2016_ICM_Problem_D

基于主动学习和二次有理核的模型无关局部解释方法
周晟昊;袁伟伟;关东海
【期刊名称】《计算机科学》
【年(卷),期】2024(51)2
【摘要】深度学习模型的广泛使用,在更大程度上使人们意识到模型的决策是亟需解决的问题,复杂难以解释的黑盒模型阻碍了算法在实际场景中部署。

LIME作为最流行的局部解释方法,生成的扰动数据却具有不稳定性,导致最终的解释产生偏差。

针对上述问题,提出了一种基于主动学习和二次有理核的模型无关局部解释方法ActiveLIME,使得局部解释模型更加忠于原始分类器。

ActiveLIME生成扰动数据后,通过主动学习的查询策略对扰动数据进行采样,筛选不确定性高的扰动集训练,使用迭代过程中准确度最高的局部模型对感兴趣实例生成解释。

并且,针对容易陷入局部过拟合的高维稀疏样本,在模型损失函数中引入了二次有理核来减少过拟合。

实验结果表明,所提出的ActiveLIME方法引比传统局部解释方法具有更高的局部保真度和解释质量。

【总页数】7页(P245-251)
【作者】周晟昊;袁伟伟;关东海
【作者单位】南京航空航天大学计算机科学与技术学院
【正文语种】中文
【中图分类】TP391
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A: 一个人充满热水的浴缸从一个单一的水龙头，并落户到浴缸里清洁和放松。

不幸的是，浴缸不是温泉式浴盆与辅助加热系统和循环飞机，而是一个简单的水安全壳。

一段时间后，在浴变得明显冷却器，所以该人增加的热水恒定涓流从龙头再加热洗澡水。

所述浴缸的设计以这样的方式，当所述桶达到其容量，过量的水逸出通过溢流漏极。

开发浴缸水的空间和时间，以确定该人在浴缸可以采用，以保持温度甚至整个浴缸和尽可能接近到初始温度，而不会浪费过多的水的最佳策略的温度的模型。

使用模型确定哪个你的策略取决于形状和桶，该人在浴缸的形状/体积/温度的体积，并且由该人在浴缸所作的运动的程度。

如果对方使用了泡泡浴的添加剂，而最初填补了浴缸，以协助清洁，怎么会变成这样影响模型的结果吗？除了必需的单页摘要您的MCM提交，你的报告必须包括一份一页纸的非技术性解释浴缸的用户描述你的策略，同时解释了为什么它是如此难以得到整个均匀保持温度浴缸里的水。

B: 小碎片在轨道上绕地球金额已日益受到关注。

据估计，超过50万件的空间碎片，也被称为轨道碎片，目前都被跟踪的潜在危害航天器。

这个问题本身变得更广泛的讨论，在新闻媒体时，俄罗斯卫星的Kosmos-2251和美国铱卫星-33相撞2009年2月10日。

有许多方法以除去碎片已被提出。

这些方法包括小的，基于空间的水射流和用于针对特定的碎片高能激光器和设计，以清扫杂物，其中包括大型卫星。

从漆片的废弃卫星的大小和质量的碎片范围。

碎片“高速轨道进行采集困难。

开发时间依赖模型来确定的替代品，私人公司可以采取作为一个商业机会，以解决空间碎片问题的最佳替代品或组合。

您的模型应该包括成本，风险，效益的定量和/或定性的估计，以及其他重要的因素。

您的模型应该能够评估独立的替代品的替代品，以及组合和能够探索各种重要的“如果？”的情景。

使用模型，确定了经济上有吸引力的机会是否存在，或没有这样的机会是可能的。

如果一个可行的商业机会的存在作为一种替代解决方案，提供了不同的选择去除杂物进行比较，并包括具体建议，以碎片应该如何去除。

扩散模型损失函数推导扩散模型（Diffusion Model）是一种用于解决信息传播、疾病传播等问题的模型。

在扩散模型中，我们通常使用损失函数来衡量模型预测结果与实际数据之间的差异。

在扩散模型中，我们可以假设信息或疾病的传播是基于一定的规则，例如传播速率、传播范围等。

基于这些假设，我们可以构建一个数学模型来描述信息或疾病的传播过程。

一般来说，扩散模型可以分为离散模型和连续模型。

对于离散模型，我们可以使用损失函数来衡量模型的预测结果与实际数据之间的差异。

常用的损失函数包括平方损失（Mean Square Error，MSE）、交叉熵损失（Cross Entropy Loss）等。

以平方损失为例，假设我们的模型预测结果为y_pred，实际数据为y_true，则平方损失为：\[L(y_{true}, y_{pred}) = (y_{true} - y_{pred})^2\]损失函数越小，表示模型预测结果与实际数据之间的差异越小，模型的准确性越高。

对于连续模型，我们可以利用微分方程来描述信息或疾病的传播过程。

在连续模型中，损失函数通常是通过最小化模型预测结果与实际数据之间的差异来实现的。

常用的损失函数包括均方差损失（Mean Squared Error）、平均绝对误差（Mean Absolute Error）等。

以均方差损失为例，假设模型的预测结果为u(x, t)，实际数据为u_true(x, t)，则均方差损失为：\[L(u_{true}, u_{pred}) = \int \int (u_{true}(x, t) - u_{pred}(x, t))^2 dx dt\]通过最小化损失函数，我们可以得到最优的模型参数，从而使得模型的预测结果与实际数据最为接近。

需要注意的是，损失函数的具体形式取决于问题的特点和模型的设计。

在实际应用中，我们需要根据具体情况选择合适的损失函数，以达到最佳的模型效果。

For office use only T1T2T3T4T eam Control Number42939Problem ChosenCFor office use onlyF1F2F3F42016Mathematical Contest in Modeling(MCM)Summary Sheet (Attach a copy of this page to each copy of your solution paper.)SummaryIn order to determine the optimal donation strategy,this paper proposes a data-motivated model based on an original definition of return on investment(ROI) appropriate for charitable organizations.First,after addressing missing data,we develop a composite index,called the performance index,to quantify students’educational performance.The perfor-mance index is a linear composition of several commonly used performance indi-cators,like graduation rate and graduates’earnings.And their weights are deter-mined by principal component analysis.Next,to deal with problems caused by high-dimensional data,we employ a lin-ear model and a selection method called post-LASSO to select variables that statis-tically significantly affect the performance index and determine their effects(coef-ficients).We call them performance contributing variables.In this case,5variables are selected.Among them,tuition&fees in2010and Carnegie High-Research-Activity classification are insusceptible to donation amount.Thus we only con-sider percentage of students who receive a Pell Grant,share of students who are part-time and student-to-faculty ratio.Then,a generalized adaptive model is adopted to estimate the relation between these3variables and donation amount.Wefit the relation across all institutions and get afitted function from donation amount to values of performance contributing variables.Then we divide the impact of donation amount into2parts:homogenous and heterogenous one.The homogenous influence is modeled as the change infit-ted values of performance contributing variables over increase in donation amount, which can be predicted from thefitted curve.The heterogenous one is modeled as a tuning parameter which adjusts the homogenous influence based on deviation from thefitted curve.And their product is increase in true values of performance over increase in donation amount.Finally,we calculate ROI,defined as increase in performance index over in-crease in donation amount.This ROI is institution-specific and dependent on in-crease in donation amount.By adopting a two-step ROI maximization algorithm, we determine the optimal investment strategy.Also,we propose an extended model to handle problems caused by time dura-tion and geographical distribution of donations.A Letter to the CFO of the Goodgrant FoundationDear Chiang,Our team has proposed a performance index quantifying the students’educational per-formance of each institution and defined the return of investment(ROI)appropriately for a charitable organization like Goodgrant Foundation.A mathematical model is built to help predict the return of investment after identifying the mechanism through which the donation generates its impact on the performance.The optimal investment strategy is determined by maximizing the estimated return of investment.More specifically,the composite performance index is developed after taking all the pos-sible performance indicators into consideration,like graduation rate and graduates’earnings. The performance index is constructed to represents the performance of the school as well as the positive effect that a college brings to students and the community.From this point of view, our definition manages to capture social benefits of donation.And then we adopt a variable selection method tofind out performance contributing vari-ables,which are variables that strongly affect the performance index.Among all the perfor-mance contributing variables we select,three variables which can be directly affected by your generous donation are kept to predict ROI:percentage of students who receive a Pell Grant, share of students who are part-time and student-to-faculty ratio.Wefitted a relation between these three variables and the donation amount to predict change in value of each performance contributing variable over your donation amount.And we calculate ROI,defined as increase in the performance index over your donation amount, by multiplying change in value of each performance contributing variable over your donation amount and each performance contributing variable’s effect on performance index,and then summing up the products of all performance contributing variables.The optimal investment strategy is decided after maximizing the return of investment according to an algorithm for selection.In conclusion,our model successfully produced an investment strategy including a list of target institutions and investment amount for each institution.(The list of year1is attached at the end of the letter).The time duration for the investment could also be determined based on our model.Since the model as well as the evaluation approach is fully data-motivated with no arbitrary criterion included,it is rather adaptable for solving future philanthropic educational investment problems.We have a strong belief that our model can effectively enhance the efficiency of philan-thropic educational investment and provides an appropriate as well as feasible way to best improve the educational performance of students.UNITID names ROI donation 197027United States Merchant Marine Academy21.85%2500000 102711AVTEC-Alaska’s Institute of Technology21.26%7500000 187745Institute of American Indian and Alaska Native Culture20.99%2000000 262129New College of Florida20.69%6500000 216296Thaddeus Stevens College of Technology20.66%3000000 229832Western Texas College20.26%10000000 196158SUNY at Fredonia20.24%5500000 234155Virginia State University20.04%10000000 196200SUNY College at Potsdam19.75%5000000 178615Truman State University19.60%3000000 199120University of North Carolina at Chapel Hill19.51%3000000 101648Marion Military Institute19.48%2500000187912New Mexico Military Institute19.31%500000 227386Panola College19.28%10000000 434584Ilisagvik College19.19%4500000 199184University of North Carolina School of the Arts19.15%500000 413802East San Gabriel Valley Regional Occupational Program19.09%6000000 174251University of Minnesota-Morris19.09%8000000 159391Louisiana State University and Agricultural&Mechanical Col-19.07%8500000lege403487Wabash Valley College19.05%1500000 Yours Sincerely,Team#42939An Optimal Strategy of Donation for Educational PurposeControl Number:#42939February,2016Contents1Introduction51.1Statement of the Problem (5)1.2Baseline Model (5)1.3Detailed Definitions&Assumptions (8)1.3.1Detailed Definitions: (8)1.3.2Assumptions: (9)1.4The Advantages of Our Model (9)2Addressing the Missing Values93Determining the Performance Index103.1Performance Indicators (10)3.2Performance Index via Principal-Component Factors (10)4Identifying Performance Contributing Variables via post-LASSO115Determining Investment Strategy based on ROI135.1Fitted Curve between Performance Contributing Variables and Donation Amount145.2ROI(Return on Investment) (15)5.2.1Model of Fitted ROIs of Performance Contributing Variables fROI i (15)5.2.2Model of the tuning parameter P i (16)5.2.3Calculation of ROI (17)5.3School Selection&Investment Strategy (18)6Extended Model186.1Time Duration (18)6.2Geographical Distribution (22)7Conclusions and Discussion22 8Reference23 9Appendix241Introduction1.1Statement of the ProblemThere exists no doubt in the significance of postsecondary education to the development of society,especially with the ascending need for skilled employees capable of complex work. Nevertheless,U.S.ranks only11th in the higher education attachment worldwide,which makes thefinancial support from large charitable organizations necessary.As it’s essential for charitable organizations to maximize the effectiveness of donations,an objective and systematic assessment model is in demand to develop appropriate investment strategies.To achieve this goal,several large foundations like Gates Foundation and Lumina Foundation have developed different evaluation approaches,where they mainly focus on spe-cific indexes like attendance and graduation rate.In other empirical literature,a Forbes ap-proach(Shifrin and Chen,2015)proposes a new indicator called the Grateful Graduates Index, using the median amount of private donations per student over a10-year period to measure the return on investment.Also,performance funding indicators(Burke,2002,Cave,1997,Ser-ban and Burke,1998,Banta et al,1996),which include but are not limited to external indicators like graduates’employment rate and internal indicators like teaching quality,are one of the most prevailing methods to evaluate effectiveness of educational donations.However,those methods also arise with widely acknowledged concerns(Burke,1998).Most of them require subjective choice of indexes and are rather arbitrary than data-based.And they perform badly in a data environment where there is miscellaneous cross-section data but scarce time-series data.Besides,they lack quantified analysis in precisely predicting or measuring the social benefits and the positive effect that the investment can generate,which serves as one of the targets for the Goodgrant Foundation.In accordance with Goodgrant Foundation’s request,this paper provides a prudent def-inition of return on investment(ROI)for charitable organizations,and develops an original data-motivated model,which is feasible even faced with tangled cross-section data and absent time-series data,to determine the optimal strategy for funding.The strategy contains selection of institutions and distribution of investment across institutions,time and regions.1.2Baseline ModelOur definition of ROI is similar to its usual meaning,which is the increase in students’educational performance over the amount Goodgrant Foundation donates(assuming other donationsfixed,it’s also the increase in total donation amount).First we cope with data missingness.Then,to quantify students’educational performance, we develop an index called performance index,which is a linear composition of commonly used performance indicators.Our major task is to build a model to predict the change of this index given a distribution of Goodgrant Foundation$100m donation.However,donation does not directly affect the performance index and we would encounter endogeneity problem or neglect effects of other variables if we solely focus on the relation between performance index and donation amount. Instead,we select several variables that are pivotal in predicting the performance index from many potential candidates,and determine their coefficients/effects on the performance index. We call these variables performance contributing variables.Due to absence of time-series data,it becomes difficult tofigure out how performance con-tributing variables are affected by donation amount for each institution respectively.Instead, wefit the relation between performance contributing variables and donation amount across all institutions and get afitted function from donation amount to values of performance contribut-ing variables.Then we divide the impact of donation amount into2parts:homogenous and heteroge-nous one.The homogenous influence is modeled as the change infitted values of performance contributing variables over increase in donation amount(We call these quotientsfitted ROI of performance contributing variable).The heterogenous one is modeled as a tuning parameter, which adjusts the homogenous influence based on deviation from thefitted function.And their product is the institution-specific increase in true values of performance contributing variables over increase in donation amount(We call these values ROI of performance contributing vari-able).The next step is to calculate the ROI of the performance index by adding the products of ROIs of performance contributing variables and their coefficients on the performance index. This ROI is institution-specific and dependent on increase in donation amount.By adopting a two-step ROI maximization algorithm,we determine the optimal investment strategy.Also,we propose an extended model to handle problems caused by time duration and geographical distribution of donations.Note:we only use data from the provided excel table and that mentioned in the pdffile.Table1:Data SourceVariable DatasetPerformance index Excel tablePerformance contributing variables Excel table and pdffileDonation amount PdffileTheflow chart of the whole model is presented below in Fig1:Figure1:Flow Chart Demonstration of the Model1.3Detailed Definitions&Assumptions 1.3.1Detailed Definitions:1.3.2Assumptions:A1.Stability.We assume data of any institution should be stable without the impact from outside.To be specific,the key factors like the donation amount and the performance index should remain unchanged if the college does not receive new donations.A2.Goodgrant Foundation’s donation(Increase in donation amount)is discrete rather than continuous.This is reasonable because each donation is usually an integer multiple of a minimum amount,like$1m.After referring to the data of other foundations like Lumina Foundation,we recommend donation amount should be one value in the set below:{500000,1000000,1500000, (10000000)A3.The performance index is a linear composition of all given performance indicators.A4.Performance contributing variables linearly affect the performance index.A5.Increase in donation amount affects the performance index through performance con-tributing variables.A6.The impact of increase in donation amount on performance contributing variables con-tains2parts:homogenous one and heterogenous one.The homogenous influence is repre-sented by a smooth function from donation amount to performance contributing variables.And the heterogenous one is represented by deviation from the function.1.4The Advantages of Our ModelOur model exhibits many advantages in application:•The evaluation model is fully data based with few subjective or arbitrary decision rules.•Our model successfully identifies the underlying mechanism instead of merely focusing on the relation between donation amount and the performance index.•Our model takes both homogeneity and heterogeneity into consideration.•Our model makes full use of the cross-section data and does not need time-series data to produce reasonable outcomes.2Addressing the Missing ValuesThe provided datasets suffer from severe data missing,which could undermine the reliabil-ity and interpretability of any results.To cope with this problem,we adopt several different methods for data with varied missing rate.For data with missing rate over50%,any current prevailing method would fall victim to under-or over-randomization.As a result,we omit this kind of data for simplicity’s sake.For variables with missing rate between10%-50%,we use imputation techniques(Little and Rubin,2014)where a missing value was imputed from a randomly selected similar record,and model-based analysis where missing values are substituted with distribution diagrams.For variables with missing rate under10%,we address missingness by simply replace miss-ing value with mean of existing values.3Determining the Performance IndexIn this section,we derive a composite index,called the performance index,to evaluate the educational performance of students at every institution.3.1Performance IndicatorsFirst,we need to determine which variables from various institutional performance data are direct indicators of Goodgrant Foundation’s major concern–to enhance students’educational performance.In practice,other charitable foundations such as Gates Foundation place their focus on core indexes like attendance and graduation rate.Logically,we select performance indicators on the basis of its correlation with these core indexes.With this method,miscellaneous performance data from the excel table boils down to4crucial variables.C150_4_P OOLED_SUP P and C200_L4_P OOLED_SUP P,as completion rates for different types of institutions,are directly correlated with graduation rate.We combine them into one variable.Md_earn_wne_p10and gt_25k_p6,as different measures of graduates’earnings,are proved in empirical studies(Ehren-berg,2004)to be highly dependent on educational performance.And RP Y_3Y R_RT_SUP P, as repayment rate,is also considered valid in the same sense.Let them be Y1,Y2,Y3and Y4.For easy calculation and interpretation of the performance index,we apply uniformization to all4variables,as to make sure they’re on the same scale(from0to100).3.2Performance Index via Principal-Component FactorsAs the model assumes the performance index is a linear composition of all performance indicators,all we need to do is determine the weights of these variables.Here we apply the method of Customer Satisfaction Index model(Rogg et al,2001),where principal-component factors(pcf)are employed to determine weights of all aspects.The pcf procedure uses an orthogonal transformation to convert a set of observations of pos-sibly correlated variables into a set of values of linearly uncorrelated variables called principal-component factors,each of which carries part of the total variance.If the cumulative proportion of the variance exceeds80%,it’s viable to use corresponding pcfs(usually thefirst two pcfs)to determine weights of original variables.In this case,we’ll get4pcfs(named P CF1,P CF2,P CF3and P CF4).First,the procedure provides the linear coefficients of Y m in the expression of P CF1and P CF2.We getP CF1=a11Y1+a12Y2+a13Y3+a14Y4P CF2=a21Y1+a22Y2+a23Y3+a24Y4(a km calculated as corresponding factor loadings over square root of factor k’s eigenvalue) Then,we calculate the rough weights c m for Y m.Let the variance proportions P CF1and P CF2 represent be N1and N2.We get c m=(a1m N1+a2m N2)/(N1+N2)(This formulation is justifiedbecause the variance proportions can be viewed as the significance of pcfs).If we let perfor-mance index=(P CF 1N 1+P CF 2N 2)/(N 1+N 2),c m is indeed the rough weight of Y m in terms of variance)Next,we get the weights by adjusting the sum of rough weights to 1:c m =c m /(c 1+c 2+c 3+c 4)Finally,we get the performance index,which is the weighted sum of the 4performance indicator.Performance index= m (c m Y m )Table 2presents the 10institutions with largest values of the performance index.This rank-ing is highly consistent with widely acknowledged rankings,like QS ranking,which indicates the validity of the performance index.Table 2:The Top 10Institutions in Terms of Performance IndexInstitutionPerformance index Los Angeles County College of Nursing and Allied Health79.60372162Massachusetts Institute of Technology79.06066895University of Pennsylvania79.05044556Babson College78.99269867Georgetown University78.90468597Stanford University78.70586395Duke University78.27719116University of Notre Dame78.15843964Weill Cornell Medical College 78.143341064Identifying Performance Contributing Variables via post-LASSO The next step of our model requires identifying the factors that may exert an influence on the students’educational performance from a variety of variables mentioned in the excel table and the pdf file (108in total,some of which are dummy variables converted from categorical variables).To achieve this purpose,we used a model called LASSO.A linear model is adopted to describe the relationship between the endogenous variable –performance index –and all variables that are potentially influential to it.We assign appropriate coefficient to each variable to minimize the square error between our model prediction and the actual value when fitting the data.min β1J J j =1(y j −x T j β)2where J =2881,x j =(1,x 1j ,x 2j ,...,x pj )THowever,as the amount of the variables included in the model is increasing,the cost func-tion will naturally decrease.So the problem of over fitting the data will arise,which make the model we come up with hard to predict the future performance of the students.Also,since there are hundreds of potential variables as candidates.We need a method to identify the variables that truly matter and have a strong effect on the performance index.Here we take the advantage of a method named post-LASSO (Tibshirani,1996).LASSO,also known as the least absolute shrinkage and selection operator,is a method used for variableselection and shrinkage in medium-or high-dimensional environment.And post-LASSO is to apply ordinary least squares(OLS)to the model selected byfirst-step LASSO procedure.In LASSO procedure,instead of using the cost function that merely focusing on the square error between the prediction and the actual value,a penalty term is also included into the objective function.We wish to minimize:min β1JJj=1(y j−x T jβ)2+λ||β||1whereλ||β||1is the penalty term.The penalty term takes the number of variables into con-sideration by penalizing on the absolute value of the coefficients and forcing the coefficients of many variables shrink to zero if this variable is of less importance.The penalty coefficient lambda determines the degree of penalty for including variables into the model.After min-imizing the cost function plus the penalty term,we couldfigure out the variables of larger essence to include in the model.We utilize the LARS algorithm to implement the LASSO procedure and cross-validation MSE minimization(Usai et al,2009)to determine the optimal penalty coefficient(represented by shrinkage factor in LARS algorithm).And then OLS is employed to complete the post-LASSO method.Figure2:LASSO path-coefficients as a function of shrinkage factor sFigure3:Cross-validated MSEFig2.displays the results of LASSO procedure and Fig3displays the cross-validated MSE for different shrinkage factors.As specified above,the cross-validated MSE reaches minimum with shrinkage factor between0.4-0.8.We choose0.6andfind in Fig2that6variables have nonzero coefficients via the LASSO procedure,thus being selected as the performance con-tributing variables.Table3is a demonstration of these6variables and corresponding post-LASSO results.Table3:Post-LASSO resultsDependent variable:performance_indexPCTPELL−26.453∗∗∗(0.872)PPTUG_EF−14.819∗∗∗(0.781)StudentToFaculty_ratio−0.231∗∗∗(0.025)Tuition&Fees20100.0003∗∗∗(0.00002)Carnegie_HighResearchActivity 5.667∗∗∗(0.775)Constant61.326∗∗∗(0.783)Observations2,880R20.610Adjusted R20.609Note:PCTPELL is percentage of students who receive aPell Grant;PPTUG_EF is share of students who are part-time;Carnegie_HighResearchActivity is Carnegie classifica-tion basic:High Research ActivityThe results presented in Table3are consistent with common sense.For instance,the pos-itive coefficient of High Research Activity Carnegie classification implies that active research activity helps student’s educational performance;and the negative coefficient of Student-to-Faculty ratio suggests that decrease in faculty quantity undermines students’educational per-formance.Along with the large R square value and small p-value for each coefficient,the post-LASSO procedure proves to select a valid set of performance contributing variables and describe well their contribution to the performance index.5Determining Investment Strategy based on ROIWe’ve identified5performance contributing variables via post-LASSO.Among them,tu-ition&fees in2010and Carnegie High-Research-Activity classification are quite insusceptible to donation amount.So we only consider the effects of increase in donation amount on per-centage of students who receive a Pell Grant,share of students who are part-time and student-to-faculty ratio.We denote them with F1,F2and F3,their post-LASSO coefficients withβ1,β2andβ3.In this section,wefirst introduce the procedure used tofit the relation between performance contributing variables and donation amount.Then we provide the model employed to calcu-latefitted ROIs of performance contributing variables(the homogenous influence of increase in donation amount)and the tuning parameter(the heterogenous influence of increase in dona-tion amount).Next,we introduce how to determine stly,we show how the maximiza-tion determines the investment strategy,including selection of institutions and distribution of investments.5.1Fitted Curve between Performance Contributing Variables and Donation AmountSince we have already approximated the linear relation between the performance index with the3performance contributing variables,we want to know how increase in donation changes them.In this paper,we use Generalized Adaptive Model(GAM)to smoothlyfit the relations. Generalized Adaptive Model is a generalized linear model in which the dependent variable depends linearly on unknown smooth functions of independent variables.Thefitted curve of percentage of students who receive a Pell Grant is depicted below in Fig4(see the other two fitted curves in Appendix):Figure4:GAM ApproximationA Pell Grant is money the U.S.federal government provides directly for students who needit to pay for college.Intuitively,if the amount of donation an institution receives from other sources such as private donation increases,the institution is likely to use these donations to alleviate students’financial stress,resulting in percentage of students who receive a Pell Grant. Thus it is reasonable to see afitted curve downward sloping at most part.Also,in commonsense,an increase in donation amount would lead to increase in the performance index.This downward sloping curve is consistent with the negative post-LASSO coefficient of percentage of students who receive a Pell Grant(as two negatives make a positive).5.2ROI(Return on Investment)5.2.1Model of Fitted ROIs of Performance Contributing Variables fROI iFigure5:Demonstration of fROI1Again,we usefitted curve of percentage of students who receive a Pell Grant as an example. We modeled the bluefitted curve to represent the homogeneous relation between percentage of students who receive a Pell Grant and donation amount.Recallfitted ROI of percentage of students who receive a Pell Grant(fROI1)is change in fitted values(∆f)over increase in donation amount(∆X).SofROI1=∆f/∆XAccording to assumption A2,the amount of each Goodgrant Foundation’s donation falls into a pre-specified set,namely,{500000,1000000,1500000,...,10000000}.So we get a set of possible fitted ROI of percentage of students who receive a Pell Grant(fROI1).Clearly,fROI1is de-pendent on both donation amount(X)and increase in donation amount(∆X).Calculation of fitted ROIs of other performance contributing variables is similar.5.2.2Model of the tuning parameter P iAlthough we’ve identified the homogenous influence of increase in donation amount,we shall not neglect the fact that institutions utilize donations differently.A proportion of do-nations might be appropriated by the university’s administration and different institutions allocate the donation differently.For example,university with a more convenient and well-maintained system of identifying students who needfinancial aid might be willing to use a larger portion of donations to directly aid students,resulting in a lower percentage of under-graduate students receiving Pell grant.Also,university facing lower cost of identifying and hiring suitable faculty members might be inclined to use a larger portion of donations in this direction,resulting in a lower student-to-faculty ratio.These above mentioned reasons make institutions deviate from the homogenousfitted func-tion and presents heterogeneous influence of increase in donation amount.Thus,while the homogenous influence only depends on donation amount and increase in donation amount, the heterogeneous influence is institution-specific.To account for this heterogeneous influence,we utilize a tuning parameter P i to adjust the homogenous influence.By multiplying the tuning parameter,fitted ROIs of performance con-tributing variables(fitted value changes)convert into ROI of performance contributing variable (true value changes).ROI i=fROI i·P iWe then argue that P i can be summarized by a function of deviation from thefitted curve (∆h),and the function has the shape shown in Fig6.The value of P i ranges from0to2,because P i can be viewed as an amplification or shrinkage of the homogenous influence.For example,P i=2means that the homogeneous influence is amplified greatly.P i=0means that this homogeneous influence would be entirely wiped out. The shape of the function is as shown in Fig6because of the following reasons.Intuitively,if one institution locates above thefitted line,when deviation is small,the larger it is,the larger P i is.This is because the institution might be more inclined to utilize donations to change that factor.However,when deviation becomes even larger,the institution grows less willing to invest on this factor.This is because marginal utility decreases.The discussion is similar if one institution initially lies under thefitted line.Thus,we assume the function mapping deviation to P i is similar to Fig6.deviation is on the x-axis while P i is on the y-axis.Figure6:Function from Deviation to P iIn order to simplify calculation and without loss of generality,we approximate the function。

R&D活动的就业效应基于中国数据的实证分析论文导读:：的一份失业研究报告显示。

是技术进步（胡鞍钢。

就业形势严峻。

就技术落后国家的实际情况而言。

论文关键词：R&，D，技术进步，就业，技术落后一、引言2009年，我国城镇登记失业人数首次突破900万，就业形势严峻。

CICC（中国国际金融有限公司）2010年公布的宏观经济形势预测显示，中国劳动力市场2011年劳动力供给可能增长3900万，其中包括了2500万失去与之前4万亿经济刺激计划相关联工作的临时工；与此同时，新增就业岗位可能只有800万个，就业压力明显。

奥肯定律表明，经济增长率与失业率之间存在着一种稳定的关系，诸多经验研究也证实了该关系在美国曾长期存在。

国内学者运用中国数据进行检验时，却得出奥肯定律在中国并不适用的结论；另一方面，通过对就业弹性的考察发现，我国经济增长的就业弹性自上世纪90年代开始呈现出下降趋势毕业论文模板，2005年之后的就业弹性徘徊在0.06-0.08之间，说明我国经济增长创造就业的能力在下降，中国经济进入“无就业增长”[①]时代。

归纳国内学者对“无就业增长”原因的研究，主要存在三种观点：一是经济体制改革（齐建国、常进雄；常云昆等）；二是产业结构转变（蔡昉、都阳；谌新民等）；三是技术进步（胡鞍钢；袁志刚；张军等）。

早在1994年，OECD的一份失业研究报告显示，增加就业不能从放弃技术进步，实施保护主义中寻找解决途径，而应从改进市场流动性，恢复经济与社会适应变化的能力来增加就业，在其对策建议中，首先就是加强技术知识的创造和扩散。

作为实现技术进步的最基本手段，将R&D活动纳入到分析就业问题中具有一定的理论意义和现实意义。

本文旨在明确R&D活动与就业之间是否存在一定的关系？其具体的传导途径是什么？并结合技术落后国家（中国）的实际情况进行理论与实证分析。

本文接下来的安排是：第二部分是R&D、技术进步与就业的相关文献综述；第三部分是所需变量的选取及测算；第四部分是实证分析；最后是本文的相关结论中国期刊全文数据库。

2016年美国大学生数学建模竞赛题目第５卷第２期２０１６年６月、・．．．．・‘．¨．．．‘－．．．．．’．．．Ⅲ’¨．．．．‘．．．．．．‘．．．¨．！数学建模及其应用ＭａｔｈｅｍａｔｉｃａＩＭＯｄｅ¨ｎｇａｎｄｌｔｓＡｐｐＩｉＣａｔｉＯｎｓＶＯＩ．５Ｎｏ．２Ｊｕｎ．２０１６｛竞赛论坛｝＾¨Ｉ・。

・－．．哪・．．．岫・．．．嘶・．．－‘・‘・．．．Ⅵ・‘‘“・・Ｉ２０１６年美国大学生数学建模竞赛题目韩中庚译（解放军信息工程大学四院，河南郑州４５０００１）问题Ａ：热水澡人们经常会通过用一个水龙头将浴缸注满热水，然后坐在浴缸里清洗和放松。

这个浴缸不是带有二次加热系统和循环喷流的温泉式浴缸，而是一个简单的水容器。

过一会儿，洗澡水就会明显变凉，所以洗澡的人需要不停地从水龙头注入热水，以加热洗浴的水。

该浴缸的设计是这样一种方式，即当浴缸里的水达到容量极限时，多余的水就会通过溢水口流出。

考虑空间和时间等因素，建立一个浴缸的水温控制模型，以确定最佳策略，使浴缸里的人可以利用这个策略让整个浴缸中的水保持或尽可能接近初始的温度，而且不浪费太多的水。

利用你们的模型来确定这个策略对浴缸的形状和体积，以及对浴缸中人的形状、体积、温度和活动等因素的依赖程度。

如果这个人一开始用了一种泡泡浴剂加入浴缸中以助清洗，这会对你们的模型结果有怎样的影响？除了要求提交１页的ＭＣＭ摘要之外，你们的报告必须包括１页为浴缸用户准备的非技术性的说明书，来阐述你们的策略，同时解释为什么保持洗澡水的恒温如此之难。

问题Ｂ：太空垃圾地球轨道上的小碎片数量已引起人们越来越多的关注。

据估计，目前有超过５０００００块的空间碎片，也被称为轨道碎片，由于被认为对空间飞行器是潜在的威胁而正在被跟踪。

２００９年２月１０日，俄罗斯卫星Ｋｏｓｍｏｓ一２２５１和美国卫星Ｉｒｉｄｉｕｍ一３３相撞之后，该问题受到了新闻媒体更广泛的讨论。

HospitalT A B L E O F C O N T E N T SContact Directory (1)Introduction (2)The Value of Joint Commission International Accreditation (2)Joint Commission International—Who Are We? (3)Who Is Eligible for an Academic Medical Center Accreditation Survey? (5)How to Request an Academic Medical Center Accreditation Survey (5)Survey Scheduling, Postponements, and Cancellations (6)The Standards Manual (7)Scoring Guidelines for Survey Consistency (9)Accreditation Decision Rules (15)Accreditation Preparation (17)Preparation Timeline (18)Accreditation Process Timeline (21)The On-Site Survey (22)Sample Survey Agenda (23)Tracer Methodology (28)The Accreditation Decision (31)Survey Agenda: Detailed DescriptionsOpening Conference (33)Orientation to the Hospital’s Services (35)Surveyor Planning Session (36)Document Review (38)Documents Available in English (39)Daily Briefing (43)Facility Tour (45)Sample Outline of a Facility Inspection Report (47)Individual Patient Tracer Activity (48)System Tracer: Medication Management (51)System Tracer: Infection Control (54)System Tracer: Improvement in Quality and Patient Safety (56)Quality Improvement and Patient Safety Monitoring Plan:Measures Documentation Tool (59)System Tracer: Facility Management and Safety System (61)Undetermined Survey Activities (65)Education Session: Hospital Decision Rules,Scoring Guidelines, and Strategic Improvement Plan (66)Staff Qualifications and Education Session (67)Medical Staff Qualifications Worksheet (69)Competency Assessment Process Review Form (71)Other Health Care Professional Staff Competency Assessment ProcessReview Form (72)Staff Qualifications and Education Resident Session (73)Resident Staff Qualifications Worksheet (74)Closed Medical Record Review (75)Medical Record Review Tool (76)MPE Supervision Medical Record Documentation (80)Medical Education Interview (81)Medical Professional Education Management (82)Human Subject Research Accountability Interview (83)Management of Contracted Research Studies (84)Human Subject Research Review (85)Leadership Conference (87)Surveyor Team Meeting (89)Surveyor Report Preparation (90)Leadership Exit Conference (91)Survey Planning: Reference ListsRequired Quality Monitors (93)Required Hospital Plans (94)Required Policies and Procedures, Written Documents, or Bylaws (96)Standards That Reference Laws and Regulations (114)Law and Regulation Worksheet (115)External Auditing Body Recommendation Worksheet (120)Contact DirectoryJoint Commission International Accreditation Central Office1515 West 22nd Street, Suite 1300WOak Brook, IL 60523 U.S.A.Phone: +1.630.268.4800Fax: +1.630.268.2996E-mail:jciaccreditation@Joint Commission International Joint Commission International Joint Commission International European Office Middle East Office Asia Pacific OfficeDr. Carlo Ramponi, Managing Director Dr. Ashraf Ismail, Managing Director Dr. Paul Chang, Managing Director Via Ripamonti, 44 P.O. Box 505018 37th Floor20100 Milano Dubai Healthcare City Singapore Land TowerItaly Dubai , United Arab Emirates 50 Raffles PlacePhone: +39.02.890.75.940 Phone: +971.4369.4930 Singapore 048623Fax: +39.02.999.80.695 Fax: +971.4362.4951 Phone: +6. 6829.7208E-mail: cramponi@ E-mail: aismail@ Fax: +65.6829.7070E-mail: pchang@ Contact Joint Commission International for any of the following:∙To inquire about a completed application for survey, survey date, or schedule or for assistance with specific problems or information related to accreditation∙To register for or receive information about education programs and to purchase or to inquire about publicationsJoint Commission International Website: Visit the Website to obtain any of the following:∙General information about accreditation∙Joint Commission International news∙Information about accreditation status for specific hospitals∙Application for survey∙Frequently asked questions (FAQs)∙JCInsight, JCI’s newsletter∙Revisions to standards∙Standards∙To submit a complaint about an accredited organizationJoint Commission Resources Website: Visit the Website to obtain any of the following:∙Information about upcoming education programs∙Catalog of publications∙Access to official JCI publications and e-books∙Video streaming presentationsIntroductionThe Joint Commission International Accreditation Hospital Survey Process Guide for academic medical centers is designedto help hospitals learn about Joint Commission International (JCI) standards and the survey process. This guidewill provide hospitals with important information about JCI, the hospital standards manual (expanded for academic medical centers), eligibility for accreditation, how to request accreditation, survey preparation, the on-site survey, and the accreditation decision.Hospitals should not hesitate to contact any of the JCI Accreditation Offices by telephone or e-mail using the contact directory above for any other information they may need.The Value of Joint Commission International Accreditation Accreditation may benefit hospitals by accomplishing the following:∙Giving hospitals a competitive advantageAccreditation provides evidence of high-quality patient care that helps level the playing field forhospitals performing the same types of procedures.∙Strengthening community confidenceAchieving accreditation is a visible demonstration to patients and the community that a hospital iscommitted to providing the highest-quality services.∙Obtaining recognition from insurers, associations, employers, and other stakeholders Increasingly, accreditation is becoming an eligibility prerequisite for reimbursement, associationmembership, community awareness, and contracts or grants.∙Validating high-quality care to patientsJCI standards are focused on achieving one goal: raising the safety and quality of care to the highestpossible level. Achieving accreditation is a strong validation that a hospital has taken the extra steps to meet a high level of safety and quality.∙Helping hospitals organize and strengthen their improvement effortsAccreditation encompasses state-of-the-art performance improvement concepts that help hospitalscontinuously improve quality.∙Enhancing staff educationThe survey process is designed to be educational, not punitive. JCI surveyors are trained to helphospitals improve their internal procedures and day-to-day operations.∙Improving risk managementBy enhancing risk management efforts, accreditation may improve access to or reduce the cost ofliability coverage. It can also assist in lowering adverse events or outcomes for the hospital and, moreimportantly, for the patient.∙Facilitating staff recruitmentAs staff recruitment becomes more difficult, achieving accreditation as a demonstration of a hospital’s commitment to quality and patient safety will enhance recruitment efforts and retention of staff.∙Promoting team-building skills for staffThe process of obtaining and maintaining accreditation demands a team approach to good patient care.Establishing processes and systems that support good patient care is achieved through strong teamactivities.。

数学建模知识——之新手上路一、数学模型的定义现在数学模型还没有一个统一的准确的定义，因为站在不同的角度可以有不同的定义。

不过我们可以给出如下定义：“数学模型是关于部分现实世界和为一种特殊目的而作的一个抽象的、简化的结构。

”具体来说，数学模型就是为了某种目的，用字母、数学及其它数学符号建立起来的等式或不等式以及图表、图像、框图等描述客观事物的特征及其内在联系的数学结构表达式。

一般来说数学建模过程可用如下框图来表明：数学是在实际应用的需求中产生的，要解决实际问题就必需建立数学模型，从此意义上讲数学建模和数学一样有古老历史。

例如，欧几里德几何就是一个古老的数学模型，牛顿万有引力定律也是数学建模的一个光辉典范。

今天，数学以空前的广度和深度向其它科学技术领域渗透，过去很少应用数学的领域现在迅速走向定量化，数量化，需建立大量的数学模型。

特别是新技术、新工艺蓬勃兴起，计算机的普及和广泛应用，数学在许多高新技术上起着十分关键的作用。

因此数学建模被时代赋予更为重要的意义。

二、建立数学模型的方法和步骤1. 模型准备要了解问题的实际背景，明确建模目的，搜集必需的各种信息，尽量弄清对象的特征。

2. 模型假设根据对象的特征和建模目的，对问题进行必要的、合理的简化，用精确的语言作出假设，是建模至关重要的一步。

如果对问题的所有因素一概考虑，无疑是一种有勇气但方法欠佳的行为，所以高超的建模者能充分发挥想象力、洞察力和判断力，善于辨别主次，而且为了使处理方法简单，应尽量使问题线性化、均匀化。

3. 模型构成根据所作的假设分析对象的因果关系，利用对象的内在规律和适当的数学工具，构造各个量间的等式关系或其它数学结构。

这时，我们便会进入一个广阔的应用数学天地，这里在高数、概率老人的膝下，有许多可爱的孩子们，他们是图论、排队论、线性规划、对策论等许多许多，真是泱泱大国，别有洞天。

不过我们应当牢记，建立数学模型是为了让更多的人明了并能加以应用，因此工具愈简单愈有价值。

2016 Mathematical Contest in Modeling (MCM/ICM) Summary SheetAre we heading towards a thirsty planet?SummaryWater shortage has become a global problem and has attracted universal attention for recent decades. Will the world run out of clean water?Firstly, we develop a fuzzy comprehensive evaluation model, in which we select six indicators, such as the recovery cycle of water resources and the capacity of regeneration. The weight is determined by AHP and entropy weight method. And we sort the water shortage degree into 5 grades I, II, III, IV , V (I means no scarcity problem, while V means heavily overloaded.). Besides, we carry out the sensitiveness analysis of the model to prove the reliability of the model. Then we take China for example to estimate the drought extent and analyze the reason. Using the fuzzy comprehensive evaluation method, we get the drought degree of China for the IV grade, proving that water is moderately overloaded.Secondly, we use GM (1,1) model to predict China's per capita consumption of water in the next 15 years. In the process of collecting data, we find that drought will happen when the rainfall of a year less than 26500 mm. Based on that, we build a disaster prediction model and predict that there will be two droughts in 2020 and 2030. Therefore, we give suggestions to the Chinese government to reserves sufficient fresh water to deal with the coming drought by using desalination technology and rainwater collection technology.Finally, we put forward the intervention measures from the aspects of sewage purification and population. In the model of sewage purification, we analyze the urgency of sewage purification and list the annual amount of sewage purification in next decade. In the population control models, we select the Leslie model which satisfies China's population growth model better to predict the future population of China. Ultimately, we recommend that the Chinese government should relax the policy of family planning and maintain the current urbanization process to control the population in order to buffer the water using pressure.Keywords: water scarcity, Grey Theory, Fuzzy Comprehensive Evaluation45092Team Control NumberProblem ChosenEFor office use only T1 T2 T3 T4For office use only F1 F2 F3 F4Team #45092 Page 1 of 24 1Introduction1.1 Problem BackgroundWe cannot live without water. However, water problems are becoming increasingly serious in recent years, especially water scarcity. Water use has been growing at twice the rate of population over the last century. For the fact that there are many areas of severe water shortage on the earth, effective measures should be taken to prevent water resources from drying up.1.2 Our WorkFirstly, we are asked to build a model that provides a measure of the ability of a region to provide clean water to meet the demands of its population, of which the dynamic nature of the factors is taken into consideration. Secondly, we take China for example to explain why and how water is scarce. We then use our model to predict what the water situation will be in 15 years and analyze how this situation impact the lives of citizens in China, in which the environmental drivers’ effects on the model components is incorporated. Next, an intervention plan taking all the drivers of water scarcity into account is built. Finally, we use the intervention and model designed to project water availability into the future.2 Model One: Water demand and supply analysis2.1 IntroductionThe water supply ability is determined by many factors and differs from region to region. To describe the ability of a region to provide clean water to meet the demand of its residents，In view of the fuzziness and uncertainty of the indexes of water shortage risk evaluation, we build up a water resources comprehensive evaluation model based on entropy weight, with entropy theory applied into it.2.2 Fuzzy Synthetic Evaluation Model2.2.1 The analytic hierarchy processWe select five indicators[1,2] to evaluate the quality of water. According to paired comparisons, we can get a pairwise comparison matrix A, which is shown following table.Table 1. Pairwise comparison matrixRisk Ratio Vulnerability RecoverabilityRecurrencePeriodRiskDegreeRisk Ratio 1.0000 1.1445 0.9066 0.8461 1.1153 Vulnerability0.8738 1.0000 0.7921 0.7393 0.9745 Recoverability1.1338 1.2976 1.0000 0.9593 1.2645 RecurrencePeriod1.1484 1.3143 1.0411 1.0000 1.2808RiskDegree0.8966 1.0261 0.8129 0.7587 1.0000A=1.0000 1.14450.90660.8461 1.11530.8738 1.00000.79210.73930.97451.1338 1.2976 1.00000.9593 1.2645 1.1484 1.3143 1.0411 1.0000 1.2808 0.8966 1.02610.81290.7587 1.0000It is obviously that this matrix is a consistent matrix, so there is no need for consistency check. After that, we used the method as follows to calculate weight:a.normalize Each column vectorww�ii ii=aa ii ii∑aa ii ii nn ii=1 ;b.sum according to row for ww�ii iiww�ii=�ww�ii ii nn ii=1;c.normalize ww�iiww ii=ww�ii∑ww�ii nn ii=1 ,ww=(ww1,ww2,⋯,ww nn)TT is used to approximate the feature vector;d.calculate λ=1nn∑(AAAA)ii AA ii nn ii=1as the approximate value of maximum eigenvalue.The weight of each evaluation index schemed out is#(0.198,0.173,0.224,0.227,0.177)w=.2.3 Entropy Method2.3.1Modeling ProcessThe steps of comprehensive evaluation are to establish factor set U={uu 1,uu 2,⋯,uu nn } for evaluation, to establish evaluation set V ={VV 1,VV 2,⋯,VV nn } and set a fuzzy relationship matrix to represent the relationship between evaluation set and factor set. The matrix isR =�rr 11rr 12⋯rr 1mmrr 21⋯rr22⋯⋯⋯rr 2mm ⋯rr nn1rr nn2⋯rr nnmm �.The key of the fuzzy comprehensive evaluation model is to get a result set, where we can choose the optimal one as the final result. The result set isB =�bb ii �1xxmm =ww ∙RR ,where:1) W is the weight of each factors on the water shortage risk indicators ，W =(ww 1,ww 2,⋯,ww nn )and the elements satisfy�ww ii =1nnii=1;2) "∙" is the fuzzy synthetic operator, of which we choose the weighted average operator for comprehensive evaluation;3) B is the result sets of water shortage risk evaluation,bb ii =�ww ii rr ii ii (jj =1,2,⋯,mm )nnii=1,and the maximum of bb ii was selected as the final evaluation results.2.3.2 The determination of relative membership degreeThe value to assess the risk of water scarcity has no obvious boundaries, thus, we can use fuzzy set theory to describe the continuous changes of the indicators. According to the theory of fuzzy mathematics, the evaluation indexes can be divided into several levels. We sorted the evaluations into five levels, corresponding to the five standard values respectively, namely lower, low, medium, high and higher.The water shortage risk evaluation indexes can be described by the following function:rr ii ii (xx )=�1−mmmmxx {mm ii 1−xx ,xx−mm ii 2}mmmmxx {mm ii 1−mmii nn xx ,mmmmxx−mm ii 2}xx∉[aa ii 1,aa ii 2]1 xx ϵϵ [aa ii 1,aa ii 2]ii =1,2,⋯,nn ;jj =1,2,⋯,mm(3.1)Table 2. The evaluation index and classificationRisk level uu 1uu 2uu 3uu 4uu 5vv 1(ll ll wwllrr ) ≤0.200 ≤0.200 ≥0.800 ≥0.900 ≤0.200vv 2(ll ll ww )0.200~0.400 0.200~0.400 0.600~0.800 6.000~9.000 0.200~0.600vv 3(mmllmmii uu mm ) 0.400~0.600 0.400~0.600 0.400~0.600 3.000~6.000 0.200~0.600 vv 4(ℎii ii ℎ)0.600~0.800 0.600~0.800 0.200~0.400 1.000~3.000 1.000~1.200 vv 5(ℎii ii ℎllrr ) ≥0.800≥0.800≤0.200≤1.000≥2.0002.3.3 The entropy value method to determine the weight coefficientIn information theory, the weight of index is determined by the judgment matrix made up of evaluation index, instead of subjective factors, which makes the evaluation more approximate to physical truth. The process is demonstrated below:(1) build a judgement matrix of several evaluation objects, the number is m, and several judgment matrixes of evaluation indexes, the number is n: R =�rr ii ii �mm∙nn (ii =1,2,⋯,mm )(2) normalize the judgment matrix, then a normalized matrix B was got. The elements of B is defined by following equation:bb ii ii =rr ii ii −rr mm ii nnrr mm mmxx −rr mm ii nnwhere:rr mmmmxx is the optimal value of different objects under the same indicators; rr mm ii nn is the worst value of different objects under the same indicators; (3) The entropy of evaluation indexes can be written asHH ii =−1ln mm�ff ii ii mmii=1ll nn ff ii ii(3.2)where ，ff ii ii =bb ii ii∑bb ii iimm ii=1 ii =1,2,⋯,mm ;jj =1,2,⋯,mm0≤HH ii ≤1obviously, ln ff ii ii makes no sense when ff ii ii =0, thus, ff ii ii need to be redefine asff ii ii =1+bb ii ii∑�1+bb ii ii �mm ii=1(4) Calculate the entropy weight of each evaluation index by using the entropy;ww ∗=ww ii ∗=1−HH iinn −∑HH iinn ii−1(3.3)where, i =1,2,⋯,n , also, ∑ww ii ∗=1nn ii=1;It can be seen from equations above that the smaller the entropy value is, the bigger the entropy weight will be.(5) Analyze the comprehensive weights of evaluation indexes. The comprehensive weight of an evaluation index can be written as,ww ii =ww ii ∗ww ii #∑ww ii ∗ww ii #nn ii=1(3.4)where, ww ii # is weight determined by the AHP method.2.4 Model calculationFigures about the national water consumption we getting from China Statistical Yearbook in 2014[3] are listed in Table 3.Table 3. Figures on the national water consumptionYearWaterconsumptionAgriculture water consumption Industrial water consumption Domesticwater consumption20005497.6 3783.5 1139.1 574.9 20015567.4 3825.7 1141.8 599.9 20025497.3 3736.2 1142.4 618.7 20035320.4 3432.8 1177.2 630.9 20045547.8 3585.7 1228.9 651.2 20055633.0 3580.0 1285.2 675.1 20065795.0 3664.4 1343.8 693.8 20075818.7 3599.5 1403.0 710.4 20085910.0 3663.5 1397.1 729.3 20095965.2 3723.1 1390.9 748.2 20106022.0 3689.1 1447.3 765.8 20116107.2 3743.6 1461.8 789.9 20126141.8 3880.3 1423.9 728.8 20136183.4 3921.5 1406.4 750.1Then we can get another table .Table 4．Water shortage risk evaluation index value of some provinces in China . Province Risk RatioVulnerability RecoverabilityRecurrence PeriodRisk DegreeHebei 0.734 0.384 0.384 3.624 0.297 Shandong 0.984 0.403 0.000 3.104 0.457 Sichuan 0.615 0.295 0.451 3.721 0.235 Zhejiang 0.428 0.127 0.507 3.887 0.196 Xinjiang 0.892 0.315 0.106 3.419 0.425 Shanxi 0.402 0.113 0.619 4.079 0.127 Fujian 0.547 0.287 0.473 4.000 0.224 Jilin0.7820.2940.3383.5420.315After normalization, we can get a matrix F:F =⎣⎢⎢⎢⎢⎢⎢⎡0.534440.81030.62040.53220.48551.00001.00000.00000.00000.94830.36000.04470.84190.00000.24910.65290.62760.04830.69660.00000.60000.62410.72860.81910.17121.00000.76410.54600.63150.80140.32240.99800.68680.44830.31030.19830.85630.00000.27870.5042⎦⎥⎥⎥⎥⎥⎥⎤ After normalize the Equation (3.1), we can get the entropy of each evaluation index,H =(0.9870 0.9886 0.9899 0.9915 0.9896).Besides, according to Equation (3.2), we are able to figure out the weight of each evaluation index,ww ∗=(0.2428 0.2138 0.1896 0.1590 0.1949)The weights calculated by the AHP method are#(0.198,0.173,0.224,0.227,0.177)w =Finally, according to Equation (3.3), the combination weight vector is figured out as w =(0.181 0.183 0.233 0.237 0.165 )Based on the data of Table 3 and Founction (3.1), we can schemed out the relative membership degree and estabilish the fuzzy relation matrix R,R=0.34570.46401.00000.70570.78140.2708 1.00000.48250.81880.34120.38400.4088 1.00000.64000.37960.00000.08820.1478 1.00000.11150.62261.00000.19470.35940.0908,and the final comprehensive evaluation results are:B=(0.317 0.587 0.562 0.713 0.335 ).For the reason that the maximum is 0.731, so the degree of drought in China is grade IV, namely overload. For limits of economic scarcity, China is still moderately loaded though has abundant water resources,2.5 Sensitivity analysisThe fuzzy membership matrix is related to people's subjectivity for the fact that it is voted by experts. Therefore, sensitivity analysis should be taken with the fuzzy membership matrix.Based on data we have got, we can draw a table as shown below:After calculation, we were able to get a weight vectorB=(0.317 0.587 0.562 0.713 0.335 ).As can be seen from the result that, water shortage degree in China belongs to the fourth grade, namely overloaded.Assume that analysis of the risk ratio is far from sufficient for some reason, if the risk ratio of sensitivity matrix R reduced by 10%, the new weight vector will beB=(0.340 0.601 0.605 0.690 0.327).There is no obvious change in value of the matrix B and China is still in the high degree. Then we analyze the sensitivity of the other four indicators in the same way, therefore, we can come to the conclusion that some small fluctuations in a single index will not produce significant effect on the results of fuzzy comprehensive evaluation, suggesting the validity of our model.3 Model Two: Forecasting3.1 Assumptions(1) Assume that China’s population and related policies have not changed dramatically.(2) Assume that China’s economy is growing steadily.3.2 DGM(2,1) Model[4]Defining XX0as the original data sequence of water consumption per capita:xx(0)=(xx(0)(1),xx(0)(2),⋯,xx(0)(nn))where:xx(1)is 1-AGO data sequence;αα(1)xx(0)is 1-IAGO data sequence.Then we define αα(1)xx(0)(kk)+aaxx(0)(kk)=bb as DGM(2,1) model and dd2xx(1)dddd+addxx (1)dddd=b is the albinism differential equation of DGM(2,1) model.The least squares criterion of the parameters in the DGM(2,1) model satisfyuu �=�aa � ,bb��TT =(BB TT BB )−1BB TT YY . The solution to the albinism differential equationdd 2xx (1)dddd +addxx (1)dddd=b isxx �(1)(tt )=�bb �aa �2−xx (0)(1)aa ��ll −mm�dd +bb �aa �tt +1+aa �aa �xx (0)(1)−bb �aa �2 .And the reduced value is xx �(0)(kk +1)=αα(1)xx �(1)(kk +1)=xx �(1)(kk +1)−xx �(1)(kk ) .Table 5. Annual water consumption per capita in China Year 2000 2001 2002 2003 2004 2005 2006 Waterconsumption435.4 437.7 429.3 412.9 428.0 432.1 442.0 Year 2007 2008 2009 2010 2011 2012 2013 Water consumption 441.5446.2448.0450.2454.4454.7455.5Based on figures in the table, the original data sequence can be written as^(0)(443.4,437.5,437.8,438.2,438.6,439.2,439.9,440.9,442,443.5,445.4,447.8,450.9,454.8)x=The respective time response sequence of DGM model is^(1)0.238146k(1)436.469*k+3.90908*+439.491k xe+=therefore,^(1)(443.4,880.9,1318.7,1756.9,2195.5,2634.7,3074.6,3515.5,3957.5,4401,4846.5,5294.3,5745.2,6200)x=And the reduced can be obtained by repeated decreasing:^^^(0)(1)(1)()()(1)k k k xxx =−−By applying the figures into equation above, we can get annual water consumption per capita:^(0)(443.4,437.5,437.8,438.2,438.6,439.2,439.9,440.9,442,443.5,445.4,447.8,450.9,454.8)x=Graph 1. Predicted water consumption per capital by DGM(2,1).3.3 GM(1,1) ModelThe reference sequence has been known asxx (0)=�xx (0)(1),xx (0)(2),⋯,xx (0)(nn )� .Besides, 1-AGO is a once accumulation generation sequence, in which zz (1)=0.5xx (1)(kk )+0.5xx (1)(kk −1),kk =2,3,⋯,nn .So the estimated value can be gotddxx (1)dddd+a xx (1)(t)=b . Let Y =�xx (0)(2),xx (0)(3),⋯,xx (0)(nn )�TT , B =⎣⎢⎢⎡−zz (1)(2)1−zz (1)(3)⋮1⋮−zz (1)(nn )1⎦⎥⎥⎤,we can get an estimated value of u to obtain the minimum of J(u) through the least square method, J (u )=(Y −Bu )TT (YY −BBuu ). At last, the estimated value of u is uu �=�aa �,bb��TT =(BB TT BB )−1BB TT YY . Then solve the albinism differential equation ddxx (1)dddd+a xx(1)(t)=b , the result is xx �(1)(kk +1)=�xx(0)(1)−bb�mm ��ll −mm�dd +bb�mm �tt ,kk =0,1,⋯,nn −1,⋯. And we can get the reducedxx �(0) by repeated decreasing:^^^(0)(1)(1)(1)(1)()1,2,,1,k k k k n xxx +=+−=− ，.Time sequence is as followsxx (0)=�xx (0)(1),xx (0)(2),⋯,xx (0)(7)�=(435.4,437.7,429.3,412.9,428.0,432.1,442.0,441.5,446.2,448.0,450.2,454.4,454.7,455.5).By applying data into the albinism differential equation,xx�(1)(kk+1)=�xx(0)(1)−bb�aa��ll−mm�dd+bb�aa�tt=67459.3×ll0.00627374×kk−67023.9 Then, we can get annual water consumption per capita:xx(0)=�xx(0)(1),xx(0)(2),⋯,xx(0)(7)�=(435.4,424.5,427.2,429.9,432.6,435.3,438.0,440.8,443.6,449.2,452.0,Graph 2. Water consumption per capita predicted by GM(1,1)It can be seen from the graph 1 and graph 2 that the data predicted by DGM(2,1) agree with the actual data well at the beginning, but the growth trend changes too dramatic and it is far from actual situation. While, the GM(1,1) model fit the actual situation well.Then we analyze the Residual Error, Relative Error of GM(1,1) prediction, and the following table shows the results.Table 6. Analysis outcomeyearActualconsumptionper capitalPredictedConsumptionper capitalResidualErrorrelativeError2000 435.4027 435.4000 0.0000 0.000%2001 437.7427 424.5525 -13.1902 3.013%2002 429.3408 427.2244 -2.1164 0.493%2003 412.9463 429.9131 16.9668 4.109%2004 428.0000 432.6187 4.61874 1.079%2005 432.0698 435.3414 3.2716 0.757%2006 442.0199 438.0812 -3.9386 0.891%2007 441.5157 440.8383 -0.6774 0.153%2008 446.1501 443.6127 -2.5374 0.569%2009 448.0430 446.4045 -1.6384 0.366%2010 450.1736 449.214 -0.9596 0.213%2011 454.4000 452.0411 -2.3589 0.519%2012 454.7142 454.8860 0.1717 0.038%2013 455.5430 457.7488 2.2057 0.484%As we can see from table 6, we can prove the consistency between GM(1,1) prediction and the actual water consumption per capital.Table 7. Water consumption per capital predictionyear 2015 2016 2017 2018 2019 2020 2021 2022water consumptionper capital463.5 466.4 469.3 472.3 475.3 478.2 481.3 484.3 year 2023 2024 2025 2026 2027 2028 2029 2030 water consumptionper capital487.3 490.4 493.5 496.6 499.7 502.9 506.0 509.23.4 Disaster Prediction modelIn order to predict the time of the next drought, we have gathered a lot of information about annual rainfall in each province in China from 2003 to 2013. The national annual rainfall can be revealed by the sum of annual rainfall of each province. (unit: mm)Table 8. Annual rainfallYear 2003 2004 2005 2006 2007 2008Annual27345.3 28149.8 24421.9 29706.9 27445.2 30162.5 rainfallYear 2009 2010 2011 2012 2013Annual28613 24395.8 30345.4 28981 26443.4rainfallAccording to the situation in China, we found that drought will happen when the rainfall of a year less than 26500 mm. In order to forecast the drought years,Rainfall sequence is written as follows:xx(0)=(27345,28149,24421,29706,27445,30162,28613,24395,30345,28981,26443).If annual rainfall satisfies xx(0)(ii)<26500, we can regard the year as a drought year. With 2003 marked as the first year, it is not difficult to get the dry year sequencet=(3,8,11).we developed a GM(1,1) model to predict the drought years. The result is showed in the following table:Table 9. The result of disaster prediction model3.0003 7.921962 10.86366 14.89778 20.42994 28.01641From the figures in Table 8, we are easy to draw a conclution that the next two droughts will take place in 20 years and 28 years later after 2003. In other words, the next two troughts will occur in 2022 and 2030 in the next 15 years. While, if annual rainfall is over than 29000mm, we will say that this year's rainfall is abundant.After taking all these into account, we suggest the Chinese government to collect rainwater in 2021 by the rainwater collecting technology. Then desalination technology should be adopted to get enough fresh water to provent China from the drought in 2030.The results show that the per capita water consumption will increase and drought year and flood year will occur frequently, which will make some consquences.4. Model Three: Intervention Plan4.1 Pollution intervention plan4.1.1 IntroductionThe Yangtze river is the longest river in China and third longest in the world, which has been seriously polluted. After consideration, we take the Yangtze river for example.4.1.2 One-dimensional water quality mathematical modelThe differential equation of one dimensional steady decline rule was used to describe the concentration of river pollution. The equation is shown as follows:dc u Kc dx =− in which the discrete function is ignored., after integration, /0Kx uC C e−= .where:u is the average flow velocity distribution in the cross section of the river ( the unit is meters per second );x is the distance along the river ( the unit is km );K is the comprehensive degradation coefficient ( the unit is 1/d ); C is the pollutants concentration along the river ( the unit is mg/L ); cc 0is the concentration of the pollutants after a node (the unit is mg/L ).4.1.3: The river diluted mixture model under the condition of constant designIn terms of point source, the dilution equation of the mixture of water and wastewater isPEPEPEC Q Q C C Q Q+=+ ,where:C is the concentration of completely mixed water ( the unit is mg/L );QQ pp ,CC pp are the water designed from upstream and water quality concentration designed respectively. ( The units are cubic meters per second and milligrams per liter.)QQ EE ,CC EE are sewage flow designed and wastewater discharge concentration designed respectively. ( The units are cubic meters per second and milligrams per liter.) For the fact that the pollution source function fit linear superposition well, we set Single point source discharge formula as;where:ww cc is water load ( the unit is g/L );S is water quality standards of the control section ( the unit is mg/L ). While multipoint sources discharge formula is shown as follows:where:pp E p C C Q Q Q S W ⋅−+⋅=)(pp Ei ni p C C Q Q Q S W ⋅−+⋅=∑=)(1QQ EEii is the sewage flow designed from sewage outlet i; n is the number of sewage outlet.4.1.4 Water environment capacity(1) overview of the river(2) Environmental capacity calculationBy applying into the model, we can get a formula of one-dimensional water environmental capacity:where:ww ii is the emissions from sewage outlet which is number i. ( t/a ) cc ii is background concentration around node i in the river. ( mg/L ) C is the concentration along the river. ( mg/L ) QQ ii is flow after the node. (Cubic meters per second)QQ ii is the quantity of waste water into the river around node i. (Cubic meters per second)u is flow velocity of river i. (meters per second)x is the distance between calculated point and node i.The water environment capacity value of environment function is.According to the analysis above, measures should be taken to prevent the Yangtze river from being polluted more seriously.4.1.5 Water category proportion in the total basin analysisji i Q Q Wi C C +=54.31/＋)(*)*(*54.31*4.86j i i uKx Q Q C eC Wi +−=∑=ni WiWGraph 3. Variation of water quality in the Yangtze River BasinIf no effective measures is taken, water in level I would tend to scarcity. While, water in level IV would fluctuate to 14%. At the same time water in level V would get further increased. Therefore the treatment of Yangtze river is of great urgency.4.1.6 Analysis of the changes of total sewage[5]Table 8. The changes of total sewageYear 2004 2005 2006 2007 2008Total discharge of174 179 183 189 207 waste water(Onehundred million tons)Year 2009 2010 2011 2012 2013 Total discharge of234 220.5 256 270 285 waste water(Onehundred million tons)The simulation is accomplished by the least square method,QQ FF=167.375+3.68×tt+0.835×tt2in which, t is increase in the number of years.Wastewater emissions in the next 10 years are listed in the follow table：Year 2004 2005 2006 2007 2008Total discharge of308.9 331.8 356.3 382.5 410.4 waste water(Onehundred million tons)Year 2009 2010 2011 2012 2013 Total discharge of440.0 471.3 504.2 538.7 575.0 waste water(Onehundred million tons)By far, we have gathered all the information that we need to predict the quantity of sewage should be treated every year, with the proportion of water belonging to level IV and V stay below 20%.We simulate the percentage of water of IV, V and VI respectively, and get three prediction model.II VV FF=4.3−2.18×tt+0.06×tt2+10.5×ln ttVV FF=2.15+0.23×tt+0.001×tt2−0.3×ln ttVVII FF=2.3+0.5×tt−0.005×tt2−1.14×ln ttin which, t is the year.The final predicted results are listed in the following table:Year 2004 2005 2006 2007 2008308.9 331.8 356.3 381.5 410.4 Total discharge of wastewater(One hundredmillion tons)Year 2009 2010 2011 2012 2013 Total discharge of waste440.0 471.3 504.2 538.7 575.0 water(One hundredmillion tons)The extra sewage need to treat every year is showed in the following table:Year2004 2005 2006 2007 2008Extra sewage need to treat(One hundred million tons)24 47 71 98 125Year2009 2010 2011 2012 2013Extra sewage need to treat(One hundred million tons) 155 186 219 254 1954.2 Population intervention plan4.2.1 Logistic modelAssuming that the population in a particular time t is x(t) and environment capacities of population is xx mm , the net population growth rate can be written as)1.()(mx xr t r −=Establish the differential equation of the block type population:0)0(,).1(x x x x xr dt dx m =−= Therefore,t r mme x x x t x .0.11)(−−+=,which is regarded as Logistic model.The result is obtained by simulating the data,e ty .0342.05656.212939.18−+=. The root-mean-square-error 1346.0=RMSE .4.2.2 Leslie modelThe outcome we get by estimating the parameters in the Leslie model are shown as follows:1212001000(1)0100()10n n X t X t λλλµµµ −+=−−Graph 4. The comparison of Logistic and Leslie model.As showed in graph 4, the Leslie model match China's population growth model well, so we select the Leslie model to predict the future population of China. Prediction of population growth in ChinaThe long term forecast of population growth must take the impact of policy into consideration. Policies that can be taken containing Family Planning and Urbanization.Next we analyze the impact on population of five policies. Policy one: keep the existing policy not change;Policy two: Strengthen the work of family planning and limit the process of Urbanization;Policy three: unwind the work of family planning and limit the process of Urbanization;Policy four: unwind the work of family planning and keep the process of Urbanization;Policy five: unwind the work of family planning and limit the process of Urbanization strictly.In this paper, we carry on predict and estimate aiming at the five policies, in order to provide useful information to the government of China.。

非线性规划问题的混合整数模型及求解算法研究非线性规划（Nonlinear Programming，NLP）问题是指目标函数或约束条件中至少存在一个非线性函数的优化问题。

而混合整数规划（Mixed Integer Programming，MIP）问题是指在线性规划的基础上，还包含了整数（或整数和0-1变量）的优化问题。

在实际应用中，很多问题涉及到同时考虑连续变量和离散变量的情况，即混合整数非线性规划（Mixed Integer Nonlinear Programming，MINLP）问题。

解决MINLP问题具有很高的理论和实际意义，但由于其复杂性，一直以来都是计算最困难的类型之一。

针对非线性规划问题的混合整数模型及其求解算法的研究，可以从下面几个方面展开：1. 混合整数非线性规划问题的数学建模混合整数非线性规划问题的数学建模是研究的基础，通过将实际问题转化为数学模型，可以更好地理解和解决问题。

在建模过程中，需要考虑目标函数、约束条件和决策变量等因素，确保模型的准确性和可行性。

2. 混合整数非线性规划问题的求解算法针对混合整数非线性规划问题的求解算法，有许多经典的方法可以利用。

比较常用的方法包括分支定界法、割平面法、列生成法、松弛法等。

这些算法可以根据实际问题的特点选择合适的方法进行求解，并提高求解效率和准确性。

3. 混合整数非线性规划问题的应用领域混合整数非线性规划问题的应用领域广泛，包括生产计划、资源分配、供应链优化、网络设计等。

对于不同的应用领域，需要结合实际情况对模型和算法进行特定的定制和优化，以更好地解决实际问题。

4. 混合整数非线性规划问题的软件工具和案例分析市场上有许多专门用于求解混合整数非线性规划问题的软件工具，比如GAMS、AMPL等。

通过对这些工具的学习和实际案例的分析，可以更好地理解混合整数非线性规划问题的求解方法和技巧。

5. 混合整数非线性规划问题的研究前景和挑战对于混合整数非线性规划问题的研究还存在许多挑战，如精确解和近似解的求解、多目标优化、不确定性建模等。

扩散模型处理缺失值-概述说明以及解释1.引言1.1 概述在数据分析和机器学习领域中，缺失值是一个常见的问题。

缺失值指的是数据集中某些变量或观测值缺失的情况。

缺失值可能因为各种原因产生，例如设备失效、人为错误、数据收集过程中的问题等。

处理缺失值是数据分析和建模过程中不可或缺的步骤。

因为许多机器学习算法无法直接处理含有缺失值的数据集，而且忽视缺失值可能导致结果的偏差和不准确性。

因此，为了更好地分析和应用数据，我们需要有效地处理和填充缺失值。

扩散模型是一种常见的处理缺失值的方法。

它基于数据集中已有的观测值之间的相似性，通过传播和扩散信息来填充缺失值。

扩散模型的核心思想是利用数据的内在关联性来预测缺失值，并根据已有的观测值进行插补。

在本文中，我们将讨论扩散模型的原理和使用。

我们将介绍不同类型的扩散模型，包括基于距离的扩散模型、基于相关性的扩散模型以及基于邻居的扩散模型。

我们还将讨论扩散模型的优缺点，并提出一些改进和应用的展望。

最后，在本文的结论部分，我们将总结扩散模型处理缺失值的效果和局限性。

我们还将探讨未来研究的方向，包括改进扩散模型的算法和应用领域的拓展。

通过本文的研究，我们希望能够为处理缺失值提供一种新颖有效的方法，从而提高数据分析和建模的准确性和可靠性。

1.2文章结构文章结构是指文章的组织框架和内容分布方式。

在本文中，为了解决缺失值问题，我们将采用扩散模型的方法进行处理。

为了使读者更好地理解文章的内容，本文的结构如下：第一部分是引言。

首先，我们将概述本文的主题和背景，介绍扩散模型在处理缺失值方面的应用。

然后，我们将说明全文的结构和各部分的内容安排，以便读者更好地理解文章。

第二部分是正文。

首先，我们将详细介绍扩散模型的基本原理和具体应用。

我们将阐述扩散模型作为一种处理缺失值的有效方法的优势和局限性。

然后，我们将探讨不同的缺失值处理方法，包括插补法、删除法和模型估计法等。

我们将详细介绍每种方法的原理、适用场景和优缺点。

目录2000 年美国大学生数学建模竞赛MCM、ICM 试题 (3)2000 MCM A: Air Traffic Control (3)2000 MCM B: Radio Channel Assignments (3)2000 ICM: Elephants: When is Enough, Enough? (5)2001 年美国大学生数学建模竞赛MCM、ICM 试题 (7)2001 MCM A: Choosing a Bicycle Wheel (7)2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...). (8)2001 ICM: Our Waterways - An Uncertain Future (10)2002 年美国大学生数学建模竞赛MCM、ICM 试题 (14)2002 MCM A: Wind and Waterspray (14)2002 MCM B: Airline Overbooking (14)2002 ICM: Scrub Lizards (15)2003 年美国大学生数学建模竞赛MCM、ICM 试题 (19)2003 MCM A: The Stunt Person (19)2003 MCM B: Gamma Knife Treatment Planning (19)2003 ICM: Aviation Baggage Screening Strategies: To Screen or Not to Screen, that is the Question (20)2004 年美国大学生数学建模竞赛MCM、ICM 试题 (24)2004 MCM A: Are Fingerprints Unique? (24)2004 MCM B: A Faster QuickPass System (24)2004 ICM: To Be Secure or Not to Be? (24)2005 年美国大学生数学建模竞赛MCM、ICM 试题 (25)2005 MCM A: Flood Planning (25)2005 MCM B: Tollbooths (25)2005 ICM: Nonrenewable Resources (25)2006 年美国大学生数学建模竞赛MCM、ICM 试题 (27)2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation (27)2006 MCM B: Wheel Chair Access at Airports (27)2006 ICM: Trade-offs in the fight against HIV/AIDS (28)2007 年美国大学生数学建模竞赛MCM、ICM 试题 (32)2007 MCM A: Gerrymandering (32)2007 MCM B: The Airplane Seating Problem (32)2007 ICM: Organ Transplant: The Kidney Exchange Problem (33)2008 年美国大学生数学建模竞赛MCM、ICM 试题 (38)2008 MCM A: Take a Bath (38)2008 MCM B: Creating Sudoku Puzzles (38)2008 ICM: Finding the Good in Health Care Systems (38)2009 年美国大学生数学建模竞赛MCM、ICM 试题 (40)2009 MCM A: Designing a Traffic Circle (40)2009 MCM B: Energy and the Cell Phone (40)2009 ICM: Creating Food Systems: Re-Balancing Human-Influenced Ecosystems41 2010年美国大学生数学建模竞赛 MCM、ICM 试题 (42)2010 MCM A: The Sweet Spot (42)2010 MCM B: Criminology (43)2010 ICM: The Great Pacific Ocean Garbage Patch (44)2011年美国大学生数学建模竞赛 MCM、ICM 试题 (45)2011 MCM A: Snowboard Course (45)2011 MCM B: Repeater Coordination (45)2011 ICM: Environmentally and Economically Sound (46)2012年美国大学生数学建模竞赛 MCM、ICM 试题 (48)2012 MCM A: The Leaves of a Tree (48)2012 MCM B: Camping along the Big Long River (50)2012 ICM: Modeling for Crime Busting (51)2013年美国大学生数学建模竞赛 MCM、ICM 试题 (59)2013 MCM A: The Ultimate Brownie Pan (59)2013 MCM B: Water, Water, Everywhere (61)2013 ICM: NetworkModeling of Earth's Health (62)2000 年美国大学生数学建模竞赛MCM、ICM 试题2000 MCM A: Air Traffic ControlTo improve safety and reduce air traffic controller workload, the Federal Aviation Agency (FAA) is considering adding software to the air traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analysit at the FAA has posed the following problems.Requirement A: Given two airplanes flying in space, when should the air traffic controller consider the objects to be too close and to require intervention? Requirement B: And airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how do we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of aircraft simultaneously passing through that sector1.at any one instant?2.during any given interval of time?3.during a particular time of day?How does the number of potential conflicts arising during those periods affect complexity? Does the presence of additional software tools to automatically predict conflicts and alert the controller reduce or add to this complexity? In addition to the guidelines for your report, write a summary (no more than two pages) that the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your conclusions.2000 MCM B: Radio Channel AssignmentsWe seek to model the assignment of radio channels to a symmetric network of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grix (honeycomb-style), as shown in Figure 1, where a transmitter is located at the center of each hexagon.An interval of the frequency spectrum is to be alloted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1,2,3, … . Each transmitter wil be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided.Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assugn channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span.Let s be the length of a side of one of the hexagons. We concentrate on the case that there are two levels of interference.Requirement A: There are several contrainsts on the frequency assignments. First, no two transmitters within distance 4s of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these contraints, what can we say about the span in Figure 1?Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in all directions.Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance 2s differ by at least some given integer k, while those at distance at most 4s must still differ by at least one. What cna we say about the span and about efficient strategies for designing assignments, as a function of k?Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider?Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining your findings.2000 ICM: Elephants: When is Enough, Enough?“Ultimately, if a habitat is undesirably changed by elephants, then their removal should be considered -even by culling.”National Geographic (Earth Almanac) –December 1999 A large National Park in South Africa contains approximately 11,000 elephants. Management policy requires a healthy environment that can maintain a stable herf of 11,000 elephants. Each year park rangers count the elephant population. During the past 20 years whole herds have been removed to keep the population as close to 11,000 as possible. The process involved shooting (for the most part) and occasionally relocating approximately 600 to 800 elephants per year.Recently, there has been a public outcry against the shooting of these elephants. In addition, it is no longer feasible to relocate even a small population of elephants each year. A contraceptive dart, however, has been developed that can prevent a mature elephant cow from conceiving for a period of two years.Here is some information about eh elephants in the Park:∙There is very little emigration of immigration of elephants.∙The gender ratio is very close to 1:1 and control measures have endeavored to maintain parity.∙The gender ratio of newborn calves is also about 1:1. Twins are born about 1.35% of the time.∙Cows first conceive between the ages of 10 and 12 and produce, on average, a calf every 3.5 years until they reach an age of about 60.Gestation is approximately 22 months.∙The contraceptive dart causes an elephant cow to come into oestrus every month (but not conceiving). Elephants usually have courtship only once in 3.5 years, so the monthly cycle can cause additional stress.∙ A cow can be darted every year without additional detrimental effects. A mature elephant cow will not be able to conceive for 2 years after thelast darting.∙Between 70% and 80% of newborn calves survive to age 1 year.Thereafter, the survival rate is uniform across all ages and is very high(over 95%), until about age 60; it is a good assumption that elephantsdie before reading age 70.There is no hunting and negligible poaching in the Park.The park management has a rough data file of the approximate ages and gender of the elephants they have transported out of the region during the past 2 years. This data is available on website: icm2000data.xls. Unfortunately no data is available for the elephants that have been shot or remain in the Park.Your overall task is to develop and use models to investigate how the contraceptive dart might be used for population control. Specifically:Task 1: Develop and use a model to speculate about the likely survival rate for elephants aged 2 to 60. Also speculate about the current age structure of the elephant population.Task 2: Estimate how many cows would need to be darted each year to keep the population fixed at approximately 11,000 elephants. Show how the uncertainty in the data at your disposal affects your estimate. Comment on any changes in the age structure of the population and how this might affect tourists. (You may want to look ahead about 30-60 years.)Task 3: If it were feasible to relocate between 50 and 300 elephants per year, how would this reduce the number of elephants to be darted? Comment on the trade-off between darting and relocation.Task 4: Some opponents of darting argue that if there were a sudden loss of a large number of elephants (due to disease or uncontrolled poaching), even if darting stopped immediately, the ability of the population to grow again would be seriously impeded. Investigate and respond to this concer.Task 5: The management in the Park is skeptical about modeling. In particular, they argue that a lack of complete data makes a mockery of any attempt to use models to guide their decision. In addition to your technical report, include a carefully crafted report (3-page maximum) written explicitly for the park management that responds to their concerns and provides advice. Also suggest ways to increase the park managers confidence in your model and your conclusions.Task 6: If your model works, other elephant parks in Africa would be interested in using it. Prepare a darting plan for parks of various sizes (300-25,000 elephants), with slightly different survival rates and transportation possibilities.2001 年美国大学生数学建模竞赛MCM、ICM 试题2001 MCM A: Choosing a Bicycle WheelCyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike.Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course.Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the right.The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear.Task 1. Provide a table giving the wind speed at which the power required for a solid rear wheel is less than for a spoked rear wheel. The table should include the wind speeds for different road grades startingfrom zero percent to ten percent in one percent increments. (Roadgrade is defined to be the ratio of the total rise of a hill divided by thelength of the road. If the hill is viewed as a triangle, the grade is the sine of the angle at the bottom of the hill.) A rider starts at the bottom of the hill at a speed of 45 kph, and the deceleration of the rider is proportionalto the road grade. A rider will lose about 8 kph for a five percent grade over 100 meters.∙Task 2. Provide an example of how the table could be used for a specific time trial course.∙Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision.2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...)Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare.The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading inland from Myrtle Beach and Hilton Head is also planned.A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of the coastal strip. (More accurate data, if sought, are widely available.)The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three. Columbia, another metro area of around 500,000 people, does not have sufficient hotel space to accommodate the evacuees (including some coming from farther north by other routes), so some traffic continues outbound on I-26 towards Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. In 1999, traffic leaving Columbia going northwest was moving only very slowly. Construct a model for the problem to investigate what strategies may reduce the congestion observed in 1999. Here are the questions that need to be addressed:1.Under what conditions does the plan for turning the two coastal-boundlanes of I-26 into two lanes of Columbia-bound traffic, essentiallyturning the entire I-26 into one-way traffic, significantly improveevacuation traffic flow?2.In 1999, the simultaneous evacuation of the state's entire coastal regionwas ordered. Would the evacuation traffic flow improve under analternative strategy that staggers the evacuation, perhapscounty-by-county over some time period consistent with the pattern of how hurricanes affect the coast?3.Several smaller highways besides I-26 extend inland from the coast.Under what conditions would it improve evacuation flow to turn around traffic on these?4.What effect would it have on evacuation flow to establish moretemporary shelters in Columbia, to reduce the traffic leaving Columbia?5.In 1999, many families leaving the coast brought along their boats,campers, and motor homes. Many drove all of their cars. Under whatconditions should there be restrictions on vehicle types or numbers ofvehicles brought in order to guarantee timely evacuation?6.It has been suggested that in 1999 some of the coastal residents ofGeorgia and Florida, who were fleeing the earlier predicted landfalls ofHurricane Floyd to the south, came up I-95 and compounded the traffic problems. How big an impact can they have on the evacuation trafficflow? Clearly identify what measures of performance are used tocompare strategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.Clearly identify what measures of performance are used to compare strategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.2001 ICM: Our Waterways - An Uncertain FutureZebra mussels, Dreissena polymorpha, are small, fingernail-sized, freshwater mollusks unintentionally introduced to North America via ballast water from a transoceanic vessel. Since their introduction in the mid 1980s, they have spread through all of the Great Lakes and to an increasing number of inland waterways in the United States and Canada. Zebra mussels colonize on various surfaces,such as docks, boat hulls, commercial fishing nets, water intake pipes and valves, native mollusks and other zebra mussels. Their only known predators, some diving ducks, freshwater drum, carp, and sturgeon, are not numerous enough to have a significant effect on them. Zebra mussels have significantly impacted the Great Lakes ecosystem and economy. Many communities are trying to control or eliminate these aquatic pests. SOURCE: Great Lakes Sea Grant Network /.Researchers are attempting to identify the environmental variables related to the zebra mussel infestation in North American waterways. The relevant factors that may limit or prevent the spread of the zebra mussel are uncertain. You will have access to some reference data to include listings of several chemicals and substances in the water system that may affect the spread of the zebra mussel throughout waterways. Additionally, you can assume individual zebra mussels grow at a rate of 15 millimeters per year with a life span between 4 - 6 years. The typical mussel can filter 1 liter of water each day.Requirement A: Discuss environmental factors that could influence the spread of zebra mussels.Requirement B: Utilizing the chemical data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAChem1.xls, and the mussel population data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAPopulation 1.xls model the population growth of zebra mussels in Lake A. Be sure to review the Information about the collection of the zebra mussel data. Requirement C: Utilizing additional data on Lake A from another scientist provided at :ap/undergraduate/contests/icm/imagesdata/LakeAChem2.xls and additional mussel population data provided at:ap/undergraduate/contests/icm/imagesdata/LakeAPopulation 2.xls corroborate the reasonableness of your model from Requirement B. As a result of this additional data, adjust your earlier model. Analyze the performance of your model. Discuss the sensitivity of your model. Requirement D: Utilizing the Chemical data from two lakes (Lake B and Lake C) in the United States provided atap/undergraduate/contests/icm/imagesdata/LakeB.xls and ap/undergraduate/contests/icm/imagesdata/LakeC.xls determine if these lakes are vulnerable to the spread of zebra mussels. Discuss your prediction.Requirement E: The community in the vicinity of Lake B (in requirement D) is considering specific policies for the de-icing of roadways near the lake duringthe winter season. Provide guidance to the local government officials regarding a policy on “de-icing agents.”In your guidance include predictions on the long-term impact of de-icing on the zebra mussel population. Requirement F: It has been recommended by a local community in the United States to introduce round goby fish. Zebra mussels are not often eaten by native fish species so they represent a dead end ecologically. However, round gobies greater than 100 mm feed almost exclusively on zebra mussels. Ironically, because of habitat destruction, the goby is endangered in its native habitat of the Black and Caspian Seas in Russia. In addition to your technical report, include a carefully crafted report (3-page maximum) written explicitly for the local community leaders that responds to their recommendation to introduce the round goby. Also suggest ways to help reduce the growth of the mussel within and among waterways.Information about the collection of the zebra mussel dataThe developmental state of the Zebra mussel is categorized by three stages: veligers (larvae), settling juveniles, and adults. Veligers (microscopic zebra mussel larvae) are free-swimming, suspended in the water for one to three weeks, after which they begin searching for a hard surface to attach to and begin their adult life. Looking for zebra mussel veligers is difficult because they are not easily visible by the naked eye. Settled juvenile zebra mussels can be felt on smooth surfaces like boats and motors. An advanced zebra mussel infestation can cover a surface, even forming thick mats sometimes reaching very high densities. The density of juveniles was determined along the lake using three 15×15 cm settling plates. The top plate remained in the water for the entire sampling season (S - seasonal) to estimate seasonal accumulation. The middle and bottom plates are collected after specific periods (A –alternating ) of time denoted by “Lake Days”in the data files.The settling plates are placed under the microscope and all juveniles on the undersides of the plate are counted and densities are reported as juveniles/m^2.2002 年美国大学生数学建模竞赛MCM、ICM 试题2002 MCM A: Wind and WatersprayAn ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area. Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.2002 MCM B: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat.Historically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook-that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations. Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.Consider the overbooking issue in light of the current situation: Less flights by airlines from point A to point B Heightened security at and around airports Passengers' fear Loss of billions of dollars in revenue by airlines to dateBuild a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the company's revenue is maximized. Insure that your model reflects the issues above, andconsider alternatives for handling “bumped”passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.2002 ICM: Scrub LizardsThe Florida scrub lizard is a small, gray or gray-brown lizard that lives throughout upland sandy areas in the Central and Atlantic coast regions of Florida. The Florida Committee on Rare and Endangered Plants classified the scrub lizard as endangered.You will find a fact sheet on the Florida Scrub Lizard at/undergraduate/contests/mcm/contests/2002/problem s/icm2002data/scrublizard.pdfThe long-term survival of the Florida scrub lizard is dependent upon preservation of the proper spatial configuration and size of scrub habitat patches.Task 1: Discuss factors that may contribute to the loss of appropriate habitat for scrub lizards in Florida. What recommendations would you make to the state of Florida to preserve these habitats and discuss obstacles to the implementation of your recommendations?Task 2: Utilize the data provided in Table 1 to estimate the value for Fa (the average fecundity of adult lizards); Sj (the survivorship of juvenile lizards- between birth and the first reproductive season); and Sa (the average adult survivorship).Table 1Summary data for a cohort of scrub lizards captured and followed for 4 consecutive years. Hatchling lizards (age 0) do not produce eggs during the summer they are born. Average clutch size for all other females is proportional to body size according to the function y = 0.21*(SVL)-7.5, where y is the clutch size and SVL is the snout-to-vent length in mm.Year Age Total NumberLivingNumber of LivingFemalesAvg. Female Size(mm)1 0 972 495 30.32 1 180 92 45.83 2 20 11 55.84 3 2 2 56.0Task 3: It has been conjectured that the parameters Fa , Sj , and Sa , are related to the size and amount of open sandy area of a scrub patch. Utilize the data provided in Table 2 to develop functions that estimate Fa, Sj , and Sa for different patches. In addition, develop a function that estimates C, the carrying capacity of scrub lizards for a given patch.Table 2Summary data for 8 scrub patches including vital rate data for scrub lizards. Annual female fecundity (Fa), juvenile survivorship (Sj), and adult survivorship (Sa) are presented for each patch along with patch size and the amount of open sandy habitat.Patch Patch Size (ha) Sandy Habitat (ha) Fa Sj Sa Density (lizards/ha)a 11.31 4.80 5.6 0.12 0.06 58b 35.54 11.31 6.6 0.16 0.10 60c 141.76 51.55 9.5 0.17 0.13 75d 14.65 7.55 4.8 0.15 0.09 55e 63.24 20.12 9.7 0.17 0.11 80f 132.35 54.14 9.9 0.18 0.14 82g 8.46 1.67 5.5 0.11 0.05 40h 278.26 84.32 11.0 0.19 0.15 115Task 4: There are many animal studies that indicate that food, space, shelter, or even reproductive partners may be limited within a habitat patch causing individuals to migrate between patches. There is no conclusive evidence on why scrub lizards migrate. However, about 10 percent of juvenile lizards do migrate between patches and this immigration can influence the size of the population within a patch. Adult lizards apparently do not migrate. Utilizing the data provided in the histogram below estimate the probability of lizards surviving the migration between any two patches i and patch j.Table 3HistogramMigration data for juvenile lizards marked, released, and recaptured up to 6 months later. Surveys for recapture were conducted up to 750m from release sites.Task 5: Develop a model to estimate the overall population size of scrub lizards for the landscape given in Table 3. Also, determine which patches are suitable for occupation by scrub lizards and which patches would not support a viable population.Patch size and amount of open sandy habitat for a landscape of 29 patches located on the Avon Park Air Force Range. See:/undergraduate/contests/icm/2002problem/map.jpg for a map of the landscape.Patch Identification Patch Size (ha) Sandy Habitat (ha)1 13.66 5.382 32.74 11.913 1.39 0.234 2.28 0.765 7.03 3.626 14.47 4.387 2.52 1.998 5.87 2.499 22.27 8.44。

2012 Contest ProblemsPROBLEM A: The Leaves of a Tree"How much do the leaves on a tree weigh?" How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical model to describe and classify leaves. Consider and answer the following:• Why do leaves have the various shapes that they have?• Do the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the “volume” of the tree and its branches effect the shape?• Speaking of profiles, is leaf shape (general characteristics) related to tree profile/branching structure?• How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)?In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings.2012美赛A题：一棵树的叶子（数学中国翻译）“一棵树的叶子有多重？”怎么能估计树的叶子（或者树的任何其它部分）的实际重量？怎样对叶子进行分类？建立一个数学模型来对叶子进行描述和分类。