2011年数学建模美赛论文
2011数学建模竞赛C题论文
企业退休职工养老金制度的改革研究摘要近年来，随着我国快速进入老龄化社会，退休后的养老金问题已经成为了人们的焦点问题。
本文基于山东省的一系列统计数据，对养老保险中的替代率及资金缺口问题进行了分析。
针对问题一，根据我国经济发展的实际情况并结合经济发展的中长期发展目标，我们认为工资增长率今后应该是逐年递减的，并在某个时间达到较稳定的状态，故我们采用了阻滞增长模型，利用MATLAB对问题所提供的山东省职工历年平均工资统计表中的数据进行拟合，预测出了2011年至2035年的山东省职工的年平均工资。
针对问题二，我们利用EXCEL这个工具来进行计算，对各年龄段工资进行分析统计，计算出了题目提出的各种情况下的替代率，分别是30岁到55岁为34%，到60岁为42%，到65岁为60%；40岁到55岁为21%，到60岁为29%，到65岁为39% ，并对结果进行分析，得出当开始缴费的年龄相同时缴费年限越高，替代率越高；当缴费年限相同时，开始缴费的年龄越晚，替代率越高。
针对问题三，根据该企业某职工不同的退休年龄的情况，同样利用EXCEL进行计算并得到结果，当他是55岁或60岁时退休，这两种情况就会存在缺口问题，当他是65岁退休时就不存在缺口问题，我们同时也计算出该职工若55岁退休，到69岁的时候，其缴存的养老保险基金与其领取的养老金之间达到收支平衡，若是60岁退休则会推迟到73岁达到收支平衡。
结合问题二，我们可知，当替代率越高，则缺口越小。
针对问题四，我们给出了替代率与工资增长率、资金收益率、缴费率及缴费年限等影响因素的函数表达式，由该函数很容易看出替代率是缴费年限及资金收益率的增函数，从而可以通过提高投资收益率或增加缴费年限的方式来达到预期目标。
关键词养老金替代率阻滞增长模型收支平衡一、问题重述1.1养老金简介养老金也成退休金，是一种根据劳动者对社会所贡献及其所具备享受养老保险的资格，以货币形式支付的保险待遇，用于保障职工退休后的基本生活需要。
2011年数学建模大赛优秀论文
交巡警服务平台的设置与调度的数学模型摘要针对交巡警服务平台的设置与调度问题，本文主要考虑出警速度和各服务平台的工作量来建立合理方案。
对于A区的20个交巡警服务平台分配管辖范围的问题，我们采用Dijkstra算法，分别求得在3分钟内从服务台可以到达的路口。
根据就近原则，每个路口归它最近的服务台管辖。
对进出A区的13个交通要道进行快速全封锁，我们采用目标规划进行建模，运用MATLAB软件编程，先找出13个交通要道到20个服务台的所有路径。
然后在保证全封锁时间最短的前提下，再考虑局部区域的封锁效率，即总封锁时间最短，封锁过程中总路程最小，从而得到一个较优的封锁方案。
为解决前面问题中3分钟内交巡警不能到达的路口问题，并减少工作量大的地区的负担，这里工作量以第一小问中20个服务台覆盖的路口发案率之和以及区域内的距离的和来衡量。
对此我们计划增加四个交巡警服务台。
避免有些地方出警时间过长和服务台工作量不均衡的情况。
对全市六个区交警平台设计是否合理，主要以单位服务台所管节点数，单位服务台所覆盖面积，以及单位服务台处理案件频率这些因素进行研究分析。
以A 区的指标作为参考，来检验交警服务平台设置是否合理。
对于发生在P点的刑事案件，采用改进的深度搜索和树的生成相结合的方法，对逃亡的犯罪嫌疑人进行可能的逃逸路径搜索。
由于警方是在案发后3分钟才接到报警，因此需知道疑犯在这3分钟内可能的路线。
要想围堵嫌疑犯，服务台必须要在嫌疑犯到达某节点之前到达。
用MATLAB编程，搜索出嫌疑犯可能逃跑的路线，然后调度附近的服务台对满足条件的节点进行封锁，从而实现对疑犯的围堵。
关键词：Dijkstra算法；目标规划；搜索；一、问题重述近十年来，我国科技带动生产力不断发展，我国的经济实力不断增强，而另一方面安全生产形式却相当严峻。
每年因各类生产事故造成大量的人员伤亡、经济损失。
尤其是一些大目标点，作为人类经济、政治、文化、科技信息的中心，由于其“人口集中、建筑集中、生产集中、财富集中”的特点，一旦发生重大事故，将会引起惨重的损失。
2011年大学数学建模论文
承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。
我们知道，抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。
我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。
如有违反竞赛规则的行为，我们将受到严肃处理。
我们参赛选择的题号是（从A/B/C/D中选择一项填写）： B我们的参赛报名号为（如果赛区设置报名号的话）：所属学校（请填写完整的全名）：陇东学院参赛队员(打印并签名) ：1. 任耀辉2. 魏斌3. 邵文娟指导教师或指导教师组负责人(打印并签名)：冯积社董文瑾日期： 2011 年 9 月 12日赛区评阅编号（由赛区组委会评阅前进行编号）：编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）：交巡警服务平台的设置与调度的设计方案摘要由于平面区域不能用圆来不重不漏地覆盖，我们采用正六边形来覆盖，但根据道路口疏密状况和人口数量、区域面积的差异，最后决定对于不同区域采用不同尺寸的正六边形，来覆盖城区各道路的方法来进行各个交巡警服务平台的管辖范围的初次预分配，对于初次预分配的结果按各个道路口到各服务平台的最短路的里程乘以发案率的和，得到各个服务平台的总工作量。
按相邻区域内服务平台工作量相当的标准进行调整，用Dijkstra算法求得那些重复包含和漏掉的路口，及那些距离邻近的路口节点重新分配到邻近的正六边形内，从而得到最终的管理范围。
由于要力求工作量均衡，就目前的服务平台而言数量不足，通过采用最短路的选址方法来确定，应就需再增加3个服务平台，具体位置分别在标号为29、72、90的三个路口。
2011数学建模A题神经网络优秀论文,带代码
图 1 该城区的地形分布图
首先，我们根据样本点的位置和海拔绘制出该城区的地貌，见图 1。我们运 用 matlab 软件，根据各个网格区域中的重金属含量，用三角形线性插值的方法 得到各种重金属含量在空间上分布的等值线图。
1 图 2-1
2
1 图 2-2
2
图 2-1 给出了 As 在该区域的空间分布：图中可以观察到 As 有两个明显的高 值中心，我们标记为区域 1 和 2。这两个区域都处于工业区分布范围内，并以该 两个区域作为中心向外延伸， 浓度逐渐减少，同时我们注意到在山区的很多区域
Ni
(3211,5686) (24001,12366)
Pb
(1991,3329) (4508,5412)
Zn
(1699,2867) (3725,5487) (9583,4512) (13653,9655)
综合分析所得污染源所在位置，发现不同金属的污染源有同源现象，依据 同源性汇聚污染源，绘制了八种重金属的污染源汇总图。 问题四：神经网络模型的优点是具有较强的自组织、自学习能力、泛化能 力和充分利用了海拔高度的信息；缺点是训练要求样本点容量较大。可以通过搜 集前几年该城区八种重金属浓度的采样数据和近几年工厂分布多少位置的变化、 交通路段车流量的变化、 人口及生活区分布变化与植被分布多少位置的变化等数 据，进一步拓展神经网络模型，得到该城市地质环境的演变模式。
符号
意义
k i j
x ij
xi
表示不同功能区 表示金属的种类 表示不同的样本 表示样本 j 中金属 i 的浓度 表示金属 i 背景值的平均值 表示金属 i 背景值的标准差
表示 x i j 标准化后的值
i
Y ij
i
Ik
2011 ICM 优秀论文
Electric Vehicles as a Widespread Means ofTransportationSummaryWith the development of economy and the consumption of energy, electric vehicles are becoming more and more important. So it will be of great significance to study the environmental, social, economic, and health impacts of the widespread use of electric vehicles on a region, and analysis the optimal distribution of the power stations which matching the electric vehicles. It also promotes the development of the electric vehicles industry and provides the government and vehicle manufacturers with useful suggestions.Firstly, this paper establishes a dual network model based on analytic hierarchy process (AHP) to provide a comprehensive evaluation to a region. We establish six basic nodes firstly, and use AHP to evaluate the four secondary node respectively. These form the primary network of our model. Then, we use four secondary nodes to evaluate a region to get the aggregative indicator D. Next, we make a comparison of the D of fuel vehicles and electric vehicles.This paper makes a specific discussion of America, China, Netherlands and Sub-Saharan Africa. We can draw a conclusion that the regions except for the economically backward ones can gain more by promoting electric vehicles. As for the economically backward regions, they can gain earnings after promoting electric vehicles for a period of time.Then we assume the meaning of widely used is that the new vehicles are all EVs when pr edicting the amount of oil saved.Furthermore,we think there are two aspects when consideri ng the oil saved:the saved oil due to the improvement of energy efficiency and the oil saved b y using alternative energy.After that,we use least square method to match and get the number s of new vehicles in one year around the world.And the amount of oil saved is3×1011（L）.Next, a multi-objective optimization model is set up in this paper to discuss the optimal plan of power station. We firstly attribute the problem to a multi-objective optimization problem which can make the maximum benefit of environment, society, business and person. Then we simplify the multi-objective and summarize the four aspects of income into the minimum social cost and the output power which can meet the biggest demand of the electric vehicles. Finally, the optimal plan of power station is shown in the table.At last, the strength and weakness of our model are discussed and the future work is pointed as well.Keywords: dual network model; analytic hierarchy process; multi-objective optimization; electric vehicles;1. IntroductionWith the development of the energy consumption and economy, emerging electric cars are in a strong position step by step. First of all, we introduce the basic concepts of electric cars. We mainly study pure electric vehicles in this research which use the charging or battery powered, and do not use other energy driven by motor vehicles. Compared with traditional fuel vehicle, this kind of car in the energy utilization, pollutant emission, etc. has greatly improved. It is introduced the cost of fuel vehicles and electric Vehicles in the chart below.Figure 1 the cost of the two kinds of vehiclesWe study the widespread use of electric cars to the environment, the influence of the social, economic, and health and build a model to evaluate the influence. Then, we use this model to put forward suggestions for the government and the electric car manufacturers. Furthermore, we will calculate the amount of gasoline saved. In addition, based on the previous model, we also need to set up a model to plan the numbers of power station and its form in order to achieve the environmental, social, and business and personal best interests. We decompose the problem into several small problems:●Evaluate basic indicators for four aspects environmental, social, economy and health in a region.● Analyze the weight of basic indicators when evaluating every aspect and establish a model. ● Calculate the total amount of oil saved around the world using the model established. ●Tackle the problem of the establishment of charging station combining with the evaluation modelWe regard this problem the as an evaluation considering the effects of electric car widely used on the geographical environment, social, economic and health and establish the dual network model based on analytic hierarchy process (AHP). At first we find out 6 basic indicators to evaluate every secondary nodes using AHP to analyze the weight. Then we compare the traditional fuel cars and electric vehicles and get a comprehensive index for a region to suggest our government and the electric vehicles manufacturers. As for the problem to establish electricity generation station, we set up a multi-objective optimal model withC o s tminimum of personal cost and business cost.2. Assumptions and justifications●We assume that electric cars researched in the passage are all pure electrical.We only consider the effect of common kinds of fuel cars and electric cars in the model 1.●Assuming that fuel vehicle only cause environmental pollution in the process of using by the exhaust emission and electric vehicles cause environmental pollution in the process of power generation.● We only consider the outside impact of noise pollution to the environment.●When considering the social cost of energy, we only consider the social cost of fossil fuels without the cost of such as wind, water, and solar energy.3. Models3.1 Dual network model based on analytic hierarchy process3.1.1 Introduction of model and method ➢ Network modelWe set up a dual network model based on analytic hierarchy process (AHP). Network model consists of master node: the study area; secondary nodes: economy, society, environment and health; basic nodes: these connections between six basic evaluation index. We use six basic index to evaluate secondary node, and then use four secondary nodes to evaluate each region widely used in electric vehicles under the comprehensive influence. The figure below briefly introduces the double network model:Figure 2 Dual network; six basic nodes of one secondary node➢Analytic hierarchy processAnalytic hierarchy process (AHP) is a complicated multi-objective decision-making problem as a system, the target is to decompose the problem into multiple objectives or principles, or rules, constraints, and multiple index of several levels, through qualitative index fuzzy quantification method to calculate hierarchical single sort (weight) and total ordering, as the target (index), scheme optimization decision method of system.4.1.2 Model BuildingIn order to unify the degree of impact indicators, we will convert them for the value of the degree of influence, in US dollars.4.1.2.1 Six index (six basic nodes)1.Social cost of pollution P1（1）Social cost of air pollution p1Convert the emissions caused by air pollution into social costs:The text indicates air pollution caused by two types of cars in their life cycles as following:From the assumption, we only consider air pollution of fuel vehicles caused by emissions and air pollution of electric vehicles caused by power generation process.The amount of four gas emissions from fuel vehicles and electric vehicles is in the following chart:Table 1 Comparison of emissions of fuel vehicles and electric vehiclesThe date in table 1 are from document[1].The scaling factor of CO,HC,NO x,CO2are respectivelyω1、ω2、ω3、ω4 . Thecalculation formula of P1is:p1=0.256∙n∙(ω1∙a1+ω2∙a2+ω3∙a3+ω4∙a4)(1)（2）social costs of noise pollution p2We convert noise pollution into social costs. From the assumption, we only consider the impact of noise pollution generated by car on the environment.Table 2 Noise of fuel vehicles and electric vehicles under different speed (unit: dB)Take the average of noise according to different speeds, and convert noise pollution into social costs:d1=72(db)(2)p2=0.037∙n(3)d2=66(db)(4))∙n(5)p2′=0.037∙(d2d10.037dollars/km is social cost of pollution per kilometer of fuel vehicles;d1、d2are respectively the noise average of fuel vehicles and electric vehicles;p2、p2′are respectively social costs of noise pollution caused by fuel vehicles and electricvehicles.P1=p1+p2(6)2. Electric vehicles profits P2P2=P0×n(7) P0is the net profit of an ordinary electric car.Table 3 several kinds of cost about vehiclesserial number projectelectric vehiclefuelvehicleThe ratio ofcost between them2 energy costs 3151924.15%3costs ofmaintenance5967 6316 94.48%4Basically thesame costs943 943 100.00%5A total costin the life cycle208182985169.74%In a similar way P2′is a fuel vehicles profit of equal quality.Some Analysis:This profit includes many aspects, including the promotion of science and technology by electric vehicles and the promotion of automobile industry.3. Social cost of annual energy consumption P3According to the assumption, there is no other alternative energy.P3 is the initial energy value converted from annual energy consumption of different cars.It is a comparison figure between fuel vehicles and electrical vehicles about overall efficiency below:Figure 3 Energy conversion efficiency of fuel vehicles and Electric Vehicles Through the Figure 3, we can get:η1=12% (8) η2=20.8% (9) P 3=Eη1∙n ∙ε (10)P 3′=Eη2∙n ∙ε (11)η1、η2 are overall efficiency of fuel vehicles and electric vehicles. Suppose each energyefficiency in their life cycle is q i .”E” is a car’s total energy consumption in one year. ε is US current electricity price, unit: dollars/kW ∙h.4. Social costs of energy consumption when there is alternative energyAccording to the assumption, we only consider the social costs of fossil fuels, excluding social cost of wind energy, hydropower, solar energy and other alternative energy.We suppose that fuel vehicles have no alternative energy and only get energy from fossil fuels; electric power in electric vehicles depends on energy distribution characteristics of each region:E ′=∑(αi ∙λi ∙E) (12)And we can get the formula:P 4=Eη1∙n ∙ε (13)87%83%83%83%12%100%95%36%34%28%20.80%Fuel vehiclesElectric VehiclesP4′=E−E′η2∙n∙ε(14)E represents a total amount of energy;E′is the total amount of energy consumed when there is alternative energy;αi represents the utilization of various energy sources;λi represents energy structure for electric power generation in each region.5. Recovery cost P5Fuel vehicles include the cost recovery of vehicle non-recyclable solid waste on the environment P5;electric vehicles include the cost recovery of vehicle non-recyclable solid waste on the environment and waste batteries P5′.6. The impact on employment P6Table 5 Employment impact of the deployment of electric vehiclesIndustryBaselinescenarioHighpricescenarioOperatorsubsidiesscenarioEmployment growthCharginginfrastructure178163411579472778Batterymanufacturing600658063081168 Overallgrowth238229492209553946Employment lossesGasstationattendant-23152-37353-42906Partsmanufacturer-39287-63386-72809Skilledworker-46605-75192-86370 Totallosses-109043-175931-202085The netemploymentimpact1291853162873518614.1.2.2 Evaluation for 4 main aspects using six indicatorsWe just use several indexes that have the larger correlation with these four nodes to evaluate them respectively, the method is AHP (Analytic Hierarchy Progress). The procedures are as follow:Step1 Build hierarchical structureThe hierarchical structure that AHP required is always constituted by the following three structures:1. Target hierarchical (the highest level): The intended target of issue.2. Criterion hierarchical (the intermediate layer): The criterion that influences the achieving the goal of the issue.3. Measure hierarchical (the lowest level): The measures that achieve the goal of the issue.The hierarchical structure is the figure below:Figure 4 the hierarchical structureStep2 Build pairwise comparison matrix➢Confirm the importance of scale valuesContra posing the norms of judging matrix, proceed the pairwise comparison to find which is more important and how much it is. Use 1-6 to assign the importance degree (the importance scale value can be seen in the form below)Table 6 Importance meaning scale tableTheimportancescale valueMeaning1 Indicating that when comparing two elements, they sharethe same importance. 3 Indicating that when comparing two elements, the former isslightly more important than the latter. 5 Indicating that when comparing two elements, the former isobviously more important than the latter. 7 Indicating that when comparing two elements, the former isintensively more important than the latter. 9 Indicating that when comparing two elements, the former isextremely more important than the later. 2，4，6，8 Indicate the median of the above-mentioned judgment. ReciprocalIf the ratio of importance of element i and element j is a ij , then the ratio of importance of these two elements isa ji =1a ij ⁄.➢ Pairwise comparison matrixCompare the various factors of the importance with the weight of the object. And we assume that there are n objects (A 1,A 2,A 3…A n ) and the weight of them respectively are w 1,w 2,w 3…w n .If proceed the pairwise comparison of their weight, the ratio (relative weight) can constitute a n ×n pairwise comparison matrix.1,11,21,111212,12,22,21222,1,2,12/////////n n n n n n n n n n n n a a a w w w w w w a a a w w w w w w A aa a w w w w w w ⎛⎫⎛⎫⎪ ⎪ ⎪ ⎪== ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭ (15)Step3 Calculate the weight vector➢ Normalize to calculate the weight vecto rThe sum of every line of the pairwise comparison is just in direct proportion to the weight vector W = (w 1, w 2,…, w n )T . That is1,12,21,j n j j n j n a w a w a w =⎛⎫⎛⎫ ⎪ ⎪ ⎪ ⎪∝ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭∑ (16)Choosing the appropriate scale factor to make the sum of the numerical value of each importance factor is 1.The normalized numerical value of importance factor is called weight. The importance vector is called weight vector. By this way, we get a weight vector.W 1=(w 1w2⋮w n) (17)The weight of elements in the formula above shows the comprehensive ranking of the importance of each factor.➢ Solve the characteristic equation to get the weight vector1,11,21,112,12,22,22,1,2,n n n n n n n n a a a w w a a a w w AW n nW a a a w w ⎛⎫⎛⎫⎛⎫⎪ ⎪⎪ ⎪ ⎪ ⎪=== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭ (18)W is the eigenvector of the pairwise comparison matrix A and its corresponding eigenvalueis n. The theory that n is the only maximal eigenvalue of A has been proved strictly. According to the analogy, we can also use the way of solving the characteristic equation to get the importance vector.W 2=(w 1′w 2′⋮w n′) (19)Step4 Consistency checkWe use n’ to indicate the maximal eigenvalue with deviation, then the difference betweenn and n’ reflects the degree of difference. Considering the influence of the number of factors, Saaty definedCI n nn =--'1 (20)As a coincidence indicator. When CI=0, the pairwise comparison matrix A goes all the way or there exists inconsistency. The degree of inconsistency is bigger as the CI is bigger. Saaty then also defined a consistency ratio CR to determine the allowed range of the degree of inconsistency. WhenCR CI RI =</.01 (21)We reckon that the consistency can be accepted and it won’t affect the qualitative results of the sort. The values of formula (20) are shown as the table below.4.1.2.3 Evaluation for D using 4 main aspectsWe calculate the impact of six basic indicators on the four aspects:C i =∑φij ∙Pj 6j=1 (22) j represents these six basic indicators, i represents four secondary nodes.Calculate the comprehensive influence value of a district produced by the four factors:D =∑C i 4i=1 (23)C 1 shows the influence on the society produced by the six basic indexes; C 2 shows the influence on the economy; C 3 shows the influence on the environment; C 4 shows the influence on the health.And widely use of vehicles has negative effect to health and environment therefore, C 3and C 4 are negatively related to D while C 1 and C 2 are positively related to D.4.1.2.3 Results and analysisUsing the analytic hierarchy process (AHP), we gain the six basic index weights of the four secondary nodes environment, economy, health, society. And six weights of health is the chart below:Table 7 Comprehensive influence on the four aspectsSo we can see that P 1, P 5 are more important to health.Consistency analysis: CR = 0.0244 < 0.1.So the consistency can be received.Then, we choose four countries and districts:USA, China, Netherlands and Sub-Saharan Africa. Substitute them into the model and then get the value of P, C, D. We use the line chart below to show the value of D. The red line is electric vehicle and the blue line is fuel vehicle.USA ChinaNetherlands Sub-Saharan AfricaFigure 5 the value of D in four countries●According to the chart above, we can find that different districts have different value ofD. But after comparing, we can easily know that in the most of districts or time, the D ofelectric vehicle always exceeds the D of fuel vehicle. That means in this passage, we assume that after the annual increase of fuel car is replaced by the electric car, on the other word, the electric car is widely used, we will get more profits than using the fuel car.●It is interesting to note that through the further analysis of the results, we can find themore advanced the economy is, the bigger the D value has, which means the profitsproduced by the popularization of the electric cars are more. Of course it more depends on the number vehicles.However, in place like Africa south to Sahara,popularizing ,in early times,can't make profit as much as automobiles using fuel.This is because fixed investment ,in earl y time,demand time to get payback but sales of cars in Africa is conspicuously at a low situation.In order to discuss what actions should varied countries andmerchants take to support ele ctrical automobiles, we in here convey a research to four aspects between Netherlands and A merica.Figure 6 the four different ratios of C in two countries● By analyzing the four different ratios of C, We can see that for the Netherlands, its greatestimpact is environment, which is accounted for 38% of the proportion. This is mainly because of its very high rate of energy alternatives of p4 and the wind power generation of Netherlands accounts for 56% of the total energy. If we implement electric vehicles, the automotive emissions of pollutants significantly will be reduced, while the environment is also the largest gains. ● For the United States, the widespread implementation of electric vehicles will make thegreatest impact on the economy and the environment, which respectively accounted for 38% and 32%. This is mainly because the US auto base, it is also significantly reduce emissions of pollutants. ● From the analysis, we can see that for the government that promoting electric vehicles bothin the environment and in the economy has more advantages than disadvantages. Also, if the government develop new energy sources and seek more green pollution-free alternative energy to power supply, it will bring great benefits to environment. For manufacturers, car sales have to take into account. In developed areas, there is no doubt that implementing electric vehicles has more profitable. But in the general area, it will sustain losses in business in initial stage.Econo mySocietyHealthEnviro nmentNLDEcono mySocietyHealthEnviron mentUSA4.2 Predicting the amount of oil saved4.2.1 Problem analysisWe will calculate the amount of oil saved in the model 1 under the circumstances of electrical vehicles widely used and there are two aspects we should consider:➢The oil saved due to the utilization ratio of energy comparing electric vehicles and fuel cars.➢The oil saved by using alternative energy to generate power.4.2.2 Model building1.Predict the number of electric vehicles promoted to use in the world every yearUsing the least square method, we fit each year the number of cars Nv̂(t)to predict the number of electric vehicles promoted to use in the world every year.∆Nv=Nv̂(t+1)−Nv̂(t)(24) 2.We can calculate the energy saved V1due to the energy utilization ratio ηand the energysaved V2using alternative energy considering P4 and P5:{V1=Yα∙η2∙∆Nv∙∑λiV2=Yα∙ΔNv∙(1−∑λi)(1η2−1η1)V=V1+V2(25)Symbol description:Y represents annual average fuel consumption of fuel cars (L);αrepresents the current international oil prices ($/L);η1，η2represent efficiency of energy utilization of electric vehicles and fuel cars;∆Nv represents the average annual growth number of EVs;λi represents the ratio of the alternative energy;V1 ，V2are the amount of oil widely using EVs.4.2.3 Results and analysisWe use least square method to get the expression of Nv(t) :Nv (t )=4.92×107×t −9.785×1010 (26)Fitting diagram is as shown below:Figure 7 the expression of Nv −tSo we can calculate the result:∆Nv =4.92×107Figure 8 the proportion of the alternative energy in the worldThrough the Figure 8, the proportion of the alternative energy in the world is ∑λi =21.8%, so we can calculate the amount of oil saved:V 1 =1.1917×1011 （L ）8tV2=1.8086×1011（L）V=3×1011（L）V equals the amount of oil assumed in China for one year so it is very considerable. But the value is based on the assumption that new cars for one years are all EVs, so if we let V plus ten years, we will get a value that match twice of the assumption of oil for one years. Not to mention the fuel car before replacement for electric cars, the oil saved is more objective.4.3 A multi-objective optimization model4.3.1 Model analysisTask three requires us to establish a model of the number and form of generators, which matches the established one of electric vehicles. We attribute the problem to a multi-objective optimization problem in order to maximize the income of the environment, society, business and individuals. To simplify the problem, we summarize the above four aspects of income as follow: social costs are minimized and the output electrical energy meets the maximum demand.4.3.2 Model building➢Social costAssuming that in different kinds of power plants, the generators are of the same quantity and differ only in types. Social costs include the cost of generating electricity by power stations and the cost of pollution during producing energy.(1) Electricity cost WElectricity cost consists of three parts: the investment cost (W1) to construct a power station, the management cost (W2) and the energy cost (W3) to produce electricity.W1=∑k i gN i(27)W2=∑a i gN i(28)W3=∑b i gN i(29)W=W1+W2+W3(30) k i represents the cost of each generator when using i of producing electricity;g represents the number of generators in a power station.N i indicates the quantity of power stations using the i generation form. i(i=1,2,3…)stands for various kinds of energy resources respectively – hydropower, wind energy, solar energy, tidal energy, etc.a i represents the management cost of each generator of different types.b i represents costs of different generators to consume equivalent energy resources while producing unit electricity.(2) Cost of pollution U (Environmental costs)Pollutants include CO2,NO x,SO2,PM.U=∑∑P∙λij ∙d j∙r ji(31)λi represents the ratio of that kind of energy in the area;d j represents the emission rates of pollutants above, j=1,2,3,4;r j indicates equivalent social costs of pollutants above;P=∑n mωmm(32)P stands for the required energy in order to meet the quantity and form of electric vehicles in task one.n m represents the number in the m-th form of electric vehicles.ωm represents the required energy of each one in the m-th form of electric vehicles.To get the total social cost:min (W+U)(33)➢Output electrical energy∑N iαi g≥P(34)➢Objective function{min(W+U)∑N iαi g≥P(35)4.3.3 Results and analysisWe select New York as our object of study, and get data from National Energy Administration. Then calculate the answer by LINGO. The answers are as follow:Table 8 the result of question threeP/Kw∙h N1N2N3N46.83×107572101N1 is the number of thermal power plantN2 is the number of wind power plantN3 is the number of atomic energy plantN4 is the number of hydroelectric power plantAccording to the answer, it is required to give priority to thermal power. But after taking environment into consideration, the number of other forms of power station has increased apparently.5. Discussion and Conclusion5.1 strengths1. Build a dual network model for the effect in economy, society, environment and health in each area.2. In the result analysis, we analyze four different areas (China, US, Holland and sub-Saharan Africa) with different characteristics and give some suggestions to our government and EV manufactures.5.2 Weakness and Sensitivity1. The models in question 1 only consider electric vehicles，ignoring other forms of electric cars can also affect the consequences2. Ignore charging power and its length's effect on grids.5.3 Contribution1. Put forward 6 basic indexes and comprehensively evaluate the four factors of every district.2. Build a model to evaluate the comprehensive influence on a district produced by the widely used of the electric car and provide reference for the government and manufactures.5.4 Generalize1. The model in question 3 can be applied to the planning and layout of the charging station to achieve the minimum cost of consumer charge and the minimum investment cost of the investors.2. The model of the electric car can be applied to other emerging industry.6. Reference[1] Wang Cheng,(2007),Research on Effect of Electric Vehicles Development to Energy and Environment[2] Zhang Chenxi,(2013), Analysis on electric vehicle integration technology and social comprehensive benefits[3] Gao Ciwei,(2011), A Survey of Influence of Electrics Vehicle Charging on Power Grid[4] Li Zhihui,(2013), The impact of electric vehicle charging stations on the grid[5] BP_Annual_Report_and_Form_20F_2013[6] National Energy Administration of ChinaAppendixTable 1 Environmental impact of fuel vehicles and electric vehicles in their life cycle（units of cars：vehicle/ per thousand people，units of populations：per thousand people）USA 263000 vehicle/ per thousandpeople 784.9208979 808.6085237 798.330NLD 15493 vehicle/ per thousandpeople 467.765591 477.636048 485.889CHN 1370536 vehicle/ per thousandpeople 12.4346384 13.83181271 15.7340SSF 569081 vehicle/ per thousandpeople 18.45607899 19.06166098 19.1686population 2006 2007 2008USA 263000 vehicle/ per thousandpeople 818.3047054 820.8465765 816.07NLD 15493 vehicle/ per thousandpeople 503.8972902 513.5851624 521.694CHN 1370536 vehicle/ per thousandpeople 27.50429284 32.24901945 37.4813SSF 569081 vehicle/ per thousandpeople 26.4024143 28.1 30.3586。
2011全国大学生数学建模竞赛A题获奖论文——一篇
城市表层土壤重金属污染分析的数学模型摘要为研究城市土壤地质环境异常的查证，以及如何应用查证获得的海量数据资料开展城市环境质量评价，研究人类活动影响下城市地质环境的演变模式。
本文通过处理和分析已给数据，给出金属的空间分布说明污染程度和主要原因；建立数学模型确定污染源位置；最后收集其他信息讨论城市地质环境的演变模式。
问题一，利用matlab软件作出位置坐标x、y与八种总金属元素浓度的空间分布图；分析采集的重金属元素浓度所在区域的大致情形。
对采集的重金属元素浓度的数据进行分析，并计算单因子和多因子污染指数，根据土壤污染分级标准判断出不同重金属元素在各功能区的污染程度和各功能区的综合污染程度，其中工业区中铜是所有元素在不同功能区中污染程度最严重的，而工业区和交通区的综合污染程度是最严重的。
问题二，首先利用SAS软件对八种重金属元素在五个城区的含量进行主成分分析，得到八种重金属对各功能区的贡献率，可初步推断出工业生产、交通设施和生活垃圾造成重金属污染。
再利用SAS软件对各城区的重金属进行因子分析，进一步判断八种不同重金属污染的原因，如汞污染的原因为工业生产中三废的排放、交通运输业中汽油的燃烧和汽车轮胎磨损产生的粉尘等。
问题三，根据所给数据，分析重金属污染传播特征，即分别是介质的迁移运动、污染物的分散运动、污染物的累积与转化、污染物被环境介质吸收或吸附、污染物的沉淀，然后利用Matlab软件，采用多元纯二次二项式回归分析方法，分别得到每种重金属元素浓度与坐标的回归方程，并根据该方程利用多元函数求极值的方法确定出污染源的可能位置分别为：As（1878.2634，6003.7263，4.5846），Cd(970.5835，3946.7518，6.5891)，Cr（1235.1956，2658.3427，8.5402），Cu（138.4682，6223.4521，3.2461），Hg （1231.5782，2561.5483，5.2478），Ni（12234.2587，5865.1656，23.2461），Pb （2310.68914145.2674，3.2651），Zn（3015.43418642.2365 5.0543）；问题四，基于前三问，分析所建模型的优缺点。
2011年美赛真题优秀论文
中继站的协调方案摘要（Abstract ）中继站是将信号进行再生、放大处理后，再转发给下一个中继站，以确保传输信号的质量。
低功耗的用户，例如移动电话用户，在不能直接与其他用户联系的地方可以通过中继站来保持联系。
然而,中继站之间会互相影响,除非彼此之间有足够远的距离或通过充分分离的频率来传送。
为了排除信号间的干扰，实现某一区域内（题中以40英里为半径的圆形区域）通信设备正常的发射和接收信号，需要利用PL 技术对中继站作合理的协调和分配。
首先本文结合香农理论的相关算法，考虑了信号供给系统的损耗、天线增益、信号的传播损耗、辐射效率因素的影响，得到中继站的辐射范围半径公式为：,10,10log ()37.23282010r outr inp P d -=在供给对象为低功率消耗设备，查资料一般发射功率为3.2W ，中继站能接收到的最弱的信号1W μ,代入数据得到每个中继站的辐射半径为15.28m iles 。
同时本文在不考虑其他因素（包括：地形、大雾、山川、建筑物等）对辐射范围和辐射强度的影响下，结合相关知识和题目中给出的条件，在不引入PL 技术时得出每个中继站所服务的用户数量为39个。
对于问题一， 我们首先定义了均衡覆盖、覆盖效率，在均衡覆盖中即用圆覆盖圆形区域，我们根据式子2(2)n k n ππ-=，得出(,)k n 的可能值有(3,6),(4,4),三种，即等效三角形、正方形、正六边形覆盖，并通过覆盖效率的比较，最终得出正六边形覆盖是最好的覆盖方法，即蜂窝拓扑网络。
在这种覆盖情况下我们，我结合中继站覆盖半径15.28m iles ，根据式子m i n 3(1)1,0,1,2,3,N K K K =++=……，求出最少需要19个中继站，并在满足单位面积覆盖同时在线人数的情况下引入PL 技术，得出此时中继站在该区域可同时服务在限人数为1292人。
对于问题二，我们在问题一模型基础上从提高中继站服务人数和减少中继站半径两方面考虑，得出在将PL 分为18层，即中继站同时在线服务人数为702的情况下，结合单位面积同时在线服务人数，得出在中继站最少的情况下，中继站半径在[]11.094，,11.68范围内都可，我们为了让同时在线服务人数最大，取11.094英里，得出服务人数为11305。
2011MCM美赛A题论文 repeater coordination
2 General observations
In this section we discuss observations of the problem statement, ambiguities, and general assumptions we have made to simplify and clarify the modeling process. Additional clarification of some assumptions is provided in the appendices.
Optimal Placement of Radio Repeater Networks
Control #10754 February 14, 2011
Abstract In this paper we consider the problem of placing radio repeaters to serve users in an area. Given a population distribution and geographical map, we use a hillclimbing algorithm to find a minimum number of repeaters required to cover an area and then a genetic algorithm to provide maximal population coverage and network connectedness. We then use hill-climbing techniques to allocate subnetworks based on population size at repeater locations so that two arbitrary users can communicate even when all other users are communicating over the maximum number of possible networks. Our resulting algorithm is capable of producing a range of repeater network allocations, from robust networks that are capable of handling worstcase usage scenarios to smaller networks that provide optimal population coverage and connectivity. On a set of real-world population and geography data, we found that the combination of the hill-climbing and genetic algorithms had 28% better population coverage than a control algorithm did, as well as higher connectivity.
2011年全国大学生数学建模竞赛A题论文优秀论文范文模板参考资料
题目
（写出较确切的题目；也要有新意、醒目）
摘要
(从总体上阐述文章要解决的问题、分析问题的主要思路、针对问题建立的模型以及最终的计算结果(主要是说明你用什么方法；解决了什么问题；主要结果是什么；有什么特色和创新点，以及其它工作。
摘要是整篇文章的高度压缩，文字精练，表达准确,内容不少于500字。
)
关键词：列出文章中出现的关键词汇及数学用语（3-5个）.
(第三页内容）
目录
(此页可有可无, 内容较多时最好有个目录.目录的页码用“Ⅰ、Ⅱ”连续编号)
(第四页开始论文主要内容,论文从此页开始编写页码，页码必须位于每页页脚中部，用阿拉伯数字从“1”开始连续编号)
一、问题重述
二、问题的分析
三、模型假设
四、符号及变量说明
五、模型的建立与求解
六、模型的检验
七、模型的应用与推广
八、模型的评价与改进
参考文献。
MCM美赛论文集
高教社杯全国大学生数学建模竞赛承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则。
我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。
我们知道，抄袭别人的成果是违反竞赛规则的,如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。
我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。
如有违反竞赛规则的行为，我们将受到严肃处理。
我们参赛选择的题号是（从A/B/C/D中选择一项填写）：A我们的参赛报名号为（如果赛区设置报名号的话）：99999所属学校（请填写完整的全名）：西安交通大学参赛队员(打印并签名）：1.一作者2.二作者3.三作者指导教师或指导教师组负责人(打印并签名)：导师日期：2011年8月1日赛区评阅编号（由赛区组委会评阅前进行编号）：2011高教社杯全国大学生数学建模竞赛编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：赛区评阅记录（可供赛区评阅时使用）：评阅人评分备注全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）:全国大学生数学建模竞赛L A T E X2ε模板摘要这是数学建模论文模板mcmthesis的示例文件。
特别地，这篇文档是“全国大学生数学建模竞赛（CUMCM）”模板的示例文件。
这个模板使用于参加高教社杯全国大学生数学竞赛的同学准备他们的建模论文，帮助他们更多的关注于论文内容而非论文的排版。
这个模板的设计是根据2010年修订的《全国大学生数学建模竞赛论文格式规范》[1]制作，完全符合该论文格式规范，但是该模板未得到官方认可，请使用者自己斟酌使用。
这个示例文档逐条展示其对[1]的实现效果，并对所有自定义命令进行说明。
这个示例文件还包含了一些对公示、插图、表格、交叉引用、参考文献、代码等的测试部分，以展示其效果，并作简要的使用说明。
2011年美国数学建模大赛一等奖获奖论文
Team #10076
Page 2 of 21
Contents
1 Introduction 2 Restatement of Problem 3 Basic Assumptions 4 Analysis of the Problem 4.1 Definitions of Zero-Potential Surface and “vertical 4.2 Turn Of Snowboarding . . . . . . . . . . . . . . . 4.3 Snowboarding on the Deck . . . . . . . . . . . . . 4.4 Snowboarding on the Quarter Arc . . . . . . . . . 5 “Vertical air” Model 6 Results and Sensitivity Analysis 6.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Model Applications And Analysis 7.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Maximum Twist Model 8.1 Analysis of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Results and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 9 Tradeoffs to Develop a ”Practical” Course 9.1 Halfpipe Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Snow in the Halfpipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 4 6 6 7 10 10 14 16 16 16 17 17 17 18 18 18 19 20 20 20
2011高教社杯全国大学生数学建模竞赛B卷论文
承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。
我们知道，抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。
我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。
如有违反竞赛规则的行为，我们将受到严肃处理。
我们参赛选择的题号是（从A/B/C/D中选择一项填写）：我们的参赛报名号为（如果赛区设置报名号的话）：所属学校（请填写完整的全名）：参赛队员(打印并签名) ：1.2.3.指导教师或指导教师组负责人(打印并签名)：日期：年___月___日赛区评阅编号（由赛区组委会评阅前进行编号）：编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）：交巡警服务平台的设置与调度一．摘要“交巡警服务平台的设置与调度”是根据城市的实际情况与需求合理地设置交巡警服务平台、分配各平台的管辖范围、调度警务资源的问题。
我们运用数学的集合、图论、运筹学等基本理论，通过CAD软件绘出A区平面图并计算出各条道路的距离，以此作为基本数据，运用集合的覆盖知识，结合交巡警服务平台到各节点所需要的时间，再联系实际每个节点的案件发生率以及人口密度来合理的分配各平台的管辖范围。
运用Dijkstra最优路径算法求出各警务平台到13条交通要道出口的最短距离，采用二部加权图设计出两种调度算法。
以每个平台点为圆心，以适当的半径画圆，在圆之外的节点或是圆内节点过多的平台点，分析处理，按照决策论和覆盖来确定所增加平台的数量和位置。
以A区为例，对全市的交巡警服务平台设置的合理性作出判断，其中考虑到工作量，工作效率，采用正态分布模型。
2011数学建模优秀获奖论文
承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。
我们知道，抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。
我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。
如有违反竞赛规则的行为，我们将受到严肃处理。
我们参赛选择的题号是（从A/B/C/D中选择一项填写）： A 我们的参赛报名号为（如果赛区设置报名号的话）：S15030 所属学校（请填写完整的全名）：河南理工大学参赛队员(打印并签名) ：1. 王景佩2. 付玉洁3. 刘争光指导教师或指导教师组负责人(打印并签名)：竞赛指导组日期： 2011 年 9 月 12 日赛区评阅编号（由赛区组委会评阅前进行编号）：编号专用页赛区评阅编号（由赛区组委会评阅前进行编号）：赛区评阅记录（可供赛区评阅时使用）：评阅人评分备注全国统一编号（由赛区组委会送交全国前编号）：全国评阅编号（由全国组委会评阅前进行编号）：城市表层土壤重金属污染分析摘要随着经济的发展，城市土壤的污染越来越严重，尤其是土壤重金属的污染。
本文就某一城区进行取样调查，要求根据调查数据来评价不同区域的重金属污染程度，并说明其主要原因，且建立模型来确定污染源的位置并进行优化。
对于问题一，首先我们应用软件surfer 9.0，采用克立格插值法分别画出8种重金属在该城区的空间分布图；然后以污染指数来表现污染程度，先根据单因子指数法求出每种金属的污染指数，再利用内梅罗指数法求解出8种重金属的综合污染指数，最后依据土壤综合污染程度分级标准，来评价每个区域的污染程度。
通过上述过程的求解，可得到如下结果：功能区生活区工业区山区交通区公园绿地区综合指数 3.1706 7.3583 1.2484 3.8290 2.7344污染程度中度污染重度污染警界线中度污染轻度污染问题二要求通过数据分析来说明污染原因，我们采用多元统计数学中的因子分析法，首先建立中金属污染浓度矩阵，进行标准化处理消除量纲的影响，进而借助matlab求得各因子对重金属污染的累积贡献率，依此数据为依据来分析重金属污染的主要原因为：工业“三废”，交通机动车尾气排放，人类生活废水的排放等。
美赛数学建模比赛论文实用模板
The Keep-Right-Except-To-Pass RuleSummaryAs for the first question, it provides a traffic rule of keep right except to pass, requiring us to verify its effectiveness. Firstly, we define one kind of traffic rule different from the rule of the keep right in order to solve the problem clearly; then, we build a Cellular automaton model and a Nasch model by collecting massive data; next, we make full use of the numerical simulation according to several influence factors of traffic flow; At last, by lots of analysis of graph we obtain, we indicate a conclusion as follow: when vehicle density is lower than 0.15, the rule of lane speed control is more effective in terms of the factor of safe in the light traffic; when vehicle density is greater than 0.15, so the rule of keep right except passing is more effective In the heavy traffic.As for the second question, it requires us to testify that whether the conclusion we obtain in the first question is the same apply to the keep left rule. First of all, we build a stochastic multi-lane traffic model; from the view of the vehicle flow stress, we propose that the probability of moving to the right is 0.7and to the left otherwise by making full use of the Bernoulli process from the view of the ping-pong effect, the conclusion is that the choice of the changing lane is random. On the whole, the fundamental reason is the formation of the driving habit, so the conclusion is effective under the rule of keep left.As for the third question, it requires us to demonstrate the effectiveness of the result advised in the first question under the intelligent vehicle control system. Firstly, taking the speed limits into consideration, we build a microscopic traffic simulator model for traffic simulation purposes. Then, we implement a METANET model for prediction state with the use of the MPC traffic controller. Afterwards, we certify that the dynamic speed control measure can improve the traffic flow .Lastly neglecting the safe factor, combining the rule of keep right with the rule of dynamical speed control is the best solution to accelerate the traffic flow overall.Key words：Cellular automaton model Bernoulli process Microscopic traffic simulator model The MPC traffic controlContentContent (2)1. Introduction (3)2. Analysis of the problem (3)3. Assumption (3)4. Symbol Definition (3)5. Models (4)5.1 Building of the Cellular automaton model (4)5.1.1 Verify the effectiveness of the keep right except to pass rule (4)5.1.2 Numerical simulation results and discussion (5)5.1.3 Conclusion (8)5.2 The solving of second question (8)5.2.1 The building of the stochastic multi-lane traffic model (9)5.2.2 Conclusion (9)5.3 Taking the an intelligent vehicle system into a account (9)5.3.1 Introduction of the Intelligent Vehicle Highway Systems (9)5.3.2 Control problem (9)5.3.3 Results and analysis (9)5.3.4 The comprehensive analysis of the result (10)6. Improvement of the model (11)6.1 strength and weakness (11)6.1.1 Strength (11)6.1.2 Weakness (11)6.2 Improvement of the model (11)7. Reference (13)1. IntroductionAs is known to all, it’s essential for us to drive automobiles, thus the driving rules is crucial important. In many countries like USA, China, drivers obey the rules which called “The Keep-Right-Except-To-Pass (that is, when driving automobiles, the rule requires drivers to drive in the right-most unless theyare passing another vehicle)”.2. Analysis of the problemFor the first question, we decide to use the Cellular automaton to build models,then analyze the performance of this rule in light and heavy traffic. Firstly,we mainly use the vehicle density to distinguish the light and heavy traffic; secondly, we consider the traffic flow and safe as the represent variable which denotes the light or heavy traffic; thirdly, we build and analyze a Cellular automaton model; finally, we judge the rule through two different driving rules,and then draw conclusions.3. AssumptionIn order to streamline our model we have made several key assumptions●The highway of double row three lanes that we study can representmulti-lane freeways.●The data that we refer to has certain representativeness and descriptive●Operation condition of the highway not be influenced by blizzard oraccidental factors●Ignore the driver's own abnormal factors, such as drunk driving andfatigue driving●The operation form of highway intelligent system that our analysis canreflect intelligent system●In the intelligent vehicle system, the result of the sampling data hashigh accuracy.4. Symbol Definitioni The number of vehiclest The time5. ModelsBy analyzing the problem, we decided to propose a solution with building a cellular automaton model.5.1 Building of the Cellular automaton modelThanks to its simple rules and convenience for computer simulation, cellular automaton model has been widely used in the study of traffic flow in recent years. Let )(t x i be the position of vehicle i at time t , )(t v i be the speed of vehicle i at time t , p be the random slowing down probability, and R be the proportion of trucks and buses, the distance between vehicle i and the front vehicle at time t is:1)()(1--=-t x t x gap i i i , if the front vehicle is a small vehicle.3)()(1--=-t x t x gap i i i , if the front vehicle is a truck or bus.5.1.1 Verify the effectiveness of the keep right except to pass ruleIn addition, according to the keep right except to pass rule, we define a new rule called: Control rules based on lane speed. The concrete explanation of the new rule as follow:There is no special passing lane under this rule. The speed of the first lane (the far left lane) is 120–100km/h (including 100 km/h)；the speed of the second lane (the middle lane) is 100–80km8/h (including80km/h)；the speed of the third lane (the far right lane) is below 80km/ h. The speeds of lanes decrease from left to right.● Lane changing rules based lane speed controlIf vehicle on the high-speed lane meets control v v <, ),1)(min()(max v t v t gap i f i +≥, safe b i gap t gap ≥)(, the vehicle will turn into the adjacent right lane, and the speed of the vehicle after lane changing remains unchanged, where control v is the minimum speed of the corresponding lane.● The application of the Nasch model evolutionLet d P be the lane changing probability (taking into account the actual situation that some drivers like driving in a certain lane, and will not takethe initiative to change lanes), )(t gap f i indicates the distance between the vehicle and the nearest front vehicle, )(t gap b i indicates the distance between the vehicle and the nearest following vehicle. In this article, we assume that the minimum safe distance gap safe of lane changing equals to the maximum speed of the following vehicle in the adjacent lanes.Lane changing rules based on keeping right except to passIn general, traffic flow going through a passing zone (Fig. 5.1.1) involves three processes: the diverging process (one traffic flow diverging into two flows), interacting process (interacting between the two flows), and merging process (the two flows merging into one) [4].Fig.5.1.1 Control plan of overtaking process(1) If vehicle on the first lane (passing lane) meets ),1)(min()(max v t v t gap i f i +≥ and safe b i gap t gap ≥)(, the vehicle will turn into the second lane, the speed of the vehicle after lane changing remains unchanged.5.1.2 Numerical simulation results and discussionIn order to facilitate the subsequent discussions, we define the space occupation rate as L N N p truck CAR ⨯⨯+=3/)3(, where CAR N indicates the number ofsmall vehicles on the driveway,truck N indicates the number of trucks and buses on the driveway, and L indicates the total length of the road. The vehicle flow volume Q is the number of vehicles passing a fixed point per unit time,T N Q T /=, where T N is the number of vehicles observed in time duration T .The average speed ∑∑⨯=T it i a v T N V 11)/1(, t i v is the speed of vehicle i at time t . Take overtaking ratio f p as the evaluation indicator of the safety of traffic flow, which is the ratio of the total number of overtaking and the number of vehicles observed. After 20,000 evolution steps, and averaging the last 2000 steps based on time, we have obtained the following experimental results. In order to eliminate the effect of randomicity, we take the systemic average of 20 samples [5].Overtaking ratio of different control rule conditionsBecause different control conditions of road will produce different overtaking ratio, so we first observe relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions.(a) Based on passing lane control (b) Based on speed control Fig.5.1.3Fig.5.1.3 Relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions.It can be seen from Fig. 5.1.3:(1) when the vehicle density is less than 0.05, the overtaking ratio will continue to rise with the increase of vehicle density; when the vehicle density is larger than 0.05, the overtaking ratio will decrease with the increase of vehicle density; when density is greater than 0.12, due to the crowding, it willbecome difficult to overtake, so the overtaking ratio is almost 0.(2) when the proportion of large vehicles is less than 0.5, the overtaking ratio will rise with the increase of large vehicles; when the proportion of large vehicles is about 0.5, the overtaking ratio will reach its peak value; when the proportion of large vehicles is larger than 0.5, the overtaking ratio will decrease with the increase of large vehicles, especially under lane-based control condition s the decline is very clear.● Concrete impact of under different control rules on overtaking ratioFig.5.1.4Fig.5.1.4 Relationships among vehicle density, proportion of large vehicles and overtaking ratio under different control conditions. (Figures in left-hand indicate the passing lane control, figures in right-hand indicate the speed control. 1f P is the overtaking ratio of small vehicles over large vehicles, 2f P is the overtaking ratio of small vehicles over small vehicles, 3f P is the overtaking ratio of large vehicles over small vehicles, 4f P is the overtaking ratio of large vehicles over large vehicles.). It can be seen from Fig. 5.1.4:(1) The overtaking ratio of small vehicles over large vehicles under passing lane control is much higher than that under speed control condition, which is because, under passing lane control condition, high-speed small vehicles have to surpass low-speed large vehicles by the passing lane, while under speed control condition, small vehicles are designed to travel on the high-speed lane, there is no low- speed vehicle in front, thus there is no need to overtake.● Impact of different control rules on vehicle speedFig. 5.1.5 Relationships among vehicle density, proportion of large vehicles and average speed under different control conditions. (Figures in left-hand indicates passing lane control, figures in right-hand indicates speed control.a X is the average speed of all the vehicles, 1a X is the average speed of all the small vehicles, 2a X is the average speed of all the buses and trucks.).It can be seen from Fig. 5.1.5:(1) The average speed will reduce with the increase of vehicle density and proportion of large vehicles.(2) When vehicle density is less than 0.15,a X ,1a X and 2a X are almost the same under both control conditions.Effect of different control conditions on traffic flowFig.5.1.6Fig. 5.1.6 Relationships among vehicle density, proportion of large vehicles and traffic flow under different control conditions. (Figure a1 indicates passing lane control, figure a2 indicates speed control, and figure b indicates the traffic flow difference between the two conditions.It can be seen from Fig. 5.1.6：(1) When vehicle density is lower than 0.15 and the proportion of large vehicles is from 0.4 to 1, the traffic flow of the two control conditions are basically the same.(2) Except that, the traffic flow under passing lane control condition is slightly larger than that of speed control condition.5.1.3 ConclusionIn this paper, we have established three-lane model of different control conditions, studied the overtaking ratio, speed and traffic flow under different control conditions, vehicle density and proportion of large vehicles.5.2 The solving of second question5.2.1 The building of the stochastic multi-lane traffic model5.2.2 ConclusionOn one hand, from the analysis of the model, in the case the stress is positive, we also consider the jam situation while making the decision. More specifically, if a driver is in a jam situation, applying ))(,2(x P B R results with a tendency of moving to the right lane for this driver. However in reality, drivers tend to find an emptier lane in a jam situation. For this reason, we apply a Bernoulli process )7.0,2(B where the probability of moving to the right is 0.7and to the left otherwise, and the conclusion is under the rule of keep left except to pass, So, the fundamental reason is the formation of the driving habit.5.3 Taking the an intelligent vehicle system into a accountFor the third question, if vehicle transportation on the same roadway was fully under the control of an intelligent system, we make some improvements for the solution proposed by us to perfect the performance of the freeway by lots of analysis.5.3.1 Introduction of the Intelligent Vehicle Highway SystemsWe will use the microscopic traffic simulator model for traffic simulation purposes. The MPC traffic controller that is implemented in the Matlab needs a traffic model to predict the states when the speed limits are applied in Fig.5.3.1. We implement a METANET model for prediction purpose[14].5.3.2 Control problemAs a constraint, the dynamic speed limits are given a maximum and minimum allowed value. The upper bound for the speed limits is 120 km/h, and the lower bound value is 40 km/h. For the calculation of the optimal control values, all speed limits are constrained to this range. When the optimal values are found, they are rounded to a multiplicity of 10 km/h, since this is more clear for human drivers, and also technically feasible without large investments.5.3.3 Results and analysisWhen the density is high, it is more difficult to control the traffic, since the mean speed might already be below the control speed. Therefore, simulations are done using densities at which the shock wave can dissolve without using control, and at densities where the shock wave remains. For each scenario, five simulations for three different cases are done, each with a duration of one hour. The results of the simulations are reported in Table 5.1, 5.2, 5.3. Table.5.1 measured results for the unenforced speed limit scenariodem q case#1 #2 #3 #4 #5 TTS:mean(std ) TPN 4700no shock 494.7452.1435.9414.8428.3445.21(6.9%) 5:4wave 3 5 8 8 0 14700nocontrolled520.42517.48536.13475.98539.58517.92(4.9%)6:364700 controlled 513.45488.43521.35479.75-486.5500.75(4.0%)6:244700 no shockwave493.9472.6492.78521.1489.43493.96(3.5%)6:034700 uncontrolled635.1584.92643.72571.85588.63604.84(5.3%)7:244700 controlled 575.3654.12589.77572.15586.46597.84(6.4%)7:19●Enforced speed limits●Intelligent speed adaptationFor the ISA scenario, the desired free-flow speed is about 100% of the speed limit. The desired free-flow speed is modeled as a Gaussian distribution, with a mean value of 100% of the speed limit, and a standard deviation of 5% of the speed limit. Based on this percentage, the influence of the dynamic speed limits is expected to be good[19].5.3.4 The comprehensive analysis of the resultFrom the analysis above, we indicate that adopting the intelligent speed control system can effectively decrease the travel times under the control of an intelligent system, in other words, the measures of dynamic speed control can improve the traffic flow.Evidently, under the intelligent speed control system, the effect of the dynamic speed control measure is better than that under the lane speed control mentioned in the first problem. Because of the application of the intelligent speed control system, it can provide the optimal speed limit in time. In addition, it can guarantee the safe condition with all kinds of detection device and the sensor under the intelligent speed system.On the whole, taking all the analysis from the first problem to the end into a account, when it is in light traffic, we can neglect the factor of safe with the help of the intelligent speed control system.Thus, under the state of the light traffic, we propose a new conclusion different from that in the first problem: the rule of keep right except to pass is more effective than that of lane speed control.And when it is in the heavy traffic, for sparing no effort to improve the operation efficiency of the freeway, we combine the dynamical speed control measure with the rule of keep right except to pass, drawing a conclusion that the application of the dynamical speed control can improve the performance ofthe freeway.What we should highlight is that we can make some different speed limit as for different section of road or different size of vehicle with the application of the Intelligent Vehicle Highway Systems.In fact, that how the freeway traffic operate is extremely complex, thereby, with the application of the Intelligent Vehicle Highway Systems, by adjusting our solution originally, we make it still effective to freeway traffic.6. Improvement of the model6.1 strength and weakness6.1.1 Strength●it is easy for computer simulating and can be modified flexibly to consideractual traffic conditions ,moreover a large number of images make the model more visual.●The result is effectively achieved all of the goals we set initially, meantimethe conclusion is more persuasive because of we used the Bernoulli equation.●We can get more accurate result as we apply Matlab.6.1.2 Weakness●The relationship between traffic flow and safety is not comprehensivelyanalysis.●Due to there are many traffic factors, we are only studied some of the factors,thus our model need further improved.6.2 Improvement of the modelWhile we compare models under two kinds of traffic rules, thereby we come to the efficiency of driving on the right to improve traffic flow in some circumstance. Due to the rules of comparing is too less, the conclusion is inadequate. In order to improve the accuracy, We further put forward a kinds of traffic rules: speed limit on different type of cars.The possibility of happening traffic accident for some vehicles is larger, and it also brings hidden safe troubles. So we need to consider separately about different or specific vehicle types from the angle of the speed limiting in order to reduce the occurrence of traffic accidents, the highway speed limit signs is in Fig.6.1.Fig .6.1Advantages of the improving model are that it is useful to improve the running condition safety of specific type of vehicle while considering the difference of different types of vehicles. However, we found that the rules may be reduce the road traffic flow through the analysis. In the implementation it should be at the 85V speed of each model as the main reference basis. In recent years, the 85V of some researchers for the typical countries from Table 6.1[ 21]: Table 6.1 Operating speed prediction modeAuthorCountry Model Ottesen andKrammes2000America LC DC L DC V C ⨯---=01.0012.057.144.10285Andueza2000Venezuel a ].[308.9486.7)/894()/2795(25.9885curve horizontal L DC Ra R V T ++--= ].[tan 819.27)/3032(69.10085gent L R V T +-= Jessen2001 America ][00239.0614.0279.080.86185LSD ADT G V V P --+=][00212.0432.010.7285NLSD ADT V V P -+=Donnell2001 America 22)2(8500724.040.10140.04.78T L G R V --+=22)3(85008369.048.10176.01.75T L G R V --+= 22)4(8500810.069.10176.05.74T L G R V --+=22)5(8500934.008.21.83T L G V --=BucchiA.BiasuzziK.And SimoneA.2005Italy DC V 124.0164.6685-= DC E V 4.046.3366.5585--= 2855.035.1119.0745.65DC E DC V ---= Fitzpatrick America KV 98.17507.11185-= Meanwhile, there are other vehicles driving rules such as speed limit in adverseweather conditions. This rule can improve the safety factor of the vehicle to some extent. At the same time, it limits the speed at the different levels.7. Reference[1] M. Rickert, K. Nagel, M. Schreckenberg, A. Latour, Two lane traffi csimulations using cellular automata, Physica A 231 (1996) 534–550.[20] J.T. Fokkema, Lakshmi Dhevi, Tamil Nadu Traffi c Management and Control inIntelligent Vehicle Highway Systems,18(2009).[21] Yang Li, New Variable Speed Control Approach for Freeway. (2011) 1-66。
2011年数学建模B题优秀论文
B题交巡警服务平台的设置与调度小组人数：3模型建立：程序编写：论文撰写：目录一．摘要二．问题重述三．问题分析与建模思路四．基本假设五．符号说明六．模型的建立与求解七．模型的评价与推广八．参考文献与附录一．摘要警察是现代社会不可或缺的角色，肩负着执法、治安、提供社会服务等重要职责。
为了更好更有效的实现这些只能，必须设立交巡警服务平台。
这些平台需要合理地分布在城市的各个地区和交通要道，这样不仅可以及时响应出警到达案发现场，在遇到重要的或者突发的事件时也能高效的通过联合调度行动起来。
该论文就交巡警服务平台的设置与调度等实际问题，针对提出的5个问题分别给出具体的解决方案并给出结果。
问题一：（1）题目要求根据已知20个交巡警服务平台的位置，为它们分别分配各自的管辖范围，使其能在3min内到达自己管辖区域内的事发地点。
对于此问题本文建立最大集合覆盖模型，建立了A区街道结点连通性的邻接矩阵。
通过对该邻接矩阵进行优化，建立了带权边邻接矩阵。
借助floyd多源最短路算法并利用数学软件MATLAB进行分配求解，最后得到A区现有每个巡警服务台的管辖范围如表1。
（2）题目要求对13条交通要道实现快速全封锁，我们以所用时间最少为目标，引入0-1变量，建立该问题的0-1规划模型，并借助数学软件LINGO进行求解，求解结果表明需要8.05分钟可以实现快速封锁。
（3）题目要求以交巡警服务平台工作量尽量均衡以及出警时间尽量短为前提，确定增设平台（2~5）的具体数目及位置。
由问题（1）的分配结果可知，在现有巡警服务台的设置下：①还有6个路口在案发时巡警不能在3min之内到达，即某些地方出警时间过长；②我们根据巡警服务台的工作量的方差定义工作量不均衡度，结果显示：此时服务台的工作量不均衡度为8.4314。
为了解决上述出警时间过长与工作量不均衡的问题。
我们建立集合覆盖的0-1规划模型，求解结果表明：在增加4个平台的情况下，可以解决出警时间过长的问题。
2011年美国数学建模大赛论文
dimension reduction(MDR) model to maximize vertical air, twist and flip. The MDR analysis model is proposed. And we define as a tradeoff fac tor to develop a “practical” course.
In requirement 3, we take key factors, including s lope angle angar radius
and the
into consideration to implement the “practical” multiple-factor
Accordingly, by setting up an objective function and using the simulation, we get the conclusion
Team # 9561
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that the optimal value of
is
and
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9561
Problem Chosen
For office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________
.
Keywords: half-pipe snowboard, variant force differential-integration model, semi-spiral track decomposition model, multi-factor dimension reduction model, vertical air, twist, flip
2011数学建模竞赛论文
房价的合理性和未来的走势的问题摘 要房价是一个国家在发展过程中与人们生活密切相关的重要指标之一，本文研究房价的合理性和未来的走势的问题，并分析其对经济等方面的影响，以上海市的房价为代表，从多个角度建立了以下三个模型：模型一----多元线性方程模型：通过查阅上海年鉴，收集人均可支配收入、人均GDP 、房屋造价和人均储蓄额四个变量的数据，运用最小二乘法、mathematics 软件求解、matlab 软件拟合等，建立了房价与这四个变量的表达式：12340.453014*0.182798*0.289857*0.426408*y x x x x =++-，通过该表达式预测2010年的房价，与实际的房价进行比较，从而判断其合理性；模型二----房价的构造模型：房地产价格可分为四大块：土地成本、开发成本、政策税费，运用层次分析、主成分分析等方法，建立了房价与这四个变量的表达式: 0P PL C T D =+++(1)()PLt d C r=++⨯+,从表达式中得到房价与他们的关系。
模型三----房价的供需模型：从建造面积和购买面积的角度，运用线性差分方程方法来分析供与求的三种关系：供大于求、供等于求和供小于求对房价的影响，建立了房价与供、求的关系式()()(0)1()d a c d P t P b b d b +⎡⎤=-'+--'⎢⎥+⎣⎦。
关键词: 多元线性方程、构造模型、层次分析、供需模型、差分方程流程图目录一、问题重述1、问题的背景2、问题的提出二、问题分析三、模型的建立、求解及预测1、模型一----多元线性方程1.1模型的假设及说明1.2模型的建立与求解1.2.1模型的建立（1）房价与人均可支配的收入之间的关系（2）房价与建房成本之间的关系（3）房价与人均GDP之间的关系（4）房价与人均储蓄存款之间的关系1.2.2模型的求解1.3模型的修正1.4模型结果的检验与分析1.5利用已建立的模型对上海市的房价进行预测1.6预测房价1.7模型的优缺点分析与改进方向2、模型二----房价的构造模型2.1楼面地价2.2开发成本2.3政策税费2.4预期利润3、模型三----房价的供需模型3.1模型的建立3.2利用已建立的模型对上海市住房的供求关系进行预测3.3关于住房供需模型的讨论和评价四、房价的合理性判断及合理措施4.1 房价的合理性判断4.2 对房价采取的合理措施五、对房价未来走势的分析六、附录一、问题重述1、问题的背景随着中国综合实力的不断发展，人们的生活质量在逐步的提高，同时民生的问题也显得愈发的重要，而房价问题事关国计民生，对国家经济发展和社会稳定有着重大的影响，因此一直是各国政府大力关注的问题。
2011年数学建模校赛论文成品
目录一、问题重述 .............................................................................................................................. - 1 -1.1问题的提出 ............................................................................................................................ - 1 -1.2问题的分析 ............................................................................................................................ - 2 -二、条件假设 .............................................................................................................................. - 2 -三、符号约定 .............................................................................................................................. - 3 -四、辐射井的地下水降落曲线数学公式的构造 ...................................................................... - 3 -4.1流态判断条件的确定............................................................................................................. - 3 - 4.1.1辐射流的流动特性 ............................................................................................................. - 3 - 4.1.2辐射流流动状态的判断 ..................................................................................................... - 4 - 4.1.3潜水流态的判定方法 ......................................................................................................... - 6 -4.2最大影响半径R的确定 ...................................................................................................... - 8 - 4.3辐射井地下水降落曲线的构造............................................................................................. - 9 - 4.3.1辐射井地下水降落曲线图的分析...................................................................................... - 9 - 4.3.2辐射井地下水降落曲线数学公式的构造........................................................................ - 10 - 五、辐射井水量计算模型的建立 ............................................................................................ - 14 -5.1积分法计算辐射井水量....................................................................................................... - 14 - 5.2等效大井法计算辐射井水量............................................................................................... - 15 - 六、对建立公式、模型的分析检验 ........................................................................................ - 16 - 七、模型的优缺点及改进方法 .............................................................................................. - 17 -7 7.1模型的优缺点 .................................................................................................................... - 17 -7 7.2模型的改进 .......................................................................................................................... - 18 - 7.2.1改进因素的分析 ............................................................................................................. - 18 -8 7.2.2利用灰色模型进行求解 ................................................................................................. - 18 -8 7.2.3利用新陈代谢GM(1, 1)模型进行求解 .......................................................................... - 20 -0 八、参考文献 .......................................................................................................................... - 21 -1一、问题重述1.1问题的提出辐射井是由一口大口径的竖井和自竖井内周围含水层任意方向、高程打进一层数条水平辐射管组成，地下水沿水平辐射管汇集到竖井中。
2011年美国大学生数学建模竞赛优秀作品
AbstractThis paper presents one case study to illustrate how probability distribution and genetic algorithm and geographical analysis of serial crime conducted within a geographic information system can assist crime investigation.Techniques are illustrated for predicting the location of future crimes and for determining the possible residence of offenders based on the geographical pattern of the existing crimes and quantitative method,which is PSO.It is found that such methods are relatively easy to implement within GIS given appropriate data but rely on many assumptions regarding offenders’behaviour.While some success has been achieved in applying the techniques it is concluded that the methods are essentially theory-less and lack evaluation.Future research into the evaluation of such methods and in the geographic behaviour of serial offenders is required in order to apply such methods to investigations with confidence in their reliability.1.IntroductionThis series of armed robberies occurred in Phoenix,Arizona between13September and5December1999and included35robberies of fast food restaurants,hotels and retail businesses.The offenders were named the“Supersonics”by the Phoenix Police Department Robbery Detail as the first two robberies were of Sonic Drive-In restaurants.After the35th robbery,the offenders appear to have desisted from their activity and at present the case remains unsolved.The MO was for the offenders to target businesses where they could easily gain entry,pull on a ski mask or bandanna, confront employees with a weapon,order them to the ground,empty the cash from a safe or cash register into a bag and flee on foot most likely to a vehicle waiting nearby. While it appears that the offenders occasionally worked alone or in pairs,the MO, weapons and witness descriptions tend to suggest a group of at least three offenders. The objective of the analysis was to use the geographic distribution of the crimes to predict the location of the next crime in an area that was small enough to be suitable for the Robbery Detail to conduct stakeouts and surveillance.After working with a popular crime analysis manual(Gottleib,Arenberg and Singh,1994)it was found that the prescribed method produced target areas so large that they were not operationally useful.However,the approach was attractive as it required only basic information and relied on simple statistical analysis.To identify areas that were more useful for the Robbery Detail,it was decided to use a similar approach combined with other measurable aspects of the spatial distribution of the crimes.As this was a“live”case, new crimes and information were integrated into the analysis as it came to hand.2.AssumptionIn order to modify the model existed,we apply serial new assumptions to the principle so that our rectified model can be much more practical.Below are the assumptions:1.C riminals prefer something about the locations where previous crimes werecommitted committed..We supposed the criminals have a greater opportunity to ran away if they choose to crime in the site they are familiar with.In addition,the criminals probably choose previous kill sites where their target potential victims live and work.2.Offenders regard it safer to crime in their previous kill site as time went by.This is true that the site would be severely monitored by police when a short term crime happened and consequently the criminal would suffer a risk of being arrested in that site.And as mentioned above ,the police would reduce the frequency of examining the previous kill sites as time went by.3.Criminals are likely to choose the site that have optimal distance .This is a reasonable assumption since it is probably insecure to crime in the site that stays far away and that costs an amount of energy to escape and adds the opportunity to be arrested in such an unfamiliar terrain.And it is also impossible to crime in the site nearby since it increases the probability of being recognized or being trapped.As a result,we can measure a optimal distance in series perpetrations.4.Crimes are committed by individual.We assume that all the case in the model are committed by individuals instead of by organized members.In this way the criminal is subject to the assumptions mentioned above due to his insufficient preparation.5.Criminals Criminals''movements unconstrained.Because of the difficulty of finding real-world distance data,we invoke the “Manhattan assumption”:There are enough streets and sidewalks in a sufficiently grid-like pattern that movements along real-world movement routes is the same as “straight-line”movement in a space be discrete into city blocks.It is demonstrated that across several types of serial crime,the Euclidean and Manhattan distances are essentially interchangeable in predicting anchor points.3.The prediction of the next crime site3.1The measure of the optimal distanceDue to the fact that the mental optimal distance of the criminal is related to whether he is a careful person or not,it is impossible for him to make a fixed constant.Besides,the optimal distance will change in different moment.However,such distance should be reflected on the distances of the former crime sites.Presume that the coordinates of the n crime sites is respectively ),(11y x 、),(22y x 、……、),(n n y x ,and define the distance between the th i crime site and the th j one as j D ,i .The distance above we first consider it as Euclid distance,which is:22,)()(j i j i j i y y x x D −+−=With that,we are able to measure the distance between the th n crime site and the th 1-n one respectively.According to the assumption 2,the criminal believes that the earlier crime sites have became saferfor him to commit a crime again,so we can define his mental optimal distance,giving the sites the weights from little to much according to when the offenses happened in time sequence,as:∑−==11,n i ni i D w SD Satisfying 121......−<<<n w w w ，111=∑−=n i i w .Presuming the th i crime happens in i t ,whichis measured by week,we can have ∑−==11n i i kk t t w .SD can reflect the criminal's mental condition to some extent,so we can use it to predict the mental optimal distance of the criminal in the th n 1+case.While referring to the th n crime site,the criminal is able to use SD to estimate the optimal distance in the next time,and while referring to the rest crime sites,the optimal distances reduce as time goes back.Thus,the optimal security of the th i crime site can be measured as the following:n ni i SD t t SD *=3.2The measure of the probability distributionGiven the crime sites and location,we can estimate tentatively the probability density distribution of the future crimes,which equals to that we add some small normal distribution to every scene of crime to produce a probability distribution estimate function.The small normal distribution uses the SD mentioned above as the mean,which is:∑=−−=n i i i SD r n y x f 122)2)(exp(211),(σσπi r is defined as the Euclid distance between the site to the th i crime site,and the standard difference of the deviation of the criminal's mental optimal distance is defined as σ,which also reflects the uncertainty of the deviation of the criminal's mental optimal distance,involves the impacts of many factors and can not be measured quantitatively.The discussion of the standard difference is as following:3.3The quantization of the standard differenceThe standard difference is identified according to the following goal,which is,every prediction of the next crime site according to the crime sites where the crimes were committed before should have the highest rate of success.When having to satisfying such optimization objective,it isimpossible to make the direct analysis and exhaustivity.Instead,we have to use the optimized solutions searching algorithm,which is genetic algorithm.\Figure1:The Distribution of the Population of the Last GenerationAccording to the figure,the population of the last generation is mostly concentrated near80, which is used as the standard distance and substituted to the*formula.With the*formula,we are able to predict the probability density of Whether the zones will be the next crime site.Case analysis:5crime site according to the4ones happened before Figure2:The prediction of theth6crime site according to the5ones happened before Figure3:The prediction of theth6crime site according to the5ones happened before Figure4:The prediction of thethAccording to the predictions happened before,the predictions of the outputs based on the models are accurate relatively,and they are able to be the references of the criminal investigations to some extent.However,when is frequency of such crime increases,the predictions of the outputs23crime site according deviated the actual sites more and more,such as the prediction of thethto the22ones happened before,which is:23crime site according to the22ones happened before Figure5:the prediction of thethConclusion according to analysis:It may not be able to predict the next crime site accurately if we use Euclid distance to measure the probability directly.So,we should analyze according to the actual related conditions.For example,we can consider the traffic commutes comprehensively based on the conveniences of the escapes,such as the facilities of the express ways network and the tunnels.According to the hidden security of the commitments,we should consider the population of the area and the distance from the police department.Thus,we should give more weights to the commute convenience,hidden security and less population.In addition,when the commitments increases,the accuracy of the model may decrease,resulted from the fact that when the criminal has more experience,he will choose the next crime sites more randomly.4.Problems and further improvementsWith23crimes in the series the predictions tended to provide large areas that included the target crime but were too large to be useful given the limited resources the police had at their disposal.At this stage,a more detailed look was taken at the directionality and distances between crimes.No significant trends could be found in the sequential distance between crimes so an attempt was made to better quantify the relationship between crimes in terms of directionality.The methodology began by calculating the geographic center of the existing crimes. The geographic center is a derived point that identifies the position at which the distance to each crime is minimized.For applications of the geographic center to crime analysis.Once constructed,the angle of each crime from the north point of the geographic center was calculated.From this it was possible to calculate the change indirection for the sequential crimes.It was found that the offenders were tending to pattern their crimes by switching direction away from the last crime.It appears that the offenders were trying to create a random pattern to avoid detection but unwittingly created a uniform pattern based upon their choice of locations.This relationship was quantified and a simple linear regression used to predict what the next direction would be.The analysis was once again applied to the data.While the area identified was reduced from previous versions and prioritized into sub-segments,the problem remained that the areas predicted were still too large to be used as more than a general guide to resource deployment.A major improvement to the methodology was to include individual targets.By this stage of the series,hotels and auto parts retailers had become the targets of choice.A geo-coded data set became available that allowed hotels and retail outlets to be plotted and compared to the predicted target areas.Ideally those businesses falling within the target areas could be prioritized as more likely targets.However,in some cases the distribution of the likely businesses appeared to contradict the area predicted.For example,few target hotels appeared in the target zone identified by the geographic analysis.In this case,more reliance was placed upon the location of individual targets. From this analysis it was possible to identify a prioritized list of individual commercial targets,which was of more use operationally.Maps were also provided to give an indication of target areas.Figure6demonstrates a map created using this methodology.It is apparent from the above discussion that the target areas identified were often too large to be used as more than a general guide by the Robbery Detail.However,by including the individual targets,it was possible to restrict the possible target areas to smaller,more useful areas,and a few prioritized targets.However,such an approach has the danger of being overly restrictive and it is not the purpose of the analysis to restrict police operations but to suggest priorities.This problem was somewhat dealt with by involving investigators in the analysis and presenting the results in an objective manner,such that investigators could make their own judgments about the results.To be more confident in using this kind of analysis a stronger theoretical background to the methods is required.What has been applied here is to simply exploit the spatial relationships in the information available without considering what the connection is to the actual behaviour of the offenders.For example,what is the reason behind a particular trend observed in the distance between crimes?Why would such a trend be expected between crimes that occur on different days and possibly involve different individuals?While some consideration was given to identifying the reason behind the pattern of directionality and while it seems reasonable to expect offender’s to look for freeway access,such reasoning has tended to follow the analysis rather than substantiate it.Without a theoretical background the analysis rests only on untested statistical relationships that do not provide an answer to the basic question:why this pattern?So next we will apply a quantitative method,which is PSO,based on a theoretical background,to locate the residence of the criminal's residence.5.The prediction of the residenceParticle Swarm Optimization is a evolutionary computation,invented by Dr.Eberhart and Dr.Kennedy.It is a tool of optimization based on iteration,resulted from the research on the behaviors of the bird predation.Initiating a series of random number,the PSO is able to catch the optimization with iteration.Like PSO,the resolution of our residence search problem is the criminal,whose serial crime sites have been abstracted into 23particles without volume and weight and extended to the 2-D space.Like bird,the criminal is presumed to go directly home when he committed a crime.So,there are 23criminals who commit the crimes in the 23sites mention before and then they will go home directly.The criminals are defined as a vector,so are their speed.All criminals have a fittness decided by the optimized functions,and every of them has a according speed which can decide their direction and distance.All the criminals know the best position (pbest,defined as the residence known by the individual),which has been discovered so far,and where they are now.Besides,every criminals also know the best position which has been found by the group (gbest,defined as the residence known by the group).Such search can be regarded as the experience of other criminals.The criminals are able to locate the residence by the experience of itself and the whole criminals.PSO computation initiates the 23criminals and then the offenders will pursue the optimized one to search in the space.In other words,they find the optimized solutions by iteration.Presume that in the 2-D space the location and speed of the ith crime site is relatively ),(2,1,i i i x x X =and ),(2,1,i i i v v V =.In every iteration,the criminals will pursue the two best positions to update themselves.The two best positions are relatively the individual peak (pbest),),(2,1,i i i p p P =,which is found by the criminal himself,and the group optimized solution (gbest),g P ,which has been found to be the optimized solution by the whole group so far.When the criminals found the two optimized solutions,they will update their speed and new position based on the following formulas.2,1),1()()1()]([)]([)()1(,,,,,22,,11,,=++=+−+−+=+j t v t x t x t x p r c t x p r c t wv t V j i j i j i j i j g j i j i j i j i In the above,the w is inertial weighted factor,21c andc are positive learning factors,21r andr are random number which are distributed uniformly between 0and 1.The learning factor can make the criminals have self-conclude ability and ability of learning from others.Here we make both of them be 2,as what they always are in PSO.The inertial weighted factor w decides the extent of the inheritance of the current speed of the crime sites.The appropriate choice can make them have balanced searching and exploring ability.For balancing the global searching ability and the local improving ability of the criminal in the PSO algorithm,here we adopt one of the self-adapted methods,which is Non-linear Dynamic Inertial Weight Coefficient to choose the inertial weight.The expression is as following:⎪⎩⎪⎨⎧=≤−−−−>avg avg avg f f f f f f w w w f f w w ,))*((,minmin min max min max In the above,the max w and min w are defined respectively as the maximum and minimum of w,f means the current functional value of the criminal,and the avg f and min f respectively means the average value and minimum value of all the current criminals.In addition,the inertial weight will change automatically according to the objective value,which gives the name self-adapted method.When the final values,which are estimations of the criminal's residence,become consistent,it will make the inertial weight increase.When they become sparser,it will make the inertial weight decrease.In the meantime,referring to the criminals whose final values are worse than the average value,its according inertial weighted factor will become smaller,which protect the crime site.Oppositely,when referring to the criminals whose final values are better than the average value,its according inertial weighted factor will become bigger,which makes the criminal nearer to the searching zone.So now,with the PSO of Non-linear Dynamic Inertial Weight Coefficient,we can calculate the minimum value of22,)()(j j j i y y x x R −+−=,j=1,2,3 (23)In the above,j ,i R is the residence of the criminal.Thus,we have the output (x,y)as(2.368260870656715,3.031739124610613).We can see the residence in the figure 7.Figure7:The residence in the map6.ConclusionThis paper has presented one case study to illustrate how probability distribution and geographical analysis of serial crime conducted can assist crime investigation. Unfortunately,in the Supersonic armed robbery investigation the areas identified were too large to have been of much use to investigators.Further,because of the number of assumptions applied the method does not inspire enough confidence to dedicate resources to comparing its results to the enormous amount of suspect data collected on the case.While the target areas predicted tended to be large,the mapping of individual commercial targets appears to offer a significant improvement to the method.However,as they stand,these methods lack a theoretical basis that would allow the results to be judged and applied in investigations.Limitations such as these can be offset to some degree by the involvement of investigators in the analysis.In the end,we used a quantitative method to locate the residence of the criminal to make the identified areas smaller.So,due to the advantages and drawbacks of the above methods,we suggest that we should use different methods to help us fight again the crimes comprehensively.。
2011美国大学生数学建模论文
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A DYNAMIC MODEL FOR OPTIMUM DESIGN OF SNOWBOARD HALF-Fra bibliotekIPE SHAPE
Abstract- Firstly the types of skiing about how a skier uses the snowboard are being discussed and we conclude that a skilled skier can perform carved turns to minimize the speed loss while turning. We assume that a skilled snowboarder can always choose the best path in a “half-pipe” game and can turn without energy reduction. Then a dynamic model studying the process of a snowboarder moving from the bottom to the air is set up, aiming at investigating the relationship between “speed loss” and “path choice” in a “half-pipe”. Applying the functional analysis technique we work out a solution to the simplified model. With this solution a new type “half-pipes” that can minimize the resistance effect are designed. When analyzing the maximum twists in the air, we recognize that the kinetics just before the snowboarder taking off is the key. We do the analysis and conclude that a steep end of the course can help to fly high. At last we simply research the “practical” snow course and remark the strengths and weaknesses of our model.
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2011 Mathematical Contest in Modeling (MCM) Summary Sheet
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The Coordination of Repeaters Based on the Frequency Division Multiplexing Theory
Abstract
A repeater is an automated device that extends the range of communications. Interference occurs if repeaters are not far enough apart or transmit on very close frequencies. Our goal is to create an optimal system which has the minimum number of repeaters while achieving high capacity with limited spectrum width.
We develop a theoretical model to give a rough estimation of the necessary repeater numbers considering factors such as the radius of the area, the radius of the repeater’s coverage, the number of the simultaneous users, and the available spectrum width.
Problem I is solved by two steps. One key issue is to cover the whole area seamlessly and to minimize the overlapping area of the adjacent repeaters’ coverage, thus reducing the co-frequency interference. Regular hexagons are chosen to fill the given area seamlessly instead of circles. Another vital issue is to accommodate 1000 users at the same time. We establish frequency division multiplexing model to solve the insufficiency of the spectrum width. By dividing the available spectrum width, we attain channels that only meet 24.6% of the total demand. We improve the user capacity by divide the area into cells and multiplex the spectrum width in cells far apart to meet the demand. We calculate that at least 13 repeaters are needed.
In problem II, as the number of simultaneous users increases from 1,000 to 10,000, while the given spectrum width and the number of PL remain unchanged, we need further expansion of the user capacity. Our model is the combination of the frequency division multiplexing model and the cell division model, and designed to solve the conflict between the rapidly increasing demand and the limited spectrum width. We adopt cell division model to divide the original cells into a proper number of micro cells, and again apply frequency division multiplexing model to the micro cells, thus greatly improving the user capacity. So the result is that 122 repeaters are needed.
In problem III, the changes of environment in mountainous areas are analyzed on its influence in communication. We contract a simulation model utilizing MATLAB software considering factors such as the height and the gradient of the mountain and apply it to 3 kinds of conditions. These situations involving in communication failure caused by mountains are analyzed. We calculate and generate the placement of the repeaters to overcome the difficulties in line-of-sight propagation caused by mountainous areas.
Combining the frequency division multiplexing model and the cell splitting model, our model can be successfully adopted to solve the insufficiency of spectrum resources and meet the rapidly increasing demand for radio communication.。