: When is Enough, Enough?15. 1999 MCM A: Deep Impact1999 MCM B: Unlawful Assembly1999 ICM: Ground Pollution16. 1998 MCM A: MRI Scanners1998 MCM B: Grade Inflation17. 1997 MCM A: The Velociraptor Problem1997 MCM B: Mix Well for Fruitful Discussions 18. 1996 MCM A: Submarine Tracking1996 MCM B: Paper Judging19. 1995 MCM A: Helix Construction1995 MCM B: Faculty Compensation20. 1994 MCM A: Concrete Slab Floors1994 MCM B: Network Design21. 1993 MCM A: Optimal Composting1993 MCM B: Coal-Tipple Operations22. 1992 MCM A: Air-Traffic-Control Radar Power1992 MCM B: Emergency Power Restoration23. 1991 MCM A: Water Tank Flow1991 MCM B: The Steiner Tree Problem24. 1990 MCM A: The Brain-Drug Problem1990 MCM B: Snowplow Routing25. 1989 MCM A: The Midge Classification Problem1989 MCM B: Aircraft Queueing26. 1988 MCM A: The Drug Runner Problem1988 MCM B: Packing Railroad Flatcars27. 1987 MCM A: The Salt Storage Problem1987 MCM B: Parking Lot Design28. 1986 MCM A: Hydrographic Data1986 MCM B: Emergency-Facilities Location29. 1985 MCM A: Animal Populations1985 MCM B: Strategic Reserve Management。

美国大学生数学建模竞赛翻译必备知识

Aabsolute value 绝对值accept 接受acceptable region 接受域additivity 可加性adjusted 调整的alternative hypothesis 对立假设analysis 分析analysis of covariance 协方差分析analysis of variance 方差分析arithmetic mean 算术平均值association 相关性assumption 假设assumption checking 假设检验availability 有效度average 均值Bbalanced 平衡的band 带宽bar chart 条形图beta-distribution 贝塔分布between groups 组间的bias 偏倚binomial distribution 二项分布binomial test 二项检验Ccalculate 计算case 个案category 类别center of gravity 重心central tendency 中心趋势chi-square distribution 卡方分布chi-square test 卡方检验classify 分类cluster analysis 聚类分析coefficient 系数coefficient of correlation 相关系数collinearity 共线性column 列compare 比较comparison 对照components 构成，分量compound 复合的confidence interval 置信区间consistency 一致性constant 常数continuous variable 连续变量control charts 控制图correlation 相关covariance 协方差covariance matrix 协方差矩阵critical point 临界点critical value 临界值crosstab 列联表cubic 三次的，立方的cubic term 三次项cumulative distributionfunction 累加分布函数curve estimation 曲线估计Ddata 数据default 默认的definition 定义deleted residual 剔除残差density function 密度函数dependent variable 因变量description 描述design of experiment 试验设计deviations 差异df.(degree of freedom) 自由度diagnostic 诊断dimension 维discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析distance 距离distribution 分布D-optimal design D-优化设计Eeaqual 相等effects of interaction 交互效应efficiency 有效性eigenvalue 特征值equal size 等含量equation 方程error 误差estimate 估计estimation of parameters参数估计estimations 估计量evaluate 衡量exact value 精确值expectation 期望expected value 期望值exponential 指数的exponential distributon 指数分布extreme value 极值Ffactor 因素，因子factor analysis 因子分析factor score 因子得分factorial designs 析因设计factorial experiment 析因试验fit 拟合fitted line 拟合线fitted value 拟合值fixed model 固定模型fixed variable 固定变量fractional factorial design部分析因设计frequency 频数F-test F检验full factorial design 完全析因设计function 函数Ggamma distribution 伽玛分布geometric mean 几何均值group 组Hharmomic mean 调和均值heterogeneity 不齐性histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性hypothesis 假设hypothesis test 假设检验Iindependence 独立independent variable 自变量independent-samples 独立样本index 指数index of correlation 相关指数interaction 交互作用interclass correlation 组内相关interval estimate 区间估计intraclass correlation 组间相关inverse 倒数的iterate 迭代Kkernal 核Kolmogorov-Smirnov test 柯尔莫哥洛夫-斯米诺夫检验kurtosis 峰度Llarge sample problem 大样本问题layer 层least-significant difference 最小显著差数least-square estimation 最小二乘估计least-square method 最小二乘法level 水平level of significance 显著性水平leverage value 中心化杠杆值life 寿命life test 寿命试验likelihood function 似然函数likelihood ratio test 似然比检验linear 线性的linear estimator 线性估计linear model 线性模型linear regression 线性回归linear relation 线性关系linear term 线性项logarithmic 对数的logarithms 对数logistic 逻辑的lost function 损失函数Mmain effect 主效应matrix 矩阵maximum 最大值maximum likelihoodestimation 极大似然估计mean squareddeviation(MSD) 均方差mean sum of square 均方和measure 衡量media 中位数M-estimator M估计minimum 最小值missing values 缺失值mixed model 混合模型mode 众数model 模型Monte Carle method 蒙特卡罗法moving average 移动平均值multicollinearity 多元共线性multiple comparison 多重比较multiple correlation 多重相关multiple correlationcoefficient 复相关系数multiple correlationcoefficient 多元相关系数multiple regression analysis多元回归分析multiple regressionequation 多元回归方程multiple response 多响应multivariate analysis 多元分析Nnegative relationship 负相关nonadditively 不可加性nonlinear 非线性nonlinear regression 非线性回归noparametric tests 非参数检验normal distribution 正态分布null hypothesis 零假设number of cases 个案数Oone-sample 单样本one-tailed test 单侧检验one-way ANOVA 单向方差分析one-way classification 单向分类optimal 优化的optimum allocation 最优配制order 排序order statistics 次序统计量origin 原点orthogonal 正交的outliers 异常值Ppaired observations 成对观测数据paired-sample 成对样本parameter 参数parameter estimation 参数估计partial correlation 偏相关partial correlation coefficient 偏相关系数partial regression coefficient 偏回归系数percent 百分数percentiles 百分位数pie chart 饼图point estimate 点估计poisson distribution 泊松分布polynomial curve 多项式曲线polynomial regression 多项式回归polynomials 多项式positive relationship 正相关power 幂P-P plot P-P概率图predict 预测predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析proability 概率probability density function 概率密度函数probit analysis 概率分析proportion 比例Qqadratic 二次的Q-Q plot Q-Q概率图quadratic term 二次项quality control 质量控制quantitative 数量的，度量的quartiles 四分位数Rrandom 随机的random number 随机数random number 随机数random sampling 随机取样random seed 随机数种子random variable 随机变量randomization 随机化range 极差rank 秩rank correlation 秩相关rank statistic 秩统计量regression analysis 回归分析regression coefficient 回归系数regression line 回归线reject 拒绝rejection region 拒绝域relationship 关系reliability 可靠性repeated 重复的report 报告，报表residual 残差residual sum of squares 剩余平方和response 响应risk function 风险函数robustness 稳健性root mean square 标准差row 行run 游程run test 游程检验Ssample 样本sample size 样本容量sample space 样本空间sampling 取样sampling inspection 抽样检验scatter chart 散点图S-curve S形曲线separately 单独地sets 集合sign test 符号检验significance 显著性significance level 显著性水平significance testing 显著性检验significant 显著的，有效的significant digits 有效数字skewed distribution 偏态分布skewness 偏度small sample problem 小样本问题smooth 平滑sort 排序soruces of variation 方差来源space 空间spread 扩展square 平方standard deviation 标准离差standard error of mean 均值的标准误差standardization 标准化standardize 标准化statistic 统计量statistical quality control 统计质量控制std. residual 标准残差stepwise regressionanalysis 逐步回归stimulus 刺激strong assumption 强假设stud. deleted residual 学生化剔除残差stud. residual 学生化残差subsamples 次级样本sufficient statistic 充分统计量sum 和sum of squares 平方和summary 概括，综述Ttable 表t-distribution t分布test 检验test criterion 检验判据test for linearity 线性检验test of goodness of fit 拟合优度检验test of homogeneity 齐性检验test of independence 独立性检验test rules 检验法则test statistics 检验统计量testing function 检验函数time series 时间序列tolerance limits 容许限total 总共，和transformation 转换treatment 处理trimmed mean 截尾均值true value 真值t-test t检验two-tailed test 双侧检验Uunbalanced 不平衡的unbiased estimation 无偏估计unbiasedness 无偏性uniform distribution 均匀分布Vvalue of estimator 估计值variable 变量variance 方差variance components 方差分量variance ratio 方差比various 不同的vector 向量Wweight 加权，权重weighted average 加权平均值within groups 组内的ZZ score Z分数Ⅱ.2 最优化方法词汇英汉对照表Aactive constraint 活动约束active set method 活动集法analytic gradient 解析梯度approximate 近似arbitrary 强制性的argument 变量attainment factor 达到因子Bbandwidth 带宽be equivalent to 等价于best-fit 最佳拟合bound 边界Ccoefficient 系数complex-value 复数值component 分量constant 常数constrained 有约束的constraint 约束constraint function 约束函数continuous 连续的converge 收敛cubic polynomialinterpolation method三次多项式插值法curve-fitting 曲线拟合Ddata-fitting 数据拟合default 默认的，默认的define 定义diagonal 对角的direct search method 直接搜索法direction of search 搜索方向discontinuous 不连续Eeigenvalue 特征值empty matrix 空矩阵equality 等式exceeded 溢出的Ffeasible 可行的feasible solution 可行解finite-difference 有限差分first-order 一阶GGauss-Newton method 高斯-牛顿法goal attainment problem 目标达到问题gradient 梯度gradient method 梯度法Hhandle 句柄Hessian matrix 海色矩阵Iindependent variables 独立变量inequality 不等式infeasibility 不可行性infeasible 不可行的initial feasible solution 初始可行解initialize 初始化inverse 逆invoke 激活iteration 迭代iteration 迭代JJacobian 雅可比矩阵LLagrange multiplier 拉格朗日乘子large-scale 大型的least square 最小二乘least squares sense 最小二乘意义上的Levenberg-Marquardtmethod列文伯格-马夸尔特法line search 一维搜索linear 线性的linear equality constraints线性等式约束linear programmingproblem 线性规划问题local solution 局部解Mmedium-scale 中型的minimize 最小化mixed quadratic and cubic polynomial interpolation and extrapolation method 混合二次、三次多项式内插、外插法multiobjective 多目标的Nnonlinear 非线性的norm 范数Oobjective function 目标函数observed data 测量数据optimization routine 优化过程optimize 优化optimizer 求解器over-determined system 超定系统Pparameter 参数partial derivatives 偏导数polynomial interpolation method多项式插值法Qquadratic 二次的quadratic interpolation method 二次内插法quadratic programming 二次规划Rreal-value 实数值residuals 残差robust 稳健的robustness 稳健性，鲁棒性Sscalar 标量semi-infinitely problem 半无限问题Sequential Quadratic Programming method序列二次规划法simplex search method 单纯形法solution 解sparse matrix 稀疏矩阵sparsity pattern 稀疏模式sparsity structure 稀疏结构starting point 初始点step length 步长subspace trust regionmethod 子空间置信域法sum-of-squares 平方和symmetric matrix 对称矩阵Ttermination message 终止信息termination tolerance 终止容限the exit condition 退出条件the method of steepestdescent 最速下降法transpose 转置Uunconstrained 无约束的under-determined system负定系统Vvariable 变量vector 矢量Wweighting matrix 加权矩阵Ⅱ.3 样条词汇英汉对照表Aapproximation 逼近array 数组a spline in b-form/b-splineb样条a spline of polynomial piece/ppform spline分段多项式样条Bbivariate spline function 二元样条函数break/breaks 断点Ccoefficient/coefficients 系数cubic interpolation 三次插值/三次内插cubic polynomial 三次多项式cubic smoothing spline 三次平滑样条cubic spline 三次样条cubic spline interpolation三次样条插值/三次样条内插curve 曲线Ddegree of freedom 自由度dimension 维数Eend conditions 约束条件Iinput argument 输入参数interpolation 插值/内插interval 取值区间Kknot/knots 节点Lleast-squaresapproximation 最小二乘拟合Mmultiplicity 重次multivariate function 多元函数Ooptional argument 可选参数order 阶次output argument 输出参数Ppoint/points 数据点Rrational spline 有理样条rounding error 舍入误差（相对误差）Sscalar 标量sequence 数列（数组）spline 样条spline approximation 样条逼近/样条拟合spline function 样条函数spline curve 样条曲线spline interpolation 样条插值/样条内插spline surface 样条曲面smoothing spline 平滑样条Ttolerance 允许精度Uunivariate function 一元函数Vvector 向量Wweight/weights 权重Ⅱ.4 偏微分方程数值解词汇英汉对照表Aabsolute error 绝对误差absolute tolerance 绝对容限adaptive mesh 适应性网格Bboundary condition 边界条件Ccontour plot 等值线图converge 收敛coordinate 坐标系Ddecomposed 分解的decomposed geometry matrix 分解几何矩阵diagonal matrix 对角矩阵Dirichlet boundary conditionsDirichlet边界条件Eeigenvalue 特征值elliptic 椭圆形的error estimate 误差估计exact solution 精确解Ggeneralized Neumann boundary condition推广的Neumann边界条件geometry 几何形状geometry descriptionmatrix 几何描述矩阵geometry matrix 几何矩阵graphical user interface（GUI）图形用户界面Hhyperbolic 双曲线的Iinitial mesh 初始网格Jjiggle 微调LLagrange multipliers 拉格朗日乘子Laplace equation 拉普拉斯方程linear interpolation 线性插值loop 循环Mmachine precision 机器精度mixed boundary condition混合边界条件NNeuman boundarycondition Neuman边界条件node point 节点nonlinear solver 非线性求解器normal vector 法向量PParabolic 抛物线型的partial differential equation偏微分方程plane strain 平面应变plane stress 平面应力Poisson's equation 泊松方程polygon 多边形positive definite 正定Qquality 质量Rrefined triangular mesh 加密的三角形网格relative tolerance 相对容限relative tolerance 相对容限residual 残差residual norm 残差范数Ssingular 奇异的sparce matrix 稀疏矩阵stiffness matrix 刚度矩阵subregion 子域Ttriangular mesh 三角形网格Uundetermined 未定的uniform refinement 均匀加密uniform triangle net 均匀三角形网络Wwave equation 波动方程Algebraic Equation代数方程Elementary Operations-Addition基础混算-加法ElementaryOperations-Subtaction基础混算-减法ElementaryOperations-Multiplication基础混算-乘法Elementary Operations-Division基础混算-除法Elementary Operation基础四则混算Decimal Operations 小数混算Fractional Operations分数混算Convert fractional no. intodecimal no.分数转小数Convert fractional no. intopercentage.分数转百分数Convert decimal no. intopercentage.小数转百分数Convert percentage into decimal no.百分数转小数Percentage百分数Numerals数字符号Common factors and multiples公因子及公倍数Sorting数字排序Area图形面积Perimeter图形周界Change Units : Time单位转换-时间Change Units : Weight 单位转换-重量Change Units :Length单位转换-长度Directed Numbers 有向数Fractional Operations 分数混算Decimal Operations 小数混算Convert fractional no. into decimal no.分数转小数Convert fractional no. into percentage.分数转百分数Convert decimal no. into percentage.小数转百分数Convert percentage into decimal no.百分数转小数Percentage百分数Indices指数Algebraic Substitution 代数代入Polynomials多项式Co-Geometry坐标几何学Solving Linear Equation解一元线性方程Solving Simultaneous Equation解联立方程Slope直线斜率Equation of Straight Line直线方程x-intercept ( Equation of St. Line )直线x轴截距y-intercept ( Equation of St. Line )直线y轴截距Factorization因式分解Quadratic Equation 二次方程x-intercept ( Quadratic Equation )二次曲线x轴截距Geometry几何学Inequalities不等式Rate and Ratio比和比例Bearing方位角Trigonometry三角学Probability概率Statistics-Graph统计学-统计图表Statistics-Measure of centraltendency统计学-量度集中趋势Salary Tax薪俸税Bridging Game汉英对对碰Indices指数Function函数Rate and Ratio比和比例Trigonometry三角学Inequalities不等式Linear Programming线性规划Co-Geometry坐标几何学Slope直线斜率Equation of Straight Line直线方程x-intercept ( Equation of St. Line )直线x轴截距y-intercept ( Equation of St. Line )直线y轴截距Factorization因式分解Quadratic Equation二次方程x-intercept ( Quadratic Equation )二次曲线x轴截距Method of Bisection分半方法Polynomials多项式Probability概率Statistics-Graph统计学-统计图表Statistics-Measure of centraltendency统计学-量度集中趋势Statistics-Measure of dispersion统计学-量度分布Statistics-Normal Distribution统计学-正态分布Surds根式Probability概率Statistics-Measure of dispersion统计学-量度离差Statistics-Normal Distribution统计学-正态分布Statistics-Binomial Distribution统计学Statistics-Poisson Distribution统计学Statistics-Geometric Distribution统计学Co-Geometry坐标几何学Sequence序列十万Hundred thousand三位数3-digit number千Thousand千万Ten million小数Decimal分子Numerator分母Denominator分数Fraction五位数5-digit number公因子Common factor公倍数Common multiple中国数字Chinese numeral平方Square平方根Square root古代计时工具Ancient timingdevice古代记时工具Ancienttime-recording device古代记数方法Ancient countingmethod古代数字Ancient numeral包含Grouping四位数4-digit number四则计算Mixed operations (Thefour operations)加Plus加法Addition加法交换性质Commutativeproperty of addition未知数Unknown百分数Percentage百万Million合成数Composite number多位数Large number因子Factor折扣Discount近似值Approximation阿拉伯数字Hindu-Arabic numeral定价Marked price括号Bracket计算器Calculator差Difference真分数Proper fraction退位Decomposition除Divide除法Division除数Divisor乘Multiply乘法Multiplication乘法交换性质Commutative property of multiplication乘法表Multiplication table乘法结合性质Associative property of multiplication被除数Dividend珠算Computation using Chinese abacus倍数Multiple假分数Improper fraction带分数mixed number现代计算工具Modern calculating devices售价Selling price万Ten thousand最大公因子Highest Common Factor (H.C.F.)最小公倍数Lowest Common Multiple (L.C.M.)减Minus / Subtract减少Decrease减法Subtraction等分Sharing 等于Equal进位Carrying短除法Short division单数Odd number循环小数Recurring decimal零Zero算盘Chinese abacus亿Hundred million增加Increase质数Prime number积Product整除性Divisibility双数Even number罗马数字Roman numeral数学mathematics, maths(BrE),math(AmE)公理axiom定理theorem计算calculation运算operation证明prove假设hypothesis, hypotheses(pl.)命题proposition算术arithmetic加plus(prep.), add(v.),addition(n.)被加数augend, summand加数addend和sum减minus(prep.), subtract(v.),subtraction(n.)被减数minuend减数subtrahend差remainder乘times(prep.), multiply(v.),multiplication(n.)被乘数multiplicand, faciend乘数multiplicator积product除divided by(prep.), divide(v.),division(n.)被除数dividend除数divisor商quotient等于equals, is equal to, isequivalent to大于is greater than小于is lesser than大于等于is equal or greater than小于等于is equal or lesser than运算符operator平均数mean算术平均数arithmatic mean几何平均数geometric mean n个数之积的n次方根倒数（reciprocal）x的倒数为1/x有理数rational number无理数irrational number实数real number虚数imaginary number数字digit数number自然数natural number整数integer小数decimal小数点decimal point分数fraction分子numerator分母denominator比ratio正positive负negative零null, zero, nought, nil十进制decimal system二进制binary system十六进制hexadecimal system权weight, significance进位carry截尾truncation四舍五入round下舍入round down上舍入round up有效数字significant digit无效数字insignificant digit代数algebra公式formula, formulae(pl.)单项式monomial多项式polynomial, multinomial 系数coefficient未知数unknown, x-factor, y-factor, z-factor等式，方程式equation一次方程simple equation二次方程quadratic equation三次方程cubic equation四次方程quartic equation不等式inequation阶乘factorial对数logarithm指数，幂exponent乘方power二次方，平方square三次方，立方cube四次方the power of four, the fourth powern次方the power of n, the nth power开方evolution, extraction二次方根，平方根square root 三次方根，立方根cube root四次方根the root of four, the fourth rootn次方根the root of n, the nth rootsqrt(2)=1.414sqrt(3)=1.732sqrt(5)=2.236常量constant变量variable坐标系coordinates坐标轴x-axis, y-axis, z-axis横坐标x-coordinate纵坐标y-coordinate原点origin象限quadrant截距(有正负之分)intercede（方程的）解solution几何geometry点point线line面plane 体solid线段segment射线radial平行parallel相交intersect角angle角度degree弧度radian锐角acute angle直角right angle钝角obtuse angle平角straight angle周角perigon底base边side高height三角形triangle锐角三角形acute triangle直角三角形right triangle直角边leg斜边hypotenuse勾股定理Pythagorean theorem钝角三角形obtuse triangle不等边三角形scalene triangle等腰三角形isosceles triangle等边三角形equilateral triangle四边形quadrilateral平行四边形parallelogram矩形rectangle长length宽width周长perimeter面积area相似similar全等congruent三角trigonometry正弦sine余弦cosine正切tangent余切cotangent正割secant余割cosecant反正弦arc sine反余弦arc cosine反正切arc tangent反余切arc cotangent反正割arc secant反余割arc cosecant补充：集合aggregate元素element空集void子集subset交集intersection并集union补集complement映射mapping函数function定义域domain, field ofdefinition值域range单调性monotonicity奇偶性parity周期性periodicity图象image数列，级数series微积分calculus微分differential导数derivative极限limit无穷大infinite(a.) infinity(n.)无穷小infinitesimal积分integral定积分definite integral不定积分indefinite integral复数complex number矩阵matrix行列式determinant圆circle圆心centre(BrE), center(AmE)半径radius直径diameter圆周率pi弧arc半圆semicircle扇形sector环ring椭圆ellipse圆周circumference轨迹locus, loca(pl.)平行六面体parallelepiped立方体cube七面体heptahedron八面体octahedron九面体enneahedron十面体decahedron十一面体hendecahedron十二面体dodecahedron二十面体icosahedron多面体polyhedron旋转rotation轴axis球sphere半球hemisphere底面undersurface表面积surface area体积volume空间space双曲线hyperbola抛物线parabola四面体tetrahedron五面体pentahedron六面体hexahedron菱形rhomb, rhombus, rhombi(pl.), diamond正方形square梯形trapezoid直角梯形right trapezoid等腰梯形isosceles trapezoid五边形pentagon六边形hexagon七边形heptagon八边形octagon九边形enneagon十边形decagon十一边形hendecagon十二边形dodecagon多边形polygon正多边形equilateral polygon相位phase周期period振幅amplitude内心incentre(BrE), incenter(AmE)外心excentre(BrE),excenter(AmE)旁心escentre(BrE),escenter(AmE)垂心orthocentre(BrE),orthocenter(AmE)重心barycentre(BrE),barycenter(AmE)内切圆inscribed circle外切圆circumcircle统计statistics平均数average加权平均数weighted average方差variance标准差root-mean-squaredeviation, standard deviation比例propotion百分比percent百分点percentage百分位数percentile排列permutation组合combination概率，或然率probability分布distribution正态分布normal distribution非正态分布abnormaldistribution图表graph条形统计图bar graph柱形统计图histogram折线统计图broken line graph曲线统计图curve diagram扇形统计图pie diagramEnglish Chineseabbreviation 简写符号；简写abscissa 横坐标absolute complement 绝对补集absolute error 绝对误差absolute inequality 绝不等式absolute maximum 绝对极大值absolute minimum 绝对极小值absolute monotonic 绝对单调absolute value 绝对值accelerate 加速acceleration 加速度acceleration due to gravity 重力加速度; 地心加速度accumulation 累积accumulative 累积的accuracy 准确度act on 施于action 作用; 作用力acute angle 锐角acute-angled triangle 锐角三角形add 加addition 加法addition formula 加法公式addition law 加法定律addition law(of probability) （概率）加法定律additive inverse 加法逆元; 加法反元additive property 可加性adjacent angle 邻角adjacent side 邻边adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式；代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algebraically closed 代数封闭algorithm 算法系统; 规则系统alternate angle （交）错角alternate segment 内错弓形alternating series 交错级数alternative hypothesis 择一假设;备择假设; 另一假设altitude 高；高度；顶垂线；高线ambiguous case 两义情况；二义情况amount 本利和；总数analysis 分析；解析analytic geometry 解析几何angle 角angle at the centre 圆心角angle at the circumference 圆周角angle between a line and a plane 直与平面的交角angle between two planes 两平面的交角angle bisection 角平分angle bisector 角平分线；分角线angle in the alternate segment 交错弓形的圆周角angle in the same segment 同弓形内的圆周角angle of depression 俯角angle of elevation 仰角angle of friction 静摩擦角; 极限角angle of greatest slope 最大斜率的角angle of inclination 倾斜角angle of intersection 相交角；交角angle of projection 投射角angle of rotation 旋转角angle of the sector 扇形角angle sum of a triangle 三角形内角和angles at a point 同顶角angular displacement 角移位angular momentum 角动量angular motion 角运动angular velocity 角速度annum(X% per annum) 年（年利率X%）anti-clockwise direction 逆时针方向；返时针方向anti-clockwise moment 逆时针力矩anti-derivative 反导数; 反微商anti-logarithm 逆对数；反对数anti-symmetric 反对称apex 顶点approach 接近；趋近approximate value 近似值approximation 近似；略计；逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc length 弧长arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积Argand diagram 阿根图, 阿氏图argument (1)论证; (2)辐角argument of a complex number 复数的辐角argument of a function 函数的自变量arithmetic 算术arithmetic mean 算术平均；等差中顶；算术中顶arithmetic progression 算术级数；等差级数arithmetic sequence 等差序列arithmetic series 等差级数arm 边array 数组; 数组arrow 前号ascending order 递升序ascending powers of X X 的升幂assertion 断语; 断定associative law 结合律assumed mean 假定平均数assumption 假定；假设asymmetrical 非对称asymptote 渐近asymptotic error constant 渐近误差常数at rest 静止augmented matrix 增广矩阵auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 辅助方程average 平均；平均数；平均值average speed 平均速率axiom 公理axiom of existence 存在公理axiom of extension 延伸公理axiom of inclusion 包含公理axiom of pairing 配对公理axiom of power 幂集公理axiom of specification 分类公理axiomatic theory of probability 概率公理论axis 轴axis of parabola 拋物线的轴axis of revolution 旋转轴axis of rotation 旋转轴axis of symmetry 对称轴back substitution 回代bar chart 棒形图；条线图；条形图；线条图base （1）底；（2）基；基数base angle 底角base area 底面base line 底线base number 底数；基数base of logarithm 对数的底basis 基Bayes' theorem 贝叶斯定理bearing 方位（角）；角方向（角）bell-shaped curve 钟形图belong to 属于Bernoulli distribution 伯努利分布Bernoulli trials 伯努利试验bias 偏差；偏倚biconditional 双修件式; 双修件句bijection 对射; 双射; 单满射bijective function 对射函数; 只射函数billion 十亿bimodal distribution 双峰分布binary number 二进数binary operation 二元运算binary scale 二进法binary system 二进制binomial 二项式binomial distribution 二项分布binomial expression 二项式binomial series 二项级数binomial theorem 二项式定理bisect 平分；等分bisection method 分半法；分半方法bisector 等分线；平分线Boolean algebra 布尔代数boundary condition 边界条件boundary line 界（线）；边界bounded 有界的bounded above 有上界的；上有界的bounded below 有下界的；下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度broken line graph 折线图calculation 计算calculator 计算器；计算器calculus (1) 微积分学; (2) 演算cancel 消法；相消canellation law 消去律canonical 典型; 标准capacity 容量cardioid 心脏Cartesian coordinates 笛卡儿坐标Cartesian equation 笛卡儿方程Cartesian plane 笛卡儿平面Cartesian product 笛卡儿积category 类型；范畴catenary 悬链Cauchy sequence 柯西序列Cauchy's principal value 柯西主值Cauchy-Schwarz inequality 柯西- 许瓦尔兹不等式central limit theorem 中心极限定理central line 中线central tendency 集中趋centre 中心；心centre of a circle 圆心centre of gravity 重心centre of mass 质量中心centrifugal force 离心力centripedal acceleration 向心加速度centripedal force force 向心力centroid 形心；距心certain event 必然事件chain rule 链式法则chance 机会change of axes 坐标轴的变换change of base 基的变换change of coordinates 坐标轴的变换change of subject 主项变换change of variable 换元；变量的换characteristic equation 特征(征)方程characteristic function 特征(征)函数characteristic of logarithm 对数的首数; 对数的定位部characteristic root 特征(征)根chart 图；图表check digit 检验数位checking 验算chord 弦chord of contact 切点弦circle 圆circular 圆形；圆的circular function 圆函数；三角函数circular measure 弧度法circular motion 圆周运动circular permutation 环形排列;圆形排列; 循环排列circumcentre 外心；外接圆心circumcircle 外接圆circumference 圆周circumradius 外接圆半径circumscribed circle 外接圆cissoid 蔓叶class 区；组；类class boundary 组界class interval 组区间；组距class limit 组限；区限class mark 组中点；区中点classical theory of probability 古典概率论classification 分类clnometer 测斜仪clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coaxial 共轴coaxial circles 共轴圆coaxial system 共轴系coded data 编码数据coding method 编码法co-domain 上域coefficient 系数coefficient of friction 摩擦系数coefficient of restitution 碰撞系数; 恢复系数coefficient of variation 变差系数cofactor 余因子; 余因式cofactor matrix 列矩阵coincide 迭合；重合collection of terms 并项collinear 共线collinear planes 共线面collision 碰撞column (1)列；纵行；(2) 柱column matrix 列矩阵column vector 列向量combination 组合common chord 公弦common denominator 同分母；公分母common difference 公差。

2012年美国数学建模竞赛

2012年美国数学建模竞赛（MCM/ICM）将于2.10（正月十九）—2.13（共4天）举行，我们邀请冯国灿教授介绍如何准备美国数学建模竞赛，内容包括:1、报名注册注意事项2、写作技巧3、数学建模知识的准备4、典型案例分析欢迎备战参赛的同学参加！时间：2011年12月21日（周三）晚上7：30地点：数学楼202【嘉宾简介】冯国灿：教授、博士生导师、中山大学数学与计算科学学院数学系副主任、广东省大学生数学建模竞赛组委会委员。

是十几年的数学建模竞赛的资深教练，经常参加全国、广东省大学生数学建模竞赛论文的评阅工作，指导数模美赛、国赛，获奖无数。

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数学建模美赛2012MCM B论文

Camping along the Big Long RiverSummaryIn this paper, the problem that allows more parties entering recreation system is investigated. In order to let park managers have better arrangements on camping for parties, the problem is divided into four sections to consider.The first section is the description of the process for single-party's rafting. That is, formulating a Status Transfer Equation of a party based on the state of the arriving time at any campsite. Furthermore, we analyze the encounter situations between two parties.Next we build up a simulation model according to the analysis above. Setting that there are recreation sites though the river, count the encounter times when a new party enters this recreation system, and judge whether there exists campsites available for them to station. If the times of encounter between parties are small and the campsite is available, the managers give them a good schedule and permit their rafting, or else, putting off the small interval time t until the party satisfies the conditions.Then solve the problem by the method of computer simulation. We imitate the whole process of rafting for every party, and obtain different numbers of parties, every party's schedule arrangement, travelling time, numbers of every campsite's usage, ratio of these two kinds of rafting boats, and time intervals between two parties' starting time under various numbers of campsites after several times of simulation. Hence, explore the changing law between the numbers of parties (X) and the numbers of campsites (Y) that X ascends rapidly in the first period followed by Y's increasing and the curve tends to be steady and finally looks like a S curve.In the end of our paper, we make sensitive analysis by changing parameters of simulation and evaluate the strengths and weaknesses of our model, and write a memo to river managers on the arrangements of rafting.Key words: Camping；Computer Simulation; Status Transfer Equation1 IntroductionThe number of visits to outdoor recreation areas has increased dramatically in last three decades. Among all those outdoor activities, rafting is often chose as a family get-together during May to September. Rafting or white water rafting is a kind of interesting and challenging recreational outdoor activity, which uses an inflatable raft to navigate a river or sea [1]. It is very popular in the world, especially in occidental countries. This activity is commonly considered an extreme sport that usually done to thrill and excite the raft passengers on white water or different degrees of rough water. It can be dangerous.During the peak period, there are many tourists coming to experience rafting. In order to satisfy tourists to the maximum, we must make full use of our facilities in hand, which means we must do the utmost to utilize the campsites in the best way possible. What's more, to make more people feel the wildness life, we should minimize the encounters to the best extent; meanwhile no two sets of parties can occupy the same campsite at the same time. It is naturally coming into mind that we should consider where to stop, and when to stop of a party [2].In previous studies [3-5],many researchers have simulated the outdoor creation based on real-life data, because the approach is dynamic, stochastic, and discrete-event, and most recreation systems share these traits. But there exists little research aiming at describing the way that visitors travel and distribute themselves within a recreation system [6]. Hence, in our paper, we consider the whole process of parties in detail and simulate every party ’s behavior, including the location of their campsites, and how long it will last for them to stay in a campsite to finish their itineraries. Meanwhile minimize the numbers of encounters.Aiming at showing the whole process of rafting, we firstly focus on analyzing the situation s of a single-party's rafting by using status transfer equation, then consider the problems of two parties' encounters on the river. Finally, after several times of simulation on the whole process of rafting, we obtain the optimal value of X . 2 Symbols and DefinitionsIn this section, we will give some basic symbols and definitions in the following for the convenience.Table 1. Variable Definition SymbolsDefinition i vi pj i q , S dThe velocity of oar or motor 0-1 variables on choosing rafting transportation 0-1 variables on the occupation of campsitesLength of the riverAverage distance between two campsites3 General AssumptionsIn order to have a better study on this paper, we simplify our model by the following assumptions:1) 19 : 00 to 07: 00 is people's sleeping time, during this time, people are stationedin the campsite. The total time of sleeping is 12 hours, as rafting is an exiting sport game, after a day's entertainment, people have cost a lot of energy, and nearly tired out. So in order to have a better recreation for the next day, we set that people begin their trip at 07:00, and end at 19:00 for a day's schedule.2) Oar- powered rubber rafts and motorized parties can successfully raft from FirstLaunch to Final Exit, there exist no accident over the whole trips.3) All the rubber rafts and motorized boats have the same exterior except velocities;we regard a rubber raft or a motorized boat as a party and don't consider the tourists individuals on the parties.4) There is only one entrance for parties to enter the recreation system.5) Regardless of the effects that the physical features of the river brings to oar andmotorized parties, that is to say we ignore the stream ’s propulsion and resistance to both kinds of rafting boats. Oar and motorized parties can keep the average velocity of 4 mph and 8mph.6) Divide the whole river into N segments.4 Analysis of This Rafting ProblemRafting is a very popular spots game world-wide. In the peak period of rafting, there are more people choosing to raft, it often causes congestion that not all people can raft at any time they want. Hence, it is important for managers to set an optimal schedule for every party (from our assumptions, we regard a rafting boat as a party) in advance. Meanwhile, the parties need to experience wildness life, so the managers should arrange the schedules which minimize the encounters' time between parties to the best extent. What's more, no two sets of parties can occupy the same site at the same time.Our aim is to determine an optimal mix of trips over varying duration (measured YXNj i t ,j TtK Numbers of campsites Numbers of parties Numbers of attraction sites Time of the i th party finishing the whole trip ranges from6 days to 18 days Random staying time at each campsite Delay time of rafting from beginning Threshold value of encounterin nights on the river. That is to say,we must obtain an optimal value of X through lots of trails. This optimal value represents that the campsites have a high usage while more people are available to raft.The Long Big River is 225 miles long, if we discuss the river as a whole and consider all the parties together, it will be difficult for us to have a clear recognition on parties' behaviors. Hence, we divide the river into N attraction sites. Each of the attraction sites has Y/N campsites since the campsites are uniformly distributed throughout the river corridor. So build up a model based on single-party ’s behavior of rafting in small distance. At last, we can use computer simulation to imitate more complex situations with various rafting boats and large quantities of parties. 5 Mathematic Models5.1 Rafting of the Single-party Model （Status Transfer Equation [7]）From the previous analysis, in order to have a clear recognition of the whole rafting process, we must analyze every single-party's state at any time.In this model, we consider the situation that a single-party rafts from the First Launch to the Final Exit. So we formulate a model that focus on the behavior of one single-party.For a single-party, it must satisfy the following equation: status transfer equation. it represents the relationships between its former state and the latter state. State here means: when the i th party arrives at the j th campsites, the party may occupy the j th campsite or not.As a party can choose two kinds of transportation to raft: oar- powered rubber rafts(i v =4mph) and motorized rafts(i v =8mph). i v is the velocity of the rafting boats,and i p is the 0-1 variables of the selecting for boats. Therefore, we can obtainthe following equation:)1(84i i i p p v -+= (i=1,2,…,X ). (1) where i p =0 if the i th party uses motorized boat as their rafting tool, at thistime i v =8mph ; while i p =1 when , the i th party rafts with oar- powered rubber raft with i v =4mph. In fact, Eq.(1) denotes which kind of rafting boat a party can choose.A party not only has choice on rafting boats, but also can select where to camp based on whether the campsites are occupied or not. The following formulation shows the situation whether this party chooses this campsite or not:⎩⎨⎧=p a r t y p r e v i o u s a b y o c c u pi e d is campsite the 0,party previous a by occupied not is campsite the q ij ,1 (2) where i =1,2,…,X ; j =1,2,…,Y .Where the next one can’t set their camp at this place anymore, that is to say thelatter party’s behavior is determined by the former one.As campsites are fairly uniformly distributed throughout the river corridor, hence, we discrete the whole river into segments, and regard Y campsites as Y nodes which leaves out (Y +1) intervals. Finally we get the average distance between th e j th campsite and (j+1)th campsite:1+=Y S d (3) where is the length of the river, and its value is 225 miles.What’s more, the trip -days for a party is not infinite, it has fluctuating intervals: h t h j i 432144,≤≤ (4) where is the t i ，j itinerary time for a party ranges from 144 hours to 432 hours (6 to 18 nights).From Eq.(1), (2) and (3), the status transfer equation is given as follows: ),...2,1,,...2,1(11,1,,Y j X i T q v d t t j j i i j i j i ==⨯++=--- （5）The i th party’s arriving time at the j th campsite is determined by the time when the i th arrived at （j-1）campsite, the time interval i v d , and the time T j-1 random generated by computer shown in Eq.(5). It is a dynamic process and determined by its previous behavior.5.2 The Analysis of Two Parties’ Encounter on the RiverOur goal is to making full use of the campsites. Hence, the objective of all the formulation is to maximize the quantities of trips （parties ）X while consider getting rid of the congestion. If we reduce the numbers of the encounters among parties, there will be no congestion. In order to achieve this goal, we analysis the situations of when two parties’ to encounter, and where they will enco unter.In order to create a wildness environment for parties to experience wildness life, managers arrange a schedule that can make any two parties have minimal encounters with each other. Encounter is that parties meet at the same place and at the same time. Regarding the river as a whole is not convenient to study, hence, our discussion is based on a small distance where distance=d (Eq.3), between the j th and ( j+1)th campsites. Finally the encounter problem of the whole river is transferred into small fractions. On analyzing encounter problem in d and count numbers of each encounter in d together, we get a clear recognition of the whole process and the total numbers of encounter of two parties.The following Figure 1 represents random two parties rafting in d :Figure 1. Random two parties' encounter or not on the riverThe i th party arrives at j th campsite (t j k ,-t j i ,) time earlier than the k th party reaches the j th campsite. After t time, interval distance between the i th party and the k th party can be denoted by the following function:)()(t t t v t v t S ij kj i k j +-⨯-⨯=∆ (6) Where k,i =1,2,…,X , j =1,2,…,Y . k i ≠.Whether the two parties stationed on the j th campsite and(j +1)th campsite are based on the state of the campsites’occupation, yields we obtain:⎩⎨⎧=⨯01,,j k j i q q ( i,k =1,2,…,X ; j =1,2,…,Y ; k ≠i ) (7)Note that Eq.6 is constrained by Eq.7, for different value of )(t S J ∆ andj k j i q q ,,⨯we can obtain the different cases as follows:Case 1:⎩⎨⎧=⨯=∆10)(,,j k ji j q q t S (8) Which means both the i th and k th party don’t choose the j th campsite, they arerafting on the river. Hence, when the interval distance between the two parties is 0, that is )(t S J ∆=0,they encounter at a certain place in d on the river.Cases2:⎩⎨⎧=⨯=∆00)(,,j k ji j q q t S (9) Although the interval distance between the two parties is 0, the j th campsite is occupied by the i th party or the k th party. That is one of them stop to camp at a certain place throughout the river corridor. Hence, there is no possibility for them to encounter on the river.Cases 3:⎩⎨⎧=⨯≠∆10)(,,j k j i j q q t S⎩⎨⎧=⨯≠∆00)(,,j k j i j q q t S (10) No matter the j th campsite is occupied or not for )(t S J ∆≠0 , that is at the same time, they are not at the same place. Hence, they will not encounter at any place in d .5. 3 Overview of Computer Simulation Modeling to Rafting5.3.1 Computer SimulationSimulation modeling is a kind of method to imitate the real-word process or a system. This approach is especially suited to those tasks which are too complex for direct observation, manipulation, or even analytical mathematical analysis (Banks and Carson 1984, Law and Kelton 1991, Pidd 1992).The most appropriate approach for simulating out-door recreation is dynamic, stochastic, and discrete-event model, since most recreation systems share these traits. In all, simulation models can reflect the real-world accurately.5.3.2 Simulation for the Whole Process of Parties on Rafting [8]This simulation can approximate show a party’s behavior on the river under a wide rang of conditions. From the analysis of the previous study, we have known that the next party’s behavior is affected by the former one. Hence, when the first party enters the rafting system, there is no encounter, and it can choose every campsite. then the second party comes into the rafting system , at this time, we must consider the encounter between them, and the limit on choosing the campsite. As time goes by, more and more parties enter this system to raft which lead to a more complex situation. A party who satisfies the following two conditions will be removed from the current order to the next order. So he can’t “finish his trip” right away. The two conditions are as follows:(1) He chooses a campsite where has been occupied by other parties.(2) He has two many encounters with other parties.So in order to determine typical trip itineraries for various types of rafting boat ,campsite, and time intervals （See Trip Schedule Sheet 1），we need to perform a series of trails run that can represent the real-life process of rafting based on these considerations,. A main flowchart of the program is shown in Figure 2.Figure 2. Main simulation flowchartAfter several times of simulation, we obtain the optimal X (the numbers of campsites), minimal E(Encounter) and TP (Trip Time).Followed by Figure 2, we simulate the behavior of a party whether it can enterthe rafting system or not in Figure 3.Figure 3.Sub flowchart5.3.3 The Results of SimulationAfter simulating the whole process of parties rafting on the river, we get three figures (Figure 4, Figure 5 and Figure 6) to present the results.In order to simulate the rafting process more conveniently, we divide the whole river into 31 segments (31 attraction sites), and input an initial value of Y=155(numbers of campsites), where there are 5 campsites in every attraction sites.We represent the times of campsites occupied by various parties on Figure 2 by coordinates( x, y), where x is the order of the campsites from 0 to 155（these campsites are all uniformly distributed thorough the corridor）, and y is the numbers of each campsite occupied by different parties. For example, (140，1100)represents that at the campsite, there exists nearly 1100 times of occupation in total by parties over 180days. Hence, the following Figure 4 shows the times of campsites’ usage from March to September.Figure 4.Numbers of campsites' usage during six-month period from March to SeptemberFrom Figure 4, The numbers of campsites’ usage can be identified the efficiency of every campsites’ usage. The higher usage of the campsites, the higher efficiency they are. Based on these, we give a simple suggestion to managers (see in Memo to Managers).Figure 5. the ratio of usage on campsites with time going byFigure 5 shows the changes of the ratio on campsites. when t =0, the campsites are not used , but with time going by, the ratio of the usage of campsites becomes higher and higher.We can also obtain that when t >20, the ratio keeps on a steady level of 65% ; but when t >176 , the ratio comes down, that is, there are little parties entering the recreation system. In all, these changes are rational very much, and have high coincidence with real-world.Then we obtain 1599 parties arranged into recreation system after inputting the initial value Y=155, and set orders to every party from number 0 to number 1599. Plotting every party's travelling time of the whole process on a map by simulating, as follows:Figure 6. Every party’s travelling timeFigure 6 shows the itinerary of the travelling time, most of the travelling time is fluctuating between 13days and 15.3 days, and most of travelling time are concentrated around 14 days.In order to create an outdoor life for all parties, we should minimize the numbers of encounter among different parties based on equations (6) and (7)：So we get every party’s numbers of encounter by coordinates ( x, y), where x is the order of the parties from 0 to 1600, y is the numbers of encounters. Shown in Figure 7, as follows:Figure 7. Every party’s numbers of encounterFigure 7 shows every party’s numbers of encounter at each campsite. From this figure, we can know that the numbers of their encounter are relatively less, the highest one is 8 times, and most of the parties don’t encounter during their trips, which is coincident with the real-world data.Finally, according to the travelling time of a party from March to September, we set a plan for river managers to arrange the number of parties. Hence, by simulating the model, we obtain the results by coordinate ( x, y), where y is the days of travelling time, x is the numbers of parties on every day. The figure is shown as follows:Figure 8. Simulation on travelling days versus the numbers of parties From Figure 8.we set a suitable plan for river manager, which also provide reference on his managements.6 Sensitive AnalyzeSensitive analysis is very critical in mathematical modeling, it is a way to gauge the robustness of a model with respect to assumptions about the data and parameters. We try several times of simulation to get different numbers of parties on changing the numbers of campsites ceaselessly. Thus using the simulative data, we get the relationship between the numbers of campsites and parties by fitting. On the basis of this fitting, we revise the maximal encounter times (Threshold value) continually, and can also get the results of the relationships between the numbers of campsites and parties by fitting. Finally, we obtain a Figure 9denoting the relations of Y (numbers of campsites)and X(numbers of parties), as follows:Figure 9 .Sensitive analysis under different threshold values Given the permitted maximal numbers of encounters (threshold value= K), we obtain the relationships between Y(numbers of campsites) and X( numbers of trips). For example, when K=1 , it means no encounters are allowed on the river when rafting; when K=2, there is less than 2 chances for the boats to meet. So we can define the K=4,6,8 to describe the sensitivity of our model.From Figure 9, we get the information that with the increase of K , the numbers of boats available to rafts till increase. But when K>6 , the change of the numbers of boats is inconspicuous, which is not the main factor having appreciable impact on theIn all, when the numbers of campsites ( Y ) are less than 250, they would have a great effect on the numbers of boats. But in the diverse situations, like when Y>250 , the effect caused by adding the numbers of campsites to hold more boats is not notable.When K<6 , the numbers of boats available increases with the ascending of K , While K>6 , the numbers of boats don’t have g reat change .Take all these factors into consideration, it reflects that the numbers of the boats can’t exceed its upper limit. Increase the numbers of campsites and numbers of encounter blindly can’t bring back more profits.7 Strengths and WeaknessesStrengthsOur model has achieved all of the goals we set initially effectively. It is not only fast and could handle large quantities of data, but also has the flexibility we desire.Though we don’t test all possibilities, if we had chosen to inpu t the numbers of campsites data into our program, we could have produced high-quality results with virtually no added difficulty. Aswell, our method was robust.Based on general assumptions we have made in previous task, we consider a party’s state in the first place, then simulate the whole process of rafting. It is an exact reflection of the real-world. Hence, our main model's strength is its enormousedibility and stability and there are some key strengths:(1) The flowchart represents the whole process of rafting by given different initialvalues. It not only makes it possible to develop trip itineraries that are statistically more representatives of the total population of river trips, but also eliminates the tedious task of manual writing .(2)Our m odel focuses on parties’ behavior and interactions between each other, notthe managers on the arrangement of rafting, which can also get satisfactory and high-quality results.(3) Our model makes full use of campsites, while avoid too many encounters, whichleads to rational arrangements.WeaknessesOn the one hand, although we list the model's comprehensive simulation as a strength, it is paradoxically also the most notable weakness since we don’t take into account the carrying capacity of the water when simulates, and suppose that a river can bear as much weight as possible. But in reality, that is impossible. On the other hand, our results are not optimal, but relative optimal.8 ConclusionsAfter a serial of trials, we get different values of X based on the general assumptions we make. By comparing them, we choose a relative better one. From this problem, it verifies the important use of simulation especially in complex situations. Here we consider if we change some of the assumptions, it may lead to various results. For example,(a) Let the velocity of this two kinds of boats submit to normal distribution.In this paper, the average velocity of oar- powered rubber rafts and motorized boats are 4mphand 8mph, respectively. But in real-world, the speed of the boats can’t get rid of the impacts from external force like stream’s propulsion and resistance. Hence, they keep on changing all the time.(b) Add and reduce campsites to improve the ratio of usage on campsites. By analyzing and simulating, the usage of each campsite is different which may lead to waste or congestion at a campsite. Hence, we can adjust the distribution of campsites to arrive the best use.A Memo to River ManagersOur simulation model is with high edibility and stability in many occasions. It can imitate every party’s behavior when rafting so as to make a clear recognition of the process.Internal Workings of The ModelInputsOur model needs to input initial value of Y , as well as the numbers of attraction sites. Algorithm ( Figure 2, and Figure 3)Our algorithm represents the whole process of rafting, so we can use it to simulate the process of rafting by inputting various initial values.OutputsBased on the algorithm in our paper, our model will output the relative optimalnumbers of parties X. Furthermore, we can also get other information, such as the interval time between two parties at First Launch, a detailed schedule for each party of rafting , the relationship between X and Y and so on.Summary and RecommendationsAfter 100 times of simulating, we come to two conclusions:(a) The numbers of parties (X) have relations with the numbers of campsites(Y), that is to say, with the increasing of Y , the increasing speed of X goes fast at the first place and then goes down, finally it tends to be steady. Hence, we advice river managers to adjust the numbers of campsites properly to get the optimal numbers of parties.(b) Add campsites to the high usage of the former campsites and deduce campsites at the low usage of the former campsites. From Figure 4, we know that the ratios of every campsites are different, some campsites are frequently used, but some are not. Thus we can infer that the scenic views are attractive, and have attracted lots of parties camping at the campsite. so we can add campsites to this nodes. Else the campsites with low usage have lost attractions which we should reduce the numbers of campsites at those nodes.References[1] Karlo Šimović，Wikipedia，Rafting，/wiki/Rafting.[2] C. A. Roberts and R. Gimblett，Computer Simulation for Rafting Traffic on theColorado River，COMPUTER SIMULATION FOR RAFTING TRAFFIC, 2001, 19-30.[3] C.A. Roberts, D. Stallman，J. A. Bieri. Modeling complex human-environmentinteractions: the Grand Canyon river trip simulator，Ecological Modeling153(2002) 181-196.[4] J. A. Bieri and C. A. Roberts，Using the Grand Canyon River Trip Simulator toTest New Launch Scheduleson the Colorado River，Washington DC, AWISMagazine, V ol. 29, No. 3, 2000, 6-10.[5] A.H. Underhill and A. B. Xaba，The Wilderness Simulation Model as aManagement Tool for the Colorado River in Grand Canyon National Park，NATIONAL PARK SERVICE/UNIVERSITY OF ARIZONA Unit SupportProject CONTRIBUTION NO. 034/03.[6] B. Wang and R. E. Manning，Computer Simulation Modeling for RecreationManagement: A Study on Carriage Road Use in Acadia National Park, Maine, USA, USA, Vermont 05405, 1999.[7] M. M. Meerschaert，Mathematical Modeling(Third Edition). China MachinePress publishing,2009.[8] A.H. Underhill，The Wilderness Use Simulation Model Applied to Colorado RiverBoating in Grand Canyon National Park, USA，Environmental Management V ol.10, No. 3, 1986, 367-374.。

2012年考研英语(一)翻译参考译文

2012年研究生入学考试英语（一）翻译部分参考译文2012年研究生入学考试英语一的翻译文章出自美国杂志《Nature》，题目是Universal truths。

这篇文章的理论性比较强，对于大家来说应该会感觉很有难度，大家首先要理解文章大意，依据专业性来定词义。

文章原文如下，选入2012考研英语一翻译真题时有部分删改：Since at least the days of Aristotle, a search for universal principles has characterized the scientific enterprise. In some ways, this quest for commonalities defines science: without it, there is no underlying order and pattern, merely as many explanations as there are things in the world. Newton's laws of motion, the oxygen theory of combustion and Darwinian evolution each bind a host of different phenomena into a single explicatory framework.46. In physics, one approach takes this impulse for unification to its extreme, and seeks a theory of everything —a single generative equation for all we see. It is becoming less clear, however, that such a theory would be a simplification, given the proliferation of dimensions and universes that it might entail. Nonetheless, unification of sorts remains a major goal.46. 在物理学上，一种方法就是把这种寻求统一性的冲动发挥到了极端，并努力寻找一种万能的理论，即一条唯一的为我们都看到的一切所生成的公式。

年美赛d题题目翻译

问题D：优化机场安全检查站乘客吞吐量继2001年9月11日美国发生恐怖袭击事件后，全世界的机场安全状况得到显着改善。

机场有安全检查站。

在那里，乘客及其行李被检查爆炸物和其他危险物品。

这些安全措施的目的是防止乘客劫持或摧毁飞机，并在旅行期间保持所有乘客的安全。

然而，航空公司有既得利益，通过最小化他们在安全检查站排队等候并等待他们的航班的时间，来保持乘客积极的飞行体验。

因此，在最大化安全性和最小化对乘客的不便之前存在对立。

在2016年，美国运输安全局（TSA）受到了对极长线路，特别是在芝加哥的奥黑尔国际机场的尖锐批评。

在此公众关注之后，TSA投资对其检查点设备和程序进行了若干修改，并增加了在高度拥堵的机场中的人员配置。

虽然这些修改在减少等待时间方面有一定的成功，但TSA在实施新措施和增加人员配置方面花费了多少成本尚不清楚。

除了在奥黑尔机场的问题，还有在其他机场，包括通常排队等待时间较短的机场，会出现不明原因和不可预测的排队拥挤情况的事件。

检查点排队状况的这种高度变化性对于乘客来说可能是极其不利的，因为他们面临着不必要地早到达或可能赶不上他们的预定航班的风险。

许多新闻文章，包括[1,2,3,4,5]，描述了与机场安全检查站相关的一些问题。

您的内部控制管理（ICM）团队已经与TSA签订合同，审查机场安全检查站和人员配置，以确定潜在的干扰乘客吞吐量的瓶颈。

他们特别感兴趣的解决方案是，既增加检查点吞吐量，减少等待时间的变化，同时保持相同的安全和安全标准。

美国机场安全检查点的当前流程如图1所示。

区域A：乘客随机到达检查站，并等待队列，直到安全人员可以检查他们的身份证明和登机文件。

区域B：然后乘客移动到打开检查的队列;根据机场的预期活动水平，可能开放更多或更少的线路。

一旦乘客到达这个队列的前面，他们准备所有的物品用于X射线检查。

乘客必须去除鞋子，皮带，夹克，金属物体，电子产品和带液体容器，将它们放置在单独的X射线箱中;笔记本电脑和一些医疗设备也需要从其袋中取出并放置在单独的容器中。

数学建模美赛2012MCM B论文

Camping along the Big Long RiverSummaryIn this paper,the problem that allows more parties entering recreation system is investigated.In order to let park managers have better arrangements on camping for parties,the problem is divided into four sections to consider.The first section is the description of the process for single-party's rafting.That is, formulating a Status Transfer Equation of a party based on the state of the arriving time at any campsite.Furthermore,we analyze the encounter situations between two parties.Next we build up a simulation model according to the analysis above.Setting that there are recreation sites though the river,count the encounter times when a new party enters this recreation system,and judge whether there exists campsites available for them to station.If the times of encounter between parties are small and the campsite is available,the managers give them a good schedule and permit their rafting,or else, putting off the small interval time t∆until the party satisfies the conditions.Then solve the problem by the method of computer simulation.We imitate the whole process of rafting for every party,and obtain different numbers of parties,every party's schedule arrangement,travelling time,numbers of every campsite's usage, ratio of these two kinds of rafting boats,and time intervals between two parties' starting time under various numbers of campsites after several times of simulation. Hence,explore the changing law between the numbers of parties(X)and the numbers of campsites(Y)that X ascends rapidly in the first period followed by Y's increasing and the curve tends to be steady and finally looks like a S curve.In the end of our paper,we make sensitive analysis by changing parameters of simulation and evaluate the strengths and weaknesses of our model,and write a memo to river managers on the arrangements of rafting.Key words:Camping；Computer Simulation;Status Transfer Equation1IntroductionThe number of visits to outdoor recreation areas has increased dramatically in last three decades.Among all those outdoor activities,rafting is often chose as a family get-together during May to September.Rafting or white water rafting is a kind of interesting and challenging recreational outdoor activity,which uses an inflatable raft to navigate a river or sea [1].It is very popular in the world,especially in occidental countries.This activity is commonly considered an extreme sport that usually done to thrill and excite the raft passengers on white water or different degrees of rough water.It can be dangerous.During the peak period,there are many tourists coming to experience rafting.In order to satisfy tourists to the maximum,we must make full use of our facilities in hand,which means we must do the utmost to utilize the campsites in the best way possible.What's more,to make more people feel the wildness life,we should minimize the encounters to the best extent;meanwhile no two sets of parties can occupy the same campsite at the same time.It is naturally coming into mind that we should consider where to stop,and when to stop of a party [2].In previous studies [3-5],many researchers have simulated the outdoor creation based on real-life data,because the approach is dynamic,stochastic,and discrete-event,and most recreation systems share these traits.But there exists little research aiming at describing the way that visitors travel and distribute themselves within a recreation system [6].Hence,in our paper,we consider the whole process of parties in detail and simulate every party ’s behavior,including the location of their campsites,and how long it will last for them to stay in a campsite to finish their itineraries.Meanwhile minimize the numbers of encounters.Aiming at showing the whole process of rafting,we firstly focus on analyzing the situation s of a single-party's rafting by using status transfer equation,then consider the problems of two parties'encounters on the river.Finally,after several times of simulation on the whole process of rafting,we obtain the optimal value of X .2Symbols and DefinitionsIn this section,we will give some basic symbols and definitions in the following for the convenience.Table 1.Variable Definition Symbols Definitioni v i p j i q ,S dThe velocity of oar or motor0-1variables on choosing rafting transportation0-1variables on the occupation of campsitesLength of the riverAverage distance between two campsites3General AssumptionsIn order to have a better study on this paper,we simplify our model by thefollowing assumptions:1)19:00to 07:00is people's sleeping time,during this time,people are stationedin the campsite.The total time of sleeping is 12hours,as rafting is an exiting sport game,after a day's entertainment,people have cost a lot of energy,and nearly tired out.So in order to have a better recreation for the next day,we set that people begin their trip at 07:00,and end at 19:00for a day's schedule.2)Oar-powered rubber rafts and motorized parties can successfully raft from FirstLaunch to Final Exit,there exist no accident over the whole trips.3)All the rubber rafts and motorized boats have the same exterior except velocities;we regard a rubber raft or a motorized boat as a party and don't consider the tourists individuals on the parties.4)There is only one entrance for parties to enter the recreation system.5)Regardless of the effects that the physical features of the river brings to oar andmotorized parties,that is to say we ignore the stream ’s propulsion and resistance to both kinds of rafting boats.Oar and motorized parties can keep the average velocity of 4mph and 8mph.6)Divide the whole river into N segments.4Analysis of This Rafting ProblemRafting is a very popular spots game world-wide.In the peak period of rafting,there are more people choosing to raft,it often causes congestion that not all people can raft at any time they want.Hence,it is important for managers to set an optimal schedule for every party (from our assumptions,we regard a rafting boat as a party)in advance.Meanwhile,the parties need to experience wildness life,so the managers should arrange the schedules which minimize the encounters'time between parties to the best extent.What's more,no two sets of parties can occupy the same site at the same time.Our aim is to determine an optimal mix of trips over varying duration (measured YXNj i t ,jT t∆KNumbers of campsites Numbers of parties Numbers of attraction sites Time of the i th party finishing the whole trip ranges from6days to 18daysRandom staying time at each campsiteDelay time of rafting from beginning Threshold value of encounterin nights on the river.That is to say,we must obtain an optimal value of X through lots of trails.This optimal value represents that the campsites have a high usage while more people are available to raft.The Long Big River is 225miles long,if we discuss the river as a whole and consider all the parties together,it will be difficult for us to have a clear recognition on parties'behaviors.Hence,we divide the river into N attraction sites.Each of the attraction sites has Y/N campsites since the campsites are uniformly distributed throughout the river corridor.So build up a model based on single-party ’s behavior of rafting in small distance.At last,we can use computer simulation to imitate more complex situations with various rafting boats and large quantities of parties.5Mathematic Models5.1Rafting of the Single-party Model （Status Transfer Equation [7]）From the previous analysis,in order to have a clear recognition of the whole rafting process,we must analyze every single-party's state at any time.In this model,we consider the situation that a single-party rafts from the First Launch to the Final Exit.So we formulate a model that focus on the behavior of one single-party.For a single-party,it must satisfy the following equation:status transfer equation.it represents the relationships between its former state and the latter state.State here means:when the i th party arrives at the j th campsites,the party may occupy the j th campsite or not.As a party can choose two kinds of transportation to raft:oar-powered rubber rafts(i v =4mph)and motorized rafts(i v =8mph).i v is the velocity of the rafting boats,and i p is the 0-1variables of the selecting for boats.Therefore,we can obtainthe following equation:)1(84i i i p p v −+=(i=1,2,…,X ).(1)where i p =0if the i th party uses motorized boat as their rafting tool,at thistime i v =8mph ;while i p =1when ,the i th party rafts with oar-powered rubber raft with i v =4mph.In fact,Eq.(1)denotes which kind of rafting boat a party can choose.A party not only has choice on rafting boats,but also can select where to camp based on whether the campsites are occupied or not.The following formulation shows the situation whether this party chooses this campsite or not:⎩⎨⎧=party previous a by occupied is campsite the 0,party previous a by occupied not is campsite the q ij ,1(2)where i =1,2,…,X ;j =1,2,…,Y .Where the next one can’t set their camp at this place anymore,that is to say thelatter party’s behavior is determined by the former one.As campsites are fairly uniformly distributed throughout the river corridor,hence,we discrete the whole river into segments,and regard Y campsites as Y nodes which leaves out (Y +1)intervals.Finally we get the average distance between th e j th campsite and (j+1)th campsite:1+=Y Sd (3)where is the length of the river,and its value is 225miles.What’s more,the trip-days for a party is not infinite,it has fluctuating intervals:h t h j i 432144,≤≤(4)where is the t i ，j itinerary time for a party ranges from 144hours to 432hours (6to 18nights).From Eq.(1),(2)and (3),the status transfer equation is given as follows:),...2,1,,...2,1(11,1,,Y j X i T q v d t t j j i i j i j i ==×++=−−−（5）The i th party’s arriving time at the j th campsite is determined by the time when the i th arrived at （j-1）campsite,the time interval i v d ,and the time T j-1random generated by computer shown in Eq.(5).It is a dynamic process and determined by its previous behavior.5.2The Analysis of Two Parties Parties’’Encounter on the River Our goal is to making full use of the campsites.Hence,the objective of all the formulation is to maximize the quantities of trips （parties ）X while consider getting rid of the congestion.If we reduce the numbers of the encounters among parties,there will be no congestion.In order to achieve this goal,we analysis the situations of when two parties’to encounter,and where they will encounter.In order to create a wildness environment for parties to experience wildness life,managers arrange a schedule that can make any two parties have minimal encounters with each other.Encounter is that parties meet at the same place and at the same time.Regarding the river as a whole is not convenient to study,hence,our discussion is based on a small distance where distance=d (Eq.3),between the j th and (j+1)th campsites.Finally the encounter problem of the whole river is transferred into small fractions.On analyzing encounter problem in d and count numbers of each encounter in d together,we get a clear recognition of the whole process and the total numbers of encounter of two parties.The following Figure 1represents random two parties rafting in d :Figure 1.Random two parties'encounter or not on the riverThe i th party arrives at j th campsite (t j k ,-t j i ,)time earlier than the k th party reaches the j th campsite.After t time,interval distance between the i th party and the k th party can be denoted by the following function:)()(t t t v t v t S ij kj i k j +−×−×=∆(6)Where k,i =1,2,…,X ,j =1,2,…,Y .k i ≠.Whether the two parties stationed on the j th campsite and(j +1)th campsite are based on the state of the campsites’occupation,yields we obtain:⎩⎨⎧=×01,,j k j i q q (i,k =1,2,…,X ;j =1,2,…,Y ;k ≠i )(7)Note that Eq.6is constrained by Eq.7,for different value of )(t S J ∆andj k j i q q ,,×we can obtain the different cases as follows:Case 1:⎩⎨⎧=×=∆10)(,,j k j i j q q t S (8)Which means both the i th and k th party don’t choose the j th campsite,they are rafting on the river.Hence,when the interval distance between the two parties is 0,that is )(t S J ∆=0,they encounter at a certain place in d on the river.Cases2:⎩⎨⎧=×=∆00)(,,j k j i j q q t S (9)Although the interval distance between the two parties is 0,the j th campsite is occupied by the i th party or the k th party.That is one of them stop to camp at a certain place throughout the river corridor.Hence,there is no possibility for them to encounter on the river.Cases 3:⎩⎨⎧=×≠∆10)(,,j k j i j q q t S ⎩⎨⎧=×≠∆00)(,,j k j i j q q t S (10)No matter the j th campsite is occupied or not for )(t S J ∆≠0,that is at the same time,they are not at the same place.Hence,they will not encounter at any place in d .5.3Overview of Computer Simulation Modeling to Rafting5.3.1Computer SimulationSimulation modeling is a kind of method to imitate the real-word process or a system.This approach is especially suited to those tasks which are too complex for direct observation,manipulation,or even analytical mathematical analysis (Banks and Carson 1984,Law and Kelton 1991,Pidd 1992).The most appropriate approach for simulating out-door recreation is dynamic,stochastic,and discrete-event model,since most recreation systems share these traits.In all,simulation models can reflect the real-world accurately.5.3.2Simulation for the Whole Process of Parties on Rafting [8]This simulation can approximate show a party’s behavior on the river under a wide rang of conditions.From the analysis of the previous study,we have known that the next party’s behavior is affected by the former one.Hence,when the first party enters the rafting system,there is no encounter,and it can choose every campsite.then the second party comes into the rafting system ,at this time,we must consider the encounter between them,and the limit on choosing the campsite.As time goes by,more and more parties enter this system to raft which lead to a more complex situation.A party who satisfies the following two conditions will be removed from the current order to the next order.So he can’t “finish his trip”right away.The two conditions are as follows:(1)He chooses a campsite where has been occupied by other parties.(2)He has two many encounters with other parties.So in order to determine typical trip itineraries for various types of rafting boat ,campsite,and time intervals （See Trip Schedule Sheet 1），we need to perform a series of trails run that can represent the real-life process of rafting based on these considerations,.A main flowchart of the program is shown in Figure 2.Figure2.Main simulation flowchartAfter several times of simulation,we obtain the optimal X(the numbers of campsites),minimal E(Encounter)and TP(Trip Time).Followed by Figure2,we simulate the behavior of a party whether it can enterthe rafting system or not in Figure3.Figure3.Sub flowchart5.3.3The Results of SimulationAfter simulating the whole process of parties rafting on the river,we get three figures(Figure4,Figure5and Figure6)to present the results.In order to simulate the rafting process more conveniently,we divide the whole river into31segments(31attraction sites),and input an initial value of Y=155(numbers of campsites),where there are5campsites in every attraction sites.We represent the times of campsites occupied by various parties on Figure2by coordinates(x,y),where x is the order of the campsites from0to155（these campsites are all uniformly distributed thorough the corridor）,and y is the numbers of each campsite occupied by different parties.For example,(140，1100)represents that at the campsite,there exists nearly1100times of occupation in total by parties over180days. Hence,the following Figure4shows the times of campsites’usage from March to September.Figure4.Numbers of campsites'usage during six-month period from March to SeptemberFrom Figure4,The numbers of campsites’usage can be identified the efficiency of every campsites’usage.The higher usage of the campsites,the higher efficiency they are.Based on these,we give a simple suggestion to managers(see in Memo to Managers).Figure5.the ratio of usage on campsites with time going byFigure5shows the changes of the ratio on campsites.when t=0,the campsites are not used,but with time going by,the ratio of the usage of campsites becomes higher and higher.We can also obtain that when t>20,the ratio keeps on a steady level of65%;but when t >176,the ratio comes down,that is,there are little parties entering the recreation system.In all,these changes are rational very much,and have high coincidence with real-world.Then we obtain1599parties arranged into recreation system after inputting the initial value Y=155,and set orders to every party from number0to number1599.Plotting every party's travelling time of the whole process on a map by simulating,as follows:Figure6.Every party’s travelling timeFigure6shows the itinerary of the travelling time,most of the travelling time is fluctuating between13days and15.3days,and most of travelling time are concentrated around14days.In order to create an outdoor life for all parties,we should minimize the numbers of encounter among different parties based on equations(6)and(7)：So we get every party’s numbers of encounter by coordinates(x,y),where x is the order of the parties from0to1600,y is the numbers of encounters.Shown in Figure7,as follows:Figure7.Every party’s numbers of encounterFigure7shows every party’s numbers of encounter at each campsite.From this figure,we can know that the numbers of their encounter are relatively less,the highest one is8times,and most of the parties don’t encounter during their trips,which is coincident with the real-world data.Finally,according to the travelling time of a party from March to September,we set a plan for river managers to arrange the number of parties.Hence,by simulating the model,we obtain the results by coordinate(x,y),where y is the days of travelling time,x is the numbers of parties on every day.The figure is shown as follows:Figure8.Simulation on travelling days versus the numbers of parties From Figure8.we set a suitable plan for river manager,which also provide reference on his managements.6Sensitive AnalyzeSensitive analysis is very critical in mathematical modeling,it is a way to gauge the robustness of a model with respect to assumptions about the data and parameters. We try several times of simulation to get different numbers of parties on changing the numbers of campsites ceaselessly.Thus using the simulative data,we get the relationship between the numbers of campsites and parties by fitting.On the basis of this fitting,we revise the maximal encounter times(Threshold value)continually,and can also get the results of the relationships between the numbers of campsites and parties by fitting.Finally,we obtain a Figure9denoting the relations of Y(numbers of campsites)and X(numbers of parties),as follows:Figure9.Sensitive analysis under different threshold values Given the permitted maximal numbers of encounters(threshold value=K),we obtain the relationships between Y(numbers of campsites)and X(numbers of trips). For example,when K=1,it means no encounters are allowed on the river when rafting;when K=2,there is less than2chances for the boats to meet.So we can define the K=4,6,8to describe the sensitivity of our model.From Figure9,we get the information that with the increase of K,the numbers of boats available to rafts till increase.But when K>6,the change of the numbers of boats is inconspicuous,which is not the main factor having appreciable impact on the numbers of boats.>In all,when the numbers of campsites(Y)are less than250,they would have a great effect on the numbers of boats.But in the diverse situations,like when Y>250, the effect caused by adding the numbers of campsites to hold more boats is not notable.When K<6,the numbers of boats available increases with the ascending of K, While K>6,the numbers of boats don’t have great change.Take all these factors into consideration,it reflects that the numbers of the boats can’t exceed its upper limit.Increase the numbers of campsites and numbers of encounter blindly can’t bring back more profits.7Strengths and WeaknessesStrengthsOur model has achieved all of the goals we set initially effectively.It is not only fast and could handle large quantities of data,but also has the flexibility we desire.Though we don’t test all possibilities,if we had chosen to input the numbers of campsites data into our program,we could have produced high-quality results with virtually no added difficulty.Aswell,our method was robust.Based on general assumptions we have made in previous task,we consider a party’s state in the first place,then simulate the whole process of rafting.It is an exact reflection of the real-world.Hence,our main model's strength is its enormousedibility and stability and there are some key strengths:(1)The flowchart represents the whole process of rafting by given different initialvalues.It not only makes it possible to develop trip itineraries that are statistically more representatives of the total population of river trips,but also eliminates the tedious task of manual writing.(2)Our model focuses on parties’behavior and interactions between each other,notthe managers on the arrangement of rafting,which can also get satisfactory and high-quality results.(3)Our model makes full use of campsites,while avoid too many encounters,whichleads to rational arrangements.WeaknessesOn the one hand,although we list the model's comprehensive simulation as a strength,it is paradoxically also the most notable weakness since we don’t take into account the carrying capacity of the water when simulates,and suppose that a river can bear as much weight as possible.But in reality,that is impossible.On the other hand,our results are not optimal,but relative optimal.8ConclusionsAfter a serial of trials,we get different values of X based on the general assumptions we make.By comparing them,we choose a relative better one.From this problem,it verifies the important use of simulation especially in complex situations. Here we consider if we change some of the assumptions,it may lead to various results. For example,(a)Let the velocity of this two kinds of boats submit to normal distribution.In this paper,the average velocity of oar-powered rubber rafts and motorized boats are 4mphand8mph,respectively.But in real-world,the speed of the boats can’t get rid of the impacts from external force like stream’s propulsion and resistance.Hence,they keep on changing all the time.(b)Add and reduce campsites to improve the ratio of usage on campsites.By analyzing and simulating,the usage of each campsite is different which may lead to waste or congestion at a campsite.Hence,we can adjust the distribution of campsites to arrive the best use.A Memo to River ManagersOur simulation model is with high edibility and stability in many occasions.It can imitate every party’s behavior when rafting so as to make a clear recognition of the process.Internal Workings of The ModelInputsOur model needs to input initial value of Y,as well as the numbers of attraction sites. Algorithm(Figure2,and Figure3)Our algorithm represents the whole process of rafting,so we can use it to simulate the process of rafting by inputting various initial values.OutputsBased on the algorithm in our paper,our model will output the relative optimalnumbers of parties X.Furthermore,we can also get other information,such as the interval time between two parties at First Launch,a detailed schedule for each party of rafting,the relationship between X and Y and so on.Summary and RecommendationsAfter100times of simulating,we come to two conclusions:(a)The numbers of parties(X)have relations with the numbers of campsites(Y), that is to say,with the increasing of Y,the increasing speed of X goes fast at the first place and then goes down,finally it tends to be steady.Hence,we advice river managers to adjust the numbers of campsites properly to get the optimal numbers of parties.(b)Add campsites to the high usage of the former campsites and deduce campsites at the low usage of the former campsites.From Figure4,we know that the ratios of every campsites are different,some campsites are frequently used,but some are not.Thus we can infer that the scenic views are attractive,and have attracted lots of parties camping at the campsite.so we can add campsites to this nodes.Else the campsites with low usage have lost attractions which we should reduce the numbers of campsites at those nodes.References[1]KarloŠimović，Wikipedia，Rafting，/wiki/Rafting.[2] C.A.Roberts and R.Gimblett，Computer Simulation for Rafting Traffic on theColorado River，COMPUTER SIMULATION FOR RAFTING TRAFFIC,2001, 19-30.[3] C.A.Roberts,D.Stallman，J.A.Bieri.Modeling complex human-environmentinteractions:the Grand Canyon river trip simulator，Ecological Modeling153(2002)181-196.[4]J.A.Bieri and C.A.Roberts，Using the Grand Canyon River Trip Simulator toTest New Launch Scheduleson the Colorado River，Washington DC,AWISMagazine,Vol.29,No.3,2000,6-10.[5] A.H.Underhill and A.B.Xaba，The Wilderness Simulation Model as aManagement Tool for the Colorado River in Grand Canyon National Park，NATIONAL PARK SERVICE/UNIVERSITY OF ARIZONA Unit SupportProject CONTRIBUTION NO.034/03.[6] B.Wang and R.E.Manning，Computer Simulation Modeling for RecreationManagement:A Study on Carriage Road Use in Acadia National Park,Maine,USA,USA,Vermont05405,1999.[7]M.M.Meerschaert，Mathematical Modeling(Third Edition).China MachinePress publishing,2009.[8]A.H.Underhill，The Wilderness Use Simulation Model Applied to Colorado RiverBoating in Grand Canyon National Park,USA，Environmental Management Vol.10,No.3,1986,367-374.。

2012年美国高中生数学建模竞赛特等奖论文

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.。

建模美赛C题带翻译

Problem C: “Cooperate and navigate”Traffic capacity is limited in many regions of the United States due to the number of lanes of roads. For example, in the Greater Seattle area drivers experience long delays during peak traffic hours because the volume of traffic exceeds the designed capacity of the road networks. This is particularly pronounced on Interstates 5, 90, and 405, as well as State Route 520, the roads of particular interest for this problem.Self-driving, cooperating cars have been proposed as a solution to increase capacity of highways without increasing number of lanes or roads. The behavior of these cars interacting with the existing traffic flow and each other is not well understood at this point.The Governor of the state of Washington has asked for analysis of the effects of allowing self-driving, cooperating cars on the roads listed above in Thurston, Pierce, King, and Snohomish counties. (See the provided map and Excel spreadsheet). In particular, how do the effects change as the percentage of self-driving cars increases from 10% to 50% to 90%? Do equilibria exist? Is there a tipping point where performance changes markedly? Under what conditions, if any, should lanes be dedicated to these cars? Does your analysis of your model suggest any other policy changes?Your answer should include a model of the effects on traffic flow of the number of lanes, peak and/or average traffic volume, and percentage of vehicles using self-driving, cooperating systems. Your model should address cooperation between self-driving cars as well as the interaction between self- driving and non-self-driving vehicles. Your model should then be applied to the data for the roads of interest, provided in the attached Excel spreadsheet.Your MCM submission should consist of a 1 page Summary Sheet, a 1-2 page letter to the Governor’s office, and your solution (not to exceed 20 pages) for a maximum of 23 pages. Note: The appendix and references do not count toward the 23 page limit. Some useful background information:On average, 8% of the daily traffic volume occurs during peak travel hours.•The nominal speed limit for all these roads is 60 miles per hour.•Mileposts are numbered from south to north, and west to east.•Lane widths are the standard 12 feet.•Highway 90 is classified as a state route until it intersects Interstate 5.•In case of any conflict between the data provided in this problem and any other source, use the data provided in this problem.Definitions:milepost: A marker on the road that measures distance in miles from either the start of the route or astate boundary.average daily traffic: The average number of cars per day driving on the road.interstate: A limited access highway, part of a national system.state route: A state highway that may or may not be limited access.route ID: The number of the highway.increasing direction: Northbound for N-S roads, Eastbound for E-W roads.decreasing direction: Southbound for N-S roads, Westbound for E-W roads.问题C：“合作和导航”由于道路的数量，美国许多地区的交通容量有限。

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