美国数学建模竞赛论文tex模板
美国大学生数学建模竞赛优秀论文
For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number7018Problem ChosencFor office use onlyF1________________F2________________F3________________F4________________ SummaryThe article is aimed to research the potential impact of the marine garbage debris on marine ecosystem and human beings，and how we can deal with the substantial problems caused by the aggregation of marine wastes.In task one，we give a definition of the potential long-term and short-term impact of marine plastic garbage. Regard the toxin concentration effect caused by marine garbage as long-term impact and to track and monitor it. We etablish the composite indicator model on density of plastic toxin，and the content of toxin absorbed by plastic fragment in the ocean to express the impact of marine garbage on ecosystem. Take Japan sea as example to examine our model.In ask two, we designe an algorithm, using the density value of marine plastic of each year in discrete measure point given by reference，and we plot plastic density of the whole area in varies locations. Based on the changes in marine plastic density in different years， we determine generally that the center of the plastic vortex is East—West140°W—150°W, South—North30°N—40°N. According to our algorithm, we can monitor a sea area reasonably only by regular observation of part of the specified measuring pointIn task three，we classify the plastic into three types，which is surface layer plastic，deep layer plastic and interlayer between the two. Then we analysis the the degradation mechanism of plastic in each layer. Finally，we get the reason why those plastic fragments come to a similar size.In task four， we classify the source of the marine plastic into three types，the land accounting for 80%，fishing gears accounting for 10%，boating accounting for 10%，and estimate the optimization model according to the duel-target principle of emissions reduction and management. Finally, we arrive at a more reasonable optimization strategy.In task five，we first analyze the mechanism of the formation of the Pacific ocean trash vortex， and thus conclude that the marine garbage swirl will also emerge in south Pacific，south Atlantic and the India ocean. According to the Concentration of diffusion theory, we establish the differential prediction model of the future marine garbage density，and predict the density of the garbage in south Atlantic ocean. Then we get the stable density in eight measuring point .In task six, we get the results by the data of the annual national consumption ofpolypropylene plastic packaging and the data fitting method, and predict the environmental benefit generated by the prohibition of polypropylene take-away food packaging in the next decade. By means of this model and our prediction，each nation will reduce releasing 1.31 million tons of plastic garbage in next decade.Finally, we submit a report to expediction leader，summarize our work and make some feasible suggestions to the policy- makers.Task 1:Definition:●Potential short-term effects of the plastic: the hazardeffects will be shown in the short term.●Potential long-term effects of the plastic: thepotential effects, of which hazards are great, willappear after a long time.The short- and long-term effects of the plastic on the ocean environment:In our definition, the short-term and long-term effects of the plastic on the ocean environment are as follows.Short-term effects:1）The plastic is eaten by marine animals or birds.2） Animals are wrapped by plastics, such as fishing nets, which hurt or even kill them.3）Deaden the way of the passing vessels.Long-term effects:1）Enrichment of toxins through the food chain: the waste plastic in the ocean has no natural degradation in theshort-term, which will first be broken down into tinyfragments through the role of light, waves,micro-organisms, while the molecular structure has notchanged. These "plastic sands", easy to be eaten byplankton, fish and other, are Seemingly very similar tomarine life’s food,causing the enrichment and delivery of toxins.2）Accelerate the greenhouse effect: after a long-term accumulation and pollution of plastics, the waterbecame turbid, which will seriously affect the marineplants (such as phytoplankton and algae) inphotosynthesis. A large number of plankton’s deathswould also lower the ability of the ocean to absorbcarbon dioxide, intensifying the greenhouse effect tosome extent.To monitor the impact of plastic rubbish on the marine ecosystem:According to the relevant literature, we know that plastic resin pellets accumulate toxic chemicals , such as PCBs、DDE , and nonylphenols , and may serve as a transport medium and soure of toxins to marine organisms that ingest them[]2. As it is difficult for the plastic garbage in the ocean to complete degradation in the short term, the plastic resin pellets in the water will increase over time and thus absorb more toxins, resulting in the enrichment of toxins and causing serious impact on the marine ecosystem.Therefore, we track the monitoring of the concentration of PCBs, DDE, and nonylphenols containing in the plastic resin pellets in the sea water, as an indicator to compare the extent of pollution in different regions of the sea, thus reflecting the impact of plastic rubbish on ecosystem.To establish pollution index evaluation model: For purposes of comparison, we unify the concentration indexes of PCBs, DDE, and nonylphenols in a comprehensive index.Preparations:1）Data Standardization2）Determination of the index weightBecause Japan has done researches on the contents of PCBs,DDE, and nonylphenols in the plastic resin pellets, we illustrate the survey conducted in Japanese waters by the University of Tokyo between 1997 and 1998.To standardize the concentration indexes of PCBs, DDE,and nonylphenols. We assume Kasai Sesside Park, KeihinCanal, Kugenuma Beach, Shioda Beach in the survey arethe first, second, third, fourth region; PCBs, DDE, andnonylphenols are the first, second, third indicators.Then to establish the standardized model:j j jij ij V V V V V min max min --= （1,2,3,4;1,2,3i j ==）wherej V max is the maximum of the measurement of j indicator in the four regions.j V min is the minimum of the measurement of j indicatorstandardized value of j indicator in i region.According to the literature [2], Japanese observationaldata is shown in Table 1.Table 1. PCBs, DDE, and, nonylphenols Contents in Marine PolypropyleneTable 1 Using the established standardized model to standardize, we have Table 2.In Table 2，the three indicators of Shioda Beach area are all 0, because the contents of PCBs, DDE, and nonylphenols in Polypropylene Plastic Resin Pellets in this area are the least, while 0 only relatively represents the smallest. Similarly, 1 indicates that in some area the value of a indicator is the largest.To determine the index weight of PCBs, DDE, and nonylphenolsWe use Analytic Hierarchy Process (AHP) to determine the weight of the three indicators in the general pollution indicator. AHP is an effective method which transforms semi-qualitative and semi-quantitative problems into quantitative calculation. It uses ideas of analysis and synthesis in decision-making, ideally suited for multi-index comprehensive evaluation.Hierarchy are shown in figure 1.Fig.1 Hierarchy of index factorsThen we determine the weight of each concentrationindicator in the generall pollution indicator, and the process are described as follows:To analyze the role of each concentration indicator, we haveestablished a matrix P to study the relative proportion.⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=111323123211312P P P P P P P Where mn P represents the relative importance of theconcentration indicators m B and n B . Usually we use 1，2，…，9 and their reciprocals to represent different importance. The greater the number is, the more important it is. Similarly, the relative importance of m B and n B is mn P /1（3,2,1,=n m ）.Suppose the maximum eigenvalue of P is m ax λ, then theconsistency index is1max --=n nCI λThe average consistency index is RI , then the consistencyratio isRICI CR = For the matrix P of 3≥n , if 1.0<CR the consistency isthougt to be better, of which eigenvector can be used as the weight vector.We get the comparison matrix accoding to the harmful levelsof PCBs, DDE, and nonylphenols and the requirments ofEPA on the maximum concentration of the three toxins inseawater as follows:⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=165416131431P We get the maximum eigenvalue of P by MATLAB calculation0012.3max =λand the corresponding eigenvector of it is()2393.02975.09243.0，，=W1.0042.012.1047.0<===RI CI CR Therefore,we determine the degree of inconsistency formatrix P within the permissible range. With the eigenvectors of p as weights vector, we get thefinal weight vector by normalization ()1638.02036.06326.0'，，=W . Defining the overall target of pollution for the No i oceanis i Q , among other things the standardized value of threeindicators for the No i ocean is ()321,,i i i i V V V V = and the weightvector is 'W ,Then we form the model for the overall target of marine pollution assessment, （3,2,1=i ）By the model above, we obtained the Value of the totalpollution index for four regions in Japanese ocean in Table 3T B W Q '=In Table3, the value of the total pollution index is the hightest that means the concentration of toxins in Polypropylene Plastic Resin Pellets is the hightest, whereas the value of the total pollution index in Shioda Beach is the lowest(we point up 0 is only a relative value that’s not in the name of free of plastics pollution)Getting through the assessment method above, we can monitor the concentration of PCBs, DDE and nonylphenols in the plastic debris for the sake of reflecting the influence to ocean ecosystem.The highter the the concentration of toxins,the bigger influence of the marine organism which lead to the inrichment of food chain is more and more dramatic.Above all, the variation of toxins’ concentration simultaneously reflects the distribution and time-varying of marine litter. We can predict the future development of marine litter by regularly monitoring the content of these substances, to provide data for the sea expedition of the detection of marine litter and reference for government departments to make the policies for ocean governance.Task 2:In the North Pacific, the clockwise flow formed a never-ending maelstrom which rotates the plastic garbage. Over the years, the subtropical eddy current in North Pacific gathered together the garbage from the coast or the fleet, entrapped them in the whirlpool, and brought them to the center under the action of the centripetal force, forming an area of 3.43 million square kilometers (more than one-third of Europe) .As time goes by, the garbage in the whirlpool has the trend of increasing year by year in terms of breadth, density, and distribution. In order to clearly describe the variability of the increases over time and space, according to “Count Densities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999—2008”, we analyze the data, exclude them with a great dispersion, and retain them with concentrated distribution, while the longitude values of the garbage locations in sampled regions of years serve as the x-coordinate value of a three-dimensional coordinates, latitude values as the y-coordinate value, the Plastic Count per cubic Meter of water of the position as the z-coordinate value. Further, we establish an irregular grid in the yx plane according to obtained data, and draw a grid line through all the data points. Using the inverse distance squared method with a factor, which can not only estimate the Plastic Count per cubic Meter of water of any position, but also calculate the trends of the Plastic Counts per cubic Meter of water between two original data points, we can obtain the unknown grid points approximately. When the data of all the irregular grid points are known (or approximately known, or obtained from the original data), we can draw the three-dimensional image with the Matlab software, which can fully reflect the variability of the increases in the garbage density over time and space.Preparations:First, to determine the coordinates of each year’s sampled garbage.The distribution range of garbage is about the East - West 120W-170W, South - North 18N-41N shown in the “Count Densities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”, we divide a square in the picture into 100 grids in Figure (1) as follows:According to the position of the grid where the measuring point’s center is, we can identify the latitude and longitude for each point, which respectively serve as the x- and y- coordinate value of the three-dimensional coordinates.To determine the Plastic Count per cubic Meter of water. As the “Plastic Count per cubic Meter of water” provided by “Count Densities of P lastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”are 5 density interval, to identify the exact values of the garbage density of one year’s different measuring points, we assume that the density is a random variable which obeys uniform distribution in each interval.Uniform distribution can be described as below:()⎪⎩⎪⎨⎧-=01a b x f ()others b a x ,∈We use the uniform function in Matlab to generatecontinuous uniformly distributed random numbers in each interval, which approximately serve as the exact values of the garbage density andz-coordinate values of the three-dimensional coordinates of the year’s measuring points.Assumptions(1)The data we get is accurate and reasonable.(2)Plastic Count per cubic Meter of waterIn the oceanarea isa continuous change.(3)Density of the plastic in the gyre is a variable by region.Density of the plastic in the gyre and its surrounding area is interdependent , However, this dependence decreases with increasing distance . For our discussion issue, Each data point influences the point of each unknown around and the point of each unknown around is influenced by a given data point. The nearer a given data point from the unknown point, the larger the role.Establishing the modelFor the method described by the previous,we serve the distributions of garbage density in the “Count Pensities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”as coordinates ()z y,, As Table 1:x,Through analysis and comparison, We excluded a number of data which has very large dispersion and retained the data that is under the more concentrated the distribution which, can be seen on Table 2.In this way, this is conducive for us to get more accurate density distribution map.Then we have a segmentation that is according to the arrangement of the composition of X direction and Y direction from small to large by using x co-ordinate value and y co-ordinate value of known data points n, in order to form a non-equidistant Segmentation which has n nodes. For the Segmentation we get above,we only know the density of the plastic known n nodes, therefore, we must find other density of the plastic garbage of n nodes.We only do the sampling survey of garbage density of the north pacificvortex,so only understand logically each known data point has a certain extent effect on the unknown node and the close-known points of density of the plastic garbage has high-impact than distant known point.In this respect,we use the weighted average format, that means using the adverse which with distance squared to express more important effects in close known points. There're two known points Q1 and Q2 in a line ,that is to say we have already known the plastic litter density in Q1 and Q2, then speculate the plastic litter density's affects between Q1、Q2 and the point G which in the connection of Q1 and Q2. It can be shown by a weighted average algorithm22212221111121GQ GQ GQ Z GQ Z Z Q Q G +*+*=in this formula GQ expresses the distance between the pointG and Q.We know that only use a weighted average close to the unknown point can not reflect the trend of the known points, we assume that any two given point of plastic garbage between the changes in the density of plastic impact the plastic garbage density of the unknown point and reflecting the density of plastic garbage changes in linear trend. So in the weighted average formula what is in order to presume an unknown point of plastic garbage density, we introduce the trend items. And because the greater impact at close range point, and thus the density of plastic wastes trends close points stronger. For the one-dimensional case, the calculation formula G Z in the previous example modify in the following format:2212122212212122211111112121Q Q GQ GQ GQ Q Q GQ Z GQ Z GQ Z Z Q Q Q Q G ++++*+*+*=Among them, 21Q Q known as the separation distance of the known point, 21Q Q Z is the density of plastic garbage which is the plastic waste density of 1Q and 2Q for the linear trend of point G . For the two-dimensional area, point G is not on the line 21Q Q , so we make a vertical from the point G and cross the line connect the point 1Q and 2Q , and get point P , the impact of point P to 1Q and 2Q just like one-dimensional, and the one-dimensional closer of G to P , the distant of G to P become farther, the smaller of the impact, so the weighting factor should also reflect the GP in inversely proportional to a certain way, then we adopt following format:221212222122121222211111112121Q Q GQ GP GQ GQ Q Q GQ GP Z GQ Z GQ Z Z P Q Q Q Q G ++++++*+*+*=Taken together, we speculated following roles:(1) Each known point data are influence the density of plastic garbage of each unknown point in the inversely proportional to the square of the distance;(2) the change of density of plastic garbage between any two known points data, for each unknown point are affected, and the influence to each particular point of their plastic garbage diffuse the straight line along the two known particular point; (3) the change of the density of plastic garbage between any two known data points impact a specific unknown points of the density of plastic litter depends on the three distances: a. the vertical distance to a straight line which is a specific point link to a known point;b. the distance between the latest known point to a specific unknown point;c. the separation distance between two known data points.If we mark 1Q ，2Q ，…，N Q as the location of known data points,G as an unknown node, ijG P is the intersection of the connection of i Q ，j Q and the vertical line from G to i Q ，j Q()G Q Q Z j i ,,is the density trend of i Q ，j Q in the of plasticgarbage points and prescribe ()G Q Q Z j i ,,is the testing point i Q ’ s density of plastic garbage ,so there are calculation formula:()()∑∑∑∑==-==++++*=Ni N ij ji i ijGji i ijG N i Nj j i G Q Q GQ GPQ Q GQ GP G Q Q Z Z 11222222111,,Here we plug each year’s observational data in schedule 1 into our model, and draw the three-dimensional images of the spatial distribution of the marine garbage ’s density with Matlab in Figure (2) as follows:199920002002200520062007－2008(1)It’s observed and analyzed that, from 1999 to 2008, the density of plastic garbage is increasing year by year and significantly in the region of East – West 140W-150W, south - north 30N-40N. Therefore, we can make sure that this region is probably the center of the marine litter whirlpool. Gathering process should be such that the dispersed garbage floating in the ocean move with the ocean currents and gradually close to the whirlpool region. At the beginning, the area close to the vortex will have obviously increasable about plastic litter density, because of this centripetal they keeping move to the center of the vortex ,then with the time accumulates ,the garbage density in the center of the vortex become much bigger and bigger , at last it becomes the Pacific rubbish island we have seen today.It can be seen that through our algorithm, as long as the reference to be able to detect the density in an area which has a number of discrete measuring points,Through tracking these density changes ,we Will be able to value out all the waters of the density measurement through our models to determine,This will reduce the workload of the marine expedition team monitoring marine pollution significantly, and also saving costs .Task 3:The degradation mechanism of marine plasticsWe know that light, mechanical force, heat, oxygen, water, microbes, chemicals, etc. can result in the degradation of plastics . In mechanism ,Factors result in the degradation can be summarized as optical ，biological，and chemical。
美赛数学建模比赛论文模板
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.●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 of the 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, the85V of some researchers for the typical countries from Table 6.1[ 21]:Author Country ModelOttesen and Krammes2000 AmericaLC DC L DC V C ⨯---=01.0012.057.144.10285Andueza2000Venezuela ].[308.9486.7)/894()/2795(25.9885curve horizontal L DC Ra R V T++--=].[tan 819.27)/3032(69.10085gent L R V T +-= Jessen2001America][00239.0614.0279.080.86185LSD ADT G V V P --+=][00212.0432.010.7285NLSD ADT V V P -+=Donnell2001 America22)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 DCV 124.0164.6685-= DCE V 4.046.3366.5585--=2855.035.1119.0745.65DC E DC V ---=FitzpatrickAmericaKV 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 trafficsimulations 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。
数学建模美赛写作各部分模板
第一段：写论文解决什么问题1．问题的重述a. 介绍重点词开头：例1：“Hand move” irrigation, a cheap but labor-intensive small farms, a movable pipe with sprinkler on top that can be attached to a stationary main.例2:……is a real-life common phenomenon with many complexities.例3：An (effective plan) is crucial to………b. 直接指出问题：例1：number of tollbooths in a highway toll-plaza for a given number of highway lanes: the number of tollbooths that minimizes average delay experienced by cars.例2：A brand-new university needs to balance the cost of information technology security measures with the potential cost of attacks on its systems.例3：We determine the number of sprinklers to use by analyzing the energy and motion of water in the pipe the engineering parameters of sprinklers available in the market.数学建模美赛论文例4: After mathematically analyzing the …… problem, our modeling group would like to present our conclusions, strategies, (and recommendations )to the …….例5：Our goal is... that (minimizes the time )……….2．解决这个问题的伟大意义反面说明。
2012美国数学建模大赛二等奖论文及格式——英文版
Dedicated Pipeline for Trip ArrangementSummaryIn the problem of camping, we should set reasonable schedule which can not only increase the utilization of campsites but also meet people's needs. Meanwhile, the carrying capacity of the river is also required. To solve the problem, this thesis will build optimization model with maximum campsite's utilization and river trips as the model's target function.The specific steps are as follows:step1: Determine the number of campsites Y. We use Computer Emulation Simulation to solve this problem by making full use of the given conditions that trips will spend6 to 18 nights on the river and the river is 225 miles long. We get 29 sets of data through programming, then curve fitting them by SPSS software. By comparing the value of sig. and adjusting R square and so on, the ideal number of the campsites is got .Step2: By using the number of the campsites 39 as well as the goal programming equation built in the first step, we get the number of river trips that are allowed to enter, namely the carrying capacity of the river.Step3: By using the campsites 39, we adjust the campsites of different camping program and then divide them into 4 kinds through clustering analysis using SPSS. Then we select representatives in various types of camping programs according to repetition rate and the average transfer rate. So we streamline the camping programs into the problem of goal programming for 6, 8, 11, 12, 16 nights.Step4: In those five camping programs, 39 campsites which will not repeat are distributed in 3 dedicated pipelines . The first line accounts for 12 campsites and can only be available for 6 or 12 nights trip. Each day, a couple of 6 nights trips are distributed, and the starting trip camps the campsites in turn according to the even number of the pipeline while the secondary trip camps in turn according to the odd number. The second pipeline accounting for 16 campsites is arranged just as the first one .Under the premise of guaranteeing the variety of camping project, trips start as a pipeline to make the total number of trips camping in this line the biggest and the utilization of the campsites maximum. There are 11 campsites in the third pipeline which are available for 6 to 11 nights trip.According to the above analysis, the carrying capacity of Dedicated Pipeline, namely C_line, is less than that of the river, namely C_river, within 180 days. the park managers need to grasp passenger flow(P) of the river in the following period(T) and calculate P/( C_line/T)The best distribution program: the best utirlization of campsites is P/( C_line/T) in one period.According to the best utilization of campsites, The best distribution program can be got.Key words: Cluster Analysis, Bus Rapid, Transit Pipeline System, Curve Fitting , Computer Emulation SimulationContentsI. Introduction (3)1.1Restatement of the problem (3)1.2 Theory knowledge introduction (3)II. Definitions and Key terms (4)1，The conditions given (4)2，Symbol definition (4)III. General Assumptions (4)IV Model Design (5)4.1Model Establishment (5)4.2 Model Solution (6)4.2.1.To determine Y (6)4.2.1 To Determine the Camping program (11)4.2.3 To find capacity of the river (15)4.2.3 Determine Dedicated Pipeline (15)4.3 Strength and Weakness............................................................. 错误！未定义书签。
美国数学建模论文格式
Journal Citation(to be inserted by the publisher)Copyright by Trans Tech PublicationsYour Paper's Title Starts Here:Please Centeruse Arial14First Author1,Second Author2and Others3(use Arial14)1Full address of first author,including country,email(use Arial11)2Full address of second author,including country,email3List all distinct addresses in the same wayKeywords::List the keywords covered in your paper.These keywords will also be used by the Keywordspublisher to produce a keyword index.(use Arial11)For the rest of the paper,please use Times New Roman12Abstract.This document explains and demonstrates how to prepare your camera-ready manuscript for Trans Tech Publications.The best is to read these instructions and follow the outline of this text. The text area for your manuscript must be17cm wide and25cm high(6.7and9.8inches,resp.). Do not place any text outside this e good quality,white paper of approximately21x29cm or8x11inches.Your manuscript will be reduced by approximately20%by the publisher.Please keep this in mind when designing your figures and tables etc.IntroductionAll manuscripts must be in English.Please keep a second copy of your manuscript in your office (just in case anything gets lost in the mail).When receiving the manuscript,we assume that the corresponding authors grant us the copyright to use the manuscript for the book or journal in question.Should authors use tables or figures from other Publications,they must ask the corresponding publishers to grant them the right to publish this material in their paper.Use italic for emphasizing a word or phrase.Do not use boldface typing or capital letters except for section headings(cf.remarks on section headings,below).Use a laser printer,not a matrix dot printer.Organization of the TextSection Headings.The section headings are in boldface capital and lowercase letters.Second level headings are typed as part of the succeeding paragraph(like the subsection heading of this paragraph).Page Numbers.Do not print page numbers:Please number each sheet slightly in the left corner near the bottom(outside the typing area)with a light blue pencil.Footnotes.Footnotes1should be single spaced and separated from the text.Ideally,footnotes appear on the page of their reference,and are placed at the foot of the text,separated from the text by a horizontal line.Tables.Tables(refer with:Table1,Table2,...)should be presented as part of the text,but in such a way as to avoid confusion with the text.A descriptive title should be placed above each table. The caption should be self-contained and placed below or beside the table.Units in tables should be given in square brackets[meV].If square brackets are not available,use curly{meV}or standard brackets(meV).1This is a footnote2Title of Publication(to be inserted by the publisher)Figures.Figures(refer with:Fig.1,Fig.2,...)also should be presented as part of the text, leaving enough space so that the capt-ion will not be confused with the text.The caption should be self-contained and placed below or beside the figure.Generally,only original drawings or photographic reproductions are acceptable.Only very good photocopies are acceptable.Utmost care must be taken to insert the figures in correct alignment with the text.Half-tone pictures should be in the form of glossy prints.If possible,please include your figures as graphic images in the electronic version.If TTP is required to scan and insert images,please keep the following points in mind:(a)the allotted space(for inserting illustrations)must exactly match the space made available inthe camera-ready version,so that the electronic version is identical to the hard copy with regard to page and line breaks.(b)the required positioning of any high-quality separate illustration must be clearly indicated onits reverse side.The size of the illustrations must exactly match the space left in the camera-ready manuscript.Equations.Equations(refer with:Eq.1,Eq.2,...)should be indented5mm(0.2").There should be one line of space above the equation and one line of space below it before the text continues.The equations have to be numbered sequentially,and the number put in parentheses at the right-hand edge of the text.Equations should be punctuated as if they were an ordinary part of the text.Punctuation appears after the equation but before the equation number,e.g.c2=a2+b2.(1)Literature ReferencesReferences are cited in the text just by square brackets[1].(If square brackets are not available, slashes may be used instead,e.g./2/.)Two or more references at a time may be put in one set of brackets[3,4].The references are to be numbered in the order in which they are cited in the text and are to be listed at the end of the contribution under a heading References,see our example below.SummaryOn your floppy disk,please indicate the format and word processor used.Please also provide your phone number,fax number and e_mail address for rapid communication with the publisher(will not be published).Please always send your disk along with a hard copy that must match the disk's content exactly.If you follow the foregoing,your paper will conform to the requirements of the publisher and facilitate a problem-free publication process.References[1]Dj.M.Maric,P.F.Meier and S.K.Estreicher:Mater.Sci.Forum Vol.83-87(1992),p.119[2]M.A.Green:High Efficiency Silicon Solar Cells(Trans Tech Publications,Switzerland1987).This document is available on the web at /download **Please submit your paper in hardcopy and also electronically to the conference editor.。
美赛论文LaTeX模板
\documentclass{icmmcm}\usepackage{url} % For formatting URLs and other web or% file references.\usepackage{mflogo} % Provides the METAFONT logo; you% won't need it for your report.\usepackage{graphicx} % For importing graphics.\usepackage{natbib}%%% Sample ICM/MCM Contest Submission%%%%%% Based on sample senior thesis document%%% Last modified by Jeremy Rouse%%% Summer 2000%%%%%% and on the LaTeX Hints document%%% created by C.M. Connelly <cmc@>%%% Copyright 2002-2012%%% ---------------%%% Local Command and Environment Definitions%%% If you have any local command or environment definitions, put them %%% here or in a separate style file that you load with \usepackage.% \newtheorem declarations\newtheorem{Theo1}{Theorem}\newtheorem{Theo2}{Theorem}[section]\newtheorem{Lemma}[Theo2]{Lemma}% Each of the above defines a new theorem environment.% Multiple theorems can be done in the same environment.% Theo2's number is defined by the subsection it's in.% Theo3 uses the same numbering counter and numbering system as% Theo2 (that's the meaning of [Theo2]).%%% You probably won't want any of the following commands, which are %%% here to allow various the names of commands, make examples typeset %%% properly, and so on. You can, of course, use them as examples for %%% your own user-defined commands.\newcommand{\bslash}{\symbol{'134}}%backslash\newcommand{\bsl}{{\texttt{\bslash}}}\newcommand{\com}[1]{\bsl\texttt{#1}\xspace}\newcommand{\file}[1]{\texttt{#1}\xspace}\newcommand{\pdftex}{PDF\tex}\newcommand{\pdflatex}{PDF\latex}\newcommand{\acronym}[1]{\textsc{#1}\xspace}\newcommand{\key}[1]{\textsf{\emph{#1}}\xspace}\newcommand{\class}[1]{\textsf{#1}\xspace}\newcommand{\package}[1]{\textsf{#1}\xspace}\newcommand{\env}[1]{\texttt{#1}\xspace}\newcommand{\prog}[1]{\texttt{#1}\xspace}\newcommand{\command}[1]{\texttt{\bsl{}#1}\xspace}\newcommand{\ctt}{\texttt{comp.text.tex}\xspace}\newcommand{\tex}{\TeX\xspace}\newcommand{\latex}{\LaTeX\xspace}%%% Note that the \xspace command comes from the xspace package. It %%% allows you type a command that inserts text without having to %%% worry about how you ``end'' that command.%%%%%% Without \xspace, you would need to end a command with a backslash %%% followed by a space or with an empty set of braces if you followed %%% the command with a space. For example,%%%%%% \foo is a very important algorithm.%%%%%% might produce%%%%%% The foobarbaz algorithmis a very important algorithm.%%%%%% whereas with the \xspace command, the same code would produce %%%%%% The foobarbaz algorithm is a very important algorithm.%%%%%% If you need to butt a command that produces text against a letter %%% of some sort -- say, to pluralize it -- you need to tell TeX%%% where your command name ends so that it expands the correct %%% macro. So you might do%%%%%% \bar{}s are very busy creatures.%%% TeX has an amazingly good hyphenation algorithm, but sometimes it %%% gets confused and needs some help.%%%%%% For words that only occur once or twice, you can insert hints%%% directly into your text, as in%%%%%% our data\-base system is one of the most complex ever devised %%%%%% For words that you use a lot, and that seem to keep ending up at %%% the end of a line, however, inserting the hints each time gets to %%% be a drag. You can use the \hyphenation command to globally tell %%% TeX where to hyphenate words it can't figure out on its own.\hyphenation{white-space}%%% End Local Command and Environment Definitions%%% ---------------%%% ---------------%%% Title Block\title{\latex Hints for ICM/MCM Contest Reports}%%% Which contest are you taking part in? (Just one!)\contest{ICM/MCM}%%% The question you answered. (Again, just the one.)\question{Report Sample}%%% Your Contest Team Control Number\team{21247}%%% A normal document would specify the author's name (and possibly %%% their affiliation or other information) in an \author command. %%% Because the ICM/MCM Contest rules specify that the names of the %%% team members, their advisor, and their institution should not%%% appear anywhere in the report, do *not* define an \author command.%%% Defining the \date command is optional. If you leave it blank, %%% your document will include the date that the file is typeset, in %%% the form ``Month dd, yyyy''.% \date{}%%% End Title Block%%% ---------------\begin{document}%%% ---------------%%% Summary\begin{summary}This document is meant to give you a quick introduction to \TeX\ and \LaTeX. It covers a lot of material, but still barely manages to scratch the surface. It should provide you with some inspiration and, I hope, with some useful code you can copy, modify, and use in your report.You should use the \file{blank-template.tex} file as a basis foryour report rather than this file. Be sure to change its name to something sensible (maybe your team control number), and to set the values of the \com{title}, \com{question}, and \com{team} commands to appropriate values.Good luck!\hfill{}-- Claire\end{summary}%%% End Summary%%% ---------------%%% ---------------%%% Print Title Block, Contents, et al.\maketitle\tableofcontents%%% Uncomment the following lines if you have figures or tables in %%% your report:\listoffigures\listoftables%%% End Print Title Block, Contents, et al.%%% ---------------\section{Introduction: What Is \latex?}%\label{sec:introduction}\latex is a tool that allows you to concentrate on your writing while taking advantage of the \tex typesetting system to producehigh-quality typeset documents.\latex's benefits include\begin{enumerate}\item Standardized document classes\item Structural frameworks for organizing documents\item Automatic numbering and cross-referencing of structural elements \item ``Floating'' figures and tables\item High-level programming interface for accessing \tex'stypesetting capabilities\item Access to \latex extensions through loading ``packages''\end{enumerate}\section{Structured Writing}%\label{sec:structured-writing}Like HTML,\footnote{HyperText Markup Language} \latex is a markup language rather than a \acronym{Wysiwyg}{}\footnote{What You See Is What You Get.} system. You write plain text files that use special \key{commands} and \key{environments} that govern the appearance and function of parts of your text in your final typeset document.\subsection{Document Classes}%\label{sec:document-classes}The general appearance of your document is determined by your choice of \key{document class}. Document classes also load \latex packagesto provide additional functionality.\latex provides a number of basic classes, including \class{article},\class{letter}, \class{report}, and \class{book}. There are also alarge number of other document classes available, including\class{amsart} and \class{amsbook}, created by the American Mathematical Society and providing some additional mathematically useful structures and commands; \class{foils}, \class{prosper}, and\class{seminar}, which allow you to create ``slides'' for presentations; the math department's \class{thesis} class, forformatting senior theses; and many journal- or company-specific classes that format your document to match the ``house style'' of a particular periodical or publisher.\subsection{Packages}%\label{sec:packages}%\label{sec:ctan}\latex packages, or \key{style files}, define additional commands and environments, or change the way that previously defined commands and environments work. By loading packages, you can change the fonts used in your document, write your document in a non-English language with a non-\acronym{Ascii} font encoding, include graphics, format program listings, add custom headers and footers to your document, and much more.A typical \tex installation includes hundreds of style files, andhundreds more are available from the Comprehensive \tex Archive Network (CTAN), at \url{/}.\subsection{Structural Commands}%\label{sec:structural-commands}\begin{table}\centering\begin{tabular}{ll}\topruleCommand & Notes \\ \midrule\com{part} & \class{book} \& \class{report} only \\\com{chapter} &\class{book} \& \class{report} only \\\com{section} \\\com{subsection} \\\com{subsubsection} \\\com{paragraph} \\\com{subparagraph} \\\bottomrule\end{tabular}\caption[Structural commands in \latex]{Structural commands in \latex.}% \label{tab:structural-commands}\latex provides a set of structural commands for defining sections ofyour document, as shown in Table~\ref{tab:structural-commands}.Note that the argument to structural commands are moving arguments (see Section~\ref{sec:fragile-commands}) because they can be reused in the table of contents or in page headers or footers. Structural commands can take an optional argument in which you specify nonfragile commands or a shorter version of the actual section title that fits.You'll generally know when you need to provide an optional argument by \TeX's behavior.\subsection{Labels and References}%\label{sec:labels-and-references}Sections are numbered automatically by \latex during typesetting. Ifyou change your mind and decide that a subsection should be promotedto a section, or moved to the end of your document, the sections willbe renumbered so that the numbers are consistent.Sections can also be \command{label}{}ed with a tag such as\begin{quote}\begin{verbatim}\section{Our Complicated Equations}%\label{sec:complicated-eqs}\end{verbatim}\end{quote}and referred to with a \command{ref} or \command{pageref} command, as in\begin{quote}\begin{verbatim}In Section~\ref{sec:complicated-eqs}, we pointed out...\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}On page~\pageref{fig:gordian-knot}, we illustrated...\end{verbatim}\end{quote}\latex substitutes the correct section number when typesetting yourThe same commands can be used with numbered environments such as\env{equation}, \env{theorem}, and so forth.Use \emph{meaningful} labels---labeling a section as \texttt{sec12}may seem useful, but it will be confusing if you end up moving it to a different place in the document and its number changes to Section~34.It's also easier to remember what reference you want if you use a meaningful name.You may also want to impose some additional organization through the use of \emph{namespaces}, as I've done in this document. Rather than give different types of objects undistinguished labels, I precedesection labels with \texttt{sec:}, equations with \texttt{eq:},figures with \texttt{fig:}, tables with \texttt{tab:}, and so on.Emacs with Aux\tex and Ref\tex gives you easy access to these labels,as do many other editors with \tex-specific features. It's mucheasier to find the particular label you're looking for if you havesome additional information to help you. Adding the prefixes also reminds you of what text should precede the \com{ref} command.\subsection{Commands}\latex uses commands for changes that are very limited in scope (a few words) or are unlimited in scope (the rest of a document). For example, the commands\begin{quote}\begin{verbatim}\textbf{bold}\emph{italic (emphasized)}\textsf{sans serif}\end{verbatim}\end{quote}produce the following output in a typeset document:\begin{quote}\textbf{bold} \emph{italic (emphasized)} \textsf{sans serif}\end{quote}These are ``commands with arguments''---the command itself starts witha backslash (\bsl), and its \key{argument} appears inside braces{\verb+{ }+). Some commands may also have \key{optional arguments},which are typed inside brackets (\verb+[ ]+).There are also commands that take no arguments, such as\command{noindent}, \command{raggedright}, and \command{pagebreak}.You can define your own commands, as discussed inSection~\ref{sec:customization}.\subsection{Environments}%\label{sec:environments}\latex provides a number of \key{environments} that affect the appearance of text, and are generally used for more structurally significant purposes. For example, the commands listed above are typeset inside a \env{verbatim} environment typed inside a \env{quote} environment. Their results were typeset inside a \env{quote} environment.Environments use special commands to start and close---\command{begin} and \command{end}, followed by the name of the environment in braces, as in\begin{quote}\begin{verbatim}\begin{quote}``This is disgusting---I can't eat this. That arugala is sobitter\ldots{} It's like my algebra teacher on bread.''\flushright -- Julia Roberts in \emph{Full Frontal}\end{quote}\end{verbatim}\end{quote}producing\begin{quote}``This is disgusting---I can't eat this. That arugala is sobitter\ldots{} It's like my algebra teacher on bread.''\flushright -- Julia Roberts in \emph{Full Frontal}\end{quote}Some environments may take additional arguments in braces (required)or brackets (optional).Note that the order in which environments nest is extremely important.If you type an environment inside another environment, the inner environment must be \command{end}{}ed \emph{before} the secondenvironment is closed. It's also vitally important that you have an\command{end} line for each \command{begin} line, or \latex will complain.\subsubsection{The \env{document} Environment and the Preamble}%\label{sec:document-environment}The most important environment is the \env{document} environment, which encloses the \key{body} of your document. The code before the \command{begin}\verb+{document}+ line is called the \key{preamble}, and includes the all-powerful \command{documentclass} command, which loads a particular document class (seeSection~\ref{sec:document-classes}); optional \command{usepackage} commands, which load in additional \latex packages (seeSection~\ref{sec:packages}); and other setup commands, such asuser-defined commands and environments, counter settings, and so forth.I generally also include the commands defining the title, author, anddate in my preambles, but other people include them just after\command{begin}\verb+{document}+, before the \command{maketitle} command, which creates the title block of your document.\subsubsection{Math Environments}%\label{sec:math-environments}One of the major hallmarks of \tex is its ability to typesetmathematical equations.The two primary ways of doing so are with the use of \key{inline} and\key{display math environments}. These environments are used so often that there are shorthands provided for typing them. Inline math environments, such as $a^2 + b^2 = c^2$, can be typed as\begin{quote}\begin{verbatim}\begin{math}a^{2} + b^{2} = c^{2}\end{math}\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}$a^{2} + b^{2} = c^{2}$.\end{verbatim}\end{quote}Display math environments set your equation apart from your running text. They're generally used for more complicated expressions, such as\[f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\]which can be typed as\begin{quote}\begin{verbatim}\begin{displaymath}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{displaymath}\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}\[f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\]\end{verbatim}\end{quote}Generally, you'll want to use the \verb+$+ %$ <- fool font-lock-modedelimited form for inline math, and the \com{[} \com{]} form for display math environments. [Besides being easy to type, these forms are \key{robust}, which means that they can be used in \key{moving arguments}, elements that \tex may need to typeset in more than one place (such as a table of contents) or adjust (such as footnotes).]\paragraph{The \env{equation} Environment}%\label{sec:equation-environment}You'll probably want to use the \env{equation} environment for any formula you plan to refer to. \latex not only typesets the contentsof an \env{equation} environment in display mode, it also numbers it, as in\begin{equation}\label{eq:myequation}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{equation}written as\begin{quote}\begin{verbatim}\begin{equation}\label{eq:myequation}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{equation}\end{verbatim}\end{quote}Note that you can refer to this formula asEquation~\ref{eq:myequation} with\begin{verbatim}\ref{eq:myequation}.\end{verbatim}\subsection{Fonts}%\label{sec:fonts}Generally you'll want to let \latex handle the fonts for you---Knuth's Computer Modern fonts are used by default, and include a wide range of variations that can cover most any use you can think of.If you want to get fancy (and portable; seeSection~\ref{sec:fuzzy-fonts}), you can use Type~1 PostScript fonts, such as Times, Palatino, Utopia, and so forth. These font sets are accessible with packages with names like \package{times},\package{palatino}, and \package{utopia}. There are others, aswell---a command such as \com{locate psnfss | grep sty} will find mostof them.You can also get fonts from CTAN (see Section~\ref{sec:ctan}), both bitmap and Type 1. There's even support for TrueType fonts in some\TeX\ systems.\subsubsection{Font Commands}%\label{sec:font-commands}Most of your concern about fonts is probably related to what you're writing. You might want some \emph{emphasized} or \textbf{bold} textto stress a point or highlight a key term. Filenames might be set in\texttt{typewriter text} (although you should consider using the\package{url} package to help you out---by default, text set intypewriter text isn't hyphenated, which can lead to some unattractiveline breaks).You can also set text in \textsf{sans serif} or \textsc{small caps}.Table~\ref{tab:font-commands} shows you some of the most commonly used font commands provided by \latex.\begin{table}[htbp]\centering\begin{tabular}{ll}\topruleCommand & Result\\\midrule\com{emph} & \emph{emphasized text}\\\com{textsf} & \textsf{sans-serif text}\\\com{texttt} & \texttt{typewriter text}\\\com{textbf} & \textbf{bold text}\\\com{textsc} & \textsc{small caps text}\\\com{textsl} & \textsl{slanted text}\\\com{textit} & \textit{italic text}\\\bottomrule\end{tabular}\caption[Commonly used font commands]{Commonly used font commands.} \label{tab:font-commands}\end{table}I recommend that you use \com{emph} in preference to \com{textit}, anduse \com{textbf} sparingly. \com{emph} is a smarter command than\com{textit}---it switches back to the roman font when necessary. For example, \emph{She loved \emph{Scooby Doo}.} versus \textit{He loved\textit{Titanic}.}For complicated font changes, or for special font usages that you'retyping a lot, creating a macro (Section~\ref{sec:customization}) isthe way to go. I often just write, tossing in custom commands as Igo, and waiting to define them until just before I compile thedocument.\subsection{Customization}%\label{sec:customization}The main advantage of using commands and environments is that they allow you to organize your writing. A useful side-effect is that youcan change your mind about the way an element is typeset, and change all the appearances of that element in document by editing one pieceof code. For example, in this document the names of environments have been set in ``typewriter text'', using a command I created called\command{env}, which is defined as\begin{quote}\begin{verbatim}\newcommand{\env}[1]{\texttt{#1}\xspace}\end{verbatim}\end{quote}All I have to do to make the names of all the environments in the document appear in sans-serif type instead is to change that one lineto\begin{quote}\begin{verbatim}\newcommand{\env}[1]{\textsf{#1}\xspace}\end{verbatim}\end{quote}You can do the same with almost anything you can conceptualize---key terms, people's names (especially names of people fromnon-English-speaking countries), files, functions, and so on.\section{Mathematical Notation}%\label{sec:mathematical-notation}As we saw in Section~\ref{sec:math-environments}, math is typed into one of several kinds of math environments. Choose your environment based on the context and importance of the content. Any formula you plan to refer to should be typed in an \env{equation} environment (ora similar environment that supports labels).You should punctuate your mathematics as if the formulae were normal parts of English sentences. Reading them aloud is often a useful method for ensuring that you have all the commas in the right places. Where appropriate, you should also follow a displayed formula at the end of a sentence with a period.\subsection{Sums and Products}%\label{sec:sums-n-products}It's easy to typeset sums and products. For example,\begin{equation}f(n) = \sqrt[n]{\sum_{k=1}^{n} {n \choose k} f \left( n - k \right)},~\prod_{n=2}^{\infty} \frac{n^{3}-1}{n^{3}+1} = \frac{2}{3}.\end{equation}%%% The ~ in the equation puts a nonbreaking space (equivalent to an%%% interword space in text mode) between the two halves of the equation. %%%%%% Also, note that the use of the \choose command here causes the%%% amsmath package to issue the warning%%%%%% Package amsmath Warning: Foreign command \atopwithdelims; %%% (amsmath) \frac or \genfrac should be used instead %%% (amsmath) on input line 557.%%%%%% amsmath would prefer the use of the \binom command it supplies.\subsection{Matrices}%\label{sec:matrices}It's a little more difficult to create matrices, but not too bad:%%% In LaTeX, & is the alignment tab, and separates columns. \\ is the end of %%% line marker, and separates rows. The ccc denotes that there are three %%% columns. The array environment and the tabular environment are %%% more or less identical, so what goes here also applies to a table.%%%\begin{equation}\left[ \begin{array}{ccc}2 & 1 & 2\\1 & 0 & 2\\2 & 1 & 1\end{array} \right]\left[ \begin{array}{ccc}-2 & 1 & 2\\3 & -2 & -2\\1 & 0 & -1\end{array} \right] =\left[ \begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array} \right].\end{equation}\subsection{Symbols}%\label{sec:symbols}\LaTeX provides an enormous number of symbols. Additional packages (loaded with \com{usepackage}) may provide additional symbols and fonts.For example, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ require you to load the \package{amsfonts} package (which is automatically loaded by the \texttt{icmmcm} class). These symbols are generated by \com{mathbb}, which only works in math mode.Subscripts and superscripts are easy---\verb!$a_n$! produces $a_n$,and \verb!$x^2$! produces $x^2$. Ordinal numbers, such as$3^{\textrm{rd}}$, $n^{\textrm{th}}$, and so forth,\footnote{Somefonts may include their own ordinals that can be accessed withspecial commands.} can be produced with code like\verb!$3^{\textrm{rd}}$!, \verb!$n^{\textrm{th}}$!.Equation~\ref{eq:superscript} shows a formula with a superscript.\begin{equation}\label{eq:superscript}\int_{0}^{\pi} \, \cos^{2n+1} x \, {\rm d} x = 0 \qquad\forall \, n \in \mathbb{N}.\end{equation}Notice that \com{cos} produces a nice roman ``$\cos$'' within math mode. There are similar commands for common functions like \com{log}, \com{exp}, and so forth. More can be defined with the\com{DeclareMathOperator} command provided by the \package{amsmath} package.You can stack symbols over other symbols. In math formulas,\begin{equation}m\ddot{x} + \gamma\dot{x} + kx = 0,\end{equation}or to produce diacritical accents, as in\begin{quote}Paul Erd\H{o}s s'est reveill\'{e} t\^{o}t pour enseigner lefran\c{c}ais \`{a} son fr\`{e}re et sa s\oe{}ur.\end{quote}\LaTeX{} has lots of Greek letters and ellipses too, some of which areshown in Figure~\ref{fig:greek-symbols}.\begin{figure}\begin{center}\begin{equation}\sqrt{\left[\begin{array}{cccccc}\alpha & \beta & \gamma & \delta & \epsilon & \zeta \\\eta & \theta & \iota & \kappa & \lambda & \mu \\\nu & \xi & o & \rho & \pi & \sigma \\\tau & \upsilon & \phi & \chi & \psi & \omega \\\Gamma & \Delta & \Theta & \Lambda & \Xi & \Pi \\\Sigma & \Upsilon & \Phi & \Psi & \Omega & \varphi\\\cdots & \ldots & \vdots & \ddots & : & \cdot\end{array}\right ] }.\end{equation}\end{center}\caption[Greek letters and some symbols]{Greek letters and some symbols.}% \label{fig:greek-symbols}\end{figure}See \cite{gratzer-mil}, pp.~455--474, or \cite{kopka-daly-guide},pp.~123--127, for lists of the symbols available. Intext, you mightsee some of these symbols used as\begin{quote}The Strong Induction Principle asserts that if a statement holds forthe integers $1$,~$2$,\dots,~$n$, and if whenever it holds for $n =1$, \dots,~$k$ then it also holds for $n = k+1$, then the statementholds for the integers $1$,~$2$,~$3$, $\ldots\,$ Using thisPrinciple, it can be shown that $1+2+\cdots+n = n(n+1)/2$ for allpositive integers~$n$.\end{quote}Notice that in the lists of integers, the ellipsis was made using the\com{ldots} command, and that the periods were nicely spaced betweenthe commas. In the sum, the dots were made with \com{cdots} and were centered on the line. The \package{amsmath} package provides a``smart'' \com{dots} command that can generally get things right basedon the context.So, with \com{dots} alone, the previous examples come out as\begin{quote}$1$,~$2$,~\dots,~$n$\\$n = 1$, \dots,~$k$\\$1$,~$2$,~$3$, $\dots\,$\\$1+2+\dots+n = n(n+1)/2$\end{quote}The general $n \times n$ matrix can be typeset as follows:\begin{equation}\left[\begin{array}{cccc}a_{11} & a_{12} & \ldots & a_{1n}\\a_{21} & a_{22} & \ldots & a_{2n}\\\vdots & \vdots & \ddots & \vdots\\a_{n1} & a_{n2} & \ldots & a_{nn}\\\end{array}\right].\end{equation}A fine point: lists of numbers that you're using in a mathematicalsense (as opposed to dates, numbers of objects, etc.) should be typedin math mode. For example, $341$, $541$, $561$, and $641$. The same numbers without math mode are 341, 541, 561, and 641. Depending on the fonts and packages that you're using, you may notice a little bitmore space around the first set than the second. With some packages, numbers intext may be set using old-style figures by default, as in\oldstylenums{341}, \oldstylenums{541}, \oldstylenums{561}, and\oldstylenums{641}. %%% But without the \oldstylenums commands!\subsection{More Math}In Fourier analysis, we talk about the $z$-domain.If $a$ is an even number, then\[ a + \phi(a) < \frac{3 a}{2}, \]and\[ \sigma(a) > \frac{2^{\alpha+1}-1}{2^{\alpha}} \, a \geq \frac{3a}{2}, \]where $\alpha$ is the greatest power of 2 that divides $a$, $\phi(a)$is the number of integers less than $a$ and relatively primeto $a$, and $\sigma(a)$ is the sum of the divisors of $a$ (including$1$ and $a$).。
美赛论文LaTeX模板
\documentclass{icmmcm}\usepackage{url} % For formatting URLs and other web or% file references.\usepackage{mflogo} % Provides the METAFONT logo; you% won't need it for your report.\usepackage{graphicx} % For importing graphics.\usepackage{natbib}%%% Sample ICM/MCM Contest Submission%%%%%% Based on sample senior thesis document%%% Last modified by Jeremy Rouse%%% Summer 2000%%%%%% and on the LaTeX Hints document%%% created by C.M. Connelly <cmc@>%%% Copyright 2002-2012%%% ---------------%%% Local Command and Environment Definitions%%% If you have any local command or environment definitions, put them %%% here or in a separate style file that you load with \usepackage.% \newtheorem declarations\newtheorem{Theo1}{Theorem}\newtheorem{Theo2}{Theorem}[section]\newtheorem{Lemma}[Theo2]{Lemma}% Each of the above defines a new theorem environment.% Multiple theorems can be done in the same environment.% Theo2's number is defined by the subsection it's in.% Theo3 uses the same numbering counter and numbering system as% Theo2 (that's the meaning of [Theo2]).%%% Y ou probably won't want any of the following commands, which are %%% here to allow various the names of commands, make examples typeset %%% properly, and so on. Y ou can, of course, use them as examples for %%% your own user-defined commands.\newcommand{\bslash}{\symbol{'134}}%backslash\newcommand{\bsl}{{\texttt{\bslash}}}\newcommand{\com}[1]{\bsl\texttt{#1}\xspace}\newcommand{\file}[1]{\texttt{#1}\xspace}\newcommand{\pdftex}{PDF\tex}\newcommand{\pdflatex}{PDF\latex}\newcommand{\acronym}[1]{\textsc{#1}\xspace}\newcommand{\key}[1]{\textsf{\emph{#1}}\xspace}\newcommand{\class}[1]{\textsf{#1}\xspace}\newcommand{\package}[1]{\textsf{#1}\xspace}\newcommand{\env}[1]{\texttt{#1}\xspace}\newcommand{\prog}[1]{\texttt{#1}\xspace}\newcommand{\command}[1]{\texttt{\bsl{}#1}\xspace}\newcommand{\ctt}{\texttt{comp.text.tex}\xspace}\newcommand{\tex}{\TeX\xspace}\newcommand{\latex}{\LaTeX\xspace}%%% Note that the \xspace command comes from the xspace package. It %%% allows you type a command that inserts text without having to %%% worry about how you ``end'' that command.%%%%%% Without \xspace, you would need to end a command with a backslash %%% followed by a space or with an empty set of braces if you followed %%% the command with a space. For example,%%%%%% \foo is a very important algorithm.%%%%%% might produce%%%%%% The foobarbaz algorithmis a very important algorithm.%%%%%% whereas with the \xspace command, the same code would produce %%%%%% The foobarbaz algorithm is a very important algorithm.%%%%%% If you need to butt a command that produces text against a letter %%% of some sort -- say, to pluralize it -- you need to tell TeX%%% where your command name ends so that it expands the correct %%% macro. So you might do%%%%%% \bar{}s are very busy creatures.%%% TeX has an amazingly good hyphenation algorithm, but sometimes it %%% gets confused and needs some help.%%%%%% For words that only occur once or twice, you can insert hints%%% directly into your text, as in%%%%%% our data\-base system is one of the most complex ever devised %%%%%% For words that you use a lot, and that seem to keep ending up at %%% the end of a line, however, inserting the hints each time gets to %%% be a drag. Y ou can use the \hyphenation command to globally tell %%% TeX where to hyphenate words it can't figure out on its own.\hyphenation{white-space}%%% End Local Command and Environment Definitions%%% ---------------%%% ---------------%%% Title Block\title{\latex Hints for ICM/MCM Contest Reports}%%% Which contest are you taking part in? (Just one!)\contest{ICM/MCM}%%% The question you answered. (Again, just the one.)\question{Report Sample}%%% Y our Contest Team Control Number\team{21247}%%% A normal document would specify the author's name (and possibly %%% their affiliation or other information) in an \author command. %%% Because the ICM/MCM Contest rules specify that the names of the %%% team members, their advisor, and their institution should not%%% appear anywhere in the report, do *not* define an \author command.%%% Defining the \date command is optional. If you leave it blank, %%% your document will include the date that the file is typeset, in %%% the form ``Month dd, yyyy''.% \date{}%%% End Title Block%%% ---------------\begin{document}%%% ---------------%%% Summary\begin{summary}This document is meant to give you a quick introduction to \TeX\ and \LaTeX. It covers a lot of material, but still barely manages to scratch the surface. It should provide you with some inspiration and, I hope, with some useful code you can copy, modify, and use in your report.Y ou should use the \file{blank-template.tex} file as a basis foryour report rather than this file. Be sure to change its name to something sensible (maybe your team control number), and to set the values of the \com{title}, \com{question}, and \com{team} commands to appropriate values.Good luck!\hfill{}-- Claire\end{summary}%%% End Summary%%% ---------------%%% ---------------%%% Print Title Block, Contents, et al.\maketitle\tableofcontents%%% Uncomment the following lines if you have figures or tables in %%% your report:\listoffigures\listoftables%%% End Print Title Block, Contents, et al.%%% ---------------\section{Introduction: What Is \latex?}%\label{sec:introduction}\latex is a tool that allows you to concentrate on your writing while taking advantage of the \tex typesetting system to producehigh-quality typeset documents.\latex's benefits include\begin{enumerate}\item Standardized document classes\item Structural frameworks for organizing documents\item Automatic numbering and cross-referencing of structural elements \item ``Floating'' figures and tables\item High-level programming interface for accessing \tex'stypesetting capabilities\item Access to \latex extensions through loading ``packages''\end{enumerate}\section{Structured Writing}%\label{sec:structured-writing}Like HTML,\footnote{HyperText Markup Language} \latex is a markup language rather than a \acronym{Wysiwyg}{}\footnote{What Y ou See Is What Y ou Get.} system. Y ou write plain text files that use special \key{commands} and \key{environments} that govern the appearance and function of parts of your text in your final typeset document.\subsection{Document Classes}%\label{sec:document-classes}The general appearance of your document is determined by your choice of \key{document class}. Document classes also load \latex packagesto provide additional functionality.\latex provides a number of basic classes, including \class{article},\class{letter}, \class{report}, and \class{book}. There are also alarge number of other document classes available, including\class{amsart} and \class{amsbook}, created by the American Mathematical Society and providing some additional mathematically useful structures and commands; \class{foils}, \class{prosper}, and\class{seminar}, which allow you to create ``slides'' for presentations; the math department's \class{thesis} class, forformatting senior theses; and many journal- or company-specific classes that format your document to match the ``house style'' of a particular periodical or publisher.\subsection{Packages}%\label{sec:packages}%\label{sec:ctan}\latex packages, or \key{style files}, define additional commands and environments, or change the way that previously defined commands and environments work. By loading packages, you can change the fonts used in your document, write your document in a non-English language with a non-\acronym{Ascii} font encoding, include graphics, format program listings, add custom headers and footers to your document, and much more.A typical \tex installation includes hundreds of style files, andhundreds more are available from the Comprehensive \tex Archive Network (CTAN), at \url{/}.\subsection{Structural Commands}%\label{sec:structural-commands}\begin{table}\centering\begin{tabular}{ll}\topruleCommand & Notes \\ \midrule\com{part} & \class{book} \& \class{report} only \\\com{chapter} &\class{book} \& \class{report} only \\\com{section} \\\com{subsection} \\\com{subsubsection} \\\com{paragraph} \\\com{subparagraph} \\\bottomrule\end{tabular}\caption[Structural commands in \latex]{Structural commands in \latex.}% \label{tab:structural-commands}\latex provides a set of structural commands for defining sections ofyour document, as shown in Table~\ref{tab:structural-commands}.Note that the argument to structural commands are moving arguments (see Section~\ref{sec:fragile-commands}) because they can be reused in the table of contents or in page headers or footers. Structural commands can take an optional argument in which you specify nonfragile commands or a shorter version of the actual section title that fits.Y ou'll generally know when you need to provide an optional argument by \TeX's behavior.\subsection{Labels and References}%\label{sec:labels-and-references}Sections are numbered automatically by \latex during typesetting. Ifyou change your mind and decide that a subsection should be promotedto a section, or moved to the end of your document, the sections willbe renumbered so that the numbers are consistent.Sections can also be \command{label}{}ed with a tag such as\begin{quote}\begin{verbatim}\section{Our Complicated Equations}%\label{sec:complicated-eqs}\end{verbatim}\end{quote}and referred to with a \command{ref} or \command{pageref} command, as in\begin{quote}\begin{verbatim}In Section~\ref{sec:complicated-eqs}, we pointed out...\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}On page~\pageref{fig:gordian-knot}, we illustrated...\end{verbatim}\end{quote}\latex substitutes the correct section number when typesetting yourThe same commands can be used with numbered environments such as\env{equation}, \env{theorem}, and so forth.Use \emph{meaningful} labels---labeling a section as \texttt{sec12}may seem useful, but it will be confusing if you end up moving it to a different place in the document and its number changes to Section~34.It's also easier to remember what reference you want if you use a meaningful name.Y ou may also want to impose some additional organization through the use of \emph{namespaces}, as I've done in this document. Rather than give different types of objects undistinguished labels, I precedesection labels with \texttt{sec:}, equations with \texttt{eq:},figures with \texttt{fig:}, tables with \texttt{tab:}, and so on.Emacs with Aux\tex and Ref\tex gives you easy access to these labels,as do many other editors with \tex-specific features. It's mucheasier to find the particular label you're looking for if you havesome additional information to help you. Adding the prefixes also reminds you of what text should precede the \com{ref} command.\subsection{Commands}\latex uses commands for changes that are very limited in scope (a few words) or are unlimited in scope (the rest of a document). For example, the commands\begin{quote}\begin{verbatim}\textbf{bold}\emph{italic (emphasized)}\textsf{sans serif}\end{verbatim}\end{quote}produce the following output in a typeset document:\begin{quote}\textbf{bold} \emph{italic (emphasized)} \textsf{sans serif}\end{quote}These are ``commands with arguments''---the command itself starts witha backslash (\bsl), and its \key{argument} appears inside braces{\verb+{ }+). Some commands may also have \key{optional arguments},which are typed inside brackets (\verb+[ ]+).There are also commands that take no arguments, such as\command{noindent}, \command{raggedright}, and \command{pagebreak}.Y ou can define your own commands, as discussed inSection~\ref{sec:customization}.\subsection{Environments}%\label{sec:environments}\latex provides a number of \key{environments} that affect the appearance of text, and are generally used for more structurally significant purposes. For example, the commands listed above are typeset inside a \env{verbatim} environment typed inside a \env{quote} environment. Their results were typeset inside a \env{quote} environment.Environments use special commands to start and close---\command{begin} and \command{end}, followed by the name of the environment in braces, as in\begin{quote}\begin{verbatim}\begin{quote}``This is disgusting---I can't eat this. That arugala is sobitter\ldots{} It's like my algebra teacher on bread.''\flushright -- Julia Roberts in \emph{Full Frontal}\end{quote}\end{verbatim}\end{quote}producing\begin{quote}``This is disgusting---I can't eat this. That arugala is sobitter\ldots{} It's like my algebra teacher on bread.''\flushright -- Julia Roberts in \emph{Full Frontal}\end{quote}Some environments may take additional arguments in braces (required)or brackets (optional).Note that the order in which environments nest is extremely important.If you type an environment inside another environment, the inner environment must be \command{end}{}ed \emph{before} the secondenvironment is closed. It's also vitally important that you have an\command{end} line for each \command{begin} line, or \latex will complain.\subsubsection{The \env{document} Environment and the Preamble}%\label{sec:document-environment}The most important environment is the \env{document} environment, which encloses the \key{body} of your document. The code before the \command{begin}\verb+{document}+ line is called the \key{preamble}, and includes the all-powerful \command{documentclass} command, which loads a particular document class (seeSection~\ref{sec:document-classes}); optional \command{usepackage} commands, which load in additional \latex packages (seeSection~\ref{sec:packages}); and other setup commands, such asuser-defined commands and environments, counter settings, and so forth.I generally also include the commands defining the title, author, anddate in my preambles, but other people include them just after\command{begin}\verb+{document}+, before the \command{maketitle} command, which creates the title block of your document.\subsubsection{Math Environments}%\label{sec:math-environments}One of the major hallmarks of \tex is its ability to typesetmathematical equations.The two primary ways of doing so are with the use of \key{inline} and\key{display math environments}. These environments are used so often that there are shorthands provided for typing them. Inline math environments, such as $a^2 + b^2 = c^2$, can be typed as\begin{quote}\begin{verbatim}\begin{math}a^{2} + b^{2} = c^{2}\end{math}\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}$a^{2} + b^{2} = c^{2}$.\end{verbatim}\end{quote}Display math environments set your equation apart from your running text. They're generally used for more complicated expressions, such as\[f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\]which can be typed as\begin{quote}\begin{verbatim}\begin{displaymath}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{displaymath}\end{verbatim}\end{quote}or\begin{quote}\begin{verbatim}\[f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\]\end{verbatim}\end{quote}Generally, you'll want to use the \verb+$+ %$ <- fool font-lock-modedelimited form for inline math, and the \com{[} \com{]} form for display math environments. [Besides being easy to type, these forms are \key{robust}, which means that they can be used in \key{moving arguments}, elements that \tex may need to typeset in more than one place (such as a table of contents) or adjust (such as footnotes).]\paragraph{The \env{equation} Environment}%\label{sec:equation-environment}Y ou'll probably want to use the \env{equation} environment for any formula you plan to refer to. \latex not only typesets the contentsof an \env{equation} environment in display mode, it also numbers it, as in\begin{equation}\label{eq:myequation}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{equation}written as\begin{quote}\begin{verbatim}\begin{equation}\label{eq:myequation}f(x) = \int \left( \frac{x^2 + x^3}{1} \right)dx\end{equation}\end{verbatim}\end{quote}Note that you can refer to this formula asEquation~\ref{eq:myequation} with\begin{verbatim}\ref{eq:myequation}.\end{verbatim}\subsection{Fonts}%\label{sec:fonts}Generally you'll want to let \latex handle the fonts for you---Knuth's Computer Modern fonts are used by default, and include a wide range of variations that can cover most any use you can think of.If you want to get fancy (and portable; seeSection~\ref{sec:fuzzy-fonts}), you can use Type~1 PostScript fonts, such as Times, Palatino, Utopia, and so forth. These font sets are accessible with packages with names like \package{times},\package{palatino}, and \package{utopia}. There are others, aswell---a command such as \com{locate psnfss | grep sty} will find mostof them.Y ou can also get fonts from CTAN (see Section~\ref{sec:ctan}), both bitmap and Type 1. There's even support for TrueType fonts in some\TeX\ systems.\subsubsection{Font Commands}%\label{sec:font-commands}Most of your concern about fonts is probably related to what you're writing. Y ou might want some \emph{emphasized} or \textbf{bold} textto stress a point or highlight a key term. Filenames might be set in\texttt{typewriter text} (although you should consider using the\package{url} package to help you out---by default, text set intypewriter text isn't hyphenated, which can lead to some unattractiveline breaks).Y ou can also set text in \textsf{sans serif} or \textsc{small caps}.Table~\ref{tab:font-commands} shows you some of the most commonly used font commands provided by \latex.\begin{table}[htbp]\centering\begin{tabular}{ll}\topruleCommand & Result\\\midrule\com{emph} & \emph{emphasized text}\\\com{textsf} & \textsf{sans-serif text}\\\com{texttt} & \texttt{typewriter text}\\\com{textbf} & \textbf{bold text}\\\com{textsc} & \textsc{small caps text}\\\com{textsl} & \textsl{slanted text}\\\com{textit} & \textit{italic text}\\\bottomrule\end{tabular}\caption[Commonly used font commands]{Commonly used font commands.} \label{tab:font-commands}\end{table}I recommend that you use \com{emph} in preference to \com{textit}, anduse \com{textbf} sparingly. \com{emph} is a smarter command than\com{textit}---it switches back to the roman font when necessary. For example, \emph{She loved \emph{Scooby Doo}.} versus \textit{He loved\textit{Titanic}.}For complicated font changes, or for special font usages that you'retyping a lot, creating a macro (Section~\ref{sec:customization}) isthe way to go. I often just write, tossing in custom commands as Igo, and waiting to define them until just before I compile thedocument.\subsection{Customization}%\label{sec:customization}The main advantage of using commands and environments is that they allow you to organize your writing. A useful side-effect is that youcan change your mind about the way an element is typeset, and change all the appearances of that element in document by editing one pieceof code. For example, in this document the names of environments have been set in ``typewriter text'', using a command I created called\command{env}, which is defined as\begin{quote}\begin{verbatim}\newcommand{\env}[1]{\texttt{#1}\xspace}\end{verbatim}\end{quote}All I have to do to make the names of all the environments in the document appear in sans-serif type instead is to change that one lineto\begin{quote}\begin{verbatim}\newcommand{\env}[1]{\textsf{#1}\xspace}\end{verbatim}\end{quote}Y ou can do the same with almost anything you can conceptualize---key terms, people's names (especially names of people fromnon-English-speaking countries), files, functions, and so on.\section{Mathematical Notation}%\label{sec:mathematical-notation}As we saw in Section~\ref{sec:math-environments}, math is typed into one of several kinds of math environments. Choose your environment based on the context and importance of the content. Any formula you plan to refer to should be typed in an \env{equation} environment (ora similar environment that supports labels).Y ou should punctuate your mathematics as if the formulae were normal parts of English sentences. Reading them aloud is often a useful method for ensuring that you have all the commas in the right places. Where appropriate, you should also follow a displayed formula at the end of a sentence with a period.\subsection{Sums and Products}%\label{sec:sums-n-products}It's easy to typeset sums and products. For example,\begin{equation}f(n) = \sqrt[n]{\sum_{k=1}^{n} {n \choose k} f \left( n - k \right)},~\prod_{n=2}^{\infty} \frac{n^{3}-1}{n^{3}+1} = \frac{2}{3}.\end{equation}%%% The ~ in the equation puts a nonbreaking space (equivalent to an%%% interword space in text mode) between the two halves of the equation. %%%%%% Also, note that the use of the \choose command here causes the%%% amsmath package to issue the warning%%%%%% Package amsmath Warning: Foreign command \atopwithdelims; %%% (amsmath) \frac or \genfrac should be used instead %%% (amsmath) on input line 557.%%%%%% amsmath would prefer the use of the \binom command it supplies.\subsection{Matrices}%\label{sec:matrices}It's a little more difficult to create matrices, but not too bad:%%% In LaTeX, & is the alignment tab, and separates columns. \\ is the end of %%% line marker, and separates rows. The ccc denotes that there are three %%% columns. The array environment and the tabular environment are %%% more or less identical, so what goes here also applies to a table.%%%\begin{equation}\left[ \begin{array}{ccc}2 & 1 & 2\\1 & 0 & 2\\2 & 1 & 1\end{array} \right]\left[ \begin{array}{ccc}-2 & 1 & 2\\3 & -2 & -2\\1 & 0 & -1\end{array} \right] =\left[ \begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array} \right].\end{equation}\subsection{Symbols}%\label{sec:symbols}\LaTeX provides an enormous number of symbols. Additional packages (loaded with \com{usepackage}) may provide additional symbols and fonts.For example, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ require you to load the \package{amsfonts} package (which is automatically loaded by the \texttt{icmmcm} class). These symbols are generated by \com{mathbb}, which only works in math mode.Subscripts and superscripts are easy---\verb!$a_n$! produces $a_n$,and \verb!$x^2$! produces $x^2$. Ordinal numbers, such as$3^{\textrm{rd}}$, $n^{\textrm{th}}$, and so forth,\footnote{Somefonts may include their own ordinals that can be accessed withspecial commands.} can be produced with code like\verb!$3^{\textrm{rd}}$!, \verb!$n^{\textrm{th}}$!.Equation~\ref{eq:superscript} shows a formula with a superscript.\begin{equation}\label{eq:superscript}\int_{0}^{\pi} \, \cos^{2n+1} x \, {\rm d} x = 0 \qquad\forall \, n \in \mathbb{N}.\end{equation}Notice that \com{cos} produces a nice roman ``$\cos$'' within math mode. There are similar commands for common functions like \com{log}, \com{exp}, and so forth. More can be defined with the\com{DeclareMathOperator} command provided by the \package{amsmath} package.Y ou can stack symbols over other symbols. In math formulas,\begin{equation}m\ddot{x} + \gamma\dot{x} + kx = 0,\end{equation}or to produce diacritical accents, as in\begin{quote}Paul Erd\H{o}s s'est reveill\'{e} t\^{o}t pour enseigner lefran\c{c}ais \`{a} son fr\`{e}re et sa s\oe{}ur.\end{quote}\LaTeX{} has lots of Greek letters and ellipses too, some of which areshown in Figure~\ref{fig:greek-symbols}.\begin{figure}\begin{center}\begin{equation}\sqrt{\left[\begin{array}{cccccc}\alpha & \beta & \gamma & \delta & \epsilon & \zeta \\\eta & \theta & \iota & \kappa & \lambda & \mu \\\nu & \xi & o & \rho & \pi & \sigma \\\tau & \upsilon & \phi & \chi & \psi & \omega \\\Gamma & \Delta & \Theta & \Lambda & \Xi & \Pi \\\Sigma & \Upsilon & \Phi & \Psi & \Omega & \varphi\\\cdots & \ldots & \vdots & \ddots & : & \cdot\end{array}\right ] }.\end{equation}\end{center}\caption[Greek letters and some symbols]{Greek letters and some symbols.}% \label{fig:greek-symbols}\end{figure}See \cite{gratzer-mil}, pp.~455--474, or \cite{kopka-daly-guide},pp.~123--127, for lists of the symbols available. Intext, you mightsee some of these symbols used as\begin{quote}The Strong Induction Principle asserts that if a statement holds forthe integers $1$,~$2$,\dots,~$n$, and if whenever it holds for $n =1$, \dots,~$k$ then it also holds for $n = k+1$, then the statementholds for the integers $1$,~$2$,~$3$, $\ldots\,$ Using thisPrinciple, it can be shown that $1+2+\cdots+n = n(n+1)/2$ for allpositive integers~$n$.\end{quote}Notice that in the lists of integers, the ellipsis was made using the\com{ldots} command, and that the periods were nicely spaced betweenthe commas. In the sum, the dots were made with \com{cdots} and were centered on the line. The \package{amsmath} package provides a``smart'' \com{dots} command that can generally get things right basedon the context.So, with \com{dots} alone, the previous examples come out as\begin{quote}$1$,~$2$,~\dots,~$n$\\$n = 1$, \dots,~$k$\\$1$,~$2$,~$3$, $\dots\,$\\$1+2+\dots+n = n(n+1)/2$\end{quote}The general $n \times n$ matrix can be typeset as follows:\begin{equation}\left[\begin{array}{cccc}a_{11} & a_{12} & \ldots & a_{1n}\\a_{21} & a_{22} & \ldots & a_{2n}\\\vdots & \vdots & \ddots & \vdots\\a_{n1} & a_{n2} & \ldots & a_{nn}\\\end{array}\right].\end{equation}A fine point: lists of numbers that you're using in a mathematicalsense (as opposed to dates, numbers of objects, etc.) should be typedin math mode. For example, $341$, $541$, $561$, and $641$. The same numbers without math mode are 341, 541, 561, and 641. Depending on the fonts and packages that you're using, you may notice a little bitmore space around the first set than the second. With some packages, numbers intext may be set using old-style figures by default, as in\oldstylenums{341}, \oldstylenums{541}, \oldstylenums{561}, and\oldstylenums{641}. %%% But without the \oldstylenums commands!\subsection{More Math}In Fourier analysis, we talk about the $z$-domain.If $a$ is an even number, then\[ a + \phi(a) < \frac{3 a}{2}, \]and\[ \sigma(a) > \frac{2^{\alpha+1}-1}{2^{\alpha}} \, a \geq \frac{3a}{2}, \]where $\alpha$ is the greatest power of 2 that divides $a$, $\phi(a)$is the number of integers less than $a$ and relatively primeto $a$, and $\sigma(a)$ is the sum of the divisors of $a$ (including$1$ and $a$).。
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。
美国大学生数学建模竞赛美赛--论文
Each team member must sign the statement below: (Failure to obtain signatures from each team member will result in disqualification of the entire team.)
2015 Mathematical Contest in Modeling (MCM/ICM) Control Sheet Please review this page before submitting your solution to ensure that all of the information is correct Do not make changes by hand to the information on this control sheet. If you need to change any of the information on this sheet, login via the Advisor Login link on the MCM web site, make the changes online, and print a new sheet. You may NOT photocopy this control sheet to give to a new team, nor may you assign any team a control number. Each team must have its own control number, obtained by registering via the MCM web site. Advisor Jinpeng Yu Name: Department: Control Engineering Institution: Qingdao University Address: 308 Ningxia Road,Shinan District,Qingdao,Shandong,China Qingdao, Shandong 266000 Phone: 18653250086 Fax: 053285953064 Email: zhanghaoran06@ Home Phone: 053285953064 The names of the team members will appear on your team's certificate exactly as they appear on this page, including all capitalization and punctuation, if any. Gender data is optional and will be used for statistical purposes only; it will not appear on the certificate. Team Member Haoran Zhang Yu Ma Guiying Dong Gender M M F Your team's control number is: 40906 (Place this control number on all pages of your solution paper and on any support material.) Problem Chosen: B
美赛数学建模比赛论文资料材料模板
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 (3)5.1 Building of the Cellular automaton model (3)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 (8)5.2.2 Conclusion (8)5.3 Taking the an intelligent vehicle system into a account (8)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 (9)6. Improvement of the model (10)6.1 strength and weakness (10)6.1.1 Strength (10)6.1.2 Weakness (10)6.2 Improvement of the model (10)7. Reference (12)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 they are 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 or accidental factors ● Ignore the driver's own abnormal factors, such as drunk driving and fatigue driving ● The operation form of highway intelligent system that our analysis can reflectintelligent system● In the intelligent vehicle system, the result of the sampling data has high 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 take the 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 of small 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 controlFig.5.1.3Fig.5.1.3Relationships 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 will become 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 thespeed control. 1f P is the overtaking ratio of small vehicles over large vehicles, 2f P is the overtaking ratio ofsmall 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. 5.1.6Relationships 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 BP(situation, applying ))results with a tendency of moving to the right lane for this,2(xRdriver. 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 usto 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 Table5.1, 5.2, 5.3.●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. Becauseof 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 of the 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 consider actual trafficconditions ,moreover a large number of images make the model more visual.●The result is effectively achieved all of the goals we set initially, meantime the conclusion ismore 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 comprehensively analysis.●Due to there are many traffic factors, we are only studied some of the factors, thus ourmodel 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, Wefurther 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 the85V speed of each model as the main reference basis. Inrecent years, the85V of some researchers for the typical countries from Table 6.1[ 21]:Author Country ModelOttesen andKrammes2000America LCDCLDCVC⨯---=01.0012.057.144.10285Andueza2000 Venezuela].[308.9486.7)/894()/2795(25.9885curvehorizontalLDCRaRVT++--=].[tan819.27)/3032(69.10085gentLRVT+-=Jessen2001 America][00239.0614.0279.080.86185LSDADTGVVP--+=][00212.0432.010.7285NLSDADTVVP-+=Donnell2001 America22)2(8500724.040.10140.04.78TLGRV--+=22)3(85008369.048.10176.01.75TLGRV--+=22)4(8500810.069.10176.05.74TLGRV--+=22)5(8500934.008.21.83TLGV--=BucchiA.BiasuzziK.And SimoneA.2005 ItalyDCV124.0164.6685-=DCEV4.046.3366.5585--=Meanwhile, there are other vehicles driving rules such as speed limit in adverse weather 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 traffic simulations usingcellular automata, Physica A 231 (1996) 534–550.[20] J.T. Fokkema, Lakshmi Dhevi, Tamil Nadu Traffic Management and Control inIntelligent Vehicle Highway Systems,18(2009).[21] Yang Li, New Variable Speed Control Approach for Freeway. (2011) 1-66。
美赛论文模板
T eam Control NumberFor office use only0000For office use onlyT1 F1T2 F2T3 Problem Chosen F3T4 A F42014 Mathematical Contest in Modeling (MCM) Summary Sheet(Attach a copy of this page to each copy of your solution paper.)Repeaters Coordination And DistributionFebruary 6,2015AbstractIn this paper, it aims to computing problem on Relay Strategy (repeaters coordination and distribution). According to advanced radio cellular coverage technology, usage of frequency attenuation and geometric mapping methods, Hata model， cellular coverage solution and FDM (Frequency Division Multiplexing) model were established. The algorithms used MATLAB to simulate, with the final modeling results of sensitivity analysis and improvement & promotion on models.Question one : For a circular flat area of radius 40 miles radius, determine the minimum number of repeaters necessary to accommodate 1,000 simultaneous users. Assume that the spectrum available is 145 to 148 MHz, the transmitter frequency in a repeater is either 600 kHz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available.Answer:1. Based on Frequency attenuation expression and calculation with MATLAB, it figuredout the eligible coverage radiuses, which are 30km for BS (base station), and 14.9km for repeater.2. Assuming the users in a given area under uniform distribution, using advancedcellular coverage solution, we can calculate that minimum number of required repeater is 36 under cellular features.3. Based on the US VHF spectrum allocation standard, the minimum spacing for adjacentchannels is 30kHz. And with up to 54 different PL tones, maximum 4320 channels can be allocated to provide 1000 simultaneous users to use at the same time. Conclusion:The minimum number of repeaters necessary to accommodate 1,000 simultaneous users is 36.Question Two : How does your solution change if there are 10,000 users?Answer:1. Since the given spectrum is in a fixed range, even if 54 different PL tones can not be allocated enough channels for 10,000 simultaneous users. So the number of repeaters will be increased, meanwhile, the given area will be divided into different parts.2. On the assumption that uniform distribution of the population in the given area, it will be divided into 3 sub-regions equally by analyzing the binding domain, frequency spectrum and PL tones three independent factors. And then the number of repeaters within each sub-region will be classified discussion.3. The FDM (Frequency Division Multiplexing) model is established here to improve channel efficiency to accommodate up to 10,000 simultaneous users Conclusion:The minimum number of repeaters necessary to accommodate 10,000 simultaneous users is 126.Question Three : Discuss the case where there might be defects in line-of-sight propagation caused by mountainous areas. Answer:Basically, under the same condition for question 1&2, the mountainous area will be analyzed as following:1. The function for relationship between radio attenuation x caused by obstacles and the eligible coverage radius d for repeater is 2249.354371.4110x d -=, which is to analyze the impact on the number of repeaters under full signal coverage. 2. For the mountain barrier, based on the different situation of mountains, the addition of repeaters on the suitable location will be discussed to achieve full coverage. This paper describes model established by using of cellular coverage technology and frequency attenuation expression, to achieve simple, fast, accurate algorithm. And also illustrated the effect takes the entire article. In the end, the sensitivity analysis and error calculation are applied for modeling, making the model practically.Key words: Cellular Coverage technology, frequency attenuation expression, channel allocation, MatlabRepeaters coordination and distributionContent1 Restatement of the Problem (1)1.1 Introduction (1)1.2 The Problem (1)2 Simplifying Assumption (1)3 Phrase explain (1)4 Model (2)4.1 Model I (2)4.1.1 Analysis of the Problem (2)4.1.2 Model Design (2)5 Sensitivity analysis (2)6 Model extension (2)7 Evaluating our model (2)7.1 The strengths of model (2)7.2 The weaknesses of model (2)References (3)1 Restatement of the Problem1.1 IntroductionThe VHF radio spectrum involves line-of-sight transmission and reception. This limitation can be overcome by “repeaters,” which pick up weak signals, amplify them, and retransmit them on a different frequency. Thus, using a repeater, low-power users (such as mobile stations) can communicate with one another in situations where direct user-to-user contact would not be possible. However, repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.1.2 The ProblemYour job is to:◆Design a scheme that determines the minimum number of repeaters necessaryto accommodate 1,000 simultaneous users in a circular flat area of radius40 miles radius.And assume that the spectrum available is 145 to 148 MHz,the transmitter frequency in a repeater is either 600 kHz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available.◆Change your scheme to accommodate 1,0000 simultaneous users base on yourmodel.◆Discuss the case where there might be defects in line-of-sight propagationcaused by mountainous areas.2 Simplifying Assumption3 Phrase explain4 Model4.1 Model I4.1.1 Analysis of the Problem4.1.2 Model Design5 Sensitivity analysisSymbol◆N: the number of total repeaters in the circle area ◆Q: the number of the users in the circle area◆k: the number of the red circle in figure 2最前面最好有一个Symbol List6 Model extension7 Evaluating our model7.1 The strengths of model7.2 The weaknesses of modelReferences参考文献不要引用非常差的期刊的论文，要引用比较厉害的英文期刊，证明你有足够的阅读文献量。
美赛(mcm)论文LaTeX模板
\documentclass[12pt]{article}\usepackage{CJK}\usepackage{titlesec,titletoc} \usepackage{indentfirst}\usepackage{graphicx}\usepackage{caption2}\usepackage{subfigure}\usepackage{longtable}\usepackage{slashbox}\usepackage{fancyhdr}\usepackage{times}\usepackage{amsmath}\special{papersize=21cm,29.7cm} \setlength{\textwidth}{15cm}\setlength{\textheight}{23cm} \setlength{\evensidemargin}{0.46cm}\setlength{\oddsidemargin}{0.46cm} \setlength{\topmargin}{-1.84cm} \setlength{\headheight}{2.9cm} \setlength{\headsep}{0.4cm}\newcommand{\chuhao}{\fontsize{42pt}{\baselineskip}\selectfont}\newcommand{\xiaochuhao}{\fontsize{36pt}{\baselineskip}\selectfont} \newcommand{\yihao}{\fontsize{26pt}{\baselineskip}\selectfont}\newcommand{\xiyihao}{\fontsize{24pt}{\baselineskip}\selectfont}\newcommand{\erhao}{\fontsize{22pt}{\baselineskip}\selectfont}\newcommand{\xiaoerhao}{\fontsize{18pt}{\baselineskip}\selectfont} \newcommand{\sanhao}{\fontsize{16pt}{\baselineskip}\selectfont}\newcommand{\xiaosanhao}{\fontsize{15pt}{\baselineskip}\selectfont} \newcommand{\sihao}{\fontsize{14pt}{\baselineskip}\selectfont}\newcommand{\xiaosihao}{\fontsize{12pt}{\baselineskip}\selectfont} \newcommand{\wuhao}{\fontsize{10.5pt}{\baselineskip}\selectfont} \newcommand{\xiaowuhao}{\fontsize{9pt}{\baselineskip}\selectfont} \newcommand{\liuhao}{\fontsize{7.5pt}{\baselineskip}\selectfont}\newcommand{\xiaoliuhao}{\fontsize{6.5pt}{\baselineskip}\selectfont} \newcommand{\qihao}{\fontsize{5.5pt}{\baselineskip}\selectfont}\newcommand{\bahao}{\fontsize{5pt}{\baselineskip}\selectfont}%页眉的设置, 要用到fancyhdr宏包\pagestyle{fancy} \fancyhead{} \fancyfoot{}\fancyhead[L]{\footnotesize Team \# 189}\fancyhead[R]{\footnotesize Page\ \thepage\ of\ 42}\fancypagestyle{plain}{%\fancyhead[L]{\footnotesize Team \# 189}\fancyhead[R]{\footnotesize Page\ \thepage\ of\ 42}}\setcounter{secnumdepth}{4}%更改\theparagraph的编号样式\makeatletter\renewcommand{\theparagraph}{\@arabic\c@paragraph}\makeatother%章节格式的设置\titleformat{\section}{\erhao\bf}{}{0em}{}[]\titleformat{\subsection}{\xiaoerhao\bf}{}{0em}{}[]\titleformat{\subsubsection}{\sanhao\bf}{}{0em}{}[]\titleformat{\paragraph}[hang]{\vspace*{0.5ex}\sihao\bf}{\hspace*{1em}\theparagraph)}{0.5em }{}[\vspace*{-0.5ex}]%更改插图的标题\renewcommand{\figurename}{\wuhao\bf\sf Figure}\renewcommand{\captionlabeldelim}{\ }%更改表格的标题\renewcommand{\tablename}{\wuhao\bf\sf Table}%更改图形或表格与其标题的间距\setlength{\abovecaptionskip}{10pt}\setlength{\belowcaptionskip}{10pt}%定义产生不浮动图形和表格的标题的命令\figcaption和\tabcaption\makeatletter\newcommand\figcaption{\def\@captype{figure}\caption}\newcommand\tabcaption{\def\@captype{table}\caption}\makeatother%自定义的可以调整粗细的水平线命令, 用于绘制表格, 调用格式\myhline{0.5mm}.\makeatletter\def\myhline#1{%\noalign{\ifnum0=`}\fi\hrule \@height #1 \futurelet\reserved@a\@xhline}\makeatother%第一层列表序号为带圈的阿拉伯数字\renewcommand{\labelenumi}{\textcircled{\arabic{enumi}}}%更改脚注设置\renewcommand{\thefootnote}{\fnsymbol{footnote}}\begin{document}\begin{CJK*}{GBK}{song}\CJKtilde\title{\bf\yihao Aviation Baggage Screening\\{\&} Flight Schedule}\author{}\date{}\maketitle\section{Introduction}Following the terrorist attacks on September 11, 2001, there isintense interest in improving the security screening process forairline passengers and their baggage. Airlines and airports areconsidered high-threat targets for terrorism, so aviation securityis crucial to the safety of the air-travelling public. Bombs andexplosives have been known to be introduced to aircraft by holdbaggage and cargo, carried on by passengers, and hidden withinaircraft supplies.At present To Screen or Not to Screen, that is a Hobson's choice.US Current laws mandate 100{\%} screening of all checked bags at the 429passenger airports throughout the nation by explosive detection systems(EDS) by the end of the Dec 31 2003. However, because the manufacturers arenot able to produce the expected number of EDS required to meet the federalmandate, so it is significant to determine the correct number of devicesdeploy at each airport, and to take advantage of them effectively.The Transportation Security Administration (TSA) needs a complicatedanalysis on how to allocate limited device and how to best use them.Our paper contains the mathematical models to determine the number of EDSsand flight schedules for all airports in Midwest Region. We also discuss theETD devices as the additional security measures and the future developmentof the security systems.\section{Assumption and Hypothesis}\begin{itemize}\item The passengers who will get on the same airplane will arrive uniformly, namely the distribution is flat.\item The detection systems, both EDS and ETD, operate all the time during peak hour, except downtime.\item The airline checks the passengers randomly, according to its claim.\item The passengers, who are just landing and leave out, do not have to be checked through EDS or ETD.\item According to the literature, the aircraft loads approximately equal among the sets of departing flight during the peak hour.\item The landing flight did not affect the departure of the plane.\item Once a passenger arrives, he can go to EDS to be checked, except he has to wait in line.\item Once passengers finish screening, they can broad on the plane in no time.\item During peak hours, a set of flights departs at the same time every the same minutes.\item All the runways are used as much as possible during peak hours.\item The maximum number of the baggage is two, which a passenger can carry on plane. []\item The detection machine examines the bags at the same speed.\item EDS cannot make mistakes that it detect a normal object as an explosive.\end{itemize}\section{Variable and Definition}\begin{longtable}{p{100pt}p{280pt}}\caption{Variables}\\ %第一页表头的标题\endfirsthead %第一页的标题结束\caption{(continued)}\\ %第二页的标题\endhead %第二页的标题结束\hline\hline\textbf{Symbol}&\textbf{Description}\\\hline$n_{ij}$&The airplane number of the $i^{\mathrm{th}}$ type in the $j^{\mathrm{th}}$ flight set\\\hline${NP}_i\:(i=1,2,\ldots)$&The number of passengers on each airplanes of the same type.\\\hline$\xi_{ij}\:(i,j = 1,2,\cdots)$&The number of baggage on each airplane of the $j^{\mathrm{th}}$ flights\\\hline$a$&The maximal number of airplanes type\\\hline$B_j^{set}$&The total baggage number of each set of flight\\\hline${NF}_i$&Number of airplanes of each type\\\hline$\bar{\rho}$&The mean value of passengers' baggage coming per minute in every flight set\\ \hline$N_{set}$&The number of flight sets\\\hline$B_{total}$&The total number of checked baggage during the peak hour\\\hline$H_{peak}$&The length of the peak hour\\\hline$T_{set}$&The time length during which each flight set's passengers wait to be checked\\\hline$\Delta t$&The time interval between two consecutive flight set\\\hline$N_{EDS}$&The number of all the EDSs\\\hline$N_{shadow}$&The number of flight sets whose passengers will be mixed up before being checked\\\hline$v_{EDS}$&The number of baggage checking by one EDS per minute\\\hline$\rho_j$&The number of passengers' baggage coming per minute in one flight set\\\hline$N_{runway}$&The number of an airport's runway\\\hline\\*[-2.2ex]${\bar{B}}^{set}$&The mean value of checked baggage number of every flight set\\\hline$M$&The security cost\\\hline\hline\label{tab1}\end{longtable}\subsubsection{Definition:}\begin{description}\item[Flight set] A group of flights take off at the same time\item[The length of peak hour] The time between the first set of flight and the last set\end{description}\section{Basic Model}During a peak hour, many planes and many passengers would departfrom airports. Therefore, It is difficult to arrange for thepassengers to enter airports. If there are not enough EDSs forpassengers' baggage to check, it will take too long time for themto enter. That would result in the delay of airplanes. On thecontrary, if there are too many EDSs, it will be a waste. It isour task to find a suitable number of EDSs for airport. In orderto reach this objective, we use the linear programming method tosolve it.\subsection{Base analysis}The airplanes are occupied at least partly. The passengers'baggage would be checked by EDSs before they get on the airplanes.We have assumed that every passenger carry two baggages. Thisassumption would simplify the problem. According to the data fromthe problem sheet, we can obtain the useful information thatairlines claim 20{\%} of the passengers do not check any luggage,20{\%} check one bag, and the remaining passengers check two bags.Therefore, we can gain the total number of passengers' baggagethat should be carried on one plane: $\xi_{ij}$. Moreover, we canget the equation that calculate $\xi_{ij}$:\[\xi_{ij}={NP}_i\times 20\%+{NP}_i\times 60\%\times 2\]We define the matrix below as airplane baggage number matrix:\[\overset{\rightharpoonup}{\xi}_j=\left[\xi_{1j}\quad\xi_{2j}\quad\cdots\quad\xi_{ij}\quad\cdots\ right]\]We define the matrix below as flight schedule matrix:\[\left[\begin{array}{llcl}n_{11}&n_{12}&\cdots&n_{1N_{set}}\\n_{21}&n_{22}&\cdots&n_{2N_{set}}\\\multicolumn{4}{c}\dotfill\\n_{a1}&n_{a2}&\cdots&n_{aN_{set}}\end{array}\right]\]In this matrix, $n_{ij}$ is the airplane number of the$i^{\mathrm{th}}$ type in the $j^{\mathrm{th}}$ flight set whichwill take off. Apparently, this value is an integer.We define the matrix below as flight set baggage number matrix:\[\left[B_1^{set}\quad B_2^{set}\quad\cdots\quad B_j^{set}\quad\cdots\quad B_a^{set}\right] \]It is clear that they meet the relation below:\begin{equation}\begin{array}{cl}&\left[\xi_{1j}\quad\xi_{2j}\quad\cdots\quad\xi_{ij}\quad\cdots\right]\cdot\left[\begin{array}{llcl}n_{11}&n_{12}&\cdots&n_{1N_{set}}\\n_{21}&n_{22}&\cdots&n_{2N_{set}}\\\multicolumn{4}{c}\dotfill\\n_{a1}&n_{a2}&\cdots&n_{aN_{set}}\end{array}\right]\\=&\left[B_1^{set}\quad B_2^{set}\quad\cdots\quad B_j^{set}\quad\cdots\quadB_a^{set}\right]\end{array}\label{Flight:baggage}\end{equation}Then, we know:\[B_j^{set}=\sum\limits_{i=1}^a\xi_{ij}\times n_{ij}\]There are some constraints to the equation (\ref{Flight:baggage}).First, for each set of flight, the total number of airplanesshould be less than the number of runways. Second, the totalairplane number of the same type listed in the equation(\ref{Flight:baggage}) from every set of flight should be equal tothe actual airplane number of the same type during the peak hour.We can express them like these:\[\sum\limits_{i=1}^a n_{ij}\le N_{runway}\quad\quad\sum\limits_{j=1}^b n_{ij}={NF}_i \]We should resolve the number of flight sets. According to our assumptions,during the peak hour, the airlines should make the best use of the runways.Then get the number of flight sets approximately based on the number of allthe airplanes during the peak hour and that of the runways. We use anequation below to express this relation:\begin{equation}N_{set}=\left\lceil\frac{\sum\limits_{j=1}^{N_{set}}\sum\limits_{i=1}^an_{ij}}{N_{runway}}\right\rceil\label{sets:number}\end{equation}The checked baggage numbers of each flight set are equal to eachother according to our assumption. We make it based on literature.It can also simplify our model. We define $\bar{B}^{set}$ as themean value of checked baggage number of every flight set.Moreover, We define $\bar{\rho}$ as the mean value of checkedbaggage number of every flight set per minute:\[\bar{B}^{set}=\frac{B_{total}}{N_{set}}\]\[\bar{\rho}=\frac{\bar{B}^{set}}{T_{set}}=\frac{B_{total}}{T_{set}N_{set}}=\frac{B_{total}\Delta t}{T_{set}H_{peak}}\]The course of passengers' arrival and entering airport isimportant for us to decide the number of EDSs and to make the flights schedule. Therefore, we should analyze this process carefully. Passengers will arrive between forty-five minutes andtwo hours prior to the departure time, and the passengers who will get on the same airplane will arrive uniformly. Then we can getthe flow density of all checked baggage at any time during passengers' entering. This value is the sum of numbers of passengers' checked baggage coming per minute. To calculate this value, firstly, we should obtain flow density of each flight set's checked baggage. We define $\rho_j $, namely the number of checked baggage per minute of one flight set:\[\rho_j=\frac{B_j^{set}}{T_{set}}\]Secondly, we draw graphic to help us to understand. We use rectangle to express the time length for all the passengers of one flight set to come and check bags. In the graphic, the black partis the period for them to come. During the white part, no passengers for this flight set come. According to the problem sheet, the former is 75 minute, and the latter is 45 minute. The length of rectangle is 120 minute. $T_{set}$ is the period during which all passengers of one flight set wait to be checked. Sincewe have assumed that each time interval between two consecutive flight set is same value, we define $\Delta t$ as it. Observe the section that value we want to solve is $\sum\limits_j\rho_j$. Moreover, we can get another important equation from the graphic below:\begin{equation}N_{set}=\frac{H_{peak}}{\Delta t}\label{PeakHour}\end{equation}\begin{figure}[hbtp]\centering\includegraphics[width=298.2pt,totalheight=141.6pt]{fig01.eps} \caption{}\label{fig1}\end{figure}Each EDS has certain capacity. If the number of EDSs is $N_{EDS}$and one EDS can check certain number of baggage per minute (Thatis checking velocity, marked by $v_{EDS}$), the total checkingcapacity is $N_{EDS}\cdot\frac{v_{EDS}}{60}$. $v_{EDS}$ is between160 and 210.Now we can easily decide in what condition the passengers can be checkedwithout delay:\[\sum\limits_j\rho_j\le v_{EDS}\]The passengers have to queue before being checked:$\sum\limits_j\rho_j>v_{EDS}$Well then, how can we decide how many $\rho_j$? It depends on howmany flight sets whose passengers will be mixed up before beingchecked. We note it as $N_{shadow} $. Return to the Figure\ref{fig1}, we can know:\[N_{shadow}=\left\lfloor\frac{T_{set}}{\Delta t}\right\rfloor\]\begin{figure}%[htbp]\centering\includegraphics[width=240pt,totalheight=131.4pt]{fig02.eps}\caption{}\label{fig2}\end{figure}From Figure \ref{fig1} and Figure \ref{fig2}, we can get theresult as follows:\begin{enumerate}\item If $N_{shadow}\le N_{set}$, namely $H_{peak}>T_{set}$, then $\sum\limits_{j=1}^{N_{shadow}}\rho _j\le N_{EDS}\frac{v_{EDS}}{60}$\renewcommand{\theequation}{\arabic{equation}a}That is:\begin{equation}N_{EDS}\ge\frac{60}{v_{EDS}}\sum\limits_{j=1}^{N_{shadow}}\rho_j\approx\frac{60}{v_{ EDS}}N_{shadow}\bar{\rho}=\frac{60B_{total}\Deltat}{v_{EDS}T_{set}H_{peak}}N_{shadow}\label{EDS:number:a}\end{equation}\item If $N_{shadow}>N_{set}$, namely $H_{peak}\le T_{set}$, then $\sum\limits_{j=1}^{N_{set}}\rho_j\le N_{EDS}\frac{v_{EDS}}{60}$\setcounter{equation}{3}\renewcommand{\theequation}{\arabic{equation}b}That is:\begin{equation}N_{EDS}\ge\frac{60}{v_{EDS}}\sum\limits_{j=1}^{N_{set}}\rho_j\approx\frac{60}{v_{EDS} }N_{set}\bar{\rho}=\frac{60B_{total}\Delta t}{v_{EDS}T_{set}H_{peak}}N_{set}\label{EDS:number:b}\end{equation}\end{enumerate}\subsection{The number of EDSs}Then we begin to resolve the number of EDSs assisted by the linearprogramming method.EDS is operational about 92{\%} of the time. That is to say, whenever it isduring a peak hour, there are some EDSs stopping working. Then the workingefficiency of all the EDSs is less than the level we have expected.Therefore, the airline has to add more EDSs to do the work, which can bedone with less EDSs without downtime.We use binomial distribution to solve this problem. $N$ is the number ofactual EDSs with downtime and $k$ is the number of EDSs without downtime. Ifprobability is $P$, we can get the equation below:\[\left(\begin{array}{c}N\\k\end{array}\right)\cdot98\%^k\cdot(1-98\%)^{N-k}=P\]We can obtain $N$ when we give $P$ a certain value. In this paper,$P$ is 95{\%}. The $N_{EDS}$ is the actual number we obtainthrough the equation above.Now we have assumed that passengers can be checked unless be delayed by thepeople before him once he arrives at airport. Apparently, if the time lengthbetween two sets of flight is short, the density of passengers will begreat. It will bring great stress to security check and may even make somepassengers miss their flight. To resolve this question, the airline has toinstall more EDSs to meet the demand. However, this measure will cost muchmore money. Consequently, we have to set a suitable time interval betweentwo set of flight.Based on the base analysis above. We can use the equation(\ref{sets:number}) to decide the number of flight sets $N_{set}$assuming we know the number of runways of a certain airport. Thenbased on the equation (\ref{PeakHour}), we can decide the peakhour length $H_{peak}$ when we assume a time interval between two consecutive flight sets. Then we use \textcircled{1} and\textcircled{2} to decide which to choose between equation(\ref{EDS:number:a}) and equation (\ref{EDS:number:b}). In consequence, we can obtain the minimum of EDSs number.If we choose different numbers of runways and the time intervalsbetween two flight sets, we can get different EDSs numbers. Inthis paper that followed, we gain a table of some value of$N_{runway}$ and $\Delta t$ with the corresponding EDSs numbers. Moreover, we draw some figure to reflect their relation.For a certain airport, its number of runway is known. Givencertain time interval ($\Delta t$), we can get the length of thepeak hour ($H_{peak}$). When the $N_{runway}$ is few enough,perhaps $H_{peak}$ is too long to be adopted. However, for acertain airline, they can decide the time interval of their ownpeak hour. In this given time interval, they could find theminimum of $N_{runway}$ through the Figure \ref{fig3}. We draw a sketch map to describe our steps.\begin{figure}[hbtp]\centering\includegraphics[width=352.8pt,totalheight=214.2pt]{fig03.eps}\caption{}\label{fig3}\end{figure}\subsection{The Flight Schedule }According to the base analysis, we can know that the flightschedule matrix and $\Delta t$ is one form of flight timetable. In``The number of EDSs'', we can get suitable $\Delta t$. Then weshould resolve the flight schedule matrix.Because we have assumed that the checked baggage numbers of each flight set are equal to each other. It can be described as follows:\[\left\{\begin{array}{l}\rho_j\approx\bar{\rho}\\B_j^{set}\approx\bar{B}^{set}\end{array}\right.\begin{array}{*{20}c}\hfill&{j=1,2,\cdots,N_{set}}\hfill\end{array}\]The flight schedule matrix subject to this group:\[\left\{\begin{array}{ll}\sum\limits_{j=1}^{N_{set}}n_{ij}={NF}_i&i=1,2,\cdots\\\sum\limits_{i=1}^a n_{ij}\le N_{runway}&j=1,2,\cdots,N_{set}\\n_{ij}\ge0,&\mathrm{and}\:n_{ij}\:\mathrm{is}\:\mathrm{a}\:\mathrm{Integer}\end{array}\right.\]In order to make the best use of runway, we should make$\sum\limits_{i=1}^a n_{ij}$ as great as we can unless it exceed$N_{runway}$.Then we can see that how to resolve the flight schedule matrix is a problemof divide among a group of integers. This group is all the numbers of eachflight passengers' baggage in one flight set. We program for this problemusing MA TLAB and we get at least one solution in the end. However, thematrix elements we have obtained are not integer, we have to adjust them tobe integers manually.\subsection{Results and Interpretation for Airport A and B}The number of passengers in a certain flight (${NP}_i$), the timelength of security checking ($T_{set}$), the checking velocity ofEDS ($v_{EDS}$), and the number of baggage carried by onepassenger are random.\subsubsection{Data Assumption:}\begin{itemize}\item $T_{set}$ is 110 minutes, which is reasonable for airline.\item To simplify the problem, we assume that every passenger carry 2 baggage. If some of thepassengers carry one baggage, the solution based on 2 baggages per passenger meets therequirement.\item The number of runways in airport A and airport B is 5.\end{itemize}\subsubsection{Airport A:}Once the number of runway and the number of the flights aredecided, the flight schedule matrix is decided, too. We producethis matrix using MATLAB. This matrix companied by $\Delta t$ isthe flight schedule for airport A. $\Delta t$ will be calculatedin (\ref{Flight:baggage}), (\ref{sets:number}) and(\ref{PeakHour}).We calculate $N_{EDS}$ and make the flight timetable in threeconditions. The three conditions and the solution are listed asfollowed:\paragraph{Every flight are fully occupied}The checking speed of EDS is 160 bags/hour.\begin{table}[htbp]\centering\caption{}\begin{tabular}{*{11}c}\myhline{0.4mm}$\mathbf{\Deltat(\min)}$&\textbf{2}&\textbf{4}&\textbf{6}&\textbf{8}&\textbf{10}&\textbf{12}&\textbf{14} &\textbf{16}&\textbf{18}&\textbf{20}\\\myhline{0.4mm}$N_{EDS}(\ge)$&31&31&31&31&31&29&24&22&20&17\\\hline$H_{peak}(\min)$&20&40&60&80&100&120&140&160&180&200\\\myhline{0.4mm}\end{tabular}\label{tab2}\end{table}We assume that the suitable value of $H_{peak}$ is 120 minutes.Then the suitable value of $\Delta t$ is about 12 minutes, and$N_{EDS}$ is 29 judged from Figure \ref{fig4}. Certainly, we canwork $\Delta t$ and $N_{EDS}$ out through equation.\begin{figure}[htbp]\centering\includegraphics[width=294.6pt,totalheight=253.2pt]{fig04.eps}\caption{}\label{fig4}\end{figure}\paragraph{Every flight is occupied by the minimal number of passengers onstatistics in the long run.}The checking speed of EDS is 210 bags/hour.\begin{table}[htbp]\centering\caption{}\begin{tabular}{*{11}c}\myhline{0.4mm}$\mathbf{\Deltat(\min)}$&\textbf{2}&\textbf{4}&\textbf{6}&\textbf{8}&\textbf{10}&\textbf{12}&\textbf{14} &\textbf{16}&\textbf{18}&\textbf{20}\\\myhline{0.4mm}$N_{EDS}(\ge)$&15&15&15&15&15&14&13&12&10&7\\\hline$H_{peak}(\min)$&20&40&60&80&100&120&140&160&180&200\\\myhline{0.4mm}\end{tabular}\label{tab3}\end{table}We assume that the suitable value of $H_{peak}$ is 120 minutes.Then the suitable value of $\Delta t$ is about 12 minutes, and$N_{EDS}$ is 14 judging from Figure \ref{fig5}. Certainly, we canwork $\Delta t$ and $N_{EDS}$ out through equation.\begin{figure}[htbp]\centering\includegraphics[width=294.6pt,totalheight=253.2pt]{fig05.eps}\caption{}\label{fig5}\end{figure}\paragraph{${NP}_i$ and $v_{EDS}$ are random value produced by MATLAB.}\begin{table}[htbp]\centering\caption{}\begin{tabular}{*{11}c}\myhline{0.4mm}$\mathbf{\Deltat(\min)}$&\textbf{2}&\textbf{4}&\textbf{6}&\textbf{8}&\textbf{10}&\textbf{12}&\textbf{14} &\textbf{16}&\textbf{18}&\textbf{20}\\\myhline{0.4mm}$N_{EDS}(\ge)$&15&22&21&21&15&17&21&16&13&14\\\hline$H_{peak}(\min)$&20&40&60&80&100&120&140&160&180&200\\\myhline{0.4mm}\end{tabular}\label{tab4}\end{table}We assume that the suitable value of $H_{peak}$ is 120 minutes.Then the suitable value of $\Delta t$ is about 12 minutes, and$N_{EDS}$ is 17 judging from Figure \ref{fig6}. Certainly, we canwork $\Delta t$ and $N_{EDS}$ out through equation.\begin{figure}[htbp]\centering\includegraphics[width=294.6pt,totalheight=249.6pt]{fig06.eps}\caption{}\label{fig6}\end{figure}\subsubsection{Interpretation:}By analyzing the results above, we can conclude that when$N_{EDS}$ is 29, and $\Delta t$ is 12, the flight schedule willmeet requirement at any time. The flight schedule is:\\[\intextsep]\begin{minipage}{\textwidth}\centering\tabcaption{}\begin{tabular}{c|*{8}c|c|c}\myhline{0.4mm}\backslashbox{\textbf{Set}}{\textbf{Type}}&\textbf{1}&\textbf{2}&\textbf{3}&\textbf{4}&\te xtbf{5}&\textbf{6}&\textbf{7}&\textbf{8}&\textbf{Numbers of Bags}&\textbf{Numbers of Flights}\\\myhline{0.4mm}1&2&0&0&0&2&1&0&0&766&5\\\hline2&2&0&2&0&2&0&0&0&732&4\\\hline3&0&1&1&1&2&0&0&0&762&4\\\hline4&0&1&0&0&2&1&0&0&735&4\\\hline5&0&1&0&0&2&1&0&0&735&5\\\hline6&2&0&0&0&1&0&0&1&785&5\\\hline7&2&0&0&0&2&0&1&0&795&5\\\hline8&0&1&0&0&2&1&0&0&735&4\\\hline9&2&0&0&0&2&1&0&0&766&5\\\hline10&0&0&0&2&2&0&0&0&758&5\\\hlineTotal&10&4&3&3&19&5&1&1&7569&46\\\myhline{0.4mm}\end{tabular}\label{tab5}\end{minipage}\\[\intextsep]We have produced random value for ${NP}_i$ and $v_{EDS}$. On thiscondition, the number of EDSs is 17, which is less than 29 that wedecide for the airport A. That is to say our solution can meet thereal requirement.\subsubsection{Airport B:}\paragraph{The passenger load is 100{\%}}The checking speed of EDS is 160 bags/hour.\begin{table}[htbp]\centering\caption{}\begin{tabular}{*{11}c}\myhline{0.4mm}$\mathbf{\Deltat(\min)}$&\textbf{2}&\textbf{4}&\textbf{6}&\textbf{8}&\textbf{10}&\textbf{12}&\textbf{14} &\textbf{16}&\textbf{18}&\textbf{20}\\\myhline{0.4mm}$N_{EDS}(\ge)$&33&33&33&33&33&30&27&23&21&19\\\hline$H_{peak}(\min)$&20&40&60&80&100&120&140&160&180&200\\\myhline{0.4mm}\end{tabular}\label{tab6}\end{table}We assume that the suitable value of $H_{peak}$ is 120 minutes.Then the suitable value of $\Delta t$ is about 12 minutes, and。
数学建模美赛写作模版(包含摘要、格式、总结、表格、公式、图表、假设)
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美国大学生数学建模竞赛二等奖论文
美国⼤学⽣数学建模竞赛⼆等奖论⽂The P roblem of R epeater C oordination SummaryThis paper mainly focuses on exploring an optimization scheme to serve all the users in a certain area with the least repeaters.The model is optimized better through changing the power of a repeater and distributing PL tones,frequency pairs /doc/d7df31738e9951e79b8927b4.html ing symmetry principle of Graph Theory and maximum coverage principle,we get the most reasonable scheme.This scheme can help us solve the problem that where we should put the repeaters in general cases.It can be suitable for the problem of irrigation,the location of lights in a square and so on.We construct two mathematical models(a basic model and an improve model)to get the scheme based on the relationship between variables.In the basic model,we set a function model to solve the problem under a condition that assumed.There are two variables:‘p’(standing for the power of the signals that a repeater transmits)and‘µ’(standing for the density of users of the area)in the function model.Assume‘p’fixed in the basic one.And in this situation,we change the function model to a geometric one to solve this problem.Based on the basic model,considering the two variables in the improve model is more reasonable to most situations.Then the conclusion can be drawn through calculation and MATLAB programming.We analysis and discuss what we can do if we build repeaters in mountainous areas further.Finally,we discuss strengths and weaknesses of our models and make necessary recommendations.Key words:repeater maximum coverage density PL tones MATLABContents1.Introduction (3)2.The Description of the Problem (3)2.1What problems we are confronting (3)2.2What we do to solve these problems (3)3.Models (4)3.1Basic model (4)3.1.1Terms,Definitions,and Symbols (4)3.1.2Assumptions (4)3.1.3The Foundation of Model (4)3.1.4Solution and Result (5)3.1.5Analysis of the Result (8)3.1.6Strength and Weakness (8)3.1.7Some Improvement (9)3.2Improve Model (9)3.2.1Extra Symbols (10)Assumptions (10)3.2.2AdditionalAdditionalAssumptions3.2.3The Foundation of Model (10)3.2.4Solution and Result (10)3.2.5Analysis of the Result (13)3.2.6Strength and Weakness (14)4.Conclusions (14)4.1Conclusions of the problem (14)4.2Methods used in our models (14)4.3Application of our models (14)5.Future Work (14)6.References (17)7.Appendix (17)Ⅰ.IntroductionIn order to indicate the origin of the repeater coordination problem,the following background is worth mentioning.With the development of technology and society,communications technology has become much more important,more and more people are involved in this.In order to ensure the quality of the signals of communication,we need to build repeaters which pick up weak signals,amplify them,and retransmit them on a different frequency.But the price of a repeater is very high.And the unnecessary repeaters will cause not only the waste of money and resources,but also the difficulty of maintenance.So there comes a problem that how to reduce the number of unnecessary repeaters in a region.We try to explore an optimized model in this paper.Ⅱ.The Description of the Problem2.1What problems we are confrontingThe signals transmit in the way of line-of-sight as a result of reducing the loss of the energy. As a result of the obstacles they meet and the natural attenuation itself,the signals will become unavailable.So a repeater which just picks up weak signals,amplifies them,and retransmits them on a different frequency is needed.However,repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.In addition to geographical separation,the“continuous tone-coded squelch system”(CTCSS),sometimes nicknamed“private line”(PL),technology can be used to mitigate interference.This system associates to each repeater a separate PL tone that is transmitted by all users who wish to communicate through that repeater. The PL tone is like a kind of password.Then determine a user according to the so called password and the specific frequency,in other words a user corresponds a PL tone(password)and a specific frequency.Defects in line-of-sight propagation caused by mountainous areas can also influence the radius.2.2What we do to solve these problemsConsidering the problem we are confronting,the spectrum available is145to148MHz,the transmitter frequency in a repeater is either600kHz above or600kHz below the receiver frequency.That is only5users can communicate with others without interferences when there’s noPL.The situation will be much better once we have PL.However the number of users that a repeater can serve is limited.In addition,in a flat area ,the obstacles such as mountains ,buildings don’t need to be taken into account.Taking the natural attenuation itself is reasonable.Now the most important is the radius that the signals transmit.Reducing the radius is a good way once there are more users.With MATLAB and the method of the coverage in Graph Theory,we solve this problem as follows in this paper.Ⅲ.Models3.1Basic model3.1.1Terms,Definitions,and Symbols3.1.2Assumptions●A user corresponds a PLz tone (password)and a specific frequency.●The users in the area are fixed and they are uniform distribution.●The area that a repeater covers is a regular hexagon.The repeater is in the center of the regular hexagon.●In a flat area ,the obstacles such as mountains ,buildings don’t need to be taken into account.We just take the natural attenuation itself into account.●The power of a repeater is fixed.3.1.3The Foundation of ModelAs the number of PLz tones (password)and frequencies is fixed,and a user corresponds a PLz tone (password)and a specific frequency,we can draw the conclusion that a repeater can serve the limited number of users.Thus it is clear that the number of repeaters we need relates to the density symboldescriptionLfsdfminrpµloss of transmission the distance of transmission operating frequency the number of repeaters that we need the power of the signals that a repeater transmits the density of users of the areaof users of the area.The radius of the area that a repeater covers is also related to the ratio of d and the radius of the circular area.And d is related to the power of a repeater.So we get the model of function()min ,r f p µ=If we ignore the density of users,we can get a Geometric model as follows:In a plane which is extended by regular hexagons whose side length are determined,we move a circle until it covers the least regular hexagons.3.1.4Solution and ResultCalculating the relationship between the radius of the circle and the side length of the regular hexagon.[]()()32.4420lg ()20lg Lfs dB d km f MHz =++In the above formula the unit of ’’is .Lfs dB The unit of ’’is .d Km The unit of ‘‘is .f MHz We can conclude that the loss of transmission of radio is decided by operating frequency and the distance of transmission.When or is as times as its former data,will increase f d 2[]Lfs .6dB Then we will solve the problem by using the formula mentioned above.We have already known the operating frequency is to .According to the 145MHz 148MHz actual situation and some authority material ,we assume a system whose transmit power is and receiver sensitivity is .Thus we can conclude that ()1010dBm mW +106.85dBm ?=.Substituting and to the above formula,we can get the Lfs 106.85dBm ?145MHz 148MHz average distance of transmission .()6.4d km =4mile We can learn the radius of the circle is 40mile .So we can conclude the relationship between the circle and the side length of regular hexagon isR=10d.1)The solution of the modelIn order to cover a certain plane with the least regular hexagons,we connect each regular hexagon as the honeycomb.We use A(standing for a figure)covers B(standing for another figure), only when As don’t overlap each other,the number of As we use is the smallest.Figure1According to the Principle of maximum flow of Graph Theory,the better of the symmetry ofthe honeycomb,the bigger area that it covers(Fig1).When the geometric centers of the circle andthe honeycomb which can extend are at one point,extend the honeycomb.Then we can get Fig2,Fig4:Figure2Fig3demos the evenly distribution of users.Figure4Now prove the circle covers the least regular hexagons.Look at Fig5.If we move the circle slightly as the picture,you can see three more regular hexagons are needed.Figure 52)ResultsThe average distance of transmission of the signals that a repeater transmit is 4miles.1000users can be satisfied with 37repeaters founded.3.1.5Analysis of the Result1)The largest number of users that a repeater can serveA user corresponds a PL and a specific frequency.There are 5wave bands and 54different PL tones available.If we call a code include a PL and a specific frequency,there are 54*5=270codes.However each code in two adjacent regular hexagons shouldn’t be the same in case of interfering with each other.In order to have more code available ,we can distribute every3adjacent regular hexagons 90codes each.And that’s the most optimized,because once any of the three regular hexagons have more codes,it will interfere another one in other regular hexagon.2)Identify the rationality of the basic modelNow we considering the influence of the density of users,according to 1),90*37=3330>1000,so here the number of users have no influence on our model.Our model is rationality.3.1.6Strength and Weakness●Strength:In this paper,we use the model of honeycomb-hexagon structure can maximize the use of resources,avoiding some unnecessary interference effectively.It is much more intuitive once we change the function model to the geometric model.●Weakness:Since each hexagon get too close to another one.Once there are somebuildingsor terrain fluctuations between two repeaters,it can lead to the phenomenon that certain areas will have no signals.In addition,users are distributed evenly is not reasonable.The users are moving,for example some people may get a party.3.1.7Some ImprovementAs we all know,the absolute evenly distribution is not exist.So it is necessary to say something about the normal distribution model.The maximum accommodate number of a repeater is 5*54=270.As for the first model,it is impossible that 270users are communicating in a same repeater.Look at Fig 6.If there are N people in the area 1,the maximum number of the area 2to area 7is 3*(270-N).As 37*90=3330is much larger than 1000,our solution is still reasonable to this model.Figure 63.2Improve Model3.2.1Extra SymbolsSigns and definitions indicated above are still valid.Here are some extra signs and definitions.symboldescription Ra the radius of the circular flat area the side length of a regular hexagon3.2.2Additional AdditionalAssumptionsAssumptions ●The radius that of a repeater covers is adjustable here.●In some limited situations,curved shape is equal to straight line.●Assumptions concerning the anterior process are the same as the Basic Model3.2.3The Foundation of ModelThe same as the Basic Model except that:We only consider one variable(p)in the function model of the basic model ;In this model,we consider two varibles(p and µ)of the function model.3.2.4Solution and Result1)SolutionIf there are 10,000users,the number of regular hexagons that we need is at least ,thus according to the the Principle of maximum flow of Graph Theory,the 10000111.1190=result that we draw needed to be extended further.When the side length of the figure is equal to 7Figure 7regular hexagons,there are 127regular hexagons (Fig 7).Assuming the side length of a regular hexagon is ,then the area of a regular hexagon is a .The area of regular hexagons is equal to a circlewhose radiusis 22a =1000090R.Then according to the formula below:.221000090a R π=We can get.9.5858R a =Mapping with MATLAB as below (Fig 8):Figure 82）Improve the model appropriatelyEnlarge two part of the figure above,we can get two figures below (Fig 9and Fig 10):Figure 9AREAFigure 10Look at the figure above,approximatingAREA a rectangle,then obtaining its area to getthe number of users..The length of the rectangle is approximately equal to the side length of the regular hexagon ,athe width of the rectangle is ,thus the area of AREA is ,then R ?*R awe can get the number of users in AREA is()，2**10000 2.06R a R π=????????9.5858R a =As 2.06<<10,000,2.06can be ignored ,so there is no need to set up a repeater in.There are 6suchareas(92,98,104,110,116,122)that can be ignored.At last,the number of repeaters we should set up is,1276121?=2)Get the side length of the regular hexagon of the improved modelThus we can getmile=km 40 4.1729.5858a == 1.6* 6.675a =3)Calculate the power of a repeaterAccording to the formula[]()()32.4420lg ()20lg Lfs dB d km f MHz =++We get32.4420lg 6.67520lg14592.156Los =++=32.4420lg 6.67520lg14892.334Los =++=So we get106.85-92.156=14.694106.85-92.334=14.516As the result in the basic model,we can get the conclusion the power of a repeater is from 14.694mW to 14.516mW.3.2.5Analysis of the ResultAs 10,000users are much more than 1000users,the distribution of the users is more close toevenly distribution.Thus the model is more reasonable than the basic one.More repeaters are built,the utilization of the outside regular hexagon are higher than the former one.3.2.6Strength and Weakness●Strength:The model is more reasonable than the basic one.●Weakness:Repeaters don’t cover all the area,some places may not receive signals.And thefoundation of this model is based on the evenly distribution of the users in the area,if the situation couldn’t be satisfied,the interference of signals will come out.Ⅳ.Conclusions4.1Conclusions of the problem●Generally speaking,the radius of the area that a repeater covers is4miles in our basic model.●Using the model of honeycomb-hexagon structure can maximize the use of resources,avoiding some unnecessary interference effectively.●The minimum number of repeaters necessary to accommodate1,000simultaneous users is37.The minimum number of repeaters necessary to accommodate10,000simultaneoususers is121.●A repeater's coverage radius relates to external environment such as the density of users andobstacles,and it is also determined by the power of the repeater.4.2Methods used in our models●Analysis the problem with MATLAB●the method of the coverage in Graph Theory4.3Application of our models●Choose the ideal address where we set repeater of the mobile phones.●How to irrigate reasonably in agriculture.●How to distribute the lights and the speakers in squares more reasonably.Ⅴ.Future WorkHow we will do if the area is mountainous?5.1The best position of a repeater is the top of the mountain.As the signals are line-of-sight transmission and reception.We must find a place where the signals can transmit from the repeater to users directly.So the top of the mountain is a good place.5.2In mountainous areas,we must increase the number of repeaters.There are three reasons for this problem.One reason is that there will be more obstacles in the mountainous areas. The signals will be attenuated much more quickly than they transmit in flat area.Another reason is that the signals are line-of-sight transmission and reception,we need more repeaters to satisfy this condition.Then look at Fig11and Fig12,and you will know the third reason.It can be clearly seen that hypotenuse is larger than right-angleFig11edge(R>r).Thus the radius will become smaller.In this case more repeaters are needed.Fig125.3In mountainous areas,people may mainly settle in the flat area,so the distribution of users isn’t uniform.5.4There are different altitudes in the mountainous areas.So in order to increase the rate of resources utilization,we can set up the repeaters in different altitudes.5.5However,if there are more repeaters,and some of them are on mountains,more money will be/doc/d7df31738e9951e79b8927b4.html munication companies will need a lot of money to build them,repair them when they don’t work well and so on.As a result,the communication costs will be high.What’s worse,there are places where there are many mountains but few persons. Communication companies reluctant to build repeaters there.But unexpected things often happen in these places.When people are in trouble,they couldn’t communicate well with the outside.So in my opinion,the government should take some measures to solve this problem.5.6Another new method is described as follows(Fig13):since the repeater on high mountains can beFig13Seen easily by people,so the tower which used to transmit and receive signals can be shorter.That is to say,the tower on flat areas can be a little taller..Ⅵ.References[1]YU Fei,YANG Lv-xi,"Effective cooperative scheme based on relay selection",SoutheastUniversity,Nanjing,210096,China[2]YANG Ming,ZHAO Xiao-bo,DI Wei-guo,NAN Bing-xin,"Call Admission Control Policy based on Microcellular",College of Electical and Electronic Engineering,Shijiazhuang Railway Institute,Shijiazhuang Heibei050043,China[3]TIAN Zhisheng,"Analysis of Mechanism of CTCSS Modulation",Shenzhen HYT Co,Shenzhen,518057,China[4]SHANGGUAN Shi-qing,XIN Hao-ran,"Mathematical Modeling in Bass Station Site Selectionwith Lingo Software",China University of Mining And Technology SRES,Xuzhou;Shandong Finance Institute,Jinan Shandon,250014[5]Leif J.Harcke,Kenneth S.Dueker,and David B.Leeson,"Frequency Coordination in the AmateurRadio Emergency ServiceⅦ.AppendixWe use MATLAB to get these pictures,the code is as follows:1-clc;clear all;2-r=1;3-rc=0.7;4-figure;5-axis square6-hold on;7-A=pi/3*[0:6];8-aa=linspace(0,pi*2,80);9-plot(r*exp(i*A),'k','linewidth',2);10-g1=fill(real(r*exp(i*A)),imag(r*exp(i*A)),'k');11-set(g1,'FaceColor',[1,0.5,0])12-g2=fill(real(rc*exp(i*aa)),imag(rc*exp(i*aa)),'k');13-set(g2,'FaceColor',[1,0.5,0],'edgecolor',[1,0.5,0],'EraseMode','x0r')14-text(0,0,'1','fontsize',10);15-Z=0;16-At=pi/6;17-RA=-pi/2;18-N=1;At=-pi/2-pi/3*[0:6];19-for k=1:2;20-Z=Z+sqrt(3)*r*exp(i*pi/6);21-for pp=1:6;22-for p=1:k;23-N=N+1;24-zp=Z+r*exp(i*A);25-zr=Z+rc*exp(i*aa);26-g1=fill(real(zp),imag(zp),'k');27-set(g1,'FaceColor',[1,0.5,0],'edgecolor',[1,0,0]);28-g2=fill(real(zr),imag(zr),'k');29-set(g2,'FaceColor',[1,0.5,0],'edgecolor',[1,0.5,0],'EraseMode',xor';30-text(real(Z),imag(Z),num2str(N),'fontsize',10);31-Z=Z+sqrt(3)*r*exp(i*At(pp));32-end33-end34-end35-ezplot('x^2+y^2=25',[-5,5]);%This is the circular flat area of radius40miles radius 36-xlim([-6,6]*r) 37-ylim([-6.1,6.1]*r)38-axis off;Then change number19”for k=1:2;”to“for k=1:3;”,then we get another picture:Change the original programme number19“for k=1:2;”to“for k=1:4;”,then we get another picture:。
正确写作美国大学生数学建模竞赛论文
1）、鉴别阶段： （10分钟）
所有论文在此阶段按其质量分别归入一下三类：第一类 是可以进入下一评审阶段的论文（略少于二分之一）；第二类 是满足竞赛要求，但不足以进入下一评审阶段的论文（这一类 就被定为合格论文）；第三类是不符合竞赛要求的论文（不合 格论文）。 由于在第一阶段中，评委只有10分钟左右的时间评审一 篇论文，因此评委常常只能通过阅读摘要来判断论文水平的高 低。
例如，2010年MCM竞赛中有一道赛题，要求参赛小 组根据以往的作案地点预测连环犯罪的位置。
3.1）、假设条件和解释 解答这道赛题的重点是犯罪活动方式。在一篇题为 “Centroids, Clusters, and Crime: Anchoring the Geographic Profiles of Serial Criminals”的论文中，有一条假设是“罪犯 的活动不受限制”，但罪犯是在市区的活动，实际上会受 到街道的布局及街道两旁建筑物的限制。由于街道布局通 常类似于网格，所以参赛小组对这个假设做了如下解释： Criminal’s movement is 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 discretized into city blocks…
美赛数学建模优秀论文
Why Crime Doesn’t Pay:Locating Criminals Through Geographic ProfilingControl Number:#7272February22,2010AbstractGeographic profiling,the application of mathematics to criminology, has greatly improved police efforts to catch serial criminals byfinding their residence.However,many geographic profiles either generate an extremely large area for police to cover or generates regions that are unstable with respect to internal parameters of the model.We propose,formulate,and test the Gaussian Rossmooth(GRS)Method,which takes the strongest elements from multiple existing methods and combines them into a more stable and robust model.We also propose and test a model to predict the location of the next crime.We tested our models on the Yorkshire Ripper case.Our results show that the GRS Method accurately predicts the location of the killer’s residence.Additionally,the GRS Method is more stable with respect to internal parameters and more robust with respect to outliers than the existing methods.The model for predicting the location of the next crime generates a logical and reasonable region where the next crime may occur.We conclude that the GRS Method is a robust and stable model for creating a strong and effective model.1Control number:#72722Contents1Introduction4 2Plan of Attack4 3Definitions4 4Existing Methods54.1Great Circle Method (5)4.2Centrography (6)4.3Rossmo’s Formula (8)5Assumptions8 6Gaussian Rossmooth106.1Properties of a Good Model (10)6.2Outline of Our Model (11)6.3Our Method (11)6.3.1Rossmooth Method (11)6.3.2Gaussian Rossmooth Method (14)7Gaussian Rossmooth in Action157.1Four Corners:A Simple Test Case (15)7.2Yorkshire Ripper:A Real-World Application of the GRS Method167.3Sensitivity Analysis of Gaussian Rossmooth (17)7.4Self-Consistency of Gaussian Rossmooth (19)8Predicting the Next Crime208.1Matrix Method (20)8.2Boundary Method (21)9Boundary Method in Action21 10Limitations22 11Executive Summary2311.1Outline of Our Model (23)11.2Running the Model (23)11.3Interpreting the Results (24)11.4Limitations (24)12Conclusions25 Appendices25 A Stability Analysis Images252Control number:#72723List of Figures1The effect of outliers upon centrography.The current spatial mean is at the red diamond.If the two outliers in the lower leftcorner were removed,then the center of mass would be locatedat the yellow triangle (6)2Crimes scenes that are located very close together can yield illog-ical results for the spatial mean.In this image,the spatial meanis located at the same point as one of the crime scenes at(1,1)..7 3The summand in Rossmo’s formula(2B=6).Note that the function is essentially0at all points except for the scene of thecrime and at the buffer zone and is undefined at those points..9 4The summand in smoothed Rossmo’s formula(2B=6,φ=0.5, and EPSILON=0.5).Note that there is now a region aroundthe buffer zone where the value of the function no longer changesvery rapidly (13)5The Four Corners Test Case.Note that the highest hot spot is located at the center of the grid,just as the mathematics indicates.15 6Crimes and residences of the Yorkshire Ripper.There are two residences as the Ripper moved in the middle of the case.Someof the crime locations are assaults and others are murders (16)7GRS output for the Yorkshire Ripper case(B=2.846).Black dots indicate the two residences of the killer (17)8GRS method run on Yorkshire Ripper data(B=2).Note that the major difference between this model and Figure7is that thehot zones in thisfigure are smaller than in the original run (18)9GRS method run on Yorkshire Ripper data(B=4).Note that the major difference between this model and Figure7is that thehot zones in thisfigure are larger than in the original run (19)10The boundary region generated by our Boundary Method.Note that boundary region covers many of the crimes committed bythe Sutcliffe (22)11GRS Method onfirst eleven murders in the Yorkshire Ripper Case25 12GRS Method onfirst twelve murders in the Yorkshire Ripper Case263Control number:#727241IntroductionCatching serial criminals is a daunting problem for law enforcement officers around the world.On the one hand,a limited amount of data is available to the police in terms of crimes scenes and witnesses.However,acquiring more data equates to waiting for another crime to be committed,which is an unacceptable trade-off.In this paper,we present a robust and stable geographic profile to predict the residence of the criminal and the possible locations of the next crime.Our model draws elements from multiple existing models and synthesizes them into a unified model that makes better use of certain empirical facts of criminology.2Plan of AttackOur objective is to create a geographic profiling model that accurately describes the residence of the criminal and predicts possible locations for the next attack. In order to generate useful results,our model must incorporate two different schemes and must also describe possible locations of the next crime.Addi-tionally,we must include assumptions and limitations of the model in order to ensure that it is used for maximum effectiveness.To achieve this objective,we will proceed as follows:1.Define Terms-This ensures that the reader understands what we aretalking about and helps explain some of the assumptions and limitations of the model.2.Explain Existing Models-This allows us to see how others have at-tacked the problem.Additionally,it provides a logical starting point for our model.3.Describe Properties of a Good Model-This clarifies our objectiveand will generate a sketelon for our model.With this underlying framework,we will present our model,test it with existing data,and compare it against other models.3DefinitionsThe following terms will be used throughout the paper:1.Spatial Mean-Given a set of points,S,the spatial mean is the pointthat represents the middle of the data set.2.Standard Distance-The standard distance is the analog of standarddeviation for the spatial mean.4Control number:#727253.Marauder-A serial criminal whose crimes are situated around his or herplace of residence.4.Distance Decay-An empirical phenomenon where criminal don’t traveltoo far to commit their crimes.5.Buffer Area-A region around the criminal’s residence or workplacewhere he or she does not commit crimes.[1]There is some dispute as to whether this region exists.[2]In our model,we assume that the buffer area exists and we measure it in the same spatial unit used to describe the relative locations of other crime scenes.6.Manhattan Distance-Given points a=(x1,y1)and b=(x2,y2),theManhattan distance from a to b is|x1−x2|+|y1−y2|.This is also known as the1−norm.7.Nearest Neighbor Distance-Given a set of points S,the nearestneighbor distance for a point x∈S ismin|x−s|s∈S−{x}Any norm can be chosen.8.Hot Zone-A region where a predictive model states that a criminal mightbe.Hot zones have much higher predictive scores than other regions of the map.9.Cold Zone-A region where a predictive model scores exceptionally low. 4Existing MethodsCurrently there are several existing methods for interpolating the position of a criminal given the location of the crimes.4.1Great Circle MethodIn the great circle method,the distances between crimes are computed and the two most distant crimes are chosen.Then,a great circle is drawn so that both of the points are on the great circle.The midpoint of this great circle is then the assumed location of the criminal’s residence and the area bounded by the great circle is where the criminal operates.This model is computationally inexpensive and easy to understand.[3]Moreover,it is easy to use and requires very little training in order to master the technique.[2]However,it has certain drawbacks.For example,the area given by this method is often very large and other studies have shown that a smaller area suffices.[4]Additionally,a few outliers can generate an even larger search area,thereby further slowing the police effort.5Control number:#727264.2CentrographyIn centrography ,crimes are assigned x and y coordinates and the “center of mass”is computed as follows:x center =n i =1x i ny center =n i =1y i nIntuitively,centrography finds the mean x −coordinate and the mean y -coordinate and associates this pair with the criminal’s residence (this is calledthe spatial mean ).However,this method has several flaws.First,it can be unstablewith respect to outliers.Consider the following set of points (shown in Figure 1:Figure 1:The effect of outliers upon centrography.The current spatial mean is at the red diamond.If the two outliers in the lower left corner were removed,then the center of mass would be located at the yellow triangle.Though several of the crime scenes (blue points)in this example are located in a pair of upper clusters,the spatial mean (red point)is reasonably far away from the clusters.If the two outliers are removed,then the spatial mean (yellow point)is located closer to the two clusters.A similar method uses the median of the points.The median is not so strongly affected by outliers and hence is a more stable measure of the middle.[3]6Control number:#72727 Alternatively,we can circumvent the stability problem by incorporating the 2-D analog of standard deviation called the standard distance:σSD=d center,iNwhere N is the number of crimes committed and d center,i is the distance from the spatial center to the i th crime.By incorporating the standard distance,we get an idea of how“close together”the data is.If the standard distance is small,then the kills are close together. However,if the standard distance is large,then the kills are far apart. Unfortunately,this leads to another problem.Consider the following data set (shown in Figure2):Figure2:Crimes scenes that are located very close together can yield illogical results for the spatial mean.In this image,the spatial mean is located at the same point as one of the crime scenes at(1,1).In this example,the kills(blue)are closely clustered together,which means that the centrography model will yield a center of mass that is in the middle of these crimes(in this case,the spatial mean is located at the same point as one of the crimes).This is a somewhat paradoxical result as research in criminology suggests that there is a buffer area around a serial criminal’s place of residence where he or she avoids the commission of crimes.[3,1]That is,the potential kill area is an annulus.This leads to Rossmo’s formula[1],another mathematical model that predicts the location of a criminal.7Control number:#727284.3Rossmo’s FormulaRossmo’s formula divides the map of a crime scene into grid with i rows and j columns.Then,the probability that the criminal is located in the box at row i and column j isP i,j=kTc=1φ(|x i−x c|+|y j−y c|)f+(1−φ)(B g−f)(2B−|x i−x c|−|y j−y c|)gwhere f=g=1.2,k is a scaling constant(so that P is a probability function), T is the total number of crimes,φputs more weight on one metric than the other,and B is the radius of the buffer zone(and is suggested to be one-half the mean of the nearest neighbor distance between crimes).[1]Rossmo’s formula incorporates two important ideas:1.Criminals won’t travel too far to commit their crimes.This is known asdistance decay.2.There is a buffer area around the criminal’s residence where the crimesare less likely to be committed.However,Rossmo’s formula has two drawbacks.If for any crime scene x c,y c,the equality2B=|x i−x c|+|y j−y c|,is satisfied,then the term(1−φ)(B g−f)(2B−|x i−x c|−|y j−y c|)gis undefined,as the denominator is0.Additionally,if the region associated withij is the same region as the crime scene,thenφi c j c is unde-fined by the same reasoning.Figure3illustrates this:This“delta function-like”behavior is disconcerting as it essentially states that the criminal either lives right next to the crime scene or on the boundary defined by Rossmo.Hence,the B-value becomes exceptionally important and needs its own heuristic to ensure its accuracy.A non-optimal choice of B can result in highly unstable search zones that vary when B is altered slightly.5AssumptionsOur model is an expansion and adjustment of two existing models,centrography and Rossmo’s formula,which have their own underlying assumptions.In order to create an effective model,we will make the following assumptions:1.The buffer area exists-This is a necessary assumption and is the basisfor one of the mathematical components of our model.2.More than5crimes have occurred-This assumption is importantas it ensures that we have enough data to make an accurate model.Ad-ditionally,Rossmo’s model stipulates that5crimes have occurred[1].8Control number:#72729Figure3:The summand in Rossmo’s formula(2B=6).Note that the function is essentially0at all points except for the scene of the crime and at the buffer zone and is undefined at those points3.The criminal only resides in one location-By this,we mean thatthough the criminal may change residence,he or she will not move toa completely different area and commit crimes there.Empirically,thisassumption holds,with a few exceptions such as David Berkowitz[1].The importance of this assumption is it allows us to adapt Rossmo’s formula and the centrography model.Both of these models implicitly assume that the criminal resides in only one general location and is not nomadic.4.The criminal is a marauder-This assumption is implicitly made byRossmo’s model as his spatial partition method only considers a small rectangular region that contains all of the crimes.With these assumptions,we present our model,the Gaussian Rossmooth method.9Control number:#7272106Gaussian Rossmooth6.1Properties of a Good ModelMuch of the literature regarding criminology and geographic profiling contains criticism of existing models for catching criminals.[1,2]From these criticisms, we develop the following criteria for creating a good model:1.Gives an accurate prediction for the location of the criminal-This is vital as the objective of this model is to locate the serial criminal.Obviously,the model cannot give a definite location of the criminal,but it should at least give law enforcement officials a good idea where to look.2.Provides a good estimate of the location of the next crime-Thisobjective is slightly harder than thefirst one,as the criminal can choose the location of the next crime.Nonetheless,our model should generate a region where law enforcement can work to prevent the next crime.3.Robust with respect to outliers-Outliers can severely skew predic-tions such as the one from the centrography model.A good model will be able to identify outliers and prevent them from adversely affecting the computation.4.Consitent within a given data set-That is,if we eliminate data pointsfrom the set,they do not cause the estimation of the criminal’s location to change excessively.Additionally,we note that if there are,for example, eight murders by one serial killer,then our model should give a similar prediction of the killer’s residence when it considers thefirstfive,first six,first seven,and all eight murders.5.Easy to compute-We want a model that does not entail excessivecomputation time.Hence,law enforcement will be able to get their infor-mation more quickly and proceed with the case.6.Takes into account empirical trends-There is a vast amount ofempirical data regarding serial criminals and how they operate.A good model will incorporate this data in order to minimize the necessary search area.7.Tolerates changes in internal parameters-When we tested Rossmo’sformula,we found that it was not very tolerant to changes of the internal parameters.For example,varying B resulted in substantial changes in the search area.Our model should be stable with respect to its parameters, meaning that a small change in any parameter should result in a small change in the search area.10Control number:#7272116.2Outline of Our ModelWe know that centrography and Rossmo’s method can both yield valuable re-sults.When we used the mean and the median to calculate the centroid of a string of murders in Yorkshire,England,we found that both the median-based and mean-based centroid were located very close to the home of the criminal. Additionally,Rossmo’s method is famous for having predicted the home of a criminal in Louisiana.In our approach to this problem,we adapt these methods to preserve their strengths while mitigating their weaknesses.1.Smoothen Rossmo’s formula-While the theory behind Rossmo’s for-mula is well documented,its implementation isflawed in that his formula reaches asymptotes when the distance away from a crime scene is0(i.e.point(x i,y j)is a crime scene),or when a point is exactly2B away froma crime scene.We must smoothen Rossmo’s formula so that idea of abuffer area is mantained,but the asymptotic behavior is removed and the tolerance for error is increased.2.Incorporate the spatial mean-Using the existing crime scenes,we willcompute the spatial mean.Then,we will insert a Gaussian distribution centered at that point on the map.Hence,areas near the spatial mean are more likely to come up as hot zones while areas further away from the spatial mean are less likely to be viewed as hot zones.This ensures that the intuitive idea of centrography is incorporated in the model and also provides a general area to search.Moreover,it mitigates the effect of outliers by giving a probability boost to regions close to the center of mass,meaning that outliers are unlikely to show up as hot zones.3.Place more weight on thefirst crime-Research indicates that crimi-nals tend to commit theirfirst crime closer to their home than their latter ones.[5]By placing more weight on thefirst crime,we can create a model that more effectively utilizes criminal psychology and statistics.6.3Our Method6.3.1Rossmooth MethodFirst,we eliminated the scaling constant k in Rossmo’s equation.As such,the function is no longer a probability function but shows the relative likelihood of the criminal living in a certain sector.In order to eliminate the various spikes in Rossmo’s method,we altered the distance decay function.11Control number:#727212We wanted a distance decay function that:1.Preserved the distance decay effect.Mathematically,this meant that thefunction decreased to0as the distance tended to infinity.2.Had an interval around the buffer area where the function values wereclose to each other.Therefore,the criminal could ostensibly live in a small region around the buffer zone,which would increase the tolerance of the B-value.We examined various distance decay functions[1,3]and found that the func-tions resembled f(x)=Ce−m(x−x0)2.Hence,we replaced the second term in Rossmo’s function with term of the form(1−φ)×Ce−k(x−x0)2.Our modified equation was:E i,j=Tc=1φ(|x i−x c|+|y j−y c|)f+(1−φ)×Ce−(2B−(|x i−x c|+|y j−y c|))2However,this maintained the problematic region around any crime scene.In order to eliminate this problem,we set an EPSILON so that any point within EPSILON(defined to be0.5spatial units)of a crime scene would have a weighting of a constant cap.This prevented the function from reaching an asymptote as it did in Rossmo’s model.The cap was defined asCAP=φEPSILON fThe C in our modified Rossmo’s function was also set to this cap.This way,the two maximums of our modified Rossmo’s function would be equal and would be located at the crime scene and the buffer zone.12Control number:#727213This function yielded the following curve (shown in in Figure4),which fit both of our criteria:Figure 4:The summand in smoothed Rossmo’s formula (2B =6,φ=0.5,and EPSILON =0.5).Note that there is now a region around the buffer zone where the value of the function no longer changes very rapidly.At this point,we noted that E ij had served its purpose and could be replaced in order to create a more intuitive idea of how the function works.Hence,we replaced E i,j with the following sum:Tc =1[D 1(c )+D 2(c )]where:D 1(c )=min φ(|x i −x c |+|y j −y c |),φEPSILON D 2(c )=(1−φ)×Ce −(2B −(|x i −x c |+|y j −y c |))2For equal weighting on both D 1(c )and D 2(c ),we set φto 0.5.13Control number:#7272146.3.2Gaussian Rossmooth MethodNow,in order to incorporate the inuitive method,we used centrography to locate the center of mass.Then,we generated a Gaussian function centered at this point.The Gaussian was given by:G=Ae −@(x−x center)22σ2x+(y−y center)22σ2y1Awhere A is the amplitude of the peak of the Gaussian.We determined that the optimal A was equal to2times the cap defined in our modified Rossmo’s equation.(A=2φEPSILON f)To deal with empirical evidence that thefirst crime was usually the closest to the criminal’s residence,we doubled the weighting on thefirst crime.However, the weighting can be represented by a constant,W.Hence,ourfinal Gaussian Rosmooth function was:GRS(x i,y j)=G+W(D1(1)+D2(1))+Tc=2[D1(c)+D2(c)]14Control number:#7272157Gaussian Rossmooth in Action7.1Four Corners:A Simple Test CaseIn order to test our Gaussain Rossmooth(GRS)method,we tried it against a very simple test case.We placed crimes on the four corners of a square.Then, we hypothesized that the model would predict the criminal to live in the center of the grid,with a slightly higher hot zone targeted toward the location of the first crime.Figure5shows our results,whichfits our hypothesis.Figure5:The Four Corners Test Case.Note that the highest hot spot is located at the center of the grid,just as the mathematics indicates.15Control number:#727216 7.2Yorkshire Ripper:A Real-World Application of theGRS MethodAfter the model passed a simple test case,we entered the data from the Yorkshire Ripper case.The Yorkshire Ripper(a.k.a.Peter Sutcliffe)committed a string of13murders and several assaults around Northern England.Figure6shows the crimes of the Yorkshire Ripper and the locations of his residence[1]:Figure6:Crimes and residences of the Yorkshire Ripper.There are two res-idences as the Ripper moved in the middle of the case.Some of the crime locations are assaults and others are murders.16Control number:#727217 When our full model ran on the murder locations,our data yielded the image show in Figure7:Figure7:GRS output for the Yorkshire Ripper case(B=2.846).Black dots indicate the two residences of the killer.In this image,hot zones are in red,orange,or yellow while cold zones are in black and blue.Note that the Ripper’s two residences are located in the vicinity of our hot zones,which shows that our model is at least somewhat accurate. Additionally,regions far away from the center of mass are also blue and black, regardless of whether a kill happened there or not.7.3Sensitivity Analysis of Gaussian RossmoothThe GRS method was exceptionally stable with respect to the parameter B. When we ran Rossmo’s model,we found that slight variations in B could create drastic variations in the given distribution.On many occassions,a change of 1spatial unit in B caused Rossmo’s method to destroy high value regions and replace them with mid-level value or low value regions(i.e.,the region would completely dissapper).By contrast,our GRS method scaled the hot zones.17Control number:#727218 Figures8and9show runs of the Yorkshire Ripper case with B-values of2and 4respectively.The black dots again correspond to the residence of the criminal. The original run(Figure7)had a B-value of2.846.The original B-value was obtained by using Rossmo’s nearest neighbor distance metric.Note that when B is varied,the size of the hot zone varies,but the shape of the hot zone does not.Additionally,note that when a B-value gets further away from the value obtained by the nearest neighbor distance metric,the accuracy of the model decreases slightly,but the overall search areas are still quite accurate.Figure8:GRS method run on Yorkshire Ripper data(B=2).Note that the major difference between this model and Figure7is that the hot zones in this figure are smaller than in the original run.18Control number:#727219Figure9:GRS method run on Yorkshire Ripper data(B=4).Note that the major difference between this model and Figure7is that the hot zones in this figure are larger than in the original run.7.4Self-Consistency of Gaussian RossmoothIn order to test the self-consistency of the GRS method,we ran the model on thefirst N kills from the Yorkshire Ripper data,where N ranged from6to 13,inclusive.The self-consistency of the GRS method was adversely affected by the center of mass correction,but as the case number approached11,the model stabilized.This phenomenon can also be attributed to the fact that the Yorkshire Ripper’s crimes were more separated than those of most marauders.A selection of these images can be viewed in the appendix.19Control number:#7272208Predicting the Next CrimeThe GRS method generates a set of possible locations for the criminal’s resi-dence.We will now present two possible methods for predicting the location of the criminal’s next attack.One method is computationally expensive,but more rigorous while the other method is computationally inexpensive,but more intuitive.8.1Matrix MethodGiven the parameters of the GRS method,the region analyzed will be a square with side length n spatial units.Then,the output from the GRS method can be interpreted as an n×n matrix.Hence,for any two runs,we can take the norm of their matrix difference and compare how similar the runs were.With this in mind,we generate the following method.For every point on the grid:1.Add crime to this point on the grid.2.Run the GRS method with the new set of crime points.pare the matrix generated with these points to the original matrix bysubtracting the components of the original matrix from the components of the new matrix.4.Take a matrix norm of this difference matrix.5.Remove the crime from this point on the grid.As a lower matrix norm indicates a matrix similar to our original run,we seek the points so that the matrix norm is minimized.There are several matrix norms to choose from.We chose the Frobenius norm because it takes into account all points on the difference matrix.[6]TheFrobenius norm is:||A||F=mi=1nj=1|a ij|2However,the Matrix Method has one serious drawback:it is exceptionally expensive to compute.Given an n×n matrix of points and c crimes,the GRS method runs in O(cn2).As the Matrix method runs the GRS method at each of n2points,we see that the Matrix Method runs in O(cn4).With the Yorkshire Ripper case,c=13and n=151.Accordingly,it requires a fairly long time to predict the location of the next crime.Hence,we present an alternative solution that is more intuitive and efficient.20Control number:#7272218.2Boundary MethodThe Boundary Method searches the GRS output for the highest point.Then,it computes the average distance,r,from this point to the crime scenes.In order to generate a resonable search area,it discards all outliers(i.e.,points that were several times further away from the high point than the rest of the crimes scenes.)Then,it draws annuli of outer radius r(in the1-norm sense)around all points above a certain cutoffvalue,defined to be60%of the maximum value. This value was chosen as it was a high enough percentage value to contain all of the hot zones.The beauty of this method is that essentially it uses the same algorithm as the GRS.We take all points on the hot zone and set them to“crime scenes.”Recall that our GRS formula was:GRS(x i,y j)=G+W(D1(1)+D2(1))+Tc=2[(D1(c)+D2(c))]In our boundary model,we only take the terms that involve D2(c).However, let D 2(c)be a modified D2(c)defined as follows:D 2(c)=(1−φ)×Ce−(r−(|x i−x c|+|y j−y c|))2Then,the boundary model is:BS(x i,y j)=Tc=1D 2(c)9Boundary Method in ActionThis model generates an outer boundary for the criminal’s next crime.However, our model does notfill in the region within the inner boundary of the annulus. This region should still be searched as the criminal may commit crimes here. Figure10shows the boundary generated by analyzing the Yorkshire Ripper case.21。
[DOC]-数学建模美赛论文标准格式参考--中英文对照
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数学建模美赛三等奖论文
Water, Water, EverywhereSummaryDue to population growth, economic development, rapid urbanization, large-scale industrialization and environmental concerns water stress has emerged as a real threat. [1]This paper was motivated by the increasing awareness of the need for fresh water since fresh water crisis is already evident in many areas on the world, varying in scale and intensity.Firstly, we testify water demand and supply sequence are stable by means of unit root test, then predict the freshwater demand and supply in 2025 by using ARMA Model and Malthus Population Model .Secondly, we give more concern on four aspects: Diversion Project, Desalinization, Sewage treatment and Conservation of water resources, building some models such as Cost-benefits analysis and Tiered water pricing model. Comparing the cost-benefit ratio, the sewage treatment cost-benefitratio is the smallest--0.142, that is to say it is more cost-efficient.Finally, we use our models to analyze the impacts of these strategies, we can conclude that conservation of water resources is the most feasible.Keywords:Cost-benefit analysis ARMA ModelTiered water pricing modelA Letter to a governmental leadershipFebruary 4, 2013Dear Sir,During the four days working, our team spares no effort using cost and benefits analysis determine water strategy for 2013 about how to use water efficiently to meet the need in 2025. Now, we outline our conclusion to you.z Diversion ProjectThe South-North Water Transfer Project is a multi-decade infrastructure project solved the unbalance of water resource. The cost is 6.2yuan/3m, and it will much higher while the distance is more than 40 kilometers.z DesalinizationDesalinization utilizes the enormous seawater and provides freshwater in a cheaper price. However, interior regions with water scarcity can hardly benefit from it as most desalinization manufacturers located on eastern coastal areas. The cost of production is 5.446 yuan/t, but the transport costs less the cost-efficient competitiveness. The cost can be decreased by using more advanced technology.z Sewage treatmentSewage treatment can relief the environmental impact of water pollution by removing contaminants from water, the cost of Sewage treatment is 0.5yuan/t. z Conservation of water resourcesConservation makes sure of the source of rational use of water. There are several approaches on water resources conservation, the main problem is the lack of supervision. The benefit-cost ratio is between 0.95 and 3.23, and it has a high return-investment ratio.z Each of the above water strategy has its own advantages and disadvantages, we should consider the aspects of economic, physical, environmental, geographical, and technique factors overall, then choose the optimal strategy for different area.Yours sincerely,COMAP #23052ContentI Introduction (2)II Assumptions (3)III Models (3)3.1 The prediction of freshwater shortage in 2025 (3)3.1.1 The prediction of freshwater demand (3)3.1.1.1 The description of basic model (3)3.1.1.2 Model building (4)3.1.1.3 Model prediction (5)3.1.2 The prediction of freshwater supply (7)3.1.2.1 Model building (7)3.1.2.2 Model prediction (8)3.1.3. Conclusion (9)3.2Water strategy (9)3.2.1 Diversion Project (9)3.2.2 Desalinization (14)3.2.3 Sewage Treatment (16)3.2.4 Conservation of water resources (19)3.2.4.1 Agricultural water saving (20)3.2.4.2 Life water saving (21)IV The influence of our strategy (25)4.1 The influence of Water Diversion Project (25)4.2 The influence of desalination (25)4.3 The influence of sewage treatment (26)4.4 Water-saving society construction (26)V References (27)VI Appendix (28)I IntroductionAccording to relevant data shows that 99 percent of all water on earth is unusable, which is located in oceans, glaciers, atmospheric water and other saline water. And even of the remaining fraction of 1 percent, much of that is not available for our uses. For a detailed explanation, the following bar charts show the distribution of Earth's water: The left-side bar shows where the water on Earth exists; about 97 percent of all water is in the oceans. The middle bar shows the distribution of that 3 percent of all Earth's water that is fresh water. The majority, about 69 percent, is locked up in glaciers and icecaps, mainly in Greenland and Antarctica.[2] Except for the deep groundwater which is difficult to extract, what can be really used in our daily life is just 0.26 percent of all water on earth.Figure 1 The distribution of Earth's waterFreshwater is an important natural resource necessary for the survival of all ecosystems. There is a variety of unexpected consequence due to the lack of freshwater: 6,000 children die every day from diseases associated with unsafe water and poor sanitation and hygiene; Unsafe water and sanitation leads to 80% of all the diseases in the developing world;[3]Species which live in freshwater may be extinct, thus, breaking the food chain balance severely; The development of economic slow down in no small measure.It is with these thoughts in mind, many people think freshwater is very important than ever before.So, how to use freshwater efficiently? What is the best water strategy? Readmore and you will find more.II AssumptionsIn order to streamline our model we have made several key assumptions ：1. We chose China as the object study.2. The water consumption of the whole nation could be approximate regardedas the demand of water .3. The Precipitation is in accordance with the supply of water .4. No considering about sea level rising because of global warmingIII Models3.1 The prediction of freshwater shortage in 2025How much freshwater should our strategy supply? Firstly, our work is to predict the gap between freshwater demand and supply in 2025. We obtain thefreshwater consumption data from China Statistical Yearbook. 3.1.1 The prediction of freshwater demandWe forecast the per capita demand for freshwater by building the ARMA Model .3.1.1.1 The description of basic modelThe notation ARMA(p, q) refers to the model with p autoregressive termsand q moving-average terms. This model contains the AR(p) and MA(q) models,mathematical formula is:qt q t t t p t p t t t y y y y −−−−−−−−−−+++=εθεθεθεφφφ......22112211 (1) AR(p) modelt p t p t t t y y y y εφφφ+++=−−−...2211 (2) MA(q) model q t q t t t t y −−−−−−−=εθεθεθε....2211 (3)),.....,2,1(p i i =φ ,),.....,2,1(q j j =θare undetermined coefficients of themodel, t ε is error term, t y is a stationary time series.3.1.1.2 Model buildingAll steps achieved by using EviewsStep1: ADF test stability of sequenceNull hypothesis:1:0=ρH , 1:1≠ρH , ρis unit root.Table 1 Null Hypothesis: Y has a unit root Exogenous: Constant Lag Length: 3 (Automatic based on SIC, MAXLAG=3) t-Statistic Prob. Augmented Dickey-Fuller test statistic -5.3783580.0040 Test critical values: 1% level-4.582648 5% level -3.32096910% level -2.801384We know Prob=0.0040 that we can reject the null hypothesis, and thenydoesn’t have a unit root, in other words, is stationary series. Step 2: Building the ARMA ModelThen we try to make sure of p and q by using the stationary series y .Table 2Date: 02/02/13 Time: 11:08Sample(adjusted): 2001 2011Included observations: 11 after adjusting endpointsConvergence achieved after 12 iterationsBackcast: 1998 2000Variable Coefficie nt Std. Error t-StatisticProb.AR(1) 1.0105040.005813173.8325 0.0000MA(3) 0.9454040.03650725.89639 0.0000R-squared 0.831422 Mean dependent varAdjustedR-squared 0.812692 S.D. dependent varS.E. of regression 5.085256 Akaike info criterionSo, we can get the final model, is:310.9454041.010504−−+=t t t d y y ε (4)3.1.1.3 Model predictionStep 1: The prediction of per capita freshwater demandWe use model (4) to predict the per capita demand of freshwater in the year2025, the result as Figure3.Figure 2 sequence diagram of dynamic predictionFrom the diagram, we can see the per capita freshwater demand is raising.The detailed data as Table3: Table 3 2010 2011 2012 2013 2014 2015 2016 2017 483.3584 488.4357 493.5662 498.7507503.9896509.2836514.6332 520.03892018 2019 2020 2021 2022 2023 2024 2025 525.5015 531.0214 536.5993 542.2358547.9315553.6871559.503 565.3801(cu.m/person)Through the above efforts, we get the 2025 per capita freshwater demand is565.3801 cu.mStep 2: The prediction of the whole freshwater demandThe relationship among d Q ,t N ,daverage Q is: daverage t d Q N Q ×= (5)d Q is the whole demand of freshwater, t N is the total population ,daverage Q is per capita of freshwater demand.Then we etimate the total population by the Malthus Population Model . rt e N t N 0)(=[4] (6))(t N is the population at time t,0N is the population at time 0,r is net relative growth rate of the populationrt e N N 2011)2025(= (7)By calculating, we get:(billion)42.11.347)2025(1500479.0≈=×e N (8)At last,we could get the whole demand of freshwater while the time is 2025.38.5652.14)2025(×=×=daverage d Q N Q ()cu.m million 100 8028.396= (9)3.1.2 The prediction of freshwater supplySimilarily,we predict freshwater supply using the ARMA Model. 3.1.2.1 Model buildingStep1: ADF test stability of sequenceNull hypothesis:1:0=ρH , 1:1≠ρH , ρis unit root. Table 4 Null Hypothesis: D(Y) has a unit root Exogenous: Constant Lag Length: 2 (Automatic based on SIC, MAXLAG=3)t-Statistic Prob. Augmented Dickey-Fuller test statistic-9.433708 0.0002 Test critical values: 1% level -4.803492 5% level -3.40331310% level -2.841819From the table, we find that first difference of supply data is smooth, we canreject the null hypothesis, that is ()y D is a smooth series.Step 2: Building the ARMA ModelWe use the smooth series ()y D to make sure the number of order.Table 5Date: 02/02/13 Time: 14:16Sample(adjusted): 2002 2010 Backcast: 1999 2001Variable CoefficientStd. Error t-Statistic Prob. AR(1) 0.6351030.158269 4.012804 0.0051 MA(3) -0.9923370.069186-14.34306 0.0000 R-squared 0.812690 Mean dependent var 50.51111Adjusted R-squared 0.785931 S.D. dependent var 119.1793S.E. of regression 55.14139 Akaike info criterion 11.05081Sum squared resid 21284.01 Schwarz criterion11.09464 Log likelihood -47.72864 Durbin-Watson stat 2.895553Then ,we get the final model is:)0.992337D(-)0.635103D()(31−−=t t t s y y D ε (10) 3.1.2.2 Model predictionWe use the effective model to predict freshwater supply in short-term until theyear 2025.Figure 3 sequence diagram of dynamic predictionFrom the diagram, we can see the supply remains unchanged basically .T The detailed data as Table6: Table 6 2010 2011 2012 2013 2014 2015 2016 2017 5630.203 5630.594 5630.843 5631.0015631.1025631.1655631.206 5631.2322018 2019 2020 2021 2022 2023 2024 2025 5631.248 5631.258 5631.265 5631.2695631.2725631.2735631.275 5631.275(100 million cu.m)According to the above data,we gain the supply of freshwater 2025, is5631.275(100 million cu.m)3.1.3. ConclusionFrom the above result,we find a serious issue:Table 7Year Demand offreshwater Supply of freshwater Net demand Unit2025 8028.396 5631.275 2397.121(100 million cu.m)In the year 2025, China will face the serious situation of freshwater shortage, the gap will reach 2397.121(100 million cu.m), therefore, in order to avoid this, we need to determine a series strategy to utilize freshwater efficiently.3.2Water strategy3.2.1 Diversion ProjectOn one hand, in view of Figure4, we can get information: Southeast coast is of the maximum precipitation, followed by the northern region, the western least.Figure 4 Precipitation Allocation Map of Major CitiesOn the other hand, in view of Figure 5, we can get information: The northern region and the southern coastal areas have the most water consumption, the western use less.Figure 5 Water Use MapDetailed data see to attached Table8 and Table9.South-to-North Water Diversion ProjectThe South–North Water Transfer Project is a multi-decade infrastructure project of China to better utilize water resources. This is because heavily industrialized Northern China has a much lower rainfall and its rivers are running dry. The project includes a Eastern, a Central and a Western route.Figure 6 The route of South-to-North Water Diversion ProjectHere, we take Western Route Project (WRP) as a representative, analysis the cost and benefits. As the strategic project to solve the problem of poorer water Northwest and North China, WRP will divert water from the upper reach of Yangtze River into Yellow Rive.Cost and benefits analysisThe direct quantitative economic benefits include urban water supply economic benefits, ecological environment water supply economic benefits, and the Yellow River mainstream hydroelectric economic benefits.[5]Urban water supply economic benefits:(1) Calculation MethodIn view of the water shadow price is difficult to determine, the equivalent engineering is not easy to choose, and the lack of water loss index is unpredictable, combined with the stage job characteristics, we select the method of sharing coefficient to calculate the urban water supply economic benefits.(2) Calculation ParametersThe Water consumption quota of per ten thousand yuan industrial output value is based on status quota, the predicted water consumption quota of per ten thousand yuan output value according to reach in 2 0 2 0 is :Lanzhou tom/ ten thousand yuan, gantry to Sanmenxia HeKouZhen river section for 26 3m/ ten thousand yuan. After a comprehensive analysis, set the reach for 20 3industrial water supply benefit allocation coefficient values 2.0 %.(3) Calculation ResultsAccording to (1) and (2), get table 10:Table 10water supply 3.2 billion 3.mproject benefits 20 billion yuan.8yuan /3maverage economic benefit 70z Ecological environment water supply economic benefits:(1) Calculation methodTake Forestry and animal husbandry as the representative, calculate whoseirrigation Economic benefits, and consider the allocation function of water supply. Analyse forestry benefits in reference with the increased wood savings, Animal husbandry in reference with the increased output of animals which were feeded by the incresed irrigation pasture (represented by sheep), both Forestry and animal husbandry account for half of the Ecological environment water supply.(2) Calculation parameters Set the water consumption quotas of Forestry irrigation unified as 233750hm m , the water supply sharing coefficient of Xiang irrigation as 0.60. In the calculation of forestry benefit, the increase of accumulated timber amount is ()a hm m ⋅235.22, timber price is 3300m yuan ; in the calculation of animal husbandry benefit , the increased stocking rates of unit pasture area is 25.22hm , taken a standard sheep price as yuan 260.(3) Calculation ResultsAccording to (1) and (2), the ecological environment water supply economic benefits is 714.1 billion, in which, The Yellow River replenishment economic benefits is 008.1billion yuan.z Hydroelectric economic benefits.(1) Diversion increased energy indicators:The increased electricity indicators is 306.9billion h kw ⋅, capacity enlargement the scale of 241 ten thousand kw .(2) Calculation methodTake the Optimal equivalent alternative engineering cost method, chosen fire electricity as an alternative project which can meet the power requirements of grid electricity equally. The sum of alternative engineering required annualinvestment translation and the annual running costs is increased annual power generating efficiency of the Yellow River cascade hydropower stations. (3) Calculation parametersThe power plant construction investment of kw $450, duration of five years, the investment proportion were 10%, 25%, 35%, 25%, 5%. Both the economic life of mechanical and electrical equipment and the metal structures equipment are taken as 20 years, considering the update ratio as 80% of the original investment. Standard coal price is taken as 160 dollars, standard coal consumption is taken as ()h kw g ⋅350. The fixed run rates take 4.5%, thesocial discount rate is 12%, the hydropower economic useful life of 5 years.(4) Calculation ResultsBy analysis and calculation, the first phase of water regulation produce the hydropower economic benefit is 3.087 billion.z Total economic benefits:Preliminary cost estimates of the project diversionOn the basis of economic nature classification, the total cost includes themachinery depreciation charges, wages and welfare costs, repair costs, thecost of materials,water district maintenance fees, management fees, water fees, interest expense and other . Analysis in the light of various estimates condition, the cost of water diverted into the Yellow River is 31~7.0m yuan c =The cost-benefit rate ()85.8~2.61∈=rc ω (11) 3.2.2 DesalinizationThough diversion project can balance water supply between places one has enough water and the other has water shortage, the costs will higher than desalinization when the distance more than 40 kilometers.Desalinization and comprehensive utilization of the work are increasingly taking centre stage on the problem of solving freshwater scarcity. Many countries and areas devote to optimize an effective way by enhancing the development of science and technology.According to the International Desalination Association, in 2009,14,451 desalination plants operated worldwide, producing 59.9 million cubic meters per day, a year-on-year increase of 12.3%.[6] The production was 68 million 3m in 2010, and expected to reach 120million 3m by 2020; some 40 million 3m is planned for the Middle East.[7]China has built more than 70 sets of sea water desalinization device with the design capacity of 600,000m3 and an average annual growth rate of more than 60%; technology with independent intellectual property rights of a breakthrough in the reverse osmosis seawater membranes, high pressure pumps, devices for energy recovery achieved significant progress, the desalinization rate raises from 99.2% to 99.7%; conditions of industrial development and the desalination market has been basically formed.MethodsDe-salinization refers to any of several processes that remove some amount of salt and other minerals from saline water. More generally, desalination may also refer to the removal of salts and minerals.[8] Most of the modern interest in desalination is focused on developing cost-effective ways of providing fresh water for human use.There are two main methods of desalinization:1. Extract freshwater from saline water: Distillation (Multi-stage flash distillation, Vapor compression distillation, Low temperature multi-effect distillation), Reverse osmosis, Hydrate formation process, Solvent extraction, Freezing.2. Remove salt from saline water: Ion exchange process, Pressure infiltration method, Electroosmosis demolition method.For desalination, energy consumption directly determines the level of the cost of the key. Among the above methods, reverse osmosis is more cost-effective than the other ways of providing fresh water for human use. So, reverse osmosis technology has become the dominant technology in international desalinization of seawater.The following two figures show the working principle diagram of a reverse osmosis system.Figure7 working principle diagram of a reverse osmosis systemCost and benefits analysisTable 12 general costs for a reverse osmosis systemItem Unitprice(yuan/t)Chemicals cost 0.391electric charge 2.85Wages 0.034 Labor costWelfare 0.04 Administrative expenses 0.0008maintenance costs 0.23Membrane replacement cost 0.923Depreciationexpense Fixed assets depreciation0.97expenseTotal costs 5.446Table 13 general benefit for a reverse osmosis systemItem ValueHourly output(t) 10Working hours/day24 Daily output(t)240 Working days/year 365 Yearly output(t)87600 Yearly other benefits(yuan)310980 Unit water other benefits3.55 Water Price(yuan/t)8 Unit water total benefits11.55 Unit water total benefit 55.11=rWater cost-benefit ratio 4715.055.11446.52===r c w (12) 3.2.3 Sewage TreatmentSewage treatment is an important process of water pollution treatment. It uses physical, chemical, and biological ways removing contaminants from water . Its objective is to relief the environment impact of water pollution.This diagram shows a typical sewage treatment process.Figure 8 Sewage treatment flow mapTake Sewage Treatment Plant in east china as an example to analysis the cost and benefit of sewage treatment.Suppose:Sewage treatment scale d t x 100001=,The Sewage Treatment Plant workdays in a year 300=d ,Concession period is twenty to thirty years, generally 251=t years, Construction period is one to three years, generally 32=t years.Operation period = Concession period - Construction period.Cost estimation Table 14 fixed investment estimate c1(ten thousand Yuan)number project ConstructioninvestmentEquipment investment 1 Preprocessing stage38 27 2Biological treatment section 42 134 3End-product stage 11 44 4 Sludge treatment section 6323 5 accessory equipment 456 Line instrument 687 Construction investment 3008 Unexpected expense 809 Other expense 10010 Total investment975 Table 15 Operating expense estimate c2 (ten thousand Yuan)[9]number project expenses1 maintenance expenses 6.52 wages 103 Power Consumption 404Agent cost 10 5 Small meter operating cost 66.56 Amortization of intangibles 127Amortization of Construction 6.6 8Amortization of Equipment 19.8 9Annual total cost 104.9 10 Tons of water operation cost 0.29Annual total investment 15022213=+÷=c c c ten thousand YuanAnnual amount of sewage treatment t x x 3000000100003003001=×=×= Unit sewage investment t yuan t yuan x c c 5.03000000150000034=÷=÷= Benefit analysisSewage mainly comes from domestic sewage(40%), industrial sewage(30%), and the others(including stormwater , 30%)Sewage treatment price: domestic sewage is about t yuan 8.0, industrialsewage is about t yuan 5.1, and other is about t yuan 5.2.Unit sewage treatment approximate price t yuan t yuan t yuan tyuan p 52.1%302%305.1%408.01=×+×+×=Unit Sewage treatment benefit:t yuan p p r 52.321=+= Cost-benefit ratio 142.052.35.043===r c ω (13) 3.2.4 Conservation of water resourcesTo realize the sustainable development of water resources, one of the important aspects is the conservation of water resources. Saving water is thekey of conservation, so, we the construction of water-saving society is the keyof water resources conservation strategy.To construct the water-saving society, we give more concern about two aspects:agricultural water saving and life water saving. Finally, we analysis the cost andbenefit about water-saving society by building model.3.2.4.1 Agricultural water savingStrategic suggestions of water-saving agriculture1. Strengthen the government policies and public finance support2. Mobilizing all social forces to promote water-saving agriculture development3. Innovating enterprises to improve the science and technology4. Suggesting countries to regard water saving as a basic state policy5. Implement the strategy of science and technology innovationwater saving function product research and development as the key point, the research and development of a batch of suitable for high efficiency and low energy consumption, low investment, multi-function water saving and high efficient agriculture key technology and major equipment. Micro sprinkler irrigation water saving technology and equipment is the typical technology.[10] Typical analysis: drip irrigation technologyIrrigation uniformity DU and field irrigation water utilization αE can be expressed as the technical elements of the function ：[11]),,,,,,(01co c in t F I S n L q f DU α=),,,,,,,(0SMD t F I S n L q f E co c in αα=RD SMD fc )(θθ−=in q is single discharge into earth,L is (channel) long,n is manning coefficient,0S is tiny terrain conditions,c I is soil infiltration parameters,αF is (channel) cross-sectional parameters,co t is irrigation water supply time,SMD is irrigation soil water deficit value,fc θ is the soil field capacity,θis the soil moisture content,RD is the root zone depth.According to the study we found that the use of modern surface irrigation technology such as sprinkler irrigation, micro spray irrigation and pressure irrigation system, can improve the utilization rate of water to 95%, better than common ground water saving irrigation mode, more than 1/2 ~ 2/3 of water-saving irrigation mode, therefore, advanced water saving technology is very important. 3.2.4.2 Life water saving China is a country with a large population and scarce water , so we should use water more reasonably and effectively.Tiered water pricing modelThe model is for all types of users in certain period to regulate basic water consumption, in the basic consumption, we collect fees by the basic price standard, when actual consumption beyond basic consumption, the beyond part will introduce penalty factor: the more water exceed, the higher punishment rate will be. For actual consumption is less than basic consumption, the user can get additional incentives, encouraging people to save water .[12] Three ladder water price modelWe assume that urban resident’s basic water consumption is 1q , the first stage water price is 1P , the second stage water price is 2P , by analogy, q P is used to express the water price in stage q , model formula is()()⎪⎪⎩⎪⎪⎨⎧−++−+−+=−)(11211121111m m q q q p q q p q p q q p q p q p p L L L (14) From the equation (14), that in the tiered water pricing system, as more price levers are divided, it will be more able to reflect the city water supply’s public property and public welfare, be much beneficial to motive users to save water . On the other hand, much more price levers will be bound to increase the transaction cost of both the water supplier and the water user . Seeing from practical application effects of the current step water price model , Three ladder water price model much meets the actual functional requirements of urban water supply system in our country, the specific pricing method see Figure 9.Figure 9 Taking three step level water price model, can to some extent, Contain people waste the limited water resources , promote enterprises into taking all kinds of advanced technologies to improve the Comprehensive utilization of water resources, and realize the goal of urban water conservation and limited water resources Sustainable and high-efficiency using and saving. In conclude, it’s an effective and feasible strategy at present.Cost-Benefit Analysis of water-saving society construction1. Cost-Benefit Analyses ModelThe benefit of the water-saving society construction n s B B B −= (15) :s B water use benefit of the whole society in Water-saving condition。