美赛数模论文英文版
美国大学生数学建模论文及其翻译31552
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Best all time college coach Summary
In order to select the “best all time college coach” in the last century fairly, We take selecting the best male basketball coach as an example, and establish the TOPSIS sort - Comprehensive Evaluation improved model based on entropy and Analytical Hierarchy Process. The model mainly analyzed such indicators as winning rate, coaching time, the time of winning the championship, the number of races and the ability to perceive .Firstly ， Analytical Hierarchy Process and Entropy are integratively utilized to determine the index weights of the selecting indicators Secondly，Standardized matrix and parameter matrix are combined to construct the weighted standardized decision matrix. Finally, we can get the college men's basketball com
数模美国赛总结部分英文
数模美国赛总结部分英文第一篇：数模美国赛总结部分英文Conclusions1、As our team set out to come up with a strategy on what would be the most efficient way to 我们提出了一种最有效的方法去解决……2、The first aspect that we took into major consideration was…….Other important findings through research made it apparent that the standard 首先我们考虑到……，其他重要的是我们通过研究使4、We have used mathematical modeling in a……to analyze some of the factors associated with such an activity。
为了分析这类问题的一些因素，我们运用数学模型……5、This “cannon problem” has been used in many forms in many differential equations courses in the Department of Mathematical Sciences for several years.这些年这些问题已经以不同的微分方程形式运用于自然科学部门。
6、In conclusion our team is very certain that the methods we came up with in 总之，我们很确定我们提出的方法7、We already know how well our results worked for…… 我们已经知道我们结果对……8、Now that the problem areas have been defined, we offer some ways to reduce the effect of these problems.既然已经定义了结果，我们提出一些方法减少对问题的影响。
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。
国际数学建模竞赛优秀论文英文模板
T eam Control NumberFor office use only38253For office use onlyT1F1 T2 F2 T3 Problem ChosenF3 T4 AF42015 Mathematical Contest in Modeling (MCM) Summary SheetEradicating EbolaAbstractThis paper aim at the problem which is to eradicate or inhibit the spread of Ebola, we start from three sub problem, that is: the demand for drugs, drugs delivery route and the car allocation. And establish the spreading model of Ebola, optimization model of drugs transport system and car allocation model respectively by using the differential equation method and simulated annealing algorithm. Finally, do the model extension and sensitively analysis.The first issue, figure out the demand for drugs in different regions. First, establish Ebola spread SIR model. And in the time of t, using differential equation to find the proportion of infected i (t )=1/Qln(s /s 0), then get the demand for drugs in this region H =kNi (t ).The second issue, how to find the shortest route to deliver drugs. Use Guinea, Liberia and Sierra Leone whose infection is relatively serious as the investigation object. According to the Binary classification to find the rules of iteration, which is useful to find out the nearest city to any other cities, and the result is Bombali. So we put it as the center of distribution. Then use simulated annealing algorithm and put forward two kinds of schemes for shortest path by the different ways in drugs delivery.Schemes one, asynchronous mode: put three countries as a regional countries. Using the TSP method to solve the shortest route is 54.8486, which is start from Bombali to different regions.Schemes two, synchronization method: dividing the whole area into two areas around A and B by use the longitude coordinates of Bombali as a standard. Respectively solve the shortest route is 10.1739 and 29.8075, which is start from Bombali and pass all cities in A and B, and solve the sum of the two route is 39.9814.According to the different drug delivery requirements (such as the shortest distance or transmission synchronization), can choose the asynchronous or synchronous way.The third issue, how to allocate the number of cars reasonable, and obtain the suitable speed of drug production. According to the predict number which obtained in model one, get the vehicles and drug distribution table (the results are shown Table 4.6 and Table 4.7). and obtain the speed V of drugs production is:10(ln ln )ni ii i i i k N V Q T s s =≥-∑At last, the minimum speed of drugs production is 56.14 agent/day to meet the need in three countries by calculating.Finally, use the SIR model which was optimized by using vaccination cycle control. By doing this we can know the number of susceptible and infections in crowd under the condition of the pulse vaccination significantly lower faster than without pulse vaccination. Thus, using pulse vaccination can effectively control the spread of Ebola.Keywords: SIR model; Simulated Annealing Algorithm; Pulse vaccination; EbolaEradicating EbolaContent1 Restatement of the Problem (1)1.1 Introduction (1)1.2 The Problem (1)2 General Assumptions (1)3 Variables and Abbreviations (2)4 Modeling and Solving (2)4.1 Model I (2)4.1.1 Analysis of the Problem (2)4.1.2 Model Design (2)4.2 Model II (6)4.2.1 Analysis of the Problem (6)4.2.2 Model Design (6)4.3 Model Ⅲ (8)4.3.1 Analysis of the Problem (8)4.3.2 Model Design (9)4.4 Extent our models (11)5 Sensitivity Analysis (14)5.1 Effect of Daily Contact Rate (14)5.2 Effect of inoculation rate (14)6 Model Analysis (15)6.1 The Advantages of Model (15)6.2 The Disadvantages of Model (15)7 Non-technical Explanation (16)References (18)1Restatement of the Problem1.1IntroductionEbola virus is a very rare kind of virus. It can cause humans and primates produce Ebola hemorrhagic fever virus, and has a high mortality rate. The largest and most complex Ebola outbreak appeared in the West African country in 2014. This outbreak occurred in guinea first, then through various ways to countries such as Sierra Leone, Liberia, Nigeria and Senegal. The number of cases and deaths, which occurred in this outbreak, is more than the sum of all the other epidemic. And outbreak continued to spread between countries. On August 8, 2014, the general-director of the world health organization announced the outbreak of public health emergency of international concern.In this paper, a realistic and reasonable mathematic model, which considers several aspects such as vaccine manufacturing and drug delivery, has been built.Then optimizing the model to eliminate or suppress the harm done by the Ebola virus.1.2The ProblemEstablishing a model to solve the spread of the disease, amount of drugs needed, possible feasible transportation system, transporting position, the speed of a vaccine or drug manufacturing and any other key factor. Thus, we decompose the problem into three sub-problem, modeling and finding the optimization method to face the Ebola virus.♦Building a model, which can solve the spread of the disease and the demand for drugs.♦Building a model to find the best solution.♦Using the goal programming to solve the problems of production and distribution and optimization of other factors..2General AssumptionsTo simplify the problem, we make the following basic assumptions, each of which is properly justified.♦Our assumptions is reasonable and effective.♦Vehicles only run in the path which we have simulated♦This assumption greatly simplify our model and allow us to focus on the shortest path.♦We consider the model that are enclosed.♦People who recovered, will not infected again, and exit the transmission system3Variables and AbbreviationsThe variables and abbreviations used in this paper are listed in Table 3.1.Table 3.1 Assuming variableSymbol DefinitionS the number of susceptible peopleI the number of infected personsR the number of recoveredT a vaccine or drug production cycleH the amount of drugs needed by RegionA a cycle of a vaccine or drug productionL drug reserve area to the shortest path to all affected areasV speed of vaccine or pharmaceutical productionV’vehicle speedλrate of patient contact per dayμday cure rate per dayαn rights of those infected regions weight4Modeling and Solving4.1Model I4.1.1Analysis of the ProblemAccording to the literature that different types of virus has its own different propagation process characteristics, we do not analyze the spread of viruses from a medical point of view, but from the general to analyze the propagation mechanism. So we have to analyze the spread of the Ebola virus and the requirements of drugs through the SIR[1] model.4.1.2Model DesignIn the dynamics of infectious diseases, the main follow Kermack and McKendrick SIR epidemic model which the dynamics of the established method in 1927. SIR model until now is still widely used and continue to develop. SIR model of the total population is divided into the following three categories: susceptibles, the ratio of the number denoted by s(t), at time t is not likely to be infected, but the number of infectious diseases such proportion of the total; infectives, the ratio of the number denoted by i(t), at time t become a patient has been infected and has the proportion of the total number of contagious; recovered, the ratio of the number denoted by r(t), expressed the number of those infected at time t removed from the total proportion (ie, it has quit infected systems). Assuming a total population of N(t), then there are N(t) = s(t) + i(t) + r(t).SIR model is established based on the following two assumptions:In the investigated region-wide spread of the disease is not considered during the births, deaths, population mobility and other dynamic factors. Total population N(t) remainunchanged, the population remains a constant N.The patients’ contact rate (the average number of effective contacts per patient per day) is constant λ, the cure rate (patients be cured proportion of the total number of patients a day) is a constant μ, clearly the average infectious period of 1/μ, infectious period contact number for Q = λ/μ.In the model based on the assumption that we develop a susceptible person to recover fromthe sick person in the process, such as Figure 4.1:Figure 4.1 SIR the model flowchartSIR basis differential equation model can be expressed as:disi i dt dssi dt dri dt λμλμ⎧=-⎪⎪⎪=-⎨⎪⎪=⎪⎩(5.1)But it can see that s(t), i(t) is more difficult to solve, so we use the numerical calculations to esti mate general variation. Assuming λ = 1, μ = 0.3, i(0) = 0.02, s(0) = 0.98 (at the initial time), then we borrow MATLAB software programming to get results. And according to Table 4.1 analyzed i(t), s(t) of the general variation.Figure4.2 s(t),i(t)The patient scale map Figure 4.3 i ～s Phase track diagramFrom Table 4.1 and Figure4.2, we can see that i(t) increased from the initial value to about t = 7(maximum), and then began to decrease.Based on the calculating the numerical and graphical observation, use of phase trajectories discussed i(t), s(t) in nature. Here i ~ s plane is phase plane , the domain (s, i)∈D in phase plane for:{}(,)0,0,1D s i s i s i =≥≥+≤(5.2)According to equation (5.1) and con tact number of the infectious period Q = λ / μ, we can eliminate dt, get:0011(1)(1)i s i s s sdi ds di ds Q Q =-⋅⇒=-⋅⎰⎰(5.3)Calculated using integral characteristics:0001()()ln si t s i s Q s =+-=(5.4)Curve in the domain of definition, equation(5.3) is a phase trajectory.According to equation(5.1) and equation(5.3), have to analyze the changes. If and only if the patient i(t) for some period of growth, it think that in the spread of infectious diseases , then 1/Q is a threshold. If s 0> 1/Q, infectious diseases will spread , and reduce infectious period the number of contacts with Q, namely raising the threshold 1/Q and will make s 0≤1/Q, then it will not spread diseases.And we note that Q = λ/μ in the formula, the higher the level of people's health, the smaller patients’ contact rate; the higher the level of medical, the cure rate is larger and the smaller Q. Therefore, to improve the level of hygiene and medical help to control the spread of infectious diseases. Of course, can also herd immunity and prevention, to reduce s 0.In the process, we analyzed the spread of the disease, then we are going to discuss the amount of medication needed.According to equation(5.4), you can get i(t) values, we can calculate the number of people infected with the disease who I was:()()I i t N t =⋅(5.5)And the amount of drug required, we can be expressed as: H kI =(k is a constant, w> 0)If k> 0, it indicates that the number of infections is still rising, measures to control the virus also needs to be strengthened, and the amount of drugs is a growing demand mode until fluctuation; if k≤0, it means reducing the number of people infected, the virus the measure is better, and the dose of demand is also gradually reduced.According to the data provided by the WHO, we can get the number of infections various，which areas before January 30, 2015. see Table 4.2:Table 4.2 As the number of infections January 30, 2015Region Number Proportion Region Number ProportionNzerekore 2 0.0045 Koinadugu 1 0.0022Macenta 1 0.0022 Kambia 25 0.0558Kissdougou 1 0.0022 Western Urban 105 0.2344Kankan 1 0.0022 Western Rural 64 0.1429Faranah 4 0.0089 Mali 1 0.0022Kono 28 0.0625 Boffa 4 0.0089Bo 6 0.0134 Dubreka 11 0.0246Kenema 2 0.0045 Kindia 2 0.0045Moyamba 8 0.0179 Coyah 11 0.0246Port Loko 78 0.1741 Forecariah 24 0.0536Tonkolili 18 0.0402 Conakry 20 0.0446Bombal 18 0.0402 Montserrado 13 0.029Based on the latest data Ebola virus infections in January 2015, and the regional population and the associated parameter value Ebola assumptions, the model has been solved to a time t proportion of those infected i(t) = 1/Q ln (s/s0), using MATLAB software, we have predict the number of infections each region in February, then get a weight value of those infected forecast for each region in February 2015, as can be show Table 4.3.Table 4.3 As the number of infections February 28, 2015Region Number Proportion Region Number ProportionNzerekore 1 0.00233 Koinadugu 8 0.01864Macenta 3 0.00700 Kambia 24 0.05594Kissdougou 2 0.00470 Western Urban 69 0.16083Kankan 1 0.00233 Western Rural 78 0.18182Faranah 2 0.00470 Mali 4 0.00932Kono 22 0.05130 Boffa 2 0.00470Bo 5 0.01166 Dubreka 10 0.02331 Kenema 5 0.01166 Kindia 1 0.00233Moyamba 1 0.00233 Coyah 9 0.020979Port Loko 100 0.23310 Forecariah 20 0.046620Tonkolili 12 0.02797 Conakry 18 0.041968Bombal 23 0.05361 Montserrado 9 0.020979From Table 4.2 can be known, According to the number of cases of expression，we made a rough prediction that Ebola outbreak in February. it’s provide a reference for the production of vaccines and drugs. Indeed, it have provide a theoretical basis for the relevant departments which take appropriate precautions.4.2Model II4.2.1Analysis of the ProblemBased on the model I, we obtained the equation expression of disease transmission speed and number of drugs. However, in addition to these two factors, we should also consider how to transport drugs to the demanded area quickly and effectively. Thus, it is very important to develop a good transportation system, which can greatly improve the efficiency of drug transport and reduce the cost.4.2.2Model DesignBy searching on Wikipedia, we obtain cities which have erupted Ebola, and the latitude and longitude coordinates[2]. The results are shown in Table 4.4We get the best point, which is Bombali by programming. So, we assume it as the city which produces drugs.Because these cities are breaking points, both as a place of delivery. In order to find out the optimal path, we make following assumptions:♦The demand for each city is same♦The quantity of vehicles can meet the demand of transport♦Vehicles only run in the path which we have simulated4.2.2.1SA modelSA[3] is a random algorithm which is established by imitating metal annealing principle. It can be implemented in large rough search and local fine search by controlling the changes of temperature.Basic principle of SA:♦First, generated initial solution x0 randomly, and make it as the current best solution xopt. Then calculate the value of objective function f (xopt).♦Second, make a random fluctuation on the current solution. Then calculate the value of the new objective function f (x).♦Calculating and judgingΔf = f(x) - f(xopt).IfΔf >0, accept it as the current best solution;Otherwise, accept it in the form of probability P.The calculation method of P is:10=exp[(()())]0opt i f P f x f x f ≤⎧⎨-->⎩ (5.6)In this chapter, the SA algorithm is extended by selecting Bombali as a starting point to solve the optimal path. In the extended SA algorithm.we exploits the exponential cooling strategies and controls the change of temperature, namely10k i T Apha T -=⨯(5.7)Where T i is current controlled temperature, T 0 is the initial temperature, Apha is temperature reduction coefficient, k is the iterations.Solving the initial temperature 0T by means of random iterative and setting Apha = 0.9, the results are shown in Figure 4.4Longitude coordinates of citiesP a r a l l e l v a l u e o f c i t i e sthe total distance:54.8486Figure 4.4 Path graphThe value of the shortest total distance y is 54.8486 The shortest path is presented as follow:Bombali →Tonkolili →Nzerekore →Moyamba →Kambia →Port Loko →Coyah →Mali →Bo →Kindia →Western Urban →Kono →Dubreka →Faranah →Western Rural →Kenema →Kiss-dou gou →Kankan →Forecariah →Boffa →Macenta →Conakry →Montserrado →Koinadugu → Bombali4.2.2.2 SA model refinementSA model got all the shortest path problem of the city, but transport route is single and the efficiency is not high. So we use the longitude coordinates of Bombali as the basis to divide these cities into two parts. Urban classification is shown inTable 4.5, then simulate respectively.Table 4.5 The divided city distributionClassify CitiesLeft half Conakry, Moyamba, Port Loko, Kambia, Western Urban, Western Rural, Boffa, Dubreka, Kindia, Coyah, Forecariah, Bombali.Right halfMontserrado, Nzerekore, Macenta, Kissdougou, Kankan, Faranah, Kono, Bo, Kenema, Tonkolili, Koinadugu, Mali, Bombali .Bombali appears twice, because it is the starting point.After the algorithm simulation result is shown in Figure4.5 and Figure 4.6:Longitude coordinates of citiesP a r a l l e l v a l u e o f c i t i e sLongitude coordinates of citiesP a r a l l e l v a l u e o f c i t i e sthe total distance:28.2716Figure4.5 Left half Figure 4.6 Right halfThe path of left half ：Bombali →Port Loko →Boffa →Forecariah →Dubreka →Moyamba →Kindia →Coyah →West e-rnRural →Conakry →Kambia →Western Urban →Bombali The path of right half ：Bombali →Kenema →Faranah →Mali →Nzerekore →Bo →Kissdougou →Kankan →Koinadu gu →Kono →Tonkolili →Montserrado →Macenta →Bombali The total distance is:L=10.1739+29.8075=39.9814.It is smaller than the answer before, the transport time is reduced and the efficiency of transportation is improved.4.3 Model Ⅲ4.3.1 Analysis of the ProblemAccording to the above analysis of the first model and the second model, we can learn something about the spreading of Ebola, then finding the shortest path to transport medicines or vaccines. On the basis of the spreading of Ebola, we can know the numbers of illness with Ebola, then, get the quantity demanded of illness. According to the city distribution of infected zone, we find the shortest path to transport medicines, as well as ensure the shortest transporting route.After comprehending the demand for vaccine in infected zones and its the shortest transporting route, the next problem we think about is how to transport the vaccines or drugs from storage zone to infected zone using the maximum efficiency. Besides, we also need to consider whether the production speed can keep up with the demand for drugs and delivery speed. That is to say, the quantity of medicine production must be greater than or equal to the demand for drugs. Only in this method can we give sufficient vaccines or drugs to infected zones by using the fastest speed to control the spread of Ebola. 4.3.2 Model DesignIn the second model, we consider the shortest path and find the shortest path to all infected zones, then get its occurrence of distance. Getting the basic solve of the first model and the second model, the drugs or vaccines transport system can allot cars for infected zones judging by the weight of the numbers of infections in different cities. hypothesis ：♦ All allocation cars are the same vehicle size, moreover, have sufficient cars. That is to say, the quantity of vaccines or drugs in all cars is equal.♦ All delivery routes will not block up, and the cars will not break down. That is to say, all allocation cars can reach the infected area on time.♦ In order to avoid Ebola propagate to other place, this area should be isolated immediately once this area burst Ebola.♦ The car allocation in different regions can match up with the pharmaceutical demand in different regions. That is to say, they are positively related♦ By looking for date, we can get the number of infections in different regions :I1,I2,I3….In, then get the weight of the number of infections in different regions:11,2,3nn nnn I n Iα===∑(5.8)The pharmaceutical demand in different regions is:1,2,3n n H C n α==(5.9)C is the total quantity of car ，αn is the weight of the number of infections in different regions.According to the hypothesis, we can know that the pharmaceutical demand in each infected zone is directly related to the car allocation, so, we allot all cars in the light of weight. That is to say, the bigger weight can get more cars, the smaller weight will get less cars. Thus, we not only can save time, but also cost.According to the above analysis, we can know that the model also should meet the follow conditions:123'n A H H H H L T V ≥++++⎧⎪⎨≤⎪⎩(5.10)H n is the pharmaceutical demand in different regions, V ＇is vehicle speed, T is theproduction cycle of vaccines or drugs. According to the model I solving scheme, we can get the proportion of infected is i(t)=1/Qln(s/s 0)in t time, At the same time the region's demand for drugs is H=kNi(t), Drug production speed need to meet :10(ln ln )ni ii i i i k N V QT s s =≥-∑(5.11)We seek the latest date information from WTO official website [4], and get the new casedistribution graphs of Guinea 、Sierra Leone 、Liberia .You can see on Figure 4.7Figure 4.7 Geographical distribution of new and total confirmed casesWe can get the number of infections about 24 cities in infected zones from the diagram [5], then figure out the weight of infection numbers in different regions and clear up these dates. You can see on the Table 4.1.According to the model I, it have forecast the number of infections in 2015 February, and calculate the number of infections in various regions of the weight, the allocation of all transport vehicles, and have meet the demand for drugs in February at epidemic area. so, according to the predicted values, We can get the drug distribution table show in Table 4.6 and vehicle allocation table show in Table 4.7.the future of the epidemic and how to reasonable distribution of drugs,.According to the above model analysis, after ensuring the demand for vaccines and medicines in different regions and the shortest transport route, and on the double bind of medicine production speed and medicine delivery speed. we have a discussion ,then get the car allocation in different regions to make sure the medicines or vaccines reach the infected zones by using the fastest speed. So, we can remit current epidemic situation of Ebola.4.4 Extent our modelsIn the model I, we have studied the classical SIR epidemic model, then we have an improved in the model I, the improved model is:()()()()()()()()dSN I S t dt dIS t I t I t dt dRI t R t dt λβλβλμμλ⎧=-+⎪⎪⎪=-+⎨⎪⎪=-⎪⎩(5.12)In the infectious disease model, We've added the μto the population birth rate and natural mortality, ‘β’is the coefficient of the spread of the disease, ‘N’ is the number of species number. In this model assumes that there is no population move out and the death due to illness, the number of population is constant.As mentioned above, the ‘I’ is the number of infected patients, if the ‘S’ ‘I’ ‘R’ have given the initialvalue, By solving the differential equations(5.12), can get the value of ‘I(t)’ at a certain moment. For this model, we expect the people infected can stable at a low level, this means that the spread of infectious diseases has been effectively controlled. Analyzed the infectious disease model, if we want to control effectively to ‘I’, should decrease the coefficient of the spread of the disease β, and improve disease recovery rate λ, In terms of emergency rescue, it’s should ensure that there are have adequate relief drug to patients in emergency treatment, and make the probability of recovery to increase, then , it can control effectively to the increase of ‘I’.At the beginning of the outbreak of infectious diseases, when it ’s have a pulse vaccination for the population cycle T, the spread of the corresponding SIR epidemic model [6] is shown in Figure 4.8, Propagation model expressed in equation (4.13).S λI λRλFigure 4.8 The flow chart of pulse SIR1()()()()()()()()()(1)()()()0,1,2()()()nn nn dSN I S t dt dI S t I t I t t tdt dR I t R t t t T dtS t p S t I t I t t t n R t R t pS t λβλβλμμλ+----⎧=-+⎪⎪⎪=-+≠⎪⎪⎨=-=+⎪⎪=-⎪⎪===⎪=+⎩(5.13)P is vaccination rate.Impulsive vaccination is different from traditional large-scale disposable vaccination, it can ensure to make an effective control by using the spread of lower vaccination rate. We can obtain something from the analysis of the first model that i(t) is the function which increase first and then decrease with the time. Thus, the population infected will tend to zero ultimately. If 0dIdt <, then the critical value of c S is:()(1)(1)T c TT p e pTS T p e λλλγλλβλ+--+=>-+ (5.14)Then the critical value of c p is ：()(1)()(1)T c T T e p T e λλλλμβμβλμβ+--=--+- (5.15)We can know that, if the vaccination rate p>p c , system can obtain a stable disease-free periodic solution.When the infectious disease, which is described at model(5.12), burst out at one region, we should firstly know the demand for vaccine in different rescue cycle area before doing vaccinate to the infected populations. On account of epidemical diffusion law that indicated by SIR model(5.13), which possessing the pulse vaccination, we use the following form of demand forecasting that change over time.()k k D pS T -=(5.16)We can know something from the second model that we divide the whole infected zone into two regions. The two regions are assumed to be A and B. There is a stockpile around A and B. Known about the above information, we use the suggested model to do car allocation for A and B.Given the parameters in Ebola spread model(5.13) and its initial value, as shown in the Table 4.8 and Table 4.9. If the pulse vaccination cycle T=50, we use MATLAB programming to figure out the arithmetic solution of Ebola spread model (5.8) and model(5.9), as shown in the follow form:Table 4.8 Infectious disease model parametersParameter λ β μ p T Numerical0.000060.000020.0080.150Table 4.9 A and B area initial values i r Infected area A 830 370 0 Infected area B92278daysn u m b e r sthe SIR model with pulse vaccination in the demand point Adaysn u m b e r sthe SIR model without pulse vaccination in the demand point A(a) (b)daysn u m b e r sthe SIR model with pulse vaccination in the demand point Bdaysn u m b e r sthe SIR model without pulse vaccination in the demand point B(c) (d)Figure 4.9 Numerical solution of diffusion model SIR diseaseCompare Figure 4.9(a) with Figure 4.9(b), we can see that infected people and vulnerable people are going down faster under the circumstance of pulse vaccination. The same circumstance can be seen in the comparison of Figure 4.9(c) and Figure 4.9(d), it indicate that the pulse vaccination can control the spread of Ebola more effective. Because of this, we use the pulse vaccination to make our model solve the spread of Ebola preferably.5 Sensitivity Analysis5.1 Effect of Daily Contact RateIn model Ⅰ, we get the variation of function i (t ) and s (t ) by assuming variable value. So further discuss the value of λ is 2 or 3 whether impact on the result.Based on MATLAB software programming, can get the graphics when λ=2 or λ=3.daysn u m b e r sThe rate of healthy people and patientsdaysn u m b e r sThe rate of healthy people and patientsFigure 5.1 λ=2 or λ=3Conclusion:♦ Through comparing with Figure 4.2 ( λ=1 ) in model Ⅰ, it can be seen that the growth of the I (t) section is slightly reduced.♦ Observe the Figure 5.1, you can see λ=2 or λ=3 graphics haven't changed much5.2 Effect of inoculation rateIn the model Ⅲ, we have introduced the method of pulse vaccination. At the same time drew a conclusion that pulse vaccination can effectively control the spread of the virus.。
美赛论文模板(超实用)
For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number50930Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________ 2015Mathematical Contest in Modeling (MCM/ICM) Summary SheetSummaryOur goal is a model that can use for control the water temperature through a person take a bath.After a person fills a bathtub with some hot water and then he take a bath,the water will gets cooler,it cause the person body discomfort.We construct models to analyze the temperature distribution in the bathtub space with time changing.Our basic heat transfer differential equation model focuses on the Newton cooling law and Fourier heat conduction law.We assume that the person feels comfortable in a temperature interval,consider with saving water,we decide the temperature of water first inject adopt the upper bound.The water gets more cooler with time goes by,we assume a time period and stipulation it is the temperature range,use this model can get the the first inject water volume through the temperature decline from maximum value to minimum value.Then we build a model with a partial differential equation,this model explain the water cooling after the fill bathtub.It shows the temperature distribution and water cool down feature.Wecan obtain the water temperature change with space and time by MATLAB.When the temperature decline to the lower limit,the person adds a constant trickle of hot water.At first the bathtub has a certain volume of minimum temperature of the water，in order to make the temperature after mixed with hot water more closer to the original temperature and adding hot water less,we build a heat accumulation model.In the process of adding hot water,we can calculate the temperature change function by this model until the bathtub is full.After the water fill up,water volume is a constant value,some of the water will overflow and take away some heat.Now,temperature rise didn't quickly as fill it up before,it should make the inject heat and the air convection heat difference smallest.For the movement of people, can be seen as a simple mixing movement, It plays a very good role in promoting the evenly of heat mixture. so we put the human body's degree of motion as a function, and then establish the function and the heat transfer model of the contact, and draw the relationship between them. For the impact of the size of the bathtub, due to the insulation of the wall of the bathtub, the heat radiation of the whole body is only related to the area of the water surface, So the shape and size of the bath is just the area of the water surface. Thereby affecting the amount of heat radiation, thereby affecting the amount of water added and the temperature difference,So after a long and wide bath to determine the length of the bath. The surface area is also determined, and the heattransfer rate can be solved by the heat conduction equation, which can be used to calculate the amount of hot water. Finally, considering the effect of foaming agent, after adding the foam, the foam floats on the liquid surface, which is equivalent to a layer of heat transfer medium, This layer of medium is hindered by the convective heat transfer between water and air, thereby affecting the amount of hot water added. ,ContentTitile .............................................................................................. 错误!未定义书签。
Intelligent controlled transportation system 美赛数学建模文章-交通建模方面
SummaryWith the rapidly developing of traffic, freeway gradually becomes the mainstream way of short-distance travel. In order to make the means of transportation become more perfect, we need to improve in as many aspects as possible. To measure the performance of a freeway, we must consider the following two factors: traffic flow and safety. These are the main aspects that we must take into consideration to weigh whether a freeway is good or bad.In order to better simulate the actual situation, we established a simulation model .We adopted the core ideas of the Cellular Automata Model, on whose basis, we established a new model suitable to the simulation of the performance on freeway. The key point of our model is regarding time and space to be discrete which is actually continuous. Every vehicle must be in certain discrete position. In this problem, we divide the road into many same-size rectangular grids, the vehicle must move in a fixed place. he number of grids stands for the distance, the number of the grids that a vehicle move per unit time stands for it’s speed. According to different rules, different small models are respectively established to study which rule is better. In a word the model we designed has combined the advantages of the Cellular Automata Model and the most important aspects of the actual situation on the highway.To study the performance more accurately, We have studied under the following three conditions:1. under very light traffic load;2. under a medium traffic load (normal traffic conditions)(main part);3. under a very heavy traffic load.In each case, we have analyzed the performance on freeway and discussed the traffic flow both in theory and by simulation. We have also calculated how the drivers on freeway guarantee their safety quantitatively. After that, we examined tradeoffs between traffic flow and safety, and analyzed the how each case limit the speed and overtaking ratio. Through analysis, we have got relatively reasonable conclusions. Differently, in case1, we gave an actual example to test our model. In case 2, we respectively analyzed the following and passing phenomenon in detail.Safety on freeway is so important that we have studied how much the traffic flow and speed influence it, we have calculated the two safety correlation coefficients of the traffic flow and speed and conclude that speed influence safety most.We have made comprehensive evaluation of “ the rule that requires drivers to drivein the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane” , and designed a new rule that“two lane used equally” to promote greater traffic flow while guaranteeing safe. The new rule has been tested by simulation.In countries where driving vehicles on the left is the norm, we have analyzed their performance on freeway, we found that my solution cannot be carried over with a simple change of orientation, additional requirements that the position of cab be changed should be needed.If vehicle transportation on the same roadway was fully under the control of an intelligent system, the most obvious change is the change of overtaking ratio (becomes almost 100%), this change will decrease traffic flow in our earlier analysis.ContentsAssumption and it’s Rationality (5)1. Model (5)1.1 Basic model (5)1.2 Feasibility and rationality of the model (6)1.3 How we set the parameters in the model (6)1.4 Simulation of different situation that we use in the article (6)1.4.1 Rules of the single-lane Cellular Automata model (6)1.4.2 The lane changing model (7)1.4.2.1 The lane changing rules (7)1.4.2.2 Explanations of the lane changingrules (7)1.4.3 Lane Changing Model verification (8)2. Different traffic density (8)2.1 Under very light traffic load (9)2.1.1 Traffic flow calculation and simulation (9)2.1.2 Safety guarantee (10)2.1.3 Speed limit (11)2.1.4 Overtaking ratio limit (11)2.1.5 An actual example (11)2.2 Medium traffic load (Normal traffic conditions) (12)2.2.1 Three factors influencing on traffic flow andsimulation (12)2.2.2 Safety guarantee (16)2.2.2.1 Following phenomenon (16)2.2.2.2 Overtaking phenomenon (17)2.2.3Speed limit (18)2.2.4 Overtaking ratio limit (18)2.3 Very heavy traffic load (18)2.3.1 Safety factors analysis (18)2.3.2Influence on traffic flow and simulation (19)2.3.3Speed limit(very low speed) (19)2.3.4 Overtaking ratio limit (19)2.4. Safety correlation coefficient (19)3. A better rule (20)3.1 Description (20)3.2 Simulation (21)4. For countries driving on the left (21)5. Intelligent controlled transportation system (22)Futher analysis of our model (22)Conclusion (23)Reference (24)Assumption and it’s RationalityAll the length is dispersive. Our model describes the movement of each individual vehicle according to the study for their interaction by taking vehicles as dispersive particles. Cellular Automata model divides a section of road into many cells of 2 meters in length.The time interval is one second. As length is dispersive, the time is dispersive. We make the minimum time interval is one second. One second is short enough to describe the motion of the car.The length of the car is the same. In some freeway the big truck is prohibited. The number of small cars dominates. We just study the situation that small car driving on the freeway. The length of the car is 4 meters or so. We take it as a average 2 cell length.The number of the cars on a selected section of the road is a constant. We only study a section of the road. We designed it as a closed loop which means one car gets out and one car enters. So the number of the cars on a selected section of the road is a constant. In this way, the density of the cars on the road is a constant.We ignore the factor of weather and season. Different weather may lead to different traffic. The situation is complex that we have to ignore these factors.The steering wheel is on the right side of the car. It is a common that n countries where driving vehicles on the left is the norm the steering wheel is on the right side of the car. It is also a fact in US and China.Passing is not allowed to single road. At the same time a cell can be occupied by only one car. So the car cannot pass another car in front on the same road. Analysis of the problem1. Model1.1 Basic modelThe key point of our model is regarding time and space as discrete which is actually continuous. Every vehicle must be in certain discrete position. In this problem, we divide the road into many of the same size rectangular grids, the vehicle must move in a fixed place.The number of grids stands for the distance, the number of the grids that a vehicle move per unit time stands for it’s speed.1.2 Feasibility and rationality of the modelWhen we analyze the problem, the distance we consider is long enough, and time is also long enough, dividing time and space into many small parts will not influence the results of analysis and simulation so much. On the contrary, the way we make time and space discrete will simplify the analysis and calculation process to a great extent, it can also make simulation much more easy.1.3 How we set the parameters in the modelConsidering the various aspects of factors, the Basic parameter definiteness is as follows:In this model, the length of each cell is 2 meters, per 2 successive cells contain one vehicle and these 2 successive cells are in the same state at moment t, i.e. thespeed of vehicle contained. Maximum speed of vehicle is 120km/h（33m/s）. Minimum speed of vehicle is 80km/h（22m/s）.Thus in this model, maximum speed (v m) is 16 cell length/second, minimum speed(v min) is 11 cell length/second. Speed value range is v min～v m and renewal time interval is 1 second.1.4 Simulation of different situation that we use in the article1.4.1 Rules of the single-lane Cellular Automata modelVariable symbols used in this Model are defined as follows.x n(t):the position of the vehicle at moment t;v n(t):the speed of the vehicle at moment t;a n(t):the acceleration of the vehicle at moment t~t+1;g n(t):the number of free sites ahead of the vehicle, i.e.g n(t)＝x n-1(t)－x n(t)－2.The states of all vehicles on road conduct synchronous renewal according to the following rules.Acceleration Rule: if v n(t) ≤ g n(t), the vehicle will accelerate.If g n(t) - v n(t) < 2, then a n(t) = g n(t) - v n(t).If g n(t) - v n(t) ≥ 2, then a n(t) = 2.If v n(t) = v m, then a n(t) = 0.Deceleration Rule:If v n(t) > g n(t), the vehicle will decelerate.If g n(t) - v n(t) > -2, then a n(t) = g n(t) - v n(t).If g n(t) - v n(t) ≤ -2, then a n(t) = -2.Correction Rule:If the acceleration of the vehicle is a n(t) at moment t, on the assumption that the forward vehicle is decelerated at maximum deceleration, then at moment t+1.If v n(t+1) ≤ g n(t+1), then the acceleration of the vehicle is a n(t).If v n(t+1) > g n(t+1), then the acceleration of the vehicle is a n(t)-1, and recalculate the v n(t+1) and g n(t+1), until v n(t+1) ≤ g n(t+1).Thus, the actual acceleration of the vehicle is a corrected value.1.4.2 The lane changing modelLane changing is the emphasis and difficulty of multi-lane road traffic flow simulation. A lane change decision process is assumed to have the following three steps: production of lane changing desire, feasibility analysis on lane changing activity and implementation of lane changing activity (Zou,2002).Based on the single-lane NS model, K. Nagel has put forward the multi-lane traffic simulation model, in which, the vehicles moving in each lane shall conform to the NS rule and satisfy the Lane-changing rules (Nagel,1998/ Wagner,1997) when changing lanes. This article put forward a kind of lane-changing model that is suitable for vehicle movement in order on the urban roads under the unobstructed condition, which is shown to match the real vehicle activities well through computer. Simulation.1.4.2.1 The lane changing rulesVariable symbols used in this model are defined as follows.g n(t) = x n-1(t) - x n(t) - 2 (1) Here:g n(t)--the number of free sites ahead of the vehicle on the present lane at moment t g l(t)--the number of free sites between the vehicle and the forwardvehicle on target lane at moment tg b(t)--the number of free sites between the vehicle and the backward vehicle ontarget lane at moment tv l(t) --the speed of the forward vehicle on target lane at moment tv b(t)--the speed of the backward vehicle on target lane at moment ts b(t) --the emergency braking distance of the backward vehicle on target lane at moment tThe lane changing model is as follows:(1) if g n(t) < v m, then the vehicle will produce lane changing desire(2) if g l(t) ≥ g n(t) and v l (t) > v n-1 (t) and s b(t) ≤ s b(t), then the vehicle will change lane at v n(t) at probability p changeHere, s b(t) = v b + max(v b - 2,0) + max(v b - 4,0) . (2)1.4.2.2 Explanations of the lane changing rulesIn this model, g n(t)<v m means due to the reason that the speed of forward vehicle isslower, that this vehicle will produce the desire of changing lane in order to reach faster speed and obtaining more free driving space.After producing the desire of changing lane, a vehicle will determine the feasibility of changing to adjacent lane according to observation. In general, a vehicle may chan- ge its lane only when the spaces between it and forward vehicle and it and backward vehicle are large enough. On condition of meeting g l(t)≥g n(t) and v l(t) > v n-1 (t), a vehicle can ensure that it will not collide with forward vehicle on target lane after changing its lane. On condition of meeting s b(t)≤s b(t), a vehicle will not collide with backward vehicle on target lane because the emergency braking distance of backward vehicle on target lane is less than the space between them. Only when meeting these conditions, a vehicle will implement lane changing activity at a certain probability. 1.4.3 Lane Changing Model verificationWe select 500-meter sections of two innermost lanes on the 4th Ring Road in Beijing as observation objectives to survey the lane changing condition at different time and under different flow in the condition of free flow. Observation period is 2 hours. Diamond shape points in Figure 1 are the survey number of lane changing under different volume. By linear regression fit, we can find that the relationship between number of lane changing and volume is linear.In accordance with the aforesaid lane-changing Cellular Automata model, we make a computer simulation for the lane-changing condition under the condition of free flow. During the simulation, we set up 500 cells, among which, 250 cells on the preparatory section (500m) and the other 250 cells on simulation section (500m), and the simulation time is 3900 seconds. The simulation within 0～300 seconds is the stage to clear up the bad effect, after a movement of 300 seconds, the road is full of vehicles. The simulation begins from the 301st second and simulation data is recorded after the first 250 cells, the flow diagram of lane-changing CellularAutomata model simulation is as follows.Simulations were conducted according to the above-mentioned process under different flows (i.e. 2500veh/h, 2600veh/h, 2700veh/h, 2800veh/h, 2900veh/h, 3000 veh/h), each flow is simulated for five times to acquire the average values, and thus, the lane-changing times under different flows are obtained. Comparing those simulated results (while p b=0.5 and p c=0.8)with the observed values, they are matching with each other by a large while p b and pc value are correctly selected so as to verify the validity of this Lane Changing Model.2. Different traffic densityThere’s different perform ance under different traffic load so we must analyze in three parts:2.1 Under very light traffic loadWhen in light traffic, a vehicle is almost not constrained by other vehicles (free running). Drivers will run at a speed as much as possible to get the more benefits of driving such as shortening the travel time. It may raise traffic flow in some degree, however, this psychological state will cause certain threat to the safety. So the traffic flow and safety assessment in light traffic is necessary.2.1.1 Traffic flow calculation and simulationThough under low traffic load vehicles can run at a very high speed, the very low vehicle density plays a negative role. What’s worse, the low vehicle density influence more on traffic flow in this case. In other words, the traffic flow will be very low.We assume that any vehicle can pass each other freely. The average interval of the vehicles is two thousand meters. The speed of vehicles varies from 80km/h to 120km/h. For the sake of simplicity, we choose only five kinds of speed: 80km/h, 90km/h, 100km/h, 110km/h and 120km/h. We assume that the quantity of each kind of speed of the vehicles is an equal.Compare this rule to the condition that all the vehicles run in one road without any pass.① Passing18090100110120/5052++++-=⨯=traffic flow hour (3) ②.No Passing ( the speed of almost all the cars is limited to under 80 km/h) 80/402traffic flow hour -== (4) This the result we get through calculation.Let‘s see the result of simulation:Fig1.under very low traffic loadThrough a certain tool, we can get the traffic flow (the number of vehicles through a cross-section we set in an hour).we have simulated 10 times and the data we got is as follows:42,44,49,38,45,46,44,43,47,51,45 (per hour)The average:40+42+38+44+40+41+42+45+44+43/=41.910traffic flow h -= (5) Comparing the theoretical calculation and simulation results, we can come to a similar conclusion with two methods, which has also supported our model.2.1.2 Safety guaranteeUnder very low traffic load, the main factor that influence safety is speed. Though there’re other factors that may also influence, they are negligible relative to speed.The distance from the drivers found obstacles to the vehicle to a full stop is the sum of the reaction distance and braking distance. Shown in figure 2.Fig2. brake when finding an obstacleReaction distance is the distance from the point where the driver finds an obstacle to the point where he starts to brake:1=t 3.6V S (6) Braking distance is the distance during the whole braking process:22=254(f+i)V S (7) The stopping sight distance:2=t+3.6254(f+i)V V S (8) Here:V--the speed of the vehicle (km/h);t--reaction time of the driver(s);f--coefficient of road adhesion (for dry pavement f=0.6);i--the tilt degree of the road (for level road i=0).L--the distance when the driver finds the obstacle between the obstacle and the vehicle.Suppose the safe distance is d, when +d ＞L S , t he vehicle will be in danger; when +d L S ≤, the vehicle is safe.2.1.3 Speed limitWhen the traffic load is light, vehicles can run as fast as possible under the biggest speed limit , however it will increase risk of accident; if the speed of some vehicles is t oo low, we can’t make the best of the freeway thus reducing flow. So, under the premise of ensuring safety, we can run at a relatively high speed to increase traffic flow as possible as we can. A too high speed may lead to an accident.Speed is the main factor that influence the traffic flow.2.1.4Overtaking ratio limitSince the traffic load is very low, the possibility of overtaking phenomenon is very low, and in extreme cases the overtaking ratio can even be seen as zero.That is, in this case, overtaking is a very minor factor that influence traffic flow.2.1.5 An actual exampleWe have found an actual example in this case:2011, one day in July, Shanghai-Nanjing Freeway in Jiangsu province.Considering the weather and road condition, some experts major in it confirm that the coefficient of road adhesion f is 0.40. And according to the Shanghai-Nanjing Freeway designing information, the tilt degree of the road i is 0. The visibility that day is 55 meters. The safe distance is 5 meters.Substituting the data into equation ：22.5d +53.6254(f+i)V S V +=+， (9) If the result we calculate is more than 55 meters, the vehicle can’t brake in time, which will cause a collision between the vehicle and the obstacle.So under such condition, the speed of each vehicle must be limited.For example, a vehicle ran at V=80km/h that day on Shanghai-Nanjing Freeway,22.580d 80+5123.553.6254(0.40+0)S m +=⨯+≈ (10) This result is much higher than the visibility 55 meters, so V=80km/h is a very dangerous speed.We can also calculate the highest speed that is allowed under the terrible condition:Let:22.5d+5553.6254(f+i)VS V m+=+≤, (11)We get: V≤43.52km/h (12) We can see that in this example, drivers must control the speed under 43.52km/h in order to guarantee their safety.2.2 Medium traffic load (Normal traffic conditions)Medium traffic density is the most common case, that is to say, this is the most consistent with the actual situation under normal circumstances. Therefore, we personally think that studying this case makes the most common sense. When the traffic density is not so big or so small, due to the rule requiring drivers to drive in the right-most lane unless they overtake from the left lane, the motion of vehicles is not untrammeled，each one is in interference and constraints produced by others. The performance of vehicles on freeway is mainly following and overtaking, through practical experience.2.2.1 Three factors influencing on traffic flow and simulationThe main factors that influence the traffic flow are overtaking ratio, traffic density and speed of vehicles, the three factors are not completely independent, there’s certai n mutual restraint and influence between them.“overtaking ratio--- traffic flow” relationWe choose the overtaking ratio as the main verification index when we study the process of overtaking. Through a survey that has been made and a relative simulation, the traffic flow changes with the overtaking ratio. The survey method is as figure3,the survey conclusion is as the table 1.Fig 3.Sketch map of section observatin method for field surveyTab 1. Survey of traffic flow and overtaking ratioTo get more accurate result, we have made a simulation. To make sure how thetraffic flow changes with the overtaking ratio, we set a series of overtaking ratios. Through the simulation, we get the ” traffic flow—overtaking changing curve.Fig 4. Traffic flow—overtaking changing curveComparing the results of the survey and the simulation, the rationality of the model can be made sure. According to the curve, the changing process is divided into two sections: the first section shows that in two-lane freeway, the overtaking ratio increases with the traffic flow raising, to the biggest; the second section shows that with the traffic flow raising, the overtaking ratio decreases, when the traffic flow increases to 2900pcu/h, the overtaking ratio is almost zero.“traffic density-- traffic flow” relationAccording to the equation:Q=KV(13) Here:Q---the traffic flow (pcu/h);K---traffic density (pcu/km);V---the average speed (km/h)If there’s no special situation such as rear-ending, when V is a constant ,Q ∝K, the image is as figure 5.Fig5. Q-K relationHowever, our simulation result is as figure 6.and 7Fig 6 S imulation result of “traffic density -- traffic flow” relationFig 7 . “Traffic density-- traffic flow” relation curveDifference explanation :Increased density cannot be unlimited, Q = KV is the ideal case, the actual case will be affected by external factors, our simulation result is more realistic.Analysis :When the traffic density is less than the optimum density of traffic flow,traffic flow is in th e f ree driving state,the average speed of cars is high. Traffic flow does not reach the m aximum value.The increasing of density leads to the increasing of trafficflow;when th e traffic density is equal or close to the optimum density of traffic flow, traffic team fo llowing phenomenon appears, the speed will be limited. Different kinds of car approa- ching a speed constant speed, traffic volume will reach the maximum value; when the traffic density is greater than the optimum density of traffic flow, traffic flow is in the congestion state, because of traffic density increases gradually, vehicle speed and traff ic volume decrease at the same time and traffic jam happens or even parking phenome non.From the figure, we can get the following information:① When K=0,Q=0,the curve pass O of the coordinate system;②0=dK dQ ,m j K K K ==21 ③ From the point c ，if K become larger ,Q becomes smaller ，when K= K j ，V=0 Q=0④ Drawing radius vector from the coordinate origin to any point on the curve, the slope of the radius vector stands for the average speed of the point.⑤ K ≤ Km: not crowded; K>Km: crowded.“speed -- traffic flow” re lationAccording to the equation:Q=KVIf there’s no special situation such as rear -ending, when K is a constant ,Q ∝V, the image is as figure 8.Fig 8 Q-A relationOur simulation also confirms the linear relationship.Further discussionThe equation Q=KV can be shown in an more unified way (figure 9)Fig9 .3-D image of Q=KVWe have known that )1(j f K KV V -=, so )1(jj V VK K -=,(14) So we can get a more specific form :)(2fj V V V K KV Q -==,(15) figure 9 has shown the equation.Fig 10.Q-V-K relation2.2.2 Safety guarantee2.2.2.1 Following phenomenonAccording to the accident statistics annals, rear-ending is the main part of the traffic accident, so we must guarantee the security when a vehicle follows another. We can easily come to a conclusion that the speed is the main factor influencing safety under the state of vehicle following the front one. When the front vehicle suddenly brake, whether the following vehicle can stop in time to avoid collision, and maintain at a safe distance determines the safety of the two vehicles. This process is shown in figure 11.Fig11.brake when the front vehicle brake suddenlyAccording to AASHTO parking stadia model, the distance that the front vehicle A run from starting braking to stopping completely is:2254(f+i)A A V S (16) After reaction time, vehicle B also start to brake, the distance that the front vehicle B run from starting braking to stopping completely is:2t 3.6254(f+i)B B B V V S =+ (17) Here: S A the speed of the front vehicle A (km/h);S B --the speed of the latter vehicle B (km/h);t--the reaction time of driver (s);f--coefficient of road adhesion (for dry pavement f=0.6);i--the tilt degree of the road (for level road i=0).Suppose two vehicles are L away from each other when the front one brake suddenly, set safety distance d,When ＜A B S L S d ++, or 22＜t +d 254(f+i) 3.6254(f+i)A BB V V V L ++, vehicle B can’t brakein time, A and B can’t hold a safe distance, accident may happen. Otherwise they are safe.2.2.2.2 Overtaking phenomenonWhen a freeway is in medium density, which is the most common case, overtaking often happens, in order to get rid of the limit of the slower vehicle in front of it. If an overtaking is successfully completed, the biggest driving satisfaction will be achieved. On freeway, overtaking will increase the traffic flow more or less, which is the main difference between single lane and two lanes. However, overtaking is a relatively dangerous behavior, we must make sure that we can safely finish a overtaking. This process is shown in figure 12.Fig 12. Overtaking phenomenon processVehicle P wants to overtake because there’s a vehicle C from which P will be a dangerous distance away.We can put the overtaking process down into two lane changing process, which is the key to the analysis of overtaking.In the first lane changing process, the following inequalities must be satisfied:L1≤d;L2≤d; (18)L3≤d;In the second lane changing process, with the same reason, the following inequalities must be satisfied:L”1≤d;L”2≤d;L”3≤d; (19)L”4≤d;The L”1, L”2 , L”3and L”4 can be get by the following equations:L”1=L1+V P t-V A t;L”2= L2+V B t-V P t;L”3=V P t-V C t-L3; (20)L”4= L4+V D t-V P t.Here:L1 /L2/ L3/L4—the distance between A/B/C/D and P;d--safe distance;V A/V B/V C/V D/V P--the speed of vehicle A/B/C/D/P;t--total overtaking time.Now we can use these inequalities and equations to judge whether an overtaking is safe or not in theory.2.2.3Speed limitIn this case, the vehicles must run at a moderate speed, if an vehicle runs too fast, the risk of rear-ending will increase; if an vehicle runs too slow, it will increase the number of overtaking phenomenon per unit time per unit length thus security cannot be guaranteed.2.2.4 Overtaking ratio limitBecause of the vehicles around, the overtaking ratio can’t be too low; the overtaking condition satisfied doesn’t mean the happening of a successful overtaking, we must be aware that there’s selective overtaking, that is, drivers may not overtake even if the safe condition has been ensured.2.3 Very heavy traffic loadWhen the freeway is under so heavy load that there’s few overtaking behavior, the main factor that may limit the traffic flow is the overall movement speed. The performance on the freeway is mainly following, and the intervals are relatively very small.2.3.1 Safety factors analysis。
MCM(美国数学建模论文)
the Bioaccumulation of Methylmercury in Human BodySummaryNowadays the heavy metal pollution is so common that people pay more and more attention to it. The aim of this paper is to calculate the maximum of methylmercury in human body during their lifetime and the maximum number of fish the average adult can safely eat per month. From City Officials research[1], we get information that the mean value of methylmercury in bass samples of the Neversink Reservoir is 1300 ug/kg and the average weight of bass people consume per month is 0.7 kg. According to the different consuming time in every month, we construct a discrete dynamical system model for the amount of methylmercury that will be bioaccumulated in the average adult body. In ideal conditions, we assume people consume bass at fixed term per month. Based on it, we construct fixed-ingestion model and we reach the conclusion that the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3505 ug. As methylmercury ingested is not only coming from bass but also from other food, hence, we make further revise to our model so that the model is closer to the actual situation. As a result, we figure out the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3679 ug. As a matter of fact, although we assume people consume one fish per month, the consuming time has great randomness. Taking the randomness into consideration, we construct a random-ingestion model at the basis of the first model. Through computer simulations, we obtain the maximum of methylmercury in human body is 4261 ug. We also calculate the maximum amount is 4420 ug after random-ingestion model is revised. As it is known to us, different countries and districts have different criterions for mercury toxicity. In our case, we adopt LD50 as the toxic criterions(LD50 is the dosage at which 50% of the humans exposed to a particular chemical will die. The LD50 for methylmercury is 50 mg/kg.). We speculate mercury toxicity has effect on the ability of eliminating mercury, therefore, we set up variable-elimination model at the basis of the first model. According to the first model, the amount of methylmercury in human body is 50 ug/kg, far less than 50 mg/kg, so we reach the conclusion that the fish consumption restrictions put forward by the reservoir advisories can protect the average adult. If the amount of methylmercury ingested increases, the amount of bioaccumulation will go up correspondingly. If 50 mg/kg is the maximum amount of methylmercury in human body, we can obtain the maximum number of fish that people consume safely per month is 997.Keywords: methylmercury discrete dynamical system model variable-elimination modeldiscrete uniform random distribution model random-ingestion modelIntroductionWith the development of industry, the degree of environmental pollution is also increasing. Human activities are responsible for most of the mercury emitted into the environment. Mercury, a byproduct of coal, comes from acid rain from the smokestack emissions of old, coal-fired power plants in the Midwest and South. Its particles rise on the smokestack plumes and hitch a ride on prevailing winds, which often blow northeast. After colliding with the Catskill mountain range, the particles drop to the earth. Once in the ecosystem, micro-organisms in the soil and reservoir sediment break down the mercury and produce a very toxic chemical form known as methylmercury. It has great effect on human health.Public officials are worried about the elevated levels of toxic mercury pollution in reservoirs providing drinking water to the New York City. They have asked for our assistance in analyzing the severity of the problem. As a result of the bioaccumulation of methylmercury, if the reservoir is polluted, we can make sure that the amount of methylmercury in fish is also increasing. If each person adheres to the fish consumption restrictions as published in the Neversink Reservoir advisory and consumes no more than one fish per month, through analyzing, we construct a discrete dynamical system model of time for the amount of methylmercury that will bioaccumulate in the average adult person. Then we can obtain the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime. At the same time, we can also get the time that people have taken to achieve the maximum amount of methylmercury. As we know, different countries and districts have different criterions for the mercury toxicity. In our case, we adopt the criterion of Keller Army Community Hospital. If the maximum amount of methylmercury in human body is far less than the safe criterion, we can reach the conclusion that the reservoir is not polluted by mercury or the polluted degree is very low, otherwise we can say the reservoir is great polluted by mercury. Finally, the degree of pollution is determined by the amount of methylmercury in human body.Problem Onediscrete dynamical system modelThe mean value of methylmercury in bass samples of the Neversink Reservoir is 1300 ug/kg and the average weight of bass is 0.7 kg. According to the subject, people consume no more than one fish per month. For the safety of people, we must consider the bioaccumulation of methylmercury under the worst condition that people absorb the maximum amount of methylmercury. Therefore, we assume that people consume one fish per month. Assumptions● The amount of methylmercury in fish is absorbed completely and instantly by people. ● The elimination of mercury is proportional to the amount remaining. ● People absorb fixed amount of methylmercury at fixed term per month. ● We assume the half-life of methylmercury in human body is 69.3 days. SolutionsLet 1α denote the proportion of eliminating methylmercury per month, 1β denote the accumulation proportion. As we know, methylmercury decays about 50 percent every 65 to 75 days, if no further methylmercury is ingested during that time. Consequently,111,βα=-69.3/3010.5.β=Through calculating, we get10.7408.β=L et’s define the following variables ：ω denotes the amount of methylmercury at initial time,n denotes the number of month,nω denotes the amount of methylmercury in human body at the moment people have just ingested the methylmercury in the month n ,1xdenotes the amount of methylmercury that people ingest per month and 113000.7910x ug ug =⨯=.Moreover, we assume0=0.ωThough,111,n n x ωωβ-=⋅+we get1011x ωωβ=⋅+ 2201111x x ωωββ=⋅+⋅+⋅⋅⋅10111111nn n x x x ωωβββ-=⋅+⋅+⋅⋅⋅+⋅+ 121111(1)n n n x ωβββ--=++⋅⋅⋅++⋅11111.1n n x βωβ--=-With the remaining amount of methylmercury increasing, the elimination of methylmercury is also going up. We know the amount of ingested methylmercury per mouth is a constant. Therefore, with time going by, there will be a balance between absorption and elimination. We can obtain the steady-state value of remaining methylmercury as n approaches infinity.1*1111111lim3505.11n n n x x ug βωββ-→∞-===--The value of n ω is shown by figure 1.Figure 1. merthylmercury completely coming from fish and ingested at fixed term per monthIf the difference of the remaining methylmercury between the month n and 1n - is less than five percent of the amount of methylmercury that people ingest per month, that is,115%.n n x ωω--<⋅Then we can get11=3380ug.ωAt the same time, we can work out the time that people have taken to achieve 3380 ug is 11 months.From our model, we reach the conclusion that the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3505 ug.If people ingest methylmercury every half of a month, however, the sum of methylmercury ingested per month is constant, consequently,11910405,0.86.2x ug β===As a result, we obtain the maximun amount of methylmercury in human body is 3270ug. When the difference is within 5 %, we get the time people have taken to achieve it is 11 months.Similarly, if people ingest methylmercury per day, we get the maximum amount is 3050ug, and the time is 10 months. Revising ModelAs a matter of fact, the amount of methylmercury in human body is not completely coming from fish. According to the research of Hong Kong SAR Food and Environmental Hygiene Department [1], under normal condition, about 76 percent of methylmercury comes from fish and 24 percent comes from other seafood. In order to make our model more and more in line with the actual situation, it is necessary for us to revise it. The U.S. environmental Protection Agency (USEPA) set the safe monthly dose for methylmercury at 3 microgram per kilogram (ug/kg) of body weight. If we adopt USEPA criterion, we can calculate the amount of methylmercury that the average adult ingest from seafood is 50.4 ug per month. AssumptionsThe amount of methylmercury in the seafood is absorbed completely and instantly by people.● The elimination of methylmercury is proportional to the amount remaining. ● People ingest fixed amount of methylmercury from other seafood every day. ● We assume the half-life of methylmercury in human body is 69.3 days. SolutionsLet 0ωdenote the amount of methylmercury at initial time, t denote the number of days, t ω denote the remaining amount on the day t , and 2x denote the amount of methylmercury that people ingest per day. Moreover, we assume0=0.ωIn addition, we work out2x =50.4/30=1.68 ug.The proportion of remaining methylmercury each day is 2β, then69.320.5.β=Through calculating, we get20.99.β=Because of12221,1t t x βωβ--=-we obtain steady-state value of methylmercury1*2222211lim168.11t t t x x ug βωββ-→∞-===--If the difference of remaining methylmercury between the day t and 1t - is less than five percent of the amount of methylmercury that people ingest every day, that is,125%.t t x ωω--<⋅We have301= 160 ug.ωSo we can reach the conclusion that the maximum amount of methylmercury the average adult human will bioaccumulate from seafood is 160 ug and the time that people take to achieve the maximum is 301 days.Let 1x denote the amount of methylmercury people ingest through bass at fixed term per month, so the amount of methylmercury an average adult accumulate on the day t is1221221if t is a positive integer and not divisible by 30if t is a positive integer and divisible by 30.t t t t x x x ωωβωωβ--=⋅+⎧⎨=⋅++⎩The value of t ω is shown by figure 2.Figure 2. merthylmercury coming from fish and other seafood and ingested at fixed term per day The change oftωreflects the change of the amount of methylmercury in human body. Through revising model, we can figure out the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3679 ug.Problem TwoRandom-ingestion modelAlthough people consume one fish per month, the consuming time has great randomness. We speculate the randomness has effect on the bioaccumulation of methylmercury, therefore, we construct a new model. Assumptions●The amount of methylmercury in fish is absorbed completely and instantly by people.●The elimination of methylmercury is proportional to the amount remaining.●People consume one fish per month, but the consuming time has randomness.●We assume the half-life of methylmercury in human body is 69.3 days.LetL denote the amount of methylmercury at initial time, n L denote the amount of methylmercury at the moment people have just ingested methylmercury in the month n, and x denote the amount of methylmercury that people absorb each time.We assume0=0.LWe have910.x ug=We define1βthe proportion of remaining methylmercury every day. Through69.3 10.5,β=we can get10.99.β=Let i obey discrete uniform random distribution with maximum 30 and minimum 1 and n t denote the number of days between the day1n i -of the month 1n - and the day n i of the month n , then we have-130-,n n n t i i =+(1)1.n tn n L L x β-=⋅+The value of n Lis shown by figure 3.Figure 3. merthylmercury completely coming from fish and ingested at random per monthFigure 3 shows the amount of methylmercury in human body has a great change due to the randomness of consuming time. Through the computer simulation, if we have numberless samples, n L will achieve the maximum value. That is,4261.n L ug =Revising modelIn order to make our model more accurate, we need to make further revise. We take methylmercury coming from other seafood into consideration. We know the amount of methylmercury that people ingest from other seafood every day is 1.68 ug.In that situation, we have1212.30(-1)30(-1)n n n n n n L L x if n n i L L x x if n n i ββ=⋅+≠⨯+⎧⎨=⋅++=⨯+⎩Through the computer simulation, we can get a set of data about n L shown by figure 4.Figure 4. remaining merthylmercury coming from fish consumed at random per month and other food consumed at fixedterm per dayThough the revised model, we reach the conclusion that if we have numberless samples, n L will achieve the maximum value. That is,4420.n L ug =Variable-eliminateion modelAs a matter of fact, the state of human health can affect metabolice rate so that the ability of eliminating methylmercury is not constant. We have koown the amount of methylmercury in human body will affect human health. So we can draw the conclusion that the amount of methylmercury in human body will affect the abilitity of eliminating methylmercury. Assumptions● The amount of methylmercury in fish is absorbed completely and instantly by people.● the elimination of methylmercury is not only proportional to the amount remaining, but also affected bythe change of human health which are caused by the amount of methylmercury.● People absorb fixed amount of methylmercury at fixed term per month through consuming bass. ● We assume the half-life of methylmercury in human body is 69.3 days.● In condition that no further methylmercury is ingested during a period of time, we let χ denote theeliminating proportion per month. We have known methylmercury decays about 50 percent every other day 5 to a turn 5 days, so we determine the half-life of methylmercury in human body is 69.3 days. Then we have69.3/301(1)0.5χ⋅-=. By calculating, we getχ=0.2592.We adopt LD50 as the toxic criterions, then we get the maximum amount of methylmercury in human body is 63.510⨯ ug.L et’s define the following variables ：ω denotes the amount of methylmercury at initial time,n denotes the number of month,nω denotes the amount of methylmercury in human body at the moment people have just ingested the methylmercury in the month n ,n χ denotes the ability of eliminating methylmercury in the month n . γ denotes the effect on human health caused by methylmercury toxicity.1161 3.510r n n ωχχ-⎛⎫⎡⎤=⋅- ⎪⎢⎥ ⎪⨯⎣⎦⎝⎭1(1)n n n ωωχϕ-=⋅-+Hence, we have101(1)ωωχϕ=⋅-+20212(1)(1)(1)ωωχχϕχϕ=⋅-⋅-+⋅-+[]01233(1)...(1)(1)(1)...(1)(1)...(1)...(1)1n n n n n ωωχχϕχχχχχχ=⋅--+⋅-⋅--+--++-+We define the value of γ is 0.5, then we get the maximum amount of maximum in human body is 3567 ug, that is,*=3567 ug n ωNot taking the effect on the ability of eliminating maximum caused by methylmercury toxicity into account in model one,we obtain the maximum amount is 3510 ug. The difffference proves methylmercury toxicity has effect on eliminating methylmercury. We find out through calculating when r increases, the amount of methylmercury go up correspondingly. The reason for it is that methylmercury toxicity rises as a result of r increasing. Correspondingly, the effect on human health will increase, which is in accordance with fact.Problem ThreeAccording to the first model revised, we can get the maximum amount of bioaccumulation methylmercury is 3679 ug. We assume the average weight of an adult is 70 kg and the amount of methylmercury in human body is 53 ug/kg, far less than 50 mg/kg. Therefore, according to our model, the fish consumption restrictions put forward by the reservoir advisories can protect the average adult fromreaching the LD50(LD50 is the dosage at which 50% of the humans exposed to a particular chemical will die. The LD50 for methylmercury is 50 mg/kg).We assume the lethal dosage of methylmercury is not gradually increasing. If the amount of methylmercury people ingests goes up rapidly, the bioaccumulation amount will reach to a higher value. Moreover, the value probably endangers human safety. Let LD50 be the maximum amount of methylmercury in human body, that is,*n =50 m g/kg 70 kg=3500 m g.ω⨯Let 1x denote the amount of methylmercury people ingest per month. According to the first model,1*1111111lim.11n n n x x βωββ-→∞-==--We can figure out1 x =907.2 mg.We know the mean value of methylmercury in bass samples is 1.3 mg/kg, hence, we can obtain the maximum amount of fish that people consume safely per month is1m ax 698.1.3x M kg =≈The maximum number of fish is 698/0.7=997.ConclusionIn problem one, the paper calculates the final steady-state value at the same time interval per month, per half a month and per day. Through comparing the results, we get the final bioaccumulation amount of methylmercury is less, when discrete time unit is smaller. It shows when the interval of consuming fish is smaller and the sum of methylmercury ingested is constant for a period of time, the possibility of poisoning is lower.In problem two, we analyze the change of the amount of methylmercury under the condition that consuming time is random. We find out the amount of methylmercury in human body is changing constantly in fixed range, when people have just consumed fish. Moreover, the maximum is 4261 ug, which is far bigger than 3505 ug. So we can reach the conclusion that people are more endangered when the consuming time is irregular.In order to closer to the actual situation, we construct a model in which the half-life of methylmercury in human body is not constant. Through analyzing the data of computer simulation, the maximum amount of methylmercury will increase, that is, the risk of poisoning will be higher.Control numberReferences[1] Dr.D.N.Rahni, PHD. Airborne Mercury Contamination and the NeversinkReservoir./dnabirahni/rahnidocs/Envsc/Airborne%20Mercury%20Contamination%20and %20the%20Neversink%20Reservoir.doc[2] Hu Dong Bai Ke. Bass. /wiki%E9%B2%88%E9%B1%BC.[3] Centre for Food Safety Food and Environmental Hygiene Department The Government of the HongKong Special Administrative Region. Mercury in Fish and Food Safety..hk/english/Programmme/programme_rafs/Programme_rafs_fc_01_19_mercury_in_fi sh.html.Page 11 of 11。
美国大学生数学建模竞赛MCM写作模板（各个部分）
美国⼤学⽣数学建模竞赛MCM写作模板（各个部分）摘要：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 system used on small farms, consists of 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：We find the optimal 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 and examining the engineering parameters of sprinklers available in the market.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。
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MCM 2015 Summary Sheet for Team 35565For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number35565Problem ChosenBFor office use onlyF1________________F2________________F3________________F4________________ SummaryThe lost MH370 urges us to build a universal search plan to assist searchers to locate the lost plane effi-ciently and optimize the arrangement of search plans.For the location of the search area, we divided it into two stages, respectively, to locate the splash point and the wreckage‟s sunk point. In the first stage, we consider the types of crashed aircraft, its motion and different position out of contact. We also consider the Earth‟s rotation, and other factors. Taking all these into account, we establish a model to locate the splash point. Then we apply this model to MH370. we can get the splash point in the open water is 6.813°N 103.49°E and the falling time is 52.4s. In the second stage, considering resistances of the wreckage in different shapes and its distribution affected by ocean currents, we establish a wreckage sunk point model to calculate the horizontal displacement and the angle deviation affected by the ocean currents. The result is 1517m and 0.11°respectively. Next, we extract a satellite map of submarine topography and use MATLAB to depict seabed topography map, determining the settlement of the wreckage by using dichotomy algorithm under different terrains. Finally, we build a Bayesian model and calculate the weight of corresponding area, sending aircrafts to obtain new evidence and refresh suspected wreckage area.For the assignment of the search planes, we divide it into two stages, respectively, to determine the num-ber of the aircraft and the assignment scheme of the search aircraft. In the first stage, we consider the search ability of each plane and other factors. And then we establish global optimization model. Next we use Dinkelbach algorithm to select the best n search aircrafts from all search aircrafts. In the second stage, we divide the assignment into two cases whether there are search aircrafts in the target area. If there is no search aircraft, we take the search area as an arbitrary polygon and establish the subdivision model. Considering the searching ability of each plane, we divide n small polygons into 2n sub-polygons by using NonconvexDivide algorithm, which assigns specific anchor points to these 2n sub-polygons re-spectively. If there exist search aircrafts, we divide the search area into several polygons with the search aircrafts being at the boundary of the small polygons. To improve search efficiency, we introduce” ma x-imize the minimum angle strategy” to maximize right-angle subdivision so that we can reduce the turning times of search aircraft. When we changed the speed of the crashed plane about 36m/s, the latitude of the splash point changes about 1°.When a wreck landing at 5.888m out from the initial zone, it will divorce from suspected searching area, which means our models are fairly robust to the changes in parameters. Our model is able to efficiently deal with existing data and modify some parameters basing the practical situation. The model has better versatility and stability. The weakness of our model is neglect of human factors, the search time and other uncontrollable factors that could lead to deviation compared to practical data. Therefore, we make some in-depth discussions about the model, modifying assumptions establish-Searching For a Lost PlaneControl#35565February 10, 2014Team # 35565 Page 3 of 57 Contents1 Introduction (5)1.1 Restatement of the Problem (5)1.2 Literature Review (6)2 Assumptions and Justifications (7)3 Notations (7)4 Model Overview (10)5 Modeling For Locating the Lost Plane (10)5.1 Modeling For Locating the Splash Poin t (11)5.1.1 Types of Planes (11)5.1.2 Preparation of the Model—Earth Rotation (12)5.1.3 Modeling (13)5.1.4 Solution of The Model (14)5.2 Modeling For Locating Wreckage (15)5.2.1 Assumptions of the Model (16)5.2.2 Preparation of the Model (16)5.2.3 Modeling (21)5.2.4 Solution of the Model (25)5.3 Verification of the Model (26)5.3.1 Verification of the Splash Point (26)5.3.2 Verification of the binary search algorithm (27)6 Modeling For Optimization of Search Plan (29)6.1 The Global Optimization Model (29)6.1.1 Preparation of the Model (29)6.1.2 Modeling (31)6.1.3 Solution of the Model (31)6.2 The Area Partition Algorithm (33)6.2.1 Preparation of the Model (33)6.2.2 Modeling (34)6.2.3 Solution of the Model (35)6.2.4 Improvement of the Model (36)7 Sensitivity Analysis (38)8 Further Discussions (39)9 Strengths and Weaknesses (41)9.1 Strengths (41)9.2 Weaknesses (42)10 Non-technical Paper (42)1 IntroductionAn airplane (informally plane) is a powered, fixed-wing aircraft that is propelled for-ward by thrust from a jet engine or propeller. Its main feature is fast and safe. Typi-cally, air travel is approximately 10 times safer than travel by car, rail or bus. Howev-er, when using the deaths per journey statistic, air travel is significantly more danger-ous than car, rail, or bus travel. In an aircraft crash, almost no one could survive [1]. Furthermore, the wreckage of the lost plane is difficult to find due to the crash site may be in the open ocean or other rough terrain.Thus, it will be exhilarating if we can design a model that can find the lost plane quickly. In this paper, we establish several models to find the lost plane in seawater and develop an op-timal scheme to assign search planes to model to locate the wreckage of the lost plane.1.1 Restatement of the ProblemWe are required to build a mathematical model to find the lost plane crashed in open water. We decompose the problem into three sub-problems:●Work out the position and distributions of the plane‟s wreckage●Arrange a mathematical scheme to schedule searching planesIn the first step, we seek to build a model with the inputs of altitude and other factors to locate the splash point on the sea-level. Most importantly, the model should reflect the process of the given plane. Then we can change the inputs to do some simulations. Also we can change the mechanism to apply other plane crash to our model. Finally, we can obtain the outputs of our model.In the second step, we seek to extend our model to simulate distribution of the plane wreckage and position the final point of the lost plane in the sea. We will consider more realistic factors such as ocean currents, characteristics of plane.We will design some rules to dispatch search planes to confirm the wreckage and de-cide which rule is the best.Then we attempt to adjust our model and apply it to lost planes like MH370. We also consider some further discussion of our model.1.2 Literature ReviewA model for searching the lost plane is inevitable to study the crashed point of the plane and develop a best scheme to assign search planes.According to Newton's second law, the simple types of projectile motion model can work out the splash point on the seafloor. We will analyze the motion state ofthe plane when it arrives at the seafloor considering the effect of the earth's rotation,After the types of projectile motion model was established, several scientists were devoted to finding a method to simulate the movement of wreckage. The main diffi-culty was to combine natural factors with the movement. Juan Santos-Echeandía introduced a differential equation model to simplify the difficulty [2]. Moreover,A. Boultif and D. Louër introduced a dichotomy iteration algorithm to circular compu-ting which can be borrowed to combine the motion of wreckage with underwater ter-rain [3]. Several conditions have to be fulfilled before simulating the movement: (1) Seawater density keeps unchanged despite the seawater depth. (2) The velocity of the wreck stay the same compared with velocity of the plane before it crashes into pieces.(3) Marine life will not affect our simulation. (4) Acting forceof seawater is a function of the speed of ocean currents.However the conclusion above cannot describe the wreckage zone accurately. This inaccuracy results from simplified conditions and ignoring the probability distribution of wreckage. In 1989, Stone et.al introduced a Bayesian search approach for searching problems and found the efficient search plans that maximize the probability of finding the target given a fixed time limit by maintaining an accurate target location probabil-ity density function, and by explicitly modeling the target‟s process model [4].To come up with a concrete dispatch plan. Xing Shenwei first simulated the model with different kinds of algorithm. [5] In his model, different searching planes are as-sessed by several key factors. Then based on the model established before, he use the global optimization model and an area partition algorithm to propose the number of aircrafts. He also arranged quantitative searching recourses according to the maxi-mum speed and other factors. The result shows that search operations can be ensured and effective.Further studies are carried out based on the comparison between model andreality.Some article illustrate the random error caused by assumptions.2 Assumptions and JustificationsTo simplify the problem, we make the following basic assumptions, each ofwhich is properly justified.●Utilized data is accuracy. A common modeling assumption.●We ignore the change of the gravitational acceleration. The altitude of anaircraft is less than 30 km [6]. The average radius of the earth is 6731.004km, which is much more than the altitude of an aircraft. The gravitational accele-ration changes weakly.●We assume that aeroengine do not work when a plane is out of contact.Most air crash resulted from engine failure caused by aircraft fault, bad weather, etc.●In our model, the angle of attack do not change in an air crash and thefuselage don’t wag from side to side. We neglect the impact of natural and human factors●We treat plane as a material point the moment it hit the sea-level. Thecrashing plane moves fast with a short time-frame to get into the water. The shape and volume will be negligible.●We assume that coefficient of air friction is a constant. This impact is neg-ligible compared with that of the gravity.●Planes will crash into wreckage instantly when falling to sea surface.Typically planes travel at highly speed and may happen explosion accident with water. So we ignore the short time.3 NotationsAll the variables and constants used in this paper are listed in Table 1 and Table 2.Table 1 Symbol Table–ConstantsSymbol DefinitionωRotational angular velocity of the earthg Gravitational accelerationr The average radius of the earthC D Coefficient of resistance decided by the angle of attack ρAtmospheric densityφLatitude of the lost contact pointμCoefficient of viscosityS0Area of the initial wrecking zoneS Area of the wrecking zoneS T Area of the searching zoneK Correction factorTable 2 Symbol Table-VariablesSymbol DefinitionF r Air frictionF g Inertial centrifugal forceF k Coriolis forceW Angular velocity of the crash planev r Relative velocity of the crash planev x Initial velocity of the surface layer of ocean currentsk Coefficient of fluid frictionF f Buoyancy of the wreckagef i Churning resistance of the wreckage from ocean currents f Fluid resistance opposite to the direction of motionG Gravity of the wreckageV Volume of the wreckageh Decent height of the wreckageH Marine depthS x Displacement of the wreckageS y Horizontal distance of S xα Deviation angle of factually final position of the wreckage s Horizontal distance between final point and splash point p Probability of a wreck in a given pointN The number of the searching planeTS ' The area of sea to be searched a i V ˆ The maximum speed of each planeai D The initial distance from sea to search planeai A The search ability of each plane is),(h T L i The maximum battery life of each plane isi L The mobilized times of each plane in the whole search )1(N Q Q a a ≤≤ The maximum number of search plane in the searching zone T(h) The time the whole action takes4 Model OverviewMost research for searching the lost plane can be classified as academic and practical. As practical methods are difficult to apply to our problem, we approach theproblem with academic techniques. Our study into the searching of the lost plane takes several approaches.Our basic model allows us to obtain the splash point of the lost plane. We focus on the force analysis of the plane. Then we We turn to simple types of projectile motion model. This model gives us critical data about the movement and serves as a stepping stone to our later study.The extended model views the problem based on the conclusion above. We run diffe-rential equation method and Bayesian search model to simulate the movement of wreckage. The essence of the model is the way to combine the effect of natural factors with distribution of the wreckage. Moreover, using distributing conditions, we treat size of the lost plane as “initial wreckage zone” so as to approximately describe the distribution. Thus, after considering the natural factors, we name the distribution of wreckage a “wreck zone” to minimize searching zone. While we name all the space needed to search “searching zone”.Our conclusive model containing several kinds of algorithm attempts to tackle a more realistic and more challenging problem. We add the global optimization model and an area partition algorithm to improve the efficiency of search aircrafts according to the area of search zone. An assessment of search planes consisting of search capabili-ties and other factors are also added. The Dinkelbach and NonConvexDivide algo-rithm for the solutions of the results are also added.We use the extended and conclusive model as a standard model to analyze the problem and all results have this two model at their cores.5 Modeling For Locating the Lost PlaneWe will start with the idea of the basic model. Then we present the Bayesian search model to get the position of the sinking point.5.1 Modeling For Locating the Splash PointThe basic model is a academic approach. A typical types of projectile behavior con-sists of horizontal and vertical motion. We also add another dimension consider-ing the effect of the earth's rotation. Among these actions, the force analysis is the most crucial part during descent from the point out of contact to the sea-level. Types of plane might impact trajectory of the crashing plane.5.1.1 Types of PlanesWe classify the planes into six groups [7]:●Helicopters: A helicopter is one of the most timesaving ways to transfer be-tween the city and airport, alternatively an easy way to reach remote destina-tions.●Twins Pistons: An economical aircraft range suitable for short distance flights.Aircraft seating capacity ranging from 3 to 8 passengers.●Turboprops: A wide range of aircraft suitable for short and medium distanceflights with a duration of up to 2-4 hours. Aircraft seating capacity ranging from 4 to 70 passengers.●Executive Jets:An Executive Jet is suitable for medium or long distanceflights. Aircraft seating capacity ranging from 4 to 16 passengers●Airliners:Large jet aircraft suitable for all kinds of flights. Aircraft seatingcapacity ranging from 50 to 400 passengers.●Cargo Aircrafts:Any type of cargo. Ranging from short notice flights carry-ing vital spare parts up to large cargo aircraft that can transport any volumin-ous goods.The lost plane may be one of these group. Then we extract the characteristics of planes into three essential factors: mass, maximum flying speed, volume. We use these three factors to abstract a variety of planes:●Mass: Planes of different product models have their own mass.●Maximum flying speed: Different planes are provided with kinds of me-chanical configuration, which will decide their properties such as flying speed.●Volume: Planes of distinct product models have different sizes and configura-tion, so the volume is definitive .5.1.2 Preparation of the Model —Earth RotationWhen considering the earth rotation, we should know that earth is a non-inertial run-ning system. Thus, mobile on the earth suffers two other non-inertial forces except air friction F r . They are inertial centrifugal force F g and Coriolis force F k . According to Newton ‟s second law of motion, the law of object relative motion to the earth is:Rotational angular velocity of the earth is very small, about .For a big mobile v r , it suffers far less inertial centrifugal force than Coriolis force, so we can ignore it. Thus, the equation can be approximated as follows:Now we establish a coordinate system: x axis z axis pointing to the east and south re-spectively, y axis vertical upward, then v r , ω and F r in the projection coordinate system are as follows:⎪⎪⎩⎪⎪⎨⎧++=⋅⋅-⋅⋅=++=kdt dz j dt dy i dt dx m v k j w kF j F i F F r rz ry rx r φωφωcos sinφis the latitude of the lost contact point of the lost plane. Put equation 1-3 and equa-tion 1-2 together, then the component of projectile movement in differential equation is:ma FF F k g r=++srad ⋅⨯=-5103.7ωmamv F r r =+ω2⎪⎪⎪⎩⎪⎪⎪⎨⎧+⋅=+⋅=+⎪⎭⎫ ⎝⎛+⋅-=m F dt dx w dt z d m F dt dx w dt y d m F dt dz dt dy w dtx d rz ry rx φφφφsin 2cos 2sin cos 22222225.1.3 ModelingConsidering the effect caused by earth rotation and air draught to plane when crashing to sea level, we analyze the force on the X axis by using Newton ‟s second law, the differential equation on x y and axis, we can conclude:In conclusion, we establish the earth rotation and types of projectile second order dif-ferential model:()⎪⎩⎪⎨⎧+-⋅'⋅⋅=''-⋅'+⋅'⋅⋅-=''-⋅'⋅⋅=''m gf y w m z m f z x w m y m f y w m x m obj 321cos 2cos sin 2sin 2.φφφφAccording to Coriolis theorem, we analyze the force of the plane on different direc-tions. By using the Newton ‟s laws of motion, we can work out the resultant accelera-tion on all directions:⎪⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎪⎨⎧'+'+'⋅'⋅⋅⨯+='+'+'⋅'⋅⋅⨯+='+'+'⋅'⋅⋅⨯+=⋅⋅-⋅⋅=⋅⨯=⋅'''⋅===-2222222225)()()(21)()()(21)()()(21cos sin 103.704.022z y x z c F f z y x y c F f z y x x c F f k j w s rad S y x F c D rz D ryD rx D ρρρφωφωωμφC D is the angle of attack of a plane flew in the best state, w is the angular speed of a moving object, vector j and k are the unit vector on y and z direction respectively,μisrx F y w m x m -⋅'⋅⋅⨯=''φsin 2()ry F z x w m y m -'+⋅'⋅⨯-=''φφcos sin 2mg F y w m z m rz +-⋅'⋅⋅⋅=''φcos 2the coefficient of viscosity of the object.5.1.4 Solution of the ModelWhen air flows through an object, only the air close to layer on the surface of the ob-ject in the laminar airflow is larger, whose air viscosity performance is more noticea-ble while the outer region has negligible viscous force [8]. Typically, to simplify cal-culation, we ignore the viscous force produced by plane surface caused by air resis-tance.Step 1: the examination of dimension in modelTo verify the validity of the model based on Newton ‟s second theorem, first, we standardize them respectively, turn them into the standardization of dimensionless data to diminish the influence of dimensional data. The standard equation is:Step 2: the confirmation of initial conditionsIn a space coordinate origin based on plane, we assume the earth's rotation direc-tion for the x axis, the plane's flight heading as y axis, the vertical downward di-rection for z axis. Space coordinate system are as follows:Figure 1 Space coordinate systemStep 3: the simplification and solutionAfter twice integrations of the model, ignoring some of the dimensionless in thesxx y i -=integral process, we can simplify the model and get the following:⎪⎪⎪⎩⎪⎪⎪⎨⎧+'⋅⋅⋅-⋅'⋅⨯='''-⋅⋅⋅-⋅'⋅⨯-=''⋅'⋅⨯=''g z m s c y w z y v m s c z w y y w x D D 220)(2cos 2)(2cos 2sin 2ρφρφφWe can calculate the corresponding xyz by putting in specific data to get the in-formation about the point of losing contact.Step 4: the solution of the coordinateThe distance of every latitude on the same longitude is 111km and the distance ofevery longitude on the same latitude is 111*cos (the latitude of this point) (km). Moreover, the latitude distance of two points on the same longitude is r ×cos(a ×pi/180) and the longitude distance of two points on the same latitude is: r ×sin(a ×pi/180)[9].We assume a as the clockwise angle starting with the due north direction and r as the distance between two points; X 、Y are the latitude and longitude coordinates of the known point P respectively; Lon , Lat are the latitude and longitude coordi-nates of the unknown point B respectively.Therefore, the longitude and latitude coordinates of the unknown point Q is:⎪⎪⎩⎪⎪⎨⎧⨯⨯+=⨯⨯⨯⨯+=111)180/cos()180/cos(111)180/sin(pi a r Y Lat pi Y pi a r X LonThus, we can get coordinates of the point of splash by putting in specific data.5.2 Modeling For Locating WreckageIn order to understand how the wreckage distributes in the sea, we have to understand the whole process beginning from the plane crashing into water to reaching the seaf-loor. One intuition for modeling the problem is to think of the ocean currents as astochastic process decided by water velocity. Therefore, we use a differential equation method to simulate the impact on wreckage from ocean currents.A Bayesian Searching model is a continuous model that computing a probability dis-tribution on the location of the wreckage (search object) in the presence of uncertain-ties and conflicting information that require the use of subjective probabilities. The model requires an initial searching zone and a set of the posterior distribution given failure of the search to plan the next increment of search. As the search proceeds, the subjective estimates of the detection will be more reliable.5.2.1 Assumptions of the ModelThe following general assumptions are made based on common sense and weuse them throughout our model.●Seawater density keeps unchanged despite the seawater depth.Seawater density is determined by water temperature, pressure, salinity etc.These factors are decided by or affected by the seawater density. Considering the falling height, the density changes slightly. To simplify the calculation, we consider it as a constant.●The velocity of the wreck stay the same compared with velocity of theplane before it crashes into pieces. The whole process will end quickly witha little loss of energy. Thus, we simplify the calculation.●Marine life will not affect our simulation.Most open coast habitats arefound in the deep ocean beyond the edge of the continental shelf, while the falling height of the plane cannot hit.●Acting force of seawater is a function of the speed and direction of oceancurrents. Ocean currents is a complicated element affected by temperature, wide direction, weather pattern etc. we focus on a short term of open sea.Acting force of seawater will not take this factors into consideration.5.2.2 Preparation of the Model●The resistance of objects of different shapes is different. Due to the continuityof the movement of the water, when faced with the surface of different shapes, the water will be diverted, resulting in the loss of partial energy. Thus the pressure of the surface of objects is changed. Based on this, we first consider the general object, and then revise the corresponding coefficients.●Ocean currents and influencing factorsOcean currents, also called sea currents, are large-scale seawater movements which have relatively stable speed and direction. Only in the land along the coast, due to tides, terrain, the injection of river water, and other factors, the speed and direction of ocean currents changes.Figure 2Distribution of world ocean currentsIt can be known from Figure 2 that warm and cold currents exist in the area where aircraft incidences happened. Considering the fact that the speed of ocean currents slows down as the increase of the depth of ocean, the velocity with depth sea surface currents gradually slowed down, v x is set as the initial speed of ocean currents in subsequent calculations.●Turbulent layerTurbulent flow is one kind of state of the fluid. When the flow rate is very low, the fluid is separated into different layers, called laminar flow, which do not mix with each other. As the flow speed increases, the flow line of the fluid begins to appear wavy swing. And the swing frequency and amplitude in-creases as the flow rate increases. This kind of stream regimen is called tran-sition flow. When the flow rate becomes great, the flow line is no longer clear and many small whirlpools, called turbulence, appeared in the flow field.Under the influence of ocean currents, the flow speed of the fluid changes as the water depth changes gradually, the speed and direction of the fluid is un-certain, and the density of the fluid density changes, resulting in uneven flow distribution. This indirectly causes the change of drag coefficient, and the re-sistance of the fluid is calculated as follows:2fkvGLCM texture of submarine topographyIn order to describe the impact of submarine topography, we choose a rectan-gular region from 33°33…W, 5°01…N to 31°42‟W , 3°37‟N. As texture is formed by repetitive distribution of gray in the spatial position, there is a cer-tain gray relation between two pixels which are separated by a certain dis-tance, which is space correlation character of gray in images. GLCM is a common way to describe the texture by studying the space correlation cha-racter of gray. We use correlation function of GLCM texture in MATLAB:I=imread ('map.jpg'); imshow(I);We arbitrarily select a seabed images and import seabed images to get the coordinate of highlights as follows：Table 1Coordinate of highlightsNO. x/km y/km NO. x/km y/km NO. x/km y/km1 154.59 1.365 13 91.2 22.71 25 331.42 16.632 151.25 8.19 14 40.04 18.12 26 235.77 13.93 174.6 14.02 15 117.89 14.89 27 240.22 17.754 172.38 19.23 16 74.51 12.29 28 331.42 24.455 165.71 24.82 17 45.6 8.56 29 102.32 19.486 215.75 26.31 18 103.43 5.58 30 229.1 18.247 262.46 22.96 19 48.934 3.51 31 176.83 9.188 331.42 22.34 20 212.42 2.85 32 123.45 3.239 320.29 27.55 21 272.47 2.48 33 32.252 11.7910 272.47 27.55 22 325.85 6.45 34 31.14 27.811 107.88 28.79 23 230.21 7.32 35 226.88 16.0112 25.579 27.05 24 280.26 9.93 36 291.38 5.46Then we use HDVM algorithm to get the 3D image of submarine topography, which can be simulated by MATLAB.Figure 3 3D image of submarine topographyObjects force analysis under the condition of currentsf is the resistance, f i is the disturbance resistance, F f is the buoyancy, G isgravity of object.Figure 4Force analysis of object under the conditions of currentsConsidering the impact of currents on the sinking process of objects, wheninterfered with currents, objects will sheer because of uneven force. There-。