Antarctica___美赛论文

美赛2019数模C题论文解法思路美赛2019数模C题解法思路题目:阿片类药物危机解法思路：建立阿片危机治病与毒品函数模型，二函数的交点为优化解。

阿片类药物危机问题数学模型摘要阿片类药物危机是本文要解决的数学问题，为了明确阿片类药物危机问题，本文针对阿片类药物危机问题进行了分析建模,对阿片类药物危机问题进行了参考文献研究，建立了阿片类药物危机问题的相应模型，推导出阿片类药物危机问题的计算公式，编写了阿片类药物危机问题的计算程序，经过程序运行，得到阿片类药物危机问题程序计算结果。

具体有：对于问题一，这是阿片类药物危机问题最重要的问题，根据题目，对问题一进行了分析，参考已有的资料，建立了阿片类药物危机问题一的数学模型，推导出问题一的计算公式，编写出阿片类药物危机问题一的计算程序。

求出了阿片类药物危机问题一的计算结果。

对于问题二，阿片类药物危机问题二比问题一复杂的，是阿片类药物危机问题的核心，分析的内容多，计算机的东西也多。

在阿片类药物危机问题一的基础上，根据阿片类药物危机问题，对问题二进行了分析，参考已有的资料，建立了阿片类药物危机问题二的数学模型，推导出问题二的计算公式，编写出阿片类药物危机问题二的计算程序。

求出了问题二的计算结果，并以图表形式表达结果。

对于问题三，阿片类药物危机问题三是问题一和问题二的深入。

在问题一和问题二的基础上，根据阿片类药物危机问题，对问题三进行了分析，参考已有的资料，建立了问题三的数学模型，推导出阿片类药物危机问题三的计算公式，编写出阿片类药物危机问题三的计算程序。

求出了阿片类药物危机问题三的计算结果，并以图表形式表达结果，并且进行了分析讨论。

对于问题4，阿片类药物危机问题4是问题一、问题二和问题三的扩展。

在问题一、问题二和问题三的基础上，根据阿片类药物危机问题，对阿片类药物危机问题4进行了分析，参考已有的资料，建立了阿片类药物危机数学模型，推导出阿片类药物危机问题4的计算公式，编写出问题4的计算程序。

For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number 37090Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________2015 Mathematical Contest in Modeling (MCM) Summary SheetThe advent of licensed Ebola vaccines and drugs delights the whole world while also posing a dilemma of how to allocate the needed quantity among all Ebola outbreaks and deliver them with effectiveness and efficiency.We establish comprehensive Ebola response models in three most suffering countries (Guinea, Liberia and Sierra Leone) including a prediction model generating short-term estimates of the Ebola transmission situations, an allocation-and-delivery model planning the needed quantity of medicines and the optimal delivery route, and a cellular automaton model measuring the effect of effective isolation and treatment. Besides, we also give policy making suggestions to prevent international spread to some unaffected countries.Based on the special characteristic of Ebola, we create a modified SEIR epidemic model with an added intervention factor to stand for the effect of some forms of interventions other than vaccines and drugs. We predict the potential number of future Ebola cases with or without the use of effective medicine and the result also shows that if the transmission trends continue without effective interventions, countries will undergo worse and worse situations In the next model, we first classify all outbreaks into five levels due to the different Ebola case numbers. Then we apply minimum spanning tree method, Monte Carlo method and 0-1 programming to our model to locate an optimal number of medical center and sub-centers in each country aiming to eradicate Ebola. We set one medial center in each country and one more sub-center in Guinea, three more sub-centers in Liberia and four more sub-centers in Sierra Leone. The model also calculates the minimal needed number of vaccines and drugs in every manufacturing cycle.Then, we discuss the effect of isolation and treatment by cellular automaton model and find out that if only effective isolation is conducted, the retarding effect is limited.We present a comprehensive strategy to eradicate Ebola by conducting dynamic models and as time passes, we can update the statistic data to reality which adds accuracy to our models and optimal results.An Optimal Strategy to Eradicate EbolaIntroductionEbola virus disease (EVD) is a severe, often fatal illness in humans. It has become one of the most prevalent and devastating threat for its intense transmission. Since first cases of the current West African epidemic of Ebola virus disease were reported on March 22, 2014, over 20000 new cases have been found and about 9000 patients have died from it. The western Africa areas-Guinea, Liberia and Sierra Leone in particular-are outbreaks that have suffered most [1].With the help of licensed vaccines and drugs, we aim to stop Ebola transmission in affected countries within a short period and prevent international spread. Our objectives are:●to achieve full and fast coverage with vaccines for susceptible individuals and drugs for infectious individuals among three most suffering countries (Guinea, Liberia and Sierra Leone);●to ensure emergency and immediate application of comprehensive Ebola response interventions in countries with an initial case or with localized transmission;●to strengthen preparedness of all countries to rapidly detect and response to an Ebola exposure，especially those sharing land borders with an intense transmission area and those with international transportation hubs[1].For the first objective, we create a comprehensive Ebola response models in those three countries including a prediction model of Ebola transmission, an allocation-and-delivery model for vaccines and drugs used and a cellular automaton model measuring the effect of some crucial interventions. The last two objectives are closely related to policy making and in the following part of our paper we just present detailed information of our models.Basic Assumptions1. A patient can only progress forward through the four states and can never regress(e.g. go from the incubating to the susceptible) or skip a state (e.g. go from the incubating to the recovered state, skipping the infectious state).2.Once recovered from Ebola, an individual will not be infected again in a short time.3.Populations of each country remain the same over the prediction period.4.In absence of licensed vaccines or drugs, some other interventions are used, such as effective isolation for Ebola patients and safe burial protocol.5.When vaccines and drugs are introduced to the prediction model, the incubation period and the effect of interventions other than medicine will not change.6.Building a medical center is at a high cost (e.g. storage facilities of medicines, etc.) and every medical center are capable of delivering all needed medicines.7.We ignore the potential damage to medicines when delivering.8.We calculate the distance between two sites by measuring the spherical distance and ignore the actual traffic situation.9.Once received treatment with licensed drugs, patients will no longer be infectiousindividuals, which also means that we do not take the needed recovery period into account.10.The needed vaccines or drug for an individual is one unit.11.All the data searched from the Internet are of trustworthiness and reliability.Model 1: Prediction ModelWe create a modified SEIR model [2] to estimate the potential number of future Ebola cases in countries with intense and widespread transmission- Guinea, Liberia and Sierra Leone. Not only useful in predicting future situation in absence of any licensed vaccine or drug, the modeling tool also can be used to estimate how control and prevention medicine can slow and eventually stop the epidemic.Terminology and definitionsdays is used from previous study. The resulting distribution has a mean incubation period of 6.3 days [3] and therefore, in our prediction model, patients are assumed to be infectious after a 6.3-day’s incubation period. Besides, in absence of licensed vaccines or drugs, Ebola is a disease with few cases of recovery. Thus, under this situation, we assume the recovery rate is 0.001, which is very close to zero.MethodA frightening characteristic of Ebola virus disease is that it has an incubation period ranging from 2 to 21 days before an individual exposed to the virus who finally become infectious. Thus, we create a SEIR epidemic model tracking individuals through the following four states: susceptible (at risk of contracting the disease), exposed (infected but not yet infectious), infectious (capable of transmitting the disease) and removed (recovered from the disease or dead).Moreover, based on Assumption 4, some forms of interventions other than vaccines and drugs may also reduce the spread of Ebola and death numbers, and therefore we introduce an intervention factor γ as a parameter to measure the effect. In those three intense-transmissioncountries(Guinea, Liberia and Sierra Leone),at least 20% of new Ebola infections occur during traditional burials of deceased Ebola patients when family and community members directly touching or washing the body. By conducting safe burial practice, the number of new Ebola cases may drop remarkably. Moreover, effective isolation with in-time treatment is also of significant importance in reducing transmission and deaths.In our modified SEIR model, we describe the flow of individuals between epidemiological classes as follows.Figure 1 A schematic representation of the flow of individuals between epidemiologicalclassesSusceptible individuals in class S in contact with the virus enter the exposed class E at the per-capita rate (λ-γ), where λ is transmission rate per infectious individual per day and γ is the intervention factor serves to retard the transmission. After undergoing an average incubation period of 1/α days, exposed individuals progress to the infectious class I. Infectious individuals (I) move to the R-class either recover or die at rate (μ+β+γ), where b stands for the recovery rate and d represents the fatality rate. Besides,The transmission process above is modeled by the following differential equation set: ()()()()()()()()()()()(++)()dS S t I t dt dE S t I t t E t dt dI E t I t dt dR I t dtλγλγααμβγμβγ⎧=--⎪⎪⎪=--⎪⎨⎪=-++⎪⎪⎪=⎩ (1.1)We modify SEIR model by adding intervention factor γ.Algorithm1. With known values of parameter α and μ, we solve the differential equation (1.1) by assigning certain value ranges and step values to parameter λ, β, and γ.2. We get the predicted numbers of exposed, infectious and dead individuals and these numbers can be fitted to real data by using the least square method to get the residual errors of each times’ loop iteration.3. By comparison every residual error, we find the least one and we use the corresponding values of parameters in our prediction for further prediction.ResultVia MA TLAB programming, we obtain the optimal values for parameters λ,μ,α,βand γ(Table1)and then get the estimated cumulative number of cases in Guinea, Liberia and Sierra Leone separately(Figure 2, 3 and 4). The result shows that if Ebola transmission trends continue without effective drugs and vaccines, countries will undergo worse and worse situations.Sierra Leone 0.101 0.001 0.1587 0.03 0.02Figure 2 Cumulative numbers of cases in LiberiaFigure 3 Cumulative numbers of cases in GuineaFigure 4 Cumulative numbers of cases in Sierra LeoneStability testDefinition of stabilityAn aggregation of all possible parameters’ values resulting in a downwards trend of the total number of exposed individuals and infectious individuals are defined as the stability range in our model [4].Stability range First, we draw two equations from the differential equation set (1.1):()()()()()()()()dE S t I t t E t dt dI E t I t dtλγααμβγ=--=-++ As ()E t and ()R t is relatively small, we assume that ()1()S t I t =-. Then, we sum the two equations up and get:()()[1()]()()()E I d I t I t I t tλγμβγ+=---+- In order to prevent the spread of Ebola, the total percentage of E(t) and I(t) has to present a decline trend from the first day of taking action with the licensed medicine, which also means()[]0[()]d E I d dt d I t +< . When I(t)=I(0),the inequality is equivalent to(2)()()0I t λμβγλγ-----<As ()0I t ≈, the relationship of parameters λ, μ, β and γ are(2)0λμβγ---<To conclude, the stability range for model one is (2)0λμβγ---<. When parameters’ values satisfy this inequality, the model is of stability.Model 2: Allocation-and-delivery ModelWe create an allocation-and-delivery model for vaccines and drugs used in three most suffering countries (Guinea, Liberia and Sierra Leone) and the optimal strategy is assumed to have significant effect of eradicating Ebola in 180 days.In our allocation-and-delivery model, we set medical centers and sub-centers, which serve to treat Ebola patients, inject vaccines to susceptible individuals and also store needed amount of drugs and vaccines. Besides, countries manufacturing medicines (e.g., America, Canada, etc.) are not where in need of medicines, so we set one medical center to receive drugs and vaccines from the manufacturing country and then delivers the needed amount to every sub-center once a month. For sake of the inconvenience might face when delivering medicines across borders, we model three countries desperately. In another word, we set one medical center in Guinea, one in Liberia and one in Sierra Leone respectively and drugs and vaccines are delivered from every center to the sub-centers within borders.The Figure 5 below demonstrates the model with a hypothetical scenario. The dotted arrow lines show that individuals from every Ebola outbreak (E) will go to the nearest medical center (MC) or sub-center (MSC) for treatment or injection, while the solid arrow lines represent the delivery process of medicines from manufacturing county to each medical center and then to sub-centers.Figure 5 The allocation-and-delivery mode lInstead of building new treating places, we locate our medical centers and sub-centers in some existing Ebola Treating Units (ETUs) [1]. The model shows how we choose from current ETUs, including deciding the optimal number and location.Table 3 existing ETUs their locationTerminology and definitionsGoalWe determine the number and location of medical center and sub-centers on the basis of ● Minimizing the total time-cost that an infectious individual from one outbreak spends on the way to the corresponding medical center or sub-center, while locating those center and sub-centers as few as possible, also means0min N nN ij i o j C d ===∑∑● Minimizing the total distance among one medical center to other sub-centers, also meansmin ()Nij i o D i j =≠∑● Averaging the workloads of medical center and sub-centers, also meansmin N N NSV CV AV =AlgorithmFigure 6 the flow chart for model 2Initialize parameters in previous prediction model●We do not change the value of α and γ used in Model 1.●We have deduced the relationship of parameters λ, μ, β and γ in the stability test of model 1.Estimate daily added number of infectious individualsWe use the prediction model to simulate the situation of daily added number of infectious individuals DI i in 6 months(180 days) for 10 times and choose the worst case(maximal numbers) as the final estimation of daily added number.Build geographical distribution of new added infectious individualsWe categorize all outbreaks into five levels as level I, II, III, IV and V according to the number of confirmed cases and then calculate each level’s probability of a new occurring case. According to the number of new added infectious individuals and the probability of occurring in every outbreak, we build geographical distribution among all outbreaks of new added infectious individuals.Table 5 Outbreaks and classificationSet n from 1 to kWe set n from 1 to k to conduct the process for k times and compare each optimal result as N changes.Locate sub-centers randomlyWe locate sub-centers randomly and for each sub-center, the corresponding outbreaks represent all those outbreaks with a nearer distance to this sub-center compared to others.Calculate total time-costWe define the time-cost as the period that an infectious individual from one outbreak spends on the way to the corresponding medical center or sub-center, and we add up the corresponding distance as the measurement of the time-cost. When calculating the total time-cost, the number of all potential patients is taken into account.Make comparisonWe compare the total time-cost calculated in 400 times’ loop and choose the minimal one as the optimal result.Output optimal n, C n, A V n, CV nLocate medical centerWe calculate the total distance of every medical sub-center to others and locate the one with minimal total distance as the medical center which serve to receive all needed medicine from manufacturing country and deliver the required amount to every sub-center [5].ResultWe locate medical centers and sub-centers separately in three countries as shown in Table 7 and Figure7. We get the different values of indicators (shown in Table 6) and taking total distance and margin distance into account, we choose the optimal number and location of medical sub-centersTable 6 Values of indicatorsTable 7 Location of medical center and sub-centers and their corresponding outbreaksFigure 7 Locations of medical center and sub-centers and the routesWe determine the needed amount of vaccines and drugs.We assume that the successful immune rate is 90%, the recovery rate when drugs are used is 60% and the manufacturing cycle of the licensed drug is 30 days. These rates and cycle-days can be adjusted according to reality. VaccinesIndividuals having received vaccine injection can be protected from being infectious. The larger proportion of population being injected, the lower the transmission rate is. This relationship can be measured as 1'(1)dk λλ=- and we solve this equation and get thenumber of needed vaccines (1k ) is'1dλλ-DrugsPatients will have a higher recovery rate and lower fatality rate. The shorter the course of treatment is, the greater the impact on recovery rate and fatality rate. We rewrite therelationship in mathematic equations as 2'rk D μμ=+or 2'rkDββ=-. Thus, the number of needed drugs (2k ) is (')D r μμ- or (')Drββ- .The resultWe calculate an allocation plan for vaccines and drugs in 6 months and the detailed number are present in table 8 and 9. We can see that the demand for vaccine is much larger than that of drugs because there is a wider range of individuals who need vaccine injections as an effective protection.Table 8 Allocation plan for vaccines in 6 monthsTable 9 Allocation plan for drugs in 6 monthsStability testWe make 10 times’ simulation for the three countries by the following procedures.First, we estimate the needed number of medicines for one month and supply at the first day of that month.Then, we generate added numbers of infectious individuals randomly and calculate the consumed and remaining amount of medicines.Finally, we get the line of daily reaming amount of medicines as shown in Figure 8.-100100200300400500600700Dates u r p l u sFigure 8 Surplus of medicine in Guinea, Liberia and Sierra LeoneThe figures demonstrate that the supply of medicine is sufficient except a small probability (less than 10%) of deficit at the end of the first month. Thus, the model is of high stability.Sensitivity analysisWe have estimated the cumulative number of infectious individuals based on the optimal number and location of medical center and sub-centers in model 2. Then we change the values of parameters to conduct sensitivity analysis. The results are shown in the following table. Table 10 result of sensitivity analysisThe result shows the optimal result will not change unless there is some big fluctuation of parameters’ values. Besides, the fluctuation of transmission rate will result in more significant changes to the number of infectious individuals and therefore, we should put emphasis on the generalization of vaccine injections.Dates u r p l usDates u r p l u sFigure 9 Number of daily added infectious Figure 10 Present number of infectious, exposed individuals in Sirrea,Liberia,Guinea and dead individuals in Sirrea,Liberia,GuineaModel 3: the cellular automaton modelIn model 1, we estimate the transmission trends of Ebola and then in model 2, we measure the trends when licensed vaccines and drugs are used and make an allocation-and-delivery plan of medicines. We now introduce a cellular automaton model to present a clearer dynamic simulation of the spread of Ebola in one area.Cellular automaton[6] is a model in which time, space and other variables are all discrete. lt can be expressed asCA = (Ld, S, N, f)Where Ld represents a d-dimensional cellular spaces and we set d=2, L ×L=1000×1000, S represents all finite discrete set of cell stateN represents t he set of a cell’s eight neighbors’ statef represents the transfer function of one cell and it is expressed as S t+1f(S t,N t)Figure 11 A cell and its eight neighborsThere are five states{S, E, I, Q, D, R} in our model which represent susceptible, exposed, infectious, quarantined, dead and recovered individuals. We assign them as{0, 1, 2, 3, 4, 5}. Initialize all cells state value Si j = 0, which means that all cells are susceptible individuals. We select a proportion of 0.0005’s cells in the cellular spaces randomly and set their state value Si j =2, which represent the initial infectious individuals.From t=0, we scan all cells in the cellular spaces and compare the effect of treatment and isolation. We set three situations as no treatment and no isolation, only isolation but no treatment and both isolation and treatment, and then simulate all these situations.Take the third situation (both isolation and treatment) as an example to show the renewing rules.When Si j=0, we calculate the probability p i j that a single cell C ij become infectious when contacting with its neighbors. Then we judge weather susceptible individuals will become exposed individuals with the probability p i j. If it is not the probability, they remain susceptible individuals.When S ij=1, cell C ij is exposed individuals with a probability of e to become infectious individuals (S ij=2).When S ij=2, cell C ij is infectious individuals with a probability of r1 to be isolated (S ij=3) and a probability of d to dead(S ij=4 and are moved out of the transfer).When S ij=3, cell C ij is quarantined individuals with a probability of r3 to be cured (S ij=5 andare moved out of the transfer because of high immune ability).We update the states of all cells in the cellular spaces at the same time and use the result as the initial state in the next time’s simulation.ResultWe use Matlab to realize a simulation process of 200 days and the following figures show the results.Figure 12No isolation and no treatment2040608010012014016018020020406080100120140160180200204060801001201401601802002040608010012014016018020020406080100120140160180200204060801001201401601802002040608010012014016018020020406080100120140160180200DAY 50DAY 100DAY 150DAY 200Figure 13 Only isolation and no treatmentFigure 14 Both treatment and isolation20406080100120140160180200204060801001201401601802002040608010012014016018020020406080100120140160180200204060801001201401601802002040608010012014016018020020406080100120140160180200204060801001201401601802002040608010012014016018020020406080100120140160180200204060801001201401601802002040608010012014016018020020406080100120140160180200204060801001201401601802002040608010012014016018020020406080100120140160180200DAY 50DAY 100DAY 150DAY 200DAY 50DAY 100DAY 150DAY 200The results shows that the transmission accelerates with no isolation and treatment, while slows down significantly when effective isolation is added. However, simple isolation as intervention cannot stop the spread of Ebola. Only with effective isolation and treatment, the transmission can be limited and the fatality rate is reduced.We use the cellular automaton model to simulate the spread of Ebola in three situations and illustrate that effective isolation and treatment is of significant importance,Sensitivity analysisWe assign different values to parameters λ, 12r r ⨯ and μand simulate the situation of the 100th day. The results are as follows.Figure 15 Result of sensitivity analysisThe figure demonstrates that the model is not sensitive to isolation level while sensitive to r transmission and recovery rate. The results indicate that the eradication of Ebola is rely heavily on the control of transmission and recovery rate. Besides, isolation is more effective with a relatively small scale of infectious individuals.Evaluation of the modelStrengths●The prediction model is a modified one adjusted to the unique characteristic of Ebola and this model is much more suitable for the prediction of Ebola transmission than the traditional SEIR epidemic model.●The allocation-and-delivery model is based on the real location of outbreaks and ETUs, and the resulting locations of medical centers and sub-centers are of high practical value.●The value of parameters in the allocation-and-delivery model is highly adjustable. Policy makers can change the value according to the reality or determined goals and this will not affect the modeling process.●The cellular automaton model presents a brief picture of the transmission trends. The result shows the limited retarding effect of simple isolation and indicates the crucial role of effective vaccines and drugs.Weaknesses●We use previous data and probability distribution to determine the value of some parameters in our model. Maybe they deviate from the current situation.●The models fail to take some emergent cases and their effect into account. For example, we ignore the real traffic situations and potential congestions when delivering medicines.Conclusions●We estimate the transmission trend of Ebola in (Guinea, Liberia and Sierra Leone) and present a comprehensive strategy to eradicate Ebola by planning the allocation and delivery system.●The model also presents the different effect of three kinds of interventions-injecting vaccines, treating with drugs, isolation. The best retarding method is to inject vaccines and treating with drugs can reduce deaths in a short period, while isolation is the least choice in absence of other forms of interventions.●To prevent international transmission to unaffected counties, immediate supply of vaccines and drugs should be delivered to any new initial outbreaks from the nearest available place and all unaffected counties have to establish a full Ebola surveillance preparedness and response plan.References[1] http://www.who.int/en/, Feb 2015[2] Ma J L,Ma Z E.Epidemic threshold condition for seasonally forced SEIR models. Mathematical Bio-sciences and Engineering . 2006[3] Chowell G, Hengartner NW, Castillo-Chavez C, Fenimore PW, Hyman JM. The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda. J Theor Biol 2004;229:119-26 [4]Katsuaki Koike,Setsuro Matsuda. New Indices for Characterizing Spatial Models of Ore Deposits by the Use of a Sensitivity Vector and an Influence Factor[J]. Mathematical Geology . 2006 (5)[5] Peter Kovesi.MA TLAB and Octave Functions for Computer Vision and Image Processing. Digital Image Computing:Techniques and Applications . 2012[6] rraga,,J.A.delRio,,L.Alvarez-lcaza.Cellularautomationforonelanetrafficmodeling.Transportatio researchpartC . 2005ReportTo whom it may concern:Ebola virus disease (EVD) are posing a threat to all human beings but the advent of licensed vaccines and drugs enable us to fight with Ebola. We have studied out a comprehensive strategy to stop Ebola transmission in affected countries within a short period and prevent international spread.For those unaffected countries and light Ebola outbreaks, immediate response actions to a new initial case are of significant importance. According to our model, effective isolation and treatment can prevent the widespread transmission of Ebola. Thus, immediate supply of vaccines and drugs should be delivered to any new initial outbreaks from the nearest available place and all unaffected counties have to establish a full Ebola surveillance preparedness and response plan including isolation and treatment of infectious individuals and injection of vaccines to susceptible individuals.For countries with intense and widespread transmission- Guinea, Liberia and Sierra Leone- besides the immediate isolation and treatment, a plan of allocating and delivering medicines is also crucial. We model the potential number of future Ebola cases in these three countries and estimate the goal number of transmission rate, recovery rate and fatality rate with which we can control the spread of Ebola. Meanwhile, we classify all the outbreaks in those three countries according to the number of cumulative confirmed cases. Outbreaks with different level will have a different probability of a new occurring case and we use our model to predict the possible new outbreak.Classification of outbreaksAccording to our prediction, 567 units of drugs and 2069139 units of vaccines are needed in the first manufacturing cycle, and therefore, we model an optimal delivering system with the highest efficiency. For sake of the inconvenience might face when delivering medicines across borders, we model three countries desperately. We set one medical center (MC) and a certain number of medical sub-centers (MSC), in each country which serve to treat Ebola patients, inject vaccines to susceptible individuals and also store needed amount of drugs and vaccines. Besides, the medical center serves to receive drugs and vaccines from the manufacturing country and then delivers the needed amount to every sub-center once a month.。

数学建模综述2014年美国大学生数学建模竞赛A题论文综述我们小组精读两篇14年美赛A题论文，选择了其中一篇来进行学习，总结。

1、问题分析The Keep-Right-Except-To-Pass Rule除非超车否则靠右行驶的交通规则问题：建立数学模型来分析这条规则在低负荷和高负荷状态下的交通路况的表现。

这条规则在提升车流量的方面是否有效？如果不是，提出能够提升车流量、安全系数或其他因素的替代品（包括完全没有这种规律）并加以分析。

在一些国家，汽车靠左形式是常态，探讨你的解决方案是否稍作修改即可适用，或者需要一些额外的需要。

最后，以上规则依赖于人的判断，如果相同规则的交通运输完全在智能系统的控制下，无论是部分网络还是嵌入使用的车辆的设计，在何种程度上会修改你前面的结果论文：基于元胞自动机和蒙特卡罗方法,我们建立一个模型来讨论“靠右行”规则的影响。

首先,我们打破汽车的运动过程和建立相应的子模型car-generation的流入模型,对于匀速行驶车辆，我们建立一个跟随模型,和超车模型。

然后我们设计规则来模拟车辆的运动模型。

我们进一步讨论我们的模型规则适应靠右的情况和,不受限制的情况, 和交通情况由智能控制系统的情况。

我们也设计一个道路的危险指数评价公式。

我们模拟双车道高速公路上交通(每个方向两个车道,一共四条车道),高速公路双向三车道（总共6车道)。

通过计算机和分析数据。

我们记录的平均速度,超车取代率、道路密度和危险指数和通过与不受规则限制的比较评估靠右行的性能。

我们利用不同的速度限制分析模型的敏感性和看到不同的限速的影响。

左手交通也进行了讨论。

根据我们的分析,我们提出一个新规则结合两个现有的规则(靠右的规则和无限制的规则)的智能系统来实现更好的的性能。

该论文在一开始并没有作过多分析，而是一针见血的提出了自己对于这个问题的做法。

由于题目给出的背景只有一条交通规则，而且是题目很明确的提出让我们建立模型分析。

1985~2013年美国大学生数学建模竞赛题目集锦目录1985 MCM A: Animal Populations (3)1985 MCM B: Strategic Reserve Management (3)1986 MCM A: Hydrographic Data (4)1986 MCM B: Emergency-Facilities Location (4)1987 MCM A: The Salt Storage Problem (5)1987 MCM B: Parking Lot Design (5)1988 MCM A: The Drug Runner Problem (5)1988 MCM B: Packing Railroad Flatcars (6)1989 MCM A: The Midge Classification Problem (6)1989 MCM B: Aircraft Queueing (6)1990 MCM A: The Brain-Drug Problem (6)1990 MCM B: Snowplow Routing (7)1991 MCM A: Water Tank Flow (8)1991 MCM B: The Steiner Tree Problem (8)1992 MCM A: Air-Traffic-Control Radar Power (8)1992 MCM B: Emergency Power Restoration (9)1993 MCM A: Optimal Composting (10)1993 MCM B: Coal-Tipple Operations (11)1994 MCM A: Concrete Slab Floors (11)1994 MCM B: Network Design (12)1995 MCM A: Helix Construction (13)1995 MCM B: Faculty Compensation (13)1996 MCM A: Submarine Tracking (13)1996 MCM B: Paper Judging (13)1997 MCM A: The Velociraptor Problem (14)1997 MCM B: Mix Well for Fruitful Discussions (15)1998 MCM A: MRI Scanners (16)1998 MCM B: Grade Inflation (17)1999 MCM A: Deep Impact (17)1999 MCM B: Unlawful Assembly (18)2000 MCM A: Air Traffic Control (18)2000 MCM B: Radio Channel Assignments (19)2001 MCM A: Choosing a Bicycle Wheel (20)2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...). (21)2002 MCM A: Wind and Waterspray (23)2002 MCM B: Airline Overbooking (23)2003 MCM A: The Stunt Person (24)2003 MCM B: Gamma Knife Treatment Planning (24)2004 MCM A: Are Fingerprints Unique? (25)2004 MCM B: A Faster QuickPass System (25)2005 MCM A: Flood Planning (26)2005 MCM B: Tollbooths (26)2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation (27)2006 MCM B: Wheel Chair Access at Airports (28)2007 MCM A: Gerrymandering (29)2007 MCM B: The Airplane Seating Problem (29)2008 MCM A: Take a Bath (30)2008 MCM B: Creating Sudoku Puzzles (30)2009 MCM A: Designing a Traffic Circle (30)2009 MCM B: Energy and the Cell Phone (30)2010 MCM A: The Sweet Spot (32)2010 MCM B: Criminology (32)2011 MCM A: Snowboard Course (33)2011 MCM B: Repeater Coordination (33)2012 MCM A: The Leaves of a Tree (33)2012 MCM B: Camping along the Big Long River (34)2013 MCM A: The Ultimate Brownie Pan (34)2013 MCM B: Water, Water, Everywhere (35)1985 MCM A: Animal PopulationsChoose a fish or mammal for which appropriate data are available to model it accurately. Model the animal's natural interactions with its environment by expressing population levels of different groups in terms of the significant parameters of the environment. Then adjust the model to account for harvesting in a form consistent with the actual method by which the animal is harvested. Include any outside constraints imposed by food or space limitations that are supported by the data.Consider the value of the various quantities involved, the number harvested, and the population size itself, in order to devise a numerical quantity that represents the overall value of the harvest. Find a harvesting policy in terms of population size and time that optimizes the value of the harvest over a long period of time. Check that the policy optimizes that value over a realistic range of environmental conditions.1985 MCM B: Strategic Reserve ManagementCobalt, which is not produced in the US, is essential to a number of industries. (Defense accounted for 17% of the cobalt production in 1979.) Most cobalt comes from central Africa, a politically unstable region. The Strategic and Critical Materials Stockpiling Act of 1946 requires a cobalt reserve that will carry the US through a three-year war. The government built up a stockpile in the 1950s, sold most of it off in the early 1970s, and then decided to build it up again in the late 1970s, with a stockpile goal of 85.4 million pounds. About half of this stockpile had been acquired by 1982.Build a mathematical model for managing a stockpile of the strategic metal cobalt. You will need to consider such questions as:▪How big should the stockpile be?▪At what rate should it be acquired?▪What is a reasonable price to pay for the metal?You will also want to consider such questions as:▪At what point should the stockpile be drawn down?▪At what rate should it be drawn down?▪At what price is it reasonable to sell the metal?▪How should it be allocated?Useful Information on CobaltThe government has projected a need ot 25 million pounds of cobalt in 1985.The U.S. has about 100 million pounds of proven cobalt deposits. Production becomes economically feasible when the price reaches $22/lb (as occurred in 1981). It takes four years to get operations rolling, and thsn six million pounds per year can be produced.In 1980, 1.2 million pounds of cobalt were recycled, 7% of total consumption.1986 MCM A: Hydrographic DataThe table below gives the depth Z of water in feet for surface points with rectangular coordinates X, Y in yards [table of 14 data points omitted]. The depth measurements were taken at low tide. Your ship has a draft of five feet. What region should you avoid within the rectangle (75,200) x (-50, 150)?1986 MCM B: Emergency-Facilities LocationThe township of Rio Rancho has hitherto not had its own emergency facilities. It has secured funds to erect two emergency facilities in 1986, each of which will combine ambulance, fire, and police services. Figure 1 indicates the demand [figure omitted], or number of emergencies per square block, for 1985. The ―L‖ region in the north is an obstacle, while the rectangle in the south is a part with shallow pond. It takes an emergency vehicle an average of 15 seconds to go one block in the N-S direction and 20 seconds in the E-W direction. Your task is to locate the two facilities so as to minimize the total response time.Assume that the demand is concentrated at the center of the block and that the facilities will be located on corners.▪Assume that the demand is uniformly distributed on the streets bordering each block and that the facilities may be located anywhere on the streets.1987 MCM A: The Salt Storage ProblemFor approximately 15 years, a Midwestern state has stored salt used on roads in the winter in circular domes. Figure 1 shows how salt has been stored in the past. The salt is brought into and removed from the domes by driving front-end loaders up ramps of salt leading into the domes. The salt is piled 25 to 30 ft high, using the buckets on the front-end loaders.Recently, a panel determined that this practice is unsafe. If the front-end loader gets too close to the edge of the salt pile, the salt might shift, and the loader could be thrown against the retaining walls that reinforce the dome. The panel recommended that if the salt is to be piled with the use of the loaders, then the piles should be restricted to a matimum height of 15 ft.Construct a mathematical model for this situation and find a recommended maximum height for salt in the domes.1987 MCM B: Parking Lot DesignThe owner of a paved, 100' by 200' , corner parking lot in a New England town hires you to design the layout, that is, to design how the ``lines are to be painted. You realize that squeezing as many cars into the lot as possible leads to right-angle parking with the cars aligned side by side. However, inexperienced drivers have difficulty parking their cars this way, which can give rise to expensive insurance claims. To reduce the likelihood of damage to parked vehicles, the owner might then have to hire expert drivers for ``valet parking. On the other hand, most drivers seem to have little difficulty in parking in one attempt if there is a large enough ``turning radius'' from the access lane. Of course, the wider the access lane, the fewer cars can be accommodated in the lot, leading to less revenue for the parking lot owner.1988 MCM A: The Drug Runner ProblemTwo listening posts 5.43 miles apart pick up a brief radio signal. The sensing devices were oriented at 110 degrees and 119 degrees, respectively, when the signal was detected; and they are accurate to within 2 degrees. The signal came from a region of active drug exchange, and it is inferred that there is a powerboat waiting for someone to pick up drugs. it is dusk, the weather is calm, and there are no currents. A small helicopter leaves from Post 1 and is able to fly accurately along the 110 degree angle direction. The helicopter's speed is three times the speed of the boat. The helicopter will be heard when it gets within 500 ft of the boat. This helicopter has only one detection device, a searchlight. At 200 ft, it can just illuminate a circular region with a radius of 25 ft.▪Develop an optimal search method for the helicopter.▪Use a 95% confidence level in your calculations.1988 MCM B: Packing Railroad FlatcarsTwo railroad flatcars are to be loaded with seven types of packing crates. The crates have the same width and height but varying thickness (t, in cm) and weight (w, in kg). Table 1 gives, for each crate, the thickness, weight, and number available [table omitted]. Each car has 10.2 meters of length available for packing the crates (like slices of toast) and can carry up to 40 metric tons. There is a special constraint on the total number of C_5, C_6, and C_7 crates because of a subsequent local trucking restriction: The total space (thickness) occupied by these crates must not exceed 302.7 cm. Load the two flatcars (see Figure 1) so as to minimize the wasted floor space [figure omitted].1989 MCM A: The Midge Classification ProblemTwo species of midges, Af and Apf, have been identified by biologists Grogan and Wirth on the basis of antenna and wing length (see Figure 1). It is important to be able to classify a specimen as Af of Apf, given the antenna and wing length.1. Given a midge that you know is species Af or Apf, how would you go about classifying it?2. Apply your method to three specimens with (antenna, wing) lengths(1.24,1.80),(1.28,1.84),(1.40,2.04).3. Assume that the species is a valuable pollinator and species Apf is a carrier of adebilitating disease. Would you modify your classification scheme and if so, how?1989 MCM B: Aircraft QueueingA common procedure at airports is to assign aircraft (A/C) to runways on a first-come-first-served basis. That is, as soon as an A/C is ready to leave the gate (―push-back‖), the pilot calls ground control and is added to the queue. Suppose that a control tower has access to a fast online database with the following information for each A/C:▪the time it is scheduled for pushback;▪the time it actually pushes back; the number of passengers who are scheduled to make a connection at the next stop, as well as the time to make that connection; and▪the schedule time of arrival at its next stop Assume that there are seven types of A/C with passenger capacities varying from 100 to 400 in steps of 50. Develop and analyze amathematical model that takes into account both the travelers' and airlines' satisfaction.1990 MCM A: The Brain-Drug ProblemResearches on brain disorders test the effects of the new medical drugs – for example, dopamine against Parkinson's disease – with intracerebral injections. To this end, they must estimate the size and the sape of the spatial distribution of the drug after the injection, in order to estimate accurately the region of the brain that the drug has affected.The research data consist of the measurements of the amounts of drug in each of 50 cylindrical tissue samples (see Figure 1 and Table 1). Each cylinder has length 0.76 mm and diameter 0.66 mm. The centers of the parallel cylinders lie on a grid with mesh 1mm X 0.76mm X 1mm, so that the sylinders touch one another on their circular bases but not along their sides, as shown in the accompanying figure. The injection was made near the center of the cylinder with the highest scintillation count. Naturally, one expects that there is a drug also between the cylinders and outside the region covered by the samples.Estimate the distribution in the region affected by the drug.One unit represents a scintillation count, or 4.753e-13 mole of dopamine. For example, the table shows that the middle rear sylinder contails 28353 units.Table 1. Amounts of drug in each of 50 cylindrical tissue samples.Rear vertical sectionFront vertical section1990 MCM B: Snowplow RoutingThe solid lines of the map (see Figure 1) represent paved two-lane county roads in a snow removal district in Wicomico County, Maryland [figure omitted]. The broken lines are state highways. After a snowfall, two plow-trucks are dispatched from a garage that is about 4 miles west of each of the two points (*) marked on the map. Find an efficient way to use the two trucks to sweep snow from the county roads. The trucks may use the state highways to access the county roads. Assume that the trucks neither break down nor get stuck and that the road intersections require no special plowing techniques.1991 MCM A: Water Tank FlowSome state water-right agencies require from communities data on the rate of water use, in gallons per hour, and the total amount of water used each day. Many communities do not have equipment to measure the flow of water in or out of the municipal tank. Instead, they can measure only the level of water in the tank, within 0.5% accuracy, every hour. More importantly, whenever the level in the tank drops below some minimum level L, a pump fills the tank up to the maximum level, H; however, there is no measurement of the pump flow either. Thus, one cannot readily relate the level in the tank to the amount of water used while the pump is working, which occurs once or twice per day, for a couple of hours each time. Estimate the flow out of the tank f(t) at all times, even when the pump is working, and estimate the total amount of water used during the day. Table 1 gives real data, from an actual small town, for one day[ table omitted]. The table gives the time, in, since the first measurement, and the level of water in the tank, in hundredths of a foot. For example, after 3316 seconds, the depth of water in the tank reached 31.10 feet. The tank is a vertical circular cylinder, with a height of 40 feet and a diameter of 57 feet. Usually, the pump starts filling the tank when the level drops to about 27.00 feet, and the pump stops when the level rises back to about 35.50 feet.1991 MCM B: The Steiner Tree ProblemThe cost for a communication line between two stations is proportional to the length of the line. The cost for conventional minimal spanning trees of a set of stations can often be cut by introducing―phantom‖ stations and then constructing a new Steiner tree. This device allows costs to be cut by up to 13.4% (= 1- sqrt(3/4)). Moreover, a network with n stations never requires more than n-2 points to construct the cheapest Steiner tree. Two simple cases are shown in Figure 1.For local networks, it often is necessary to use rectilinear or ―checker-board‖ distances, instead of straight Euclidean lines. Distances in this metric are computed as shown in Figure 2.Suppose you wish to design a minimum costs spanning tree for a local network with 9 stations. Their rectangular coordinates are: a(0,15), b(5,20), c(16,24), d(20,20), e(33,25), f(23,11), g(35,7), h(25,0) i(10,3). You are restricted to using rectilinear lines. Moreover, all ―phantom‖ stations must be located at lattice points (i.e., the coordinates must be integers). The cost for each line is its length.1. Find a minimal cost tree for the network.2. Suppose each stations has a cost w*d^(3/2), where d=degree of the station. If w=1.2, find aminimal cost tree.3. Try to generalize this problem1992 MCM A: Air-Traffic-Control Radar PowerYou are to determine the power to be radiated by an air-traffic-control radar at a major metropolitan airport. The airport authority wants to minimize the power of the radar consistent with safety andcost. The authority is constrained to operate with its existing antennae and receiver circuitry. The only option that they are considering is upgrading the transmitter circuits to make the radar more powerful. The question that you are to answer is what power (in watts) must be released by the radar to ensure detection of standard passenger aircraft at a distance of 100 kilometers.1992 MCM B: Emergency Power RestorationPower companies serving coastal regions must have emergency response systems for power outages due to storms. Such systems require the input of data that allow the time and cost required for restoration to be estimated and the ―value‖ of the outage judged by objective criteria. In the past, Hypothetical Electric Company (HECO) has been criticized in the media for its lack of a prioritization scheme.You are a consultant to HECO power company. HECO possesses a computerized database with real time access to service calls that currently require the following information:▪time of report,▪type of requestor,▪estimated number of people affected, and▪location (x,y).Cre sites are located at coordinates (0,0) and (40,40), where x and y are in miles. The region serviced by HECO is within -65 < x < 60 and -50 < y < 50. The region is largely metropolitan with an excellent road network. Crews must return to their dispatch site only at the beginning and end of shift. Company policy requires that no work be initiated until the storm leaves the area, unless the facility is a commuter railroad or hospital, which may be processed immediately if crews are available.HECO has hired you to develop the objective criteria and schedule the work for the storm restoration requirements listed in Table 1 using their work force described in Table 2. Note that the first call was received at 4:20 A.M. and that the storm left the area at 6:00 A.M. Also note that many outages were not reported until much later in the day.HECO has asked for a technical report for their purposes and an ―executive summary‖ i n laymen's terms that can be presented to the media. Further, they would like recommendations for the future. To determine your prioritized scheduling system, you will have to make additional assumptions. Detail those assumptions. In the future, you may desire additional data. If so, detail the information desired.Table 1. Storm restoration requirements. (table incomplete)Table 2. Crew descriptions.1993 MCM A: Optimal CompostingAn environmentally conscious institutional cafeteria is recycling customers' uneaten food into compost by means of microorganisms. Each day, the cafeteria blends the leftover food into a slurry, mixes the slurry with crisp salad wastes from the kitchen and a small amount of shredded newspaper, and feeds the resulting mixture to a culture of fungi and soil bacteria, which digest slurry, greens, and papers into usable compost. The crisp green provide pockets of oxygen for the fungi culture, and the paper absorbs excess humidity. At times, however, the fungi culture is unable or unwilling to digest as much of the leftovers as customers leave; the cafeteria does not blame the chef for the fungi culture's lack of appetite. Also, the cafeteria has received offers for the purchase of large quantities of it compost. Therefore, the cafeteria is investigating ways to increase its production of compost. Since it cannot yet afford to build a new composting facility, the cafeteria seeks methods to accelerate the fungi culture's activity, for instance, by optimizing the fungiculture's environment (currently held at about 120 F and 100% humidity), or by optimizing the composition of the moisture fed to the fungi culture, or both.Determine whether any relation exists between the proportions of slurry, greens, and paper in the mixture fed to the fungi culture, and the rate at which the fungi culture composts the mixture. if no relation exists, state so. otherwise, determine what proportions would accelerate the fungi culture's activity. In addition to the technical report following the format prescribed in the contest instructions, provide a one-page nontechnical recommendation for implementation for the cafeteria manager. Table 1 shows the composition of various mixtures in pounds of each ingredient kept in separate bins, and the time that it took the fungi to culture to compost the mixtures, from the date fed to the date completely composted [table omitted].1993 MCM B: Coal-Tipple OperationsThe Aspen-Boulder Coal Company runs a loading facility consisting of a large coal tipple. When the coal trains arrive, they are loaded from the tipple. The standard coal train takes 3 hours to load, and the tipple's capacity is 1.5 standard trainloads of coal. Each day, the railroad sends three standard trains to the loading facility, and they arrive at any time between 5 A.M. and 8 P.M. local time. Each of the trains has three engines. If a train arrives and sits idle while waiting to be loaded, the railroad charges a special fee, called a demurrage. The fee is $5,000 per engine per hour. In addition, a high-capacity train arrives once a week every Thursday between 11 A.M. and 1 P.M. This special train has five engines and holds twice as much coal as a standard train. An empty tipple can be loaded directly from the mine to its capacity in six hours by a single loading crew. This crew (and its associated equipment) cost $9,000 per hour. A second crew can be called out to increase the loading rate by conducting an additional tipple-loading operation at the cost of $12,000 per hour. Because of safety requirements, during tipple loading no trains can be loaded. Whenever train loading is interrupted to load the tipple, demurrage charges are in effect.The management of the Coal Company has asked you to determine the expected annual costs of this tipple's loading operations. Your analysis should include the following considerations:▪How often should the second crew be called out?▪What are the expected monthly demurrage costs?▪If the standard trains could be scheduled to arrive at precise times, what daily schedule would minimize loading costs? Would a third tipple-loading crew at $12,000 per hour reduce annual operations costs?▪Can this tipple support a fourth standard train every day?1994 MCM A: Concrete Slab FloorsThe U.S. Dept. of Housing and Urban Development (HUD) is considering constructing dwellings of various sizes, ranging from individual houses to large apartment complexes. A principal concern is to minimize recurring costs to occupants, especially the costs of heating and cooling. The region inwhich the construction is to take place is temperate, with a moderate variation in temperature throughout the year.Through special construction techniques, HUD engineers can build dwellings that do not need to rely on convection- that is, there is no need to rely on opening doors or windows to assist in temperature variation. The dwellings will be single-story, with concrete slab floors as the only foundation. You have been hired as a consultant to analyze the temperature variation in the concrete slab floor to determine if the temperature averaged over the floor surface can be maintained within a prescribed comfort zone throughout the year. If so, what size/shape of slabs will permit this?Part 1, Floor Temperature: Consider the temperature variation in a concrete slab given that the ambient temperature varies daily within the ranges given Table 1. Assume that the high occurs at noon and the low at midnight. Determine if slabs can be designed to maintain a temperature averaged over the floor surface within the prescribed comfort zone considering radiation only. Initially, assume that the heat transfer into the dwelling is through the exposed perimeter of the slab and that the top and bottom of the slabs are insulated. Comment on the appropriateness and sensitivity of these assumptions. If you cannot find a solution that satisfies Table 1, can you find designs that satisfy a Table 1 that you propose?Part 2, Building Temperature: Analyze the practicality of the initial assumptions and extend the analysis to temperature variation within the single-story dwelling. Can the house be kept within the comfort zone?Part 3, Cost of Construction: Suggest a design that considers HUD's objective of reducing or eliminating heating and cooling costs, considering construction restrictions and costs.1994 MCM B: Network DesignIn your company, information is shared among departments on a daily basis. This information includes the previous day's sales statistics and current production guidance. It is important to get this information out as quickly as possible. [Network diagram (with 5 nodes and 7 capacitated edges) omitted.]We are interested in scheduling transfers in an optimal way to minimize the total time it takes to complete them all. This minimum total time is called the makespan. Consider the three following situations for your company: [Three more network diagrams (on roughly 20 nodes each) omitted.]1995 MCM A: Helix ConstructionA small biotechnological company must design, prove, program and test a mathematical algorithm to locate ―in real time‖ all the intersections of a helix and a plane in general positions in space. Design, justify, program and test a method to compute all the intersections of a plane and a helix, both in general positions (at any locations and with any orientations) in space. A segment of the helix may represent, for example, a helicoidal suspension spring or a piece of tubing in a chemical or medical apparatus. Theoretical justification of the proposed algorithm is necessary to verify the solution from several points of view, for instance, through mathematical proofs of parts of the algorithm, and through tests of the final program with known examples. Such documentation and tests will be required by government agencies for medical use.1995 MCM B: Faculty CompensationAluacha Balaclava College, and undergraduate facility, has just hired a new Provost whose first priority is the institution of a fair and reasonable faculty-compensation plan. She has hired your consulting team to design a compensation system that reflects the following circumstances and principles: [Three paragraphs of details omitted] Design a new pay system, first withoutcost-of-living increases. Incorporate cost-of-living increases, and then finally, design a transition process for current faculty that will move all salaries towards your system without reducing anyone's salary. The Provost requires a detailed compensation system plan for implementation, as well as a brief, clear, executive summary outlining the model, its assumptions, strengths, weaknesses and expected results, which she can present to the Board and faculty. [A detailed table of current salaries is omitted.]1996 MCM A: Submarine TrackingThe world's oceans contain an ambient noise field. Seismic disturbances, surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large maving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, develop a method for detecting the presence of a moving submarine, its speed, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed frequency and amplitude.1996 MCM B: Paper JudgingWhen determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge. Let's say there are P=100 papers. A group of J judges is collected to accomplish the judging. Funding for the contest contrains both the number of judges that can be obtained and the amount of time they can judge. For example if P=100, then J=8 is typical.。

2003 MCM ProblemsPROBLEM A: The Stunt PersonAn exciting action scene in a movie is going to be filmed, and you are the stunt coordinator! A stunt person on a motorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. You need to protect the stunt person, and also use relatively few cardboard boxes (lower cost, not seen by camera, etc.).Your job is to:•determine what size boxes to use•determine how many boxes to use•determine how the boxes will be stacked•determine if any modifications to the boxes would help•generalize to different bined weights (stunt person & motorcycle) and different jump heightsNote that, in "Tomorrow Never Dies", the James Bond character on a motorcycle jumps over a helicopter.PROBLEM B: Gamma Knife Treatment PlanningStereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographicallywell-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. Three modalities are monly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators.The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy helmet. All 201 beams simultaneously intersect at the isocenter, resulting in a spherical (approximately) dose distribution at the effective dose levels. Irradiating the isocenter to deliver dose is termed a “shot.” Shots can be represented as different spheres. Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. For a target volume larger than one shot, multiple shots can be used to cover the entire target. In practice, most target volumes are treated with 1 to 15 shots. The target volume is a bounded, three-dimensional digital image that usually consists of millions of points.The goal of radiosurgery is to deplete tumor cells while preserving normal structures. Since there are physical limitations and biological uncertainties involved in this therapy process, a treatment plan needs to account for all those limitations and uncertainties. In general, an optimal treatment plan is designed to meet the following requirements.1.Minimize the dose gradient across the target volume.2.Match specified isodose contours to the target volumes.3.Match specified dose-volume constraints of the target and critical organ.4.Minimize the integral dose to the entire volume of normal tissues or organs.5.Constrain dose to specified normal tissue points below tolerance doses.6.Minimize the maximum dose to critical volumes.In gamma unit treatment planning, we have the following constraints:1.Prohibit shots from protruding outside the target.2.Prohibit shots from overlapping (to avoid hot spots).3.Cover the target volume with effective dosage as much as possible. But at least 90% of thetarget volume must be covered by shots.e as few shots as possible.Your tasks are to formulate the optimal treatment planning for a gamma knife unit as a sphere-packing problem, and propose an algorithm to find a solution. While designing your algorithm, you must keep in mind that your algorithm must be reasonably efficient.2002 Contest ProblemsProblem AAuthors: Tjalling YpmaTitle: Wind and WatersprayAn ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area.Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.Problem BAuthors: Bill Fox and Rich WestTitle: Airline OverbookingYou're all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat.Historically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook-that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations.Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.Consider the overbooking issue in light of the current situation:Less flights by airlines from point A to point BHeightened security at and around airportsPassengers' fearLoss of billions of dollars in revenue by airlines to dateBuild a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline pany in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the pany's revenue is maximized. Insure that your model reflects the issues above, and consider alternatives for handling "bumped" passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.MCM2000Problem A Air traffic ControlTo improve safety and reduce air traffic controller workload, the Federal Aviation Agency (FAA) is considering adding software to the air traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA r traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA has posed the following problemsRequirement A: Given two airplanes flying in space, when should the air traffic controller ld the air traffic controller consider the objects to be too close and to require intervention?Requirement B: An airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how we measure how plex it is from an air traffic workload perspective? To what extent is plexity determined by the number of we measure how plex it is from an air traffic workload perspective? To what extent is plexity determined by the number of aircraft simultaneously passing through that sector (1) at any one instant? (2) During any given interval of time? (3) During particular time of day? How does the number of potential conflicts arising during those periods affect plexity?Does the presence of additional software tools to automatically predict conflicts and alert the controller reduce or add to this plexity?In addition to the guidelines for your report, write a summary (no more than two pages) that the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your conclusionsProblem B Radio Channel AssignmentsWe seek to model the assignment of radio channels to a symmetric network of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grid (honeyb-style), as shown in Figure 1, where a transmitter is located at the center of each hexagon.An interval of the frequency spectrum is to be allotted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1, 2, 3, ... . Each transmitter will be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided. Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assign channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span.Let s be the length of a side of one of the hexagons. We concentrate on the case that there are two levels of interferenceRequirement A: There are several constraints on frequency assignments. First, no two transmitters within distance of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these constraints, what can we say about the span in,Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in all directions.Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance differ by at least some given integer k, while those at distance at most must still differ by at least one. What can we say about the span and about efficient strategies for designing assignments, as a function of k?Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider?Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining your findingsMCM2000问题A 空间交通管制为加强安全并减少空中交通指挥员的工作量，联邦航空局(FAA)考虑对空中交通管制系统添加软件，以便自动探测飞行器飞行路线可能的冲突，并提醒指挥员。