美国大学生数学建模大赛英文写作

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2011年美国大学生数学建模竞赛优秀作品

AbstractThis paper presents one case study to illustrate how probability distribution and genetic algorithm and geographical analysis of serial crime conducted within a geographic information system can assist crime investigation.Techniques are illustrated for predicting the location of future crimes and for determining the possible residence of offenders based on the geographical pattern of the existing crimes and quantitative method,which is PSO.It is found that such methods are relatively easy to implement within GIS given appropriate data but rely on many assumptions regarding offenders’behaviour.While some success has been achieved in applying the techniques it is concluded that the methods are essentially theory-less and lack evaluation.Future research into the evaluation of such methods and in the geographic behaviour of serial offenders is required in order to apply such methods to investigations with confidence in their reliability.1.IntroductionThis series of armed robberies occurred in Phoenix,Arizona between13September and5December1999and included35robberies of fast food restaurants,hotels and retail businesses.The offenders were named the“Supersonics”by the Phoenix Police Department Robbery Detail as the first two robberies were of Sonic Drive-In restaurants.After the35th robbery,the offenders appear to have desisted from their activity and at present the case remains unsolved.The MO was for the offenders to target businesses where they could easily gain entry,pull on a ski mask or bandanna, confront employees with a weapon,order them to the ground,empty the cash from a safe or cash register into a bag and flee on foot most likely to a vehicle waiting nearby. While it appears that the offenders occasionally worked alone or in pairs,the MO, weapons and witness descriptions tend to suggest a group of at least three offenders. The objective of the analysis was to use the geographic distribution of the crimes to predict the location of the next crime in an area that was small enough to be suitable for the Robbery Detail to conduct stakeouts and surveillance.After working with a popular crime analysis manual(Gottleib,Arenberg and Singh,1994)it was found that the prescribed method produced target areas so large that they were not operationally useful.However,the approach was attractive as it required only basic information and relied on simple statistical analysis.To identify areas that were more useful for the Robbery Detail,it was decided to use a similar approach combined with other measurable aspects of the spatial distribution of the crimes.As this was a“live”case, new crimes and information were integrated into the analysis as it came to hand.2.AssumptionIn order to modify the model existed,we apply serial new assumptions to the principle so that our rectified model can be much more practical.Below are the assumptions:1.C riminals prefer something about the locations where previous crimes werecommitted committed..We supposed the criminals have a greater opportunity to ran away if they choose to crime in the site they are familiar with.In addition,the criminals probably choose previous kill sites where their target potential victims live and work.2.Offenders regard it safer to crime in their previous kill site as time went by.This is true that the site would be severely monitored by police when a short term crime happened and consequently the criminal would suffer a risk of being arrested in that site.And as mentioned above ,the police would reduce the frequency of examining the previous kill sites as time went by.3.Criminals are likely to choose the site that have optimal distance .This is a reasonable assumption since it is probably insecure to crime in the site that stays far away and that costs an amount of energy to escape and adds the opportunity to be arrested in such an unfamiliar terrain.And it is also impossible to crime in the site nearby since it increases the probability of being recognized or being trapped.As a result,we can measure a optimal distance in series perpetrations.4.Crimes are committed by individual.We assume that all the case in the model are committed by individuals instead of by organized members.In this way the criminal is subject to the assumptions mentioned above due to his insufficient preparation.5.Criminals Criminals''movements unconstrained.Because of the difficulty of finding real-world distance data,we invoke the “Manhattan assumption”:There are enough streets and sidewalks in a sufficiently grid-like pattern that movements along real-world movement routes is the same as “straight-line”movement in a space be discrete into city blocks.It is demonstrated that across several types of serial crime,the Euclidean and Manhattan distances are essentially interchangeable in predicting anchor points.3.The prediction of the next crime site3.1The measure of the optimal distanceDue to the fact that the mental optimal distance of the criminal is related to whether he is a careful person or not,it is impossible for him to make a fixed constant.Besides,the optimal distance will change in different moment.However,such distance should be reflected on the distances of the former crime sites.Presume that the coordinates of the n crime sites is respectively ),(11y x 、),(22y x 、……、),(n n y x ,and define the distance between the th i crime site and the th j one as j D ,i .The distance above we first consider it as Euclid distance,which is:22,)()(j i j i j i y y x x D −+−=With that,we are able to measure the distance between the th n crime site and the th 1-n one respectively.According to the assumption 2,the criminal believes that the earlier crime sites have became saferfor him to commit a crime again,so we can define his mental optimal distance,giving the sites the weights from little to much according to when the offenses happened in time sequence,as:∑−==11,n i ni i D w SD Satisfying 121......−<<<n w w w ，111=∑−=n i i w .Presuming the th i crime happens in i t ,whichis measured by week,we can have ∑−==11n i i kk t t w .SD can reflect the criminal's mental condition to some extent,so we can use it to predict the mental optimal distance of the criminal in the th n 1+case.While referring to the th n crime site,the criminal is able to use SD to estimate the optimal distance in the next time,and while referring to the rest crime sites,the optimal distances reduce as time goes back.Thus,the optimal security of the th i crime site can be measured as the following:n ni i SD t t SD *=3.2The measure of the probability distributionGiven the crime sites and location,we can estimate tentatively the probability density distribution of the future crimes,which equals to that we add some small normal distribution to every scene of crime to produce a probability distribution estimate function.The small normal distribution uses the SD mentioned above as the mean,which is:∑=−−=n i i i SD r n y x f 122)2)(exp(211),(σσπi r is defined as the Euclid distance between the site to the th i crime site,and the standard difference of the deviation of the criminal's mental optimal distance is defined as σ,which also reflects the uncertainty of the deviation of the criminal's mental optimal distance,involves the impacts of many factors and can not be measured quantitatively.The discussion of the standard difference is as following:3.3The quantization of the standard differenceThe standard difference is identified according to the following goal,which is,every prediction of the next crime site according to the crime sites where the crimes were committed before should have the highest rate of success.When having to satisfying such optimization objective,it isimpossible to make the direct analysis and exhaustivity.Instead,we have to use the optimized solutions searching algorithm,which is genetic algorithm.\Figure1:The Distribution of the Population of the Last GenerationAccording to the figure,the population of the last generation is mostly concentrated near80, which is used as the standard distance and substituted to the*formula.With the*formula,we are able to predict the probability density of Whether the zones will be the next crime site.Case analysis:5crime site according to the4ones happened before Figure2:The prediction of theth6crime site according to the5ones happened before Figure3:The prediction of theth6crime site according to the5ones happened before Figure4:The prediction of thethAccording to the predictions happened before,the predictions of the outputs based on the models are accurate relatively,and they are able to be the references of the criminal investigations to some extent.However,when is frequency of such crime increases,the predictions of the outputs23crime site according deviated the actual sites more and more,such as the prediction of thethto the22ones happened before,which is:23crime site according to the22ones happened before Figure5:the prediction of thethConclusion according to analysis:It may not be able to predict the next crime site accurately if we use Euclid distance to measure the probability directly.So,we should analyze according to the actual related conditions.For example,we can consider the traffic commutes comprehensively based on the conveniences of the escapes,such as the facilities of the express ways network and the tunnels.According to the hidden security of the commitments,we should consider the population of the area and the distance from the police department.Thus,we should give more weights to the commute convenience,hidden security and less population.In addition,when the commitments increases,the accuracy of the model may decrease,resulted from the fact that when the criminal has more experience,he will choose the next crime sites more randomly.4.Problems and further improvementsWith23crimes in the series the predictions tended to provide large areas that included the target crime but were too large to be useful given the limited resources the police had at their disposal.At this stage,a more detailed look was taken at the directionality and distances between crimes.No significant trends could be found in the sequential distance between crimes so an attempt was made to better quantify the relationship between crimes in terms of directionality.The methodology began by calculating the geographic center of the existing crimes. The geographic center is a derived point that identifies the position at which the distance to each crime is minimized.For applications of the geographic center to crime analysis.Once constructed,the angle of each crime from the north point of the geographic center was calculated.From this it was possible to calculate the change indirection for the sequential crimes.It was found that the offenders were tending to pattern their crimes by switching direction away from the last crime.It appears that the offenders were trying to create a random pattern to avoid detection but unwittingly created a uniform pattern based upon their choice of locations.This relationship was quantified and a simple linear regression used to predict what the next direction would be.The analysis was once again applied to the data.While the area identified was reduced from previous versions and prioritized into sub-segments,the problem remained that the areas predicted were still too large to be used as more than a general guide to resource deployment.A major improvement to the methodology was to include individual targets.By this stage of the series,hotels and auto parts retailers had become the targets of choice.A geo-coded data set became available that allowed hotels and retail outlets to be plotted and compared to the predicted target areas.Ideally those businesses falling within the target areas could be prioritized as more likely targets.However,in some cases the distribution of the likely businesses appeared to contradict the area predicted.For example,few target hotels appeared in the target zone identified by the geographic analysis.In this case,more reliance was placed upon the location of individual targets. From this analysis it was possible to identify a prioritized list of individual commercial targets,which was of more use operationally.Maps were also provided to give an indication of target areas.Figure6demonstrates a map created using this methodology.It is apparent from the above discussion that the target areas identified were often too large to be used as more than a general guide by the Robbery Detail.However,by including the individual targets,it was possible to restrict the possible target areas to smaller,more useful areas,and a few prioritized targets.However,such an approach has the danger of being overly restrictive and it is not the purpose of the analysis to restrict police operations but to suggest priorities.This problem was somewhat dealt with by involving investigators in the analysis and presenting the results in an objective manner,such that investigators could make their own judgments about the results.To be more confident in using this kind of analysis a stronger theoretical background to the methods is required.What has been applied here is to simply exploit the spatial relationships in the information available without considering what the connection is to the actual behaviour of the offenders.For example,what is the reason behind a particular trend observed in the distance between crimes?Why would such a trend be expected between crimes that occur on different days and possibly involve different individuals?While some consideration was given to identifying the reason behind the pattern of directionality and while it seems reasonable to expect offender’s to look for freeway access,such reasoning has tended to follow the analysis rather than substantiate it.Without a theoretical background the analysis rests only on untested statistical relationships that do not provide an answer to the basic question:why this pattern?So next we will apply a quantitative method,which is PSO,based on a theoretical background,to locate the residence of the criminal's residence.5.The prediction of the residenceParticle Swarm Optimization is a evolutionary computation,invented by Dr.Eberhart and Dr.Kennedy.It is a tool of optimization based on iteration,resulted from the research on the behaviors of the bird predation.Initiating a series of random number,the PSO is able to catch the optimization with iteration.Like PSO,the resolution of our residence search problem is the criminal,whose serial crime sites have been abstracted into 23particles without volume and weight and extended to the 2-D space.Like bird,the criminal is presumed to go directly home when he committed a crime.So,there are 23criminals who commit the crimes in the 23sites mention before and then they will go home directly.The criminals are defined as a vector,so are their speed.All criminals have a fittness decided by the optimized functions,and every of them has a according speed which can decide their direction and distance.All the criminals know the best position (pbest,defined as the residence known by the individual),which has been discovered so far,and where they are now.Besides,every criminals also know the best position which has been found by the group (gbest,defined as the residence known by the group).Such search can be regarded as the experience of other criminals.The criminals are able to locate the residence by the experience of itself and the whole criminals.PSO computation initiates the 23criminals and then the offenders will pursue the optimized one to search in the space.In other words,they find the optimized solutions by iteration.Presume that in the 2-D space the location and speed of the ith crime site is relatively ),(2,1,i i i x x X =and ),(2,1,i i i v v V =.In every iteration,the criminals will pursue the two best positions to update themselves.The two best positions are relatively the individual peak (pbest),),(2,1,i i i p p P =,which is found by the criminal himself,and the group optimized solution (gbest),g P ,which has been found to be the optimized solution by the whole group so far.When the criminals found the two optimized solutions,they will update their speed and new position based on the following formulas.2,1),1()()1()]([)]([)()1(,,,,,22,,11,,=++=+−+−+=+j t v t x t x t x p r c t x p r c t wv t V j i j i j i j i j g j i j i j i j i In the above,the w is inertial weighted factor,21c andc are positive learning factors,21r andr are random number which are distributed uniformly between 0and 1.The learning factor can make the criminals have self-conclude ability and ability of learning from others.Here we make both of them be 2,as what they always are in PSO.The inertial weighted factor w decides the extent of the inheritance of the current speed of the crime sites.The appropriate choice can make them have balanced searching and exploring ability.For balancing the global searching ability and the local improving ability of the criminal in the PSO algorithm,here we adopt one of the self-adapted methods,which is Non-linear Dynamic Inertial Weight Coefficient to choose the inertial weight.The expression is as following:⎪⎩⎪⎨⎧=≤−−−−>avg avg avg f f f f f f w w w f f w w ,))*((,minmin min max min max In the above,the max w and min w are defined respectively as the maximum and minimum of w,f means the current functional value of the criminal,and the avg f and min f respectively means the average value and minimum value of all the current criminals.In addition,the inertial weight will change automatically according to the objective value,which gives the name self-adapted method.When the final values,which are estimations of the criminal's residence,become consistent,it will make the inertial weight increase.When they become sparser,it will make the inertial weight decrease.In the meantime,referring to the criminals whose final values are worse than the average value,its according inertial weighted factor will become smaller,which protect the crime site.Oppositely,when referring to the criminals whose final values are better than the average value,its according inertial weighted factor will become bigger,which makes the criminal nearer to the searching zone.So now,with the PSO of Non-linear Dynamic Inertial Weight Coefficient,we can calculate the minimum value of22,)()(j j j i y y x x R −+−=,j=1,2,3 (23)In the above,j ,i R is the residence of the criminal.Thus,we have the output (x,y)as(2.368260870656715,3.031739124610613).We can see the residence in the figure 7.Figure7:The residence in the map6.ConclusionThis paper has presented one case study to illustrate how probability distribution and geographical analysis of serial crime conducted can assist crime investigation. Unfortunately,in the Supersonic armed robbery investigation the areas identified were too large to have been of much use to investigators.Further,because of the number of assumptions applied the method does not inspire enough confidence to dedicate resources to comparing its results to the enormous amount of suspect data collected on the case.While the target areas predicted tended to be large,the mapping of individual commercial targets appears to offer a significant improvement to the method.However,as they stand,these methods lack a theoretical basis that would allow the results to be judged and applied in investigations.Limitations such as these can be offset to some degree by the involvement of investigators in the analysis.In the end,we used a quantitative method to locate the residence of the criminal to make the identified areas smaller.So,due to the advantages and drawbacks of the above methods,we suggest that we should use different methods to help us fight again the crimes comprehensively.。

23年美赛中英文对照版竞赛指南

美赛中英文对照版竞赛指南1. 简介23年美赛中英文对照版竞赛指南是为了帮助参与23年美国大学数学建模竞赛的学生更好地准备和参加比赛而编写的指南。

This guide is intended to help students participating in the 23rd annual MCMASpetition of the United States to prepare for and participate in thepetition.2. 竞赛概况美国大学数学建模竞赛是一项面向全球高校学生的标志性竞赛，旨在提高学生的数学建模、解决问题和团队合作能力。

比赛通常设有团队赛和个人赛两个类别，题目涉及的领域广泛，如数学、统计学、运筹学等。

The MCMAS is a prestigiouspetition for global college students, 本人ming to improve students' mathematical modeling, problem-solving, and teamwork skills. Thepetition usually consists of team and individual categories, covering a wide range of fields such as mathematics, statistics, operations research, etc.3. 竞赛时间和地点23年美国大学数学建模竞赛预计于2023年2月进行，具体的时间和地点将在冠方全球信息站上公布。

参赛学生需要提前关注并根据指定时间和地点参与比赛。

The 23rd annual MCMAS is expected to take place in February 2023, with specific dates and locations to be announced on the official website. Participating students need to pay attention in advance and participate in thepetition according to the specified time and place.4. 竞赛报名学生可以通过冠方全球信息站进行报名，需要填写个人信息并组建队伍报名参赛。

美国大学生数学建模竞赛优秀论文

For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number7018Problem ChosencFor office use onlyF1________________F2________________F3________________F4________________ SummaryThe article is aimed to research the potential impact of the marine garbage debris on marine ecosystem and human beings，and how we can deal with the substantial problems caused by the aggregation of marine wastes.In task one，we give a definition of the potential long-term and short-term impact of marine plastic garbage. Regard the toxin concentration effect caused by marine garbage as long-term impact and to track and monitor it. We etablish the composite indicator model on density of plastic toxin，and the content of toxin absorbed by plastic fragment in the ocean to express the impact of marine garbage on ecosystem. Take Japan sea as example to examine our model.In ask two, we designe an algorithm, using the density value of marine plastic of each year in discrete measure point given by reference，and we plot plastic density of the whole area in varies locations. Based on the changes in marine plastic density in different years， we determine generally that the center of the plastic vortex is East—West140°W—150°W, South—North30°N—40°N. According to our algorithm, we can monitor a sea area reasonably only by regular observation of part of the specified measuring pointIn task three，we classify the plastic into three types，which is surface layer plastic，deep layer plastic and interlayer between the two. Then we analysis the the degradation mechanism of plastic in each layer. Finally，we get the reason why those plastic fragments come to a similar size.In task four， we classify the source of the marine plastic into three types，the land accounting for 80%，fishing gears accounting for 10%，boating accounting for 10%，and estimate the optimization model according to the duel-target principle of emissions reduction and management. Finally, we arrive at a more reasonable optimization strategy.In task five，we first analyze the mechanism of the formation of the Pacific ocean trash vortex， and thus conclude that the marine garbage swirl will also emerge in south Pacific，south Atlantic and the India ocean. According to the Concentration of diffusion theory, we establish the differential prediction model of the future marine garbage density，and predict the density of the garbage in south Atlantic ocean. Then we get the stable density in eight measuring point .In task six, we get the results by the data of the annual national consumption ofpolypropylene plastic packaging and the data fitting method, and predict the environmental benefit generated by the prohibition of polypropylene take-away food packaging in the next decade. By means of this model and our prediction，each nation will reduce releasing 1.31 million tons of plastic garbage in next decade.Finally, we submit a report to expediction leader，summarize our work and make some feasible suggestions to the policy- makers.Task 1:Definition:●Potential short-term effects of the plastic: the hazardeffects will be shown in the short term.●Potential long-term effects of the plastic: thepotential effects, of which hazards are great, willappear after a long time.The short- and long-term effects of the plastic on the ocean environment:In our definition, the short-term and long-term effects of the plastic on the ocean environment are as follows.Short-term effects:1）The plastic is eaten by marine animals or birds.2） Animals are wrapped by plastics, such as fishing nets, which hurt or even kill them.3）Deaden the way of the passing vessels.Long-term effects:1）Enrichment of toxins through the food chain: the waste plastic in the ocean has no natural degradation in theshort-term, which will first be broken down into tinyfragments through the role of light, waves,micro-organisms, while the molecular structure has notchanged. These "plastic sands", easy to be eaten byplankton, fish and other, are Seemingly very similar tomarine life’s food,causing the enrichment and delivery of toxins.2）Accelerate the greenhouse effect: after a long-term accumulation and pollution of plastics, the waterbecame turbid, which will seriously affect the marineplants (such as phytoplankton and algae) inphotosynthesis. A large number of plankton’s deathswould also lower the ability of the ocean to absorbcarbon dioxide, intensifying the greenhouse effect tosome extent.To monitor the impact of plastic rubbish on the marine ecosystem:According to the relevant literature, we know that plastic resin pellets accumulate toxic chemicals , such as PCBs、DDE , and nonylphenols , and may serve as a transport medium and soure of toxins to marine organisms that ingest them[]2. As it is difficult for the plastic garbage in the ocean to complete degradation in the short term, the plastic resin pellets in the water will increase over time and thus absorb more toxins, resulting in the enrichment of toxins and causing serious impact on the marine ecosystem.Therefore, we track the monitoring of the concentration of PCBs, DDE, and nonylphenols containing in the plastic resin pellets in the sea water, as an indicator to compare the extent of pollution in different regions of the sea, thus reflecting the impact of plastic rubbish on ecosystem.To establish pollution index evaluation model: For purposes of comparison, we unify the concentration indexes of PCBs, DDE, and nonylphenols in a comprehensive index.Preparations:1）Data Standardization2）Determination of the index weightBecause Japan has done researches on the contents of PCBs,DDE, and nonylphenols in the plastic resin pellets, we illustrate the survey conducted in Japanese waters by the University of Tokyo between 1997 and 1998.To standardize the concentration indexes of PCBs, DDE,and nonylphenols. We assume Kasai Sesside Park, KeihinCanal, Kugenuma Beach, Shioda Beach in the survey arethe first, second, third, fourth region; PCBs, DDE, andnonylphenols are the first, second, third indicators.Then to establish the standardized model:j j jij ij V V V V V min max min --= （1,2,3,4;1,2,3i j ==）wherej V max is the maximum of the measurement of j indicator in the four regions.j V min is the minimum of the measurement of j indicatorstandardized value of j indicator in i region.According to the literature [2], Japanese observationaldata is shown in Table 1.Table 1. PCBs, DDE, and, nonylphenols Contents in Marine PolypropyleneTable 1 Using the established standardized model to standardize, we have Table 2.In Table 2，the three indicators of Shioda Beach area are all 0, because the contents of PCBs, DDE, and nonylphenols in Polypropylene Plastic Resin Pellets in this area are the least, while 0 only relatively represents the smallest. Similarly, 1 indicates that in some area the value of a indicator is the largest.To determine the index weight of PCBs, DDE, and nonylphenolsWe use Analytic Hierarchy Process (AHP) to determine the weight of the three indicators in the general pollution indicator. AHP is an effective method which transforms semi-qualitative and semi-quantitative problems into quantitative calculation. It uses ideas of analysis and synthesis in decision-making, ideally suited for multi-index comprehensive evaluation.Hierarchy are shown in figure 1.Fig.1 Hierarchy of index factorsThen we determine the weight of each concentrationindicator in the generall pollution indicator, and the process are described as follows:To analyze the role of each concentration indicator, we haveestablished a matrix P to study the relative proportion.⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=111323123211312P P P P P P P Where mn P represents the relative importance of theconcentration indicators m B and n B . Usually we use 1，2，…，9 and their reciprocals to represent different importance. The greater the number is, the more important it is. Similarly, the relative importance of m B and n B is mn P /1（3,2,1,=n m ）.Suppose the maximum eigenvalue of P is m ax λ, then theconsistency index is1max --=n nCI λThe average consistency index is RI , then the consistencyratio isRICI CR = For the matrix P of 3≥n , if 1.0<CR the consistency isthougt to be better, of which eigenvector can be used as the weight vector.We get the comparison matrix accoding to the harmful levelsof PCBs, DDE, and nonylphenols and the requirments ofEPA on the maximum concentration of the three toxins inseawater as follows:⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=165416131431P We get the maximum eigenvalue of P by MATLAB calculation0012.3max =λand the corresponding eigenvector of it is()2393.02975.09243.0，，=W1.0042.012.1047.0<===RI CI CR Therefore,we determine the degree of inconsistency formatrix P within the permissible range. With the eigenvectors of p as weights vector, we get thefinal weight vector by normalization ()1638.02036.06326.0'，，=W . Defining the overall target of pollution for the No i oceanis i Q , among other things the standardized value of threeindicators for the No i ocean is ()321,,i i i i V V V V = and the weightvector is 'W ,Then we form the model for the overall target of marine pollution assessment, （3,2,1=i ）By the model above, we obtained the Value of the totalpollution index for four regions in Japanese ocean in Table 3T B W Q '=In Table3, the value of the total pollution index is the hightest that means the concentration of toxins in Polypropylene Plastic Resin Pellets is the hightest, whereas the value of the total pollution index in Shioda Beach is the lowest(we point up 0 is only a relative value that’s not in the name of free of plastics pollution)Getting through the assessment method above, we can monitor the concentration of PCBs, DDE and nonylphenols in the plastic debris for the sake of reflecting the influence to ocean ecosystem.The highter the the concentration of toxins,the bigger influence of the marine organism which lead to the inrichment of food chain is more and more dramatic.Above all, the variation of toxins’ concentration simultaneously reflects the distribution and time-varying of marine litter. We can predict the future development of marine litter by regularly monitoring the content of these substances, to provide data for the sea expedition of the detection of marine litter and reference for government departments to make the policies for ocean governance.Task 2:In the North Pacific, the clockwise flow formed a never-ending maelstrom which rotates the plastic garbage. Over the years, the subtropical eddy current in North Pacific gathered together the garbage from the coast or the fleet, entrapped them in the whirlpool, and brought them to the center under the action of the centripetal force, forming an area of 3.43 million square kilometers (more than one-third of Europe) .As time goes by, the garbage in the whirlpool has the trend of increasing year by year in terms of breadth, density, and distribution. In order to clearly describe the variability of the increases over time and space, according to “Count Densities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999—2008”, we analyze the data, exclude them with a great dispersion, and retain them with concentrated distribution, while the longitude values of the garbage locations in sampled regions of years serve as the x-coordinate value of a three-dimensional coordinates, latitude values as the y-coordinate value, the Plastic Count per cubic Meter of water of the position as the z-coordinate value. Further, we establish an irregular grid in the yx plane according to obtained data, and draw a grid line through all the data points. Using the inverse distance squared method with a factor, which can not only estimate the Plastic Count per cubic Meter of water of any position, but also calculate the trends of the Plastic Counts per cubic Meter of water between two original data points, we can obtain the unknown grid points approximately. When the data of all the irregular grid points are known (or approximately known, or obtained from the original data), we can draw the three-dimensional image with the Matlab software, which can fully reflect the variability of the increases in the garbage density over time and space.Preparations:First, to determine the coordinates of each year’s sampled garbage.The distribution range of garbage is about the East - West 120W-170W, South - North 18N-41N shown in the “Count Densities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”, we divide a square in the picture into 100 grids in Figure (1) as follows:According to the position of the grid where the measuring point’s center is, we can identify the latitude and longitude for each point, which respectively serve as the x- and y- coordinate value of the three-dimensional coordinates.To determine the Plastic Count per cubic Meter of water. As the “Plastic Count per cubic Meter of water” provided by “Count Densities of P lastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”are 5 density interval, to identify the exact values of the garbage density of one year’s different measuring points, we assume that the density is a random variable which obeys uniform distribution in each interval.Uniform distribution can be described as below:()⎪⎩⎪⎨⎧-=01a b x f ()others b a x ,∈We use the uniform function in Matlab to generatecontinuous uniformly distributed random numbers in each interval, which approximately serve as the exact values of the garbage density andz-coordinate values of the three-dimensional coordinates of the year’s measuring points.Assumptions(1)The data we get is accurate and reasonable.(2)Plastic Count per cubic Meter of waterIn the oceanarea isa continuous change.(3)Density of the plastic in the gyre is a variable by region.Density of the plastic in the gyre and its surrounding area is interdependent , However, this dependence decreases with increasing distance . For our discussion issue, Each data point influences the point of each unknown around and the point of each unknown around is influenced by a given data point. The nearer a given data point from the unknown point, the larger the role.Establishing the modelFor the method described by the previous,we serve the distributions of garbage density in the “Count Pensities of Plastic Debris from Ocean Surface Samples North Pacific Gyre 1999--2008”as coordinates ()z y,, As Table 1:x,Through analysis and comparison, We excluded a number of data which has very large dispersion and retained the data that is under the more concentrated the distribution which, can be seen on Table 2.In this way, this is conducive for us to get more accurate density distribution map.Then we have a segmentation that is according to the arrangement of the composition of X direction and Y direction from small to large by using x co-ordinate value and y co-ordinate value of known data points n, in order to form a non-equidistant Segmentation which has n nodes. For the Segmentation we get above,we only know the density of the plastic known n nodes, therefore, we must find other density of the plastic garbage of n nodes.We only do the sampling survey of garbage density of the north pacificvortex,so only understand logically each known data point has a certain extent effect on the unknown node and the close-known points of density of the plastic garbage has high-impact than distant known point.In this respect,we use the weighted average format, that means using the adverse which with distance squared to express more important effects in close known points. There're two known points Q1 and Q2 in a line ,that is to say we have already known the plastic litter density in Q1 and Q2, then speculate the plastic litter density's affects between Q1、Q2 and the point G which in the connection of Q1 and Q2. It can be shown by a weighted average algorithm22212221111121GQ GQ GQ Z GQ Z Z Q Q G +*+*=in this formula GQ expresses the distance between the pointG and Q.We know that only use a weighted average close to the unknown point can not reflect the trend of the known points, we assume that any two given point of plastic garbage between the changes in the density of plastic impact the plastic garbage density of the unknown point and reflecting the density of plastic garbage changes in linear trend. So in the weighted average formula what is in order to presume an unknown point of plastic garbage density, we introduce the trend items. And because the greater impact at close range point, and thus the density of plastic wastes trends close points stronger. For the one-dimensional case, the calculation formula G Z in the previous example modify in the following format:2212122212212122211111112121Q Q GQ GQ GQ Q Q GQ Z GQ Z GQ Z Z Q Q Q Q G ++++*+*+*=Among them, 21Q Q known as the separation distance of the known point, 21Q Q Z is the density of plastic garbage which is the plastic waste density of 1Q and 2Q for the linear trend of point G . For the two-dimensional area, point G is not on the line 21Q Q , so we make a vertical from the point G and cross the line connect the point 1Q and 2Q , and get point P , the impact of point P to 1Q and 2Q just like one-dimensional, and the one-dimensional closer of G to P , the distant of G to P become farther, the smaller of the impact, so the weighting factor should also reflect the GP in inversely proportional to a certain way, then we adopt following format:221212222122121222211111112121Q Q GQ GP GQ GQ Q Q GQ GP Z GQ Z GQ Z Z P Q Q Q Q G ++++++*+*+*=Taken together, we speculated following roles:(1) Each known point data are influence the density of plastic garbage of each unknown point in the inversely proportional to the square of the distance;(2) the change of density of plastic garbage between any two known points data, for each unknown point are affected, and the influence to each particular point of their plastic garbage diffuse the straight line along the two known particular point; (3) the change of the density of plastic garbage between any two known data points impact a specific unknown points of the density of plastic litter depends on the three distances: a. the vertical distance to a straight line which is a specific point link to a known point;b. the distance between the latest known point to a specific unknown point;c. the separation distance between two known data points.If we mark 1Q ，2Q ，…，N Q as the location of known data points,G as an unknown node, ijG P is the intersection of the connection of i Q ，j Q and the vertical line from G to i Q ，j Q()G Q Q Z j i ,,is the density trend of i Q ，j Q in the of plasticgarbage points and prescribe ()G Q Q Z j i ,,is the testing point i Q ’ s density of plastic garbage ,so there are calculation formula:()()∑∑∑∑==-==++++*=Ni N ij ji i ijGji i ijG N i Nj j i G Q Q GQ GPQ Q GQ GP G Q Q Z Z 11222222111,,Here we plug each year’s observational data in schedule 1 into our model, and draw the three-dimensional images of the spatial distribution of the marine garbage ’s density with Matlab in Figure (2) as follows:199920002002200520062007－2008(1)It’s observed and analyzed that, from 1999 to 2008, the density of plastic garbage is increasing year by year and significantly in the region of East – West 140W-150W, south - north 30N-40N. Therefore, we can make sure that this region is probably the center of the marine litter whirlpool. Gathering process should be such that the dispersed garbage floating in the ocean move with the ocean currents and gradually close to the whirlpool region. At the beginning, the area close to the vortex will have obviously increasable about plastic litter density, because of this centripetal they keeping move to the center of the vortex ,then with the time accumulates ,the garbage density in the center of the vortex become much bigger and bigger , at last it becomes the Pacific rubbish island we have seen today.It can be seen that through our algorithm, as long as the reference to be able to detect the density in an area which has a number of discrete measuring points,Through tracking these density changes ,we Will be able to value out all the waters of the density measurement through our models to determine,This will reduce the workload of the marine expedition team monitoring marine pollution significantly, and also saving costs .Task 3:The degradation mechanism of marine plasticsWe know that light, mechanical force, heat, oxygen, water, microbes, chemicals, etc. can result in the degradation of plastics . In mechanism ,Factors result in the degradation can be summarized as optical ，biological，and chemical。

2014年美国大学生数学建模MCM-B题O奖论文

For ofﬁce use only T1T2T3T4T eam Control Number24857Problem ChosenBFor ofﬁce use onlyF1F2F3F42014Mathematical Contest in Modeling(MCM)Summary Sheet (Attach a copy of this page to each copy of your solution paper.)AbstractThe evaluation and selection of‘best all time college coach’is the prob-lem to be addressed.We capture the essential of an evaluation system by reducing the dimensions of the attributes by factor analysis.And we divide our modeling process into three phases:data collection,attribute clariﬁca-tion,factor model evaluation and model generalization.Firstly,we collect the data from ofﬁcial database.Then,two bottom lines are determined respectively by the number of participating games and win-loss percentage,with these bottom lines we anchor a pool with30to40 candidates,which greatly reduced data volume.And reasonably theﬁnal top5coaches should generate from this pool.Attribution clariﬁcation will be abundant in the body of the model,note that we endeavor to design an attribute to effectively evaluate the improvement of a team before and after the coach came.In phase three,we analyse the problem by following traditional method of the factor model.With three common factors indicating coaches’guiding competency,strength of guided team,competition strength,we get aﬁnal integrated score to evaluate coaches.And we also take into account the time line horizon in two aspects.On the one hand,the numbers of participating games are adjusted on the basis of time.On the other hand,we put forward a potential sub-model in our‘further attempts’concerning overlapping pe-riod of the time of two different coaches.What’s more,a‘pseudo-rose dia-gram’method is tried to show coaches’performance in different areas.Model generalization is examined by three different sports types,Foot-ball,Basketball,and Softball.Besides,our model also can be applied in all possible ball games under the frame of NCAA,assigning slight modiﬁcation according to speciﬁc regulations.The stability of our model is also tested by sensitivity analysis.Who’s who in College Coaching Legends—–A generalized Factor Analysis approach2Contents1Introduction41.1Restatement of the problem (4)1.2NCAA Background and its coaches (4)1.3Previous models (4)2Assumptions5 3Analysis of the Problem5 4Theﬁrst round of sample selection6 5Attributes for evaluating coaches86Factor analysis model106.1A brief introduction to factor analysis (10)6.2Steps of Factor analysis by SPSS (12)6.3Result of the model (14)7Model generalization15 8Sensitivity analysis189Strength and Weaknesses199.1Strengths (19)9.2Weaknesses (19)10Further attempts20 Appendices22 Appendix A An article for Sports Illustrated221Introduction1.1Restatement of the problemThe‘best all time college coach’is to be selected by Sports Illustrated,a magazine for sports enthusiasts.This is an open-ended problem—-no limitation in method of performance appraisal,gender,or sports types.The following research points should be noted:•whether the time line horizon that we use in our analysis make a difference;•the metrics for assessment are to be articulated;•discuss how the model can be applied in general across both genders and all possible sports;•we need to present our model’s Top5coaches in each of3different sports.1.2NCAA Background and its coachesNational Collegiate Athletic Association(NCAA),an association of1281institution-s,conferences,organizations,and individuals that organizes the athletic programs of many colleges and universities in the United States and Canada.1In our model,only coaches in NCAA are considered and ranked.So,why evaluate the Coaching performance?As the identity of a college football program is shaped by its head coach.Given their impacts,it’s no wonder high proﬁle athletic departments are shelling out millions of dollars per season for the services of coaches.Nick Saban’s2013total pay was$5,395,852and in the same year Coach K earned$7,233,976in total23.Indeed,every athletic director wants to hire the next legendary coach.1.3Previous modelsTraditionally,evaluation in athletics has been based on the single criterion of wins and losses.Years later,in order to reasonably evaluate coaches,many reseachers have implemented the coaching evaluation model.Such as7criteria proposed by Adams:[1] (1)the coach in the profession,(2)knowledge of and practice of medical aspects of coaching,(3)the coach as a person,(4)the coach as an organizer and administrator,(5) knowledge of the sport,(6)public relations,and(7)application of kinesiological and physiological principles.1Wikipedia:/wiki/National_Collegiate_Athletic_ Association#NCAA_sponsored_sports2USAToday:/sports/college/salaries/ncaaf/coach/ 3USAToday:/sports/college/salaries/ncaab/coach/Such models relatively focused more on some subjective and difﬁcult-to-quantify attributes to evaluate coaches,which is quite hard for sports fans to judge coaches. Therefore,we established an objective and quantiﬁed model to make a list of‘best all time college coach’.2Assumptions•The sample for our model is restricted within the scale of NCAA sports.That is to say,the coaches we discuss refers to those service for NCAA alone;•We do not take into account the talent born varying from one player to another, in this case,we mean the teams’wins or losses purely associate with the coach;•The difference of games between different Divisions in NCAA is ignored;•Take no account of the errors/amendments of the NCAA game records.3Analysis of the ProblemOur main goal is to build and analyze a mathematical model to choose the‘best all time college coach’for the previous century,i.e.from1913to2013.Objectively,it requires numerous attributes to judge and specify whether a coach is‘the best’,while many of the indicators are deemed hard to quantify.However,to put it in theﬁrst place, we consider a‘best coach’is,and supposed to be in line with several basic condition-s,which are the prerequisites.Those prerequisites incorporate attributes such as the number of games the coach has participated ever and the win-loss percentage of the total.For instance,under the conditions that either the number of participating games is below100,or the win-loss percentage is less than0.5,we assume this coach cannot be credited as the‘best’,ignoring his/her other facets.Therefore,an attempt was made to screen out the coaches we want,thus to narrow the range in ourﬁrst stage.At the very beginning,we ignore those whose guiding ses-sions or win-loss percentage is less than a certain level,and then we determine a can-didate pool for‘the best coach’of30-40in scale,according to merely two indicators—-participating games and win-loss percentage.It should be reasonably reliable to draw the top5best coaches from this candidate pool,regardless of any other aspects.One point worth mentioning is that,we take time line horizon as one of the inputs because the number of participating games is changing all the time in the previous century.Hence,it would be unfair to treat this problem by using absolute values, especially for those coaches who lived in the earlier ages when sports were less popular and games were sparse comparatively.4Theﬁrst round of sample selectionCollege Football is theﬁrst item in our research.We obtain data concerning all possible coaches since it was initiated,of which the coaches’tenures,participating games and win-loss percentage etc.are included.As a result,we get a sample of2053in scale.Theﬁrst10candidates’respective information is as below:Table1:Theﬁrst10candidates’information,here Pct means win-loss percentageCoach From To Years Games Wins Losses Ties PctEli Abbott19021902184400.5Earl Abell19281930328141220.536Earl Able1923192421810620.611 George Adams1890189233634200.944Hobbs Adams1940194632742120.185Steve Addazio20112013337201700.541Alex Agase1964197613135508320.378Phil Ahwesh19491949193600.333Jim Aiken19461950550282200.56Fred Akers19751990161861087530.589 ...........................Firstly,we employ Excel to rule out those who begun their coaching career earlier than1913.Next,considering the impact of time line horizon mentioned in the problem statement,we import our raw data into MATLAB,with an attempt to calculate the coaches’average games every year versus time,as delineated in the Figure1below.Figure1:Diagram of the coaches’average sessions every year versus time It can be drawn from theﬁgure above,clearly,that the number of each coach’s average games is related with the participating time.With the passing of time and the increasing popularity of sports,coaches’participating games yearly ascends from8to 12or so,that is,the maximum exceed the minimum for50%around.To further reﬁnethe evaluation method,we make the following adjustment for coaches’participating games,and we deﬁne it as each coach’s adjusted participating games.Gi =max(G i)G mi×G iWhere•G i is each coach’s participating games;•G im is the average participating games yearly in his/her career;and•max(G i)is the max value in previous century as coaches’average participating games yearlySubsequently,we output the adjusted data,and return it to the Excel table.Obviously,directly using all this data would cause our research a mass,and also the economy of description is hard to achieved.Logically,we propose to employ the following method to narrow the sample range.In general,the most essential attributes to evaluate a coach are his/her guiding ex-perience(which can be shown by participating games)and guiding results(shown by win-loss percentage).Fortunately,these two factors are the ones that can be quantiﬁed thus provide feasibility for our modeling.Based on our common sense and select-ed information from sports magazines and associated programs,weﬁnd the winning coaches almost all bear the same characteristics—-at high level in both the partici-pating games and the win-loss percentage.Thus we may arbitrarily enact two bottom line for these two essential attributes,so as to nail down a pool of30to40candidates. Those who do not meet our prerequisites should not be credited as the best in any case.Logically,we expect the model to yield insight into how bottom lines are deter-mined.The matter is,sports types are varying thus the corresponding features are dif-ferent.However,it should be reasonably reliable to the sports fans and commentators’perceptual intuition.Take football as an example,win-loss percentage that exceeds0.75 should be viewed as rather high,and college football coaches of all time who meet this standard are speciﬁcally listed in Wikipedia.4Consequently,we are able toﬁx upon a rational pool of candidate according to those enacted bottom lines and meanwhile, may tender the conditions according to the total scale of the coaches.Still we use Football to further articulate,to determine a pool of candidates for the best coaches,weﬁrst plot theﬁgure below to present the distributions of all the coaches.From theﬁgure2,weﬁnd that once the games number exceeds200or win-loss percentage exceeds0.7,the distribution of the coaches drops signiﬁcantly.We can thus view this group of coaches as outstanding comparatively,meeting the prerequisites to be the best coaches.4Wikipedia:/wiki/List_of_college_football_coaches_ with_a_.750_winning_percentageFigure2:Hist of the football coaches’number of games versus and average games every year versus games and win-loss percentageHence,we nail down the bottom lines for both the games number and the win-loss percentage,that is,0.7for the former and200for the latter.And these two bottom lines are used as the measure for ourﬁrst round selection.After round one,merely35 coaches are qualiﬁed to remain in the pool of candidates.Since it’s theﬁrst round sifting,rather than direct and ultimate determination,we hence believe the subjectivity to some extent in the opt of bottom lines will not cloud theﬁnal results of the best coaches.5Attributes for evaluating coachesThen anchored upon the35candidate selected,we will elaborate our coach evaluation system based on8attributes.In the indicator-select process,we endeavor to examine tradeoffs among the availability for data and difﬁculty for data quantiﬁcation.Coaches’pay,for example,though serves as the measure for coaching evaluation,the corre-sponding data are limited.Situations are similar for attributes such as the number of sportsmen the coach ever cultivated for the higher-level tournaments.Ultimately,we determine the8attributes shown in the table below:Further explanation:•Yrs:guiding years of a coach in his/her whole career•G’:Gi =max(G i)G mi×G i see it at last section•Pct:pct=wins+ties/2wins+losses+ties•SRS:a rating that takes into account average point differential and strength of schedule.The rating is denominated in points above/below average,where zeroTable2:symbols and attributessymbol attributeYrs yearsG’adjusted overall gamesPct win-lose percentageP’Adjusted percentage ratioSRS Simple Rating SystemSOS Strength of ScheduleBlp’adjusted Bowls participatedBlw’adjusted Bowls wonis the average.Note that,the bigger for this value,the stronger for the team performance.•SOS:a rating of strength of schedule.The rating is denominated in points above/below average,where zero is the average.Noted that the bigger for this value,the more powerful for the team’s rival,namely the competition is more ﬁerce.Sports-reference provides ofﬁcial statistics for SRS and SOS.5•P’is a new attribute designed in our model.It is the result of Win-loss in the coach’s whole career divided by the average of win-loss percentage(weighted by the number of games in different colleges the coach ever in).We bear in mind that the function of a great coach is not merely manifested in the pure win-loss percentage of the team,it is even more crucial to consider the improvement of the team’s win-loss record with the coach’s participation,or say,the gap between‘af-ter’and‘before’period of this team.(between‘after’and‘before’the dividing line is the day the coach take ofﬁce)It is because a coach who build a comparative-ly weak team into a much more competitive team would deﬁnitely receive more respect and honor from sports fans.To measure and specify this attribute,we col-lect the key ofﬁcial data from sports-reference,which included the independent win-loss percentage for each candidate and each college time when he/she was in the team and,the weighted average of all time win-loss percentage of all the college teams the coach ever in—-regardless of whether the coach is in the team or not.To articulate this attribute,here goes a simple physical example.Ike Armstrong (placedﬁrst when sorted by alphabetical order),of which the data can be ob-tained from website of sports-reference6.We can easily get the records we need, namely141wins,55losses,15ties,and0.704for win-losses percentage.Fur-ther,speciﬁc wins,losses,ties for the team he ever in(Utab college)can also be gained,respectively they are602,419,30,0.587.Consequently,the P’value of Ike Armstrong should be0.704/0.587=1.199,according to our deﬁnition.•Bowl games is a special event in theﬁeld of Football games.In North America,a bowl game is one of a number of post-season college football games that are5sports-reference:/cfb/coaches/6sports-reference:/cfb/coaches/ike-armstrong-1.htmlprimarily played by teams from the Division I Football Bowl Subdivision.The times for one coach to eparticipate Bowl games are important indicators to eval-uate a coach.However,noted that the total number of Bowl games held each year is changing from year to year,which should be taken into consideration in the model.Other sports events such as NCAA basketball tournament is also ex-panding.For this reason,it is irrational to use the absolute value of the times for entering the Bowl games (or NCAA basketball tournament etc.)and the times for winning as the evaluation measurement.Whereas the development history and regulations for different sports items vary from one to another (actually the differentiation can be fairly large),we here are incapable to ﬁnd a generalized method to eliminate this discrepancy ,instead,in-dependent method for each item provide a way out.Due to the time limitation for our research and the need of model generalization,we here only do root extract of blp and blw to debilitate the differentiation,i.e.Blp =√Blp Blw =√Blw For different sports items,we use the same attributes,except Blp’and Blw’,we may change it according to speciﬁc sports.For instance,we can use CREG (Number of regular season conference championship won)and FF (Number of NCAA Final Four appearance)to replace Blp and Blw in basketball games.With all the attributes determined,we organized data and show them in the table 3:In addition,before forward analysis there is a need to preprocess the data,owing to the diverse dimensions between these indicators.Methods for data preprocessing are a lot,here we adopt standard score (Z score)method.In statistics,the standard score is the (signed)number of standard deviations an observation or datum is above the mean.Thus,a positive standard score represents a datum above the mean,while a negative standard score represents a datum below the mean.It is a dimensionless quantity obtained by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation.7The standard score of a raw score x is:z =x −µσIt is easy to complete this process by statistical software SPSS.6Factor analysis model 6.1A brief introduction to factor analysisFactor analysis is a statistical method used to describe variability among observed,correlated variables in terms of a potentially lower number of unobserved variables called factors.For example,it is possible that variations in four observed variables mainly reﬂect the variations in two unobserved variables.Factor analysis searches for 7Wikipedia:/wiki/Standard_scoreTable3:summarized data for best college football coaches’candidatesCoach From To Yrs G’Pct Blp’Blw’P’SRS SOS Ike Armstrong19251949252810.70411 1.199 4.15-4.18 Dana Bible19151946313860.7152 1.73 1.0789.88 1.48 Bernie Bierman19251950242780.71110 1.29514.36 6.29 Red Blaik19341958252940.75900 1.28213.57 2.34 Bobby Bowden19702009405230.74 5.74 4.69 1.10314.25 4.62 Frank Broyles19571976202570.7 3.162 1.18813.29 5.59 Bear Bryant19451982385080.78 5.39 3.87 1.1816.77 6.12 Fritz Crisler19301947182080.76811 1.08317.15 6.67 Bob Devaney19571972162080.806 3.16 2.65 1.25513.13 2.28 Dan Devine19551980222800.742 3.16 2.65 1.22613.61 4.69 Gilmour Dobie19161938222370.70900 1.27.66-2.09 Bobby Dodd19451966222960.713 3.613 1.18414.25 6.6 Vince Dooley19641988253250.715 4.47 2.83 1.09714.537.12 Gus Dorais19221942192320.71910 1.2296-3.21 Pat Dye19741992192400.707 3.16 2.65 1.1929.68 1.51 LaVell Edwards19722000293920.716 4.69 2.65 1.2437.66-0.66 Phillip Fulmer19922008172150.743 3.87 2.83 1.08313.42 4.95 Woody Hayes19511978283290.761 3.32 2.24 1.03117.418.09 Frank Kush19581979222710.764 2.65 2.45 1.238.21-2.07 John McKay19601975162070.7493 2.45 1.05817.298.59 Bob Neyland19261952212860.829 2.65 1.41 1.20815.53 3.17 Tom Osborne19731997253340.8365 3.46 1.18119.7 5.49 Ara Parseghian19561974192250.71 2.24 1.73 1.15317.228.86 Joe Paterno19662011465950.749 6.08 4.9 1.08914.01 5.01 Darrell Royal19541976232970.7494 2.83 1.08916.457.09 Nick Saban19902013182390.748 3.74 2.83 1.12313.41 3.86 Bo Schembechler19631989273460.775 4.12 2.24 1.10414.86 3.37 Francis Schmidt19221942212670.70800 1.1928.490.16 Steve Spurrier19872013243160.733 4.363 1.29313.53 4.64 Bob Stoops19992013152070.804 3.74 2.65 1.11716.66 4.74 Jock Sutherland19191938202550.81221 1.37613.88 1.68 Barry Switzer19731988162090.837 3.61 2.83 1.16320.08 6.63 John Vaught19471973253210.745 4.24 3.16 1.33814.7 5.26 Wallace Wade19231950243070.765 2.24 1.41 1.34913.53 3.15 Bud Wilkinson19471963172220.826 2.83 2.45 1.14717.54 4.94 such joint variations in response to unobserved latent variables.The observed vari-ables are modelled as linear combinations of the potential factors,plus‘error’terms. The information gained about the interdependencies between observed variables can be used later to reduce the set of variables in a putationally this technique is equivalent to low rank approximation of the matrix of observed variables.8 Why carry out factor analyses?If we can summarise a multitude of measure-8Wikipedia:/wiki/Factor_analysisments with a smaller number of factors without losing too much information,we have achieved some economy of description,which is one of the goals of scientiﬁc investi-gation.It is also possible that factor analysis will allow us to test theories involving variables which are hard to measure directly.Finally,at a more prosaic level,factor analysis can help us establish that sets of questionnaire items(observed variables)are in fact all measuring the same underlying factor(perhaps with varying reliability)and so can be combined to form a more reliable measure of that factor.6.2Steps of Factor analysis by SPSSFirst we import the decided datasets of8attributes into SPSS,and the results can be obtained below after the software processing.[2-3]Figure3:Table of total variance explainedFigure4:Scree PlotTheﬁrst table and scree plot shows the eigenvalues and the amount of variance explained by each successive factor.The remaining5factors have small eigenvalues value.Once the top3factors are extracted,it adds up to84.3%,meaning a great as the explanatory ability for the original information.To reﬂect the quantitative analysis of the model,we obtain the following factor loading matrix,actually the loadings are in corresponding to the weight(α1,α2 (i)the set ofx i=αi1f1+αi2f2+...+αim f j+εiAnd the relative strength of the common factors and the original attribute can also be manifested.Figure5:Rotated Component MatrixThen,with Rotated Component Matrix above,weﬁnd the common factor F1main-ly expresses four attributes they are:G,Yrs,P,SRS,and logically,we deﬁne the com-mon factor generated from those four attributes as the guiding competency of the coach;similarly,the common factor F2mainly expresses two attributes,and they are: Pct and Blp,which can be de deﬁned as the integrated strength of the guided team; while the common factor F3,mainly expresses two attributes:SOS and Blw,which can be summarized into a‘latent attribute’named competition strength.In order to obtain the quantitative relation,we get the following Component Score Coefﬁcient Matrix processed by SPSS.Further,the function of common factors and the original attributes is listed as bel-low:F1=0.300x1+0.312x2+0.023x3+0.256x4+0.251x5+0.060x6−0.035x7−0.053x8F2=−0.107x1−0,054x2+0.572x3+0.103x4+0.081x5+0.280x6+0.372x7+0.142x8 F3=−0.076x1−0,098x2−0.349x3+0.004x4+0.027x5−0.656x6+0.160x7+0.400x8 Finally we calculate out the integrated factor scores,which should be the average score weighted by the corresponding proportion of variance contribution of each com-mon factor in the total variance contribution.And the function set should be:F=0.477F1+0.284F2+0.239F3Figure6:Component Score Coefﬁcient Matrix6.3Result of the modelwe rank all the coaches in the candidate pool by integrated score represented by F.Seetable4:Table4:Integrated scores for best college football coach(show15data due to the limi-tation of space)Rank coaches F1F2F3Integrated factor1Joe Paterno 3.178-0.3150.421 1.3622Bobby Bowden 2.51-0.2810.502 1.1113Bear Bryant 2.1420.718-0.142 1.0994Tom Osborne0.623 1.969-0.2390.8205Woody Hayes0.140.009 1.6130.4846Barry Switzer-0.705 2.0360.2470.4037Darrell Royal0.0460.161 1.2680.4018Vince Dooley0.361-0.442 1.3730.3749Bo Schembechler0.4810.1430.3040.32910John Vaught0.6060.748-0.870.26511Steve Spurrier0.5180.326-0.5380.18212Bob Stoops-0.718 1.0850.5230.17113Bud Wilkinson-0.718 1.4130.1050.16514Bobby Dodd0.08-0.2080.7390.16215John McKay-0.9620.228 1.870.151Based on this model,we can make a scientiﬁc rank list for US college football coach-es,the Top5coaches of our model is Joe Paterno,Bobby Bowden,Bear Bryant,TomOsborne,Woody Hayes.In order to conﬁrm our result,we get a ofﬁcial list of bestcollege football coaches from Bleacherreport99Bleacherreport:/articles/890705-college-football-the-top-50-coTable5:The result of our model in football,the last column is ofﬁcial college basketball ranking from bleacherreportRank Our model Integrated scores bleacherreport1Joe Paterno 1.362Bear Bryant2Bobby Bowden 1.111Knute Rockne3Bear Bryant 1.099Tom Osborne4Tom Osborne0.820Joe Paterno5Woody Hayes0.484Bobby Bowden By comparing thoes two ranking list,weﬁnd that four of our Top5coaches ap-peared in the ofﬁcal Top5list,which shows that our model is reasonable and effective.7Model generalizationOur coach evaluation system model,of which the feasibility of generalization is sat-isfying,can be accommodated to any possible NCAA sports concourses by assigning slight modiﬁcation concerning speciﬁc regulations.Besides,this method has nothing to do with the coach’s gender,or say,both male and female coaches can be rationally evaluated by this system.And therefore we would like to generalize this model into softball.Further,we take into account the time line horizon,making corresponding adjust-ment for the indicator of number of participating games so as to stipulate that the evaluation measure for1913and2013would be the same.To further generalize the model,ﬁrst let’s have a test in basketball,of which the data available is adequate enough as football.And the speciﬁc steps are as following:1.Obtain data from sports-reference10and rule out the coaches who begun theircoaching career earlier than1913.2.Calculate each coach’s adjusted number of participating games,and adjust theattribute—-FF(Number of NCAA Final Four appearance).3.Determine the bottom lines for theﬁrst round selection to get a pool of candidatesaccording to the coaches’participating games and win-loss percentage,and the ideal volumn of the pool should be from30to40.Hist diagrams are as below: We determine800as the bottom line for the adjusted participating games and0.7 for the win-loss percentage.Coincidently,we get a candidate pool of35in scale.4.Next,we collect the corresponding data of candidate coaches(P’,SRS,SOS etc.),as presented in the table6:5.Processed by z score method and factor analysis based on the8attributes anddata above,we get three common factors andﬁnal integrated scores.And among 10sports-reference:/cbb/coaches/Figure7:Hist of the basketball coaches’number of games versus and average gamesevery year versus games and win-loss percentagethe top5candidates,Mike Krzyzewski,Adolph Rupp,Dean SmithˇcˇnBob Knightare the same with the ofﬁcial statistics from bleacherreport.11We can say theeffectiveness of the model is pretty good.See table5.We also apply similar approach into college softball.Maybe it is because the popularity of the softball is not that high,the data avail-able is not adequate to employ ourﬁrst model.How can our model function in suchsituation?First and foremost,specialized magazines like Sports Illustrated,its com-mentators there would have more internal and conﬁdential databases,which are notexposed publicly.On the one hand,as long as the data is adequate enough,we can saythe original model is completely feasible.While under the situation that there is datadeﬁcit,we can reasonably simplify the model.The derivation of the softball data is NCAA’s ofﬁcial websites,here we only extractdata from All-Division part.12Softball is a comparatively young sports,hence we may arbitrarily neglect the re-stricted condition of‘100years’.Subsequently,because of the data deﬁcit it is hard toadjust the number of participating games.We may as well determine10as the bottomline for participating games and0.74for win-loss percentage,producing a candidatepool of33in scaleAttributed to the inadequacy of the data for attributes,it is not convenient to furtheruse the factor analysis similarly as the assessment model.Therefore,here we employsolely two of the most important attributes to evaluate a coach and they are:partic-ipating games and win-loss percentage in the coach’s whole career.Speciﬁcally,weﬁrst adopt z score to normalize all the data because of the differentiation of various dimensions,and then the integrated score of the coach can be reached by the weighted11bleacherreport:/articles/1341064-10-greatest-coaches-in-ncaa-b 12NCAA softball Coaching Record:/Docs/stats/SB_Records/2012/coaches.pdf。

2016年美国大学生数学建模E题英文版论文正稿

In this paper, a model is established to provide a measure of the ability of a region to provide clean water to meet the needs of its population, and find out the reason for the lack of water resources. Specific tasks are as follows:For Task 1: We establish a model. In the model, we think the supply of clean water depends on the amount of surface water, underground water and sewage purification. The water requirements are decided by the amount of life water, agricultural water and industrial water in the region. In water supply, surface water is affected by the annual average temperature, annual average precipitation and forest coverage rate. The groundwater is impacted by the annual average temperature, annual average precipitation. The agricultural water is affected by the population of the region and annual average precipitation. The GDP of the region influences the industrial water consumption. We use the principle of multivariate nonlinear regression to find out the regression coefficient. And then make sure its function. The ratio of water supply and water requirements is used as a measure of the ability of a region to provide clean water. We find that the ability of a region to provide clean water is good or not by comparing the ratio with 1.For Task 2: The model selects the Shandong Province of China as the testing region. We analyse the data of China's Shandong area between 2005 and 2014, and then crystallize the model through the thought of the function fitting and multivariate nonlinear regression. By the model, we think Shandong province's ability to provide clean water is weak. And then from two aspects which physical shortage and shortage of economic, this paper analyses the causes of water shortage in Shandong Province, and thus test the applicability of the model.For Task 3: We select several factors affecting water supply and water demand badly, which is annual precipitation, annual average temperature, the forest coverage rate and population forecast. We analyse the data of China's Shandong area between 2005 and 2014, according which to predict the changes of those factors in 15 years. After that this paper uses the model to analyse the situation of Shandong’s water in 15 years.For Task 4: According to the model in Task 1 and the analysis of the Task 2. We find the main factors influencing the ability to provide clean water in China's Shandong province. By these factors we make the intervention program. In view of the low average annual rainfall, increase the average annual rainfall by artificial rainfall. In view of the forest coverage rate, forest plantation and protect vegetation is came up with. For sewage purification capacity, putting forward to improve sewage treatment technology and improve the sewage conversion rate and increases daily sewage quantity. In view of the total population, we put forward the policy of family planning for water consumption per capita, putting forward to set the daily water consumption per person. And putting forward the industrial wastewater must reach the indexes of the rules, developing seawater desalination technology to increase the supply of clean water.Water has always been the hot spot in the world.The future is also not exceptional. Only finding out the problem, we can suit the remedy to the case.The model measure the ability of a region to provide clean water by analysing the cases which influence the supply and remand of water. Based on this, make a good intervention program. Offering helps to solve global water issues.1 Introduction (4)1.1 Problem Statement (4)1.2Problem Analysis (4)1.2.1Task 1 Analysis (4)1.2.2Task 2 Analysis (4)1.2.3Task 3 Analysis (5)1.2.4Task 4 Analysis (5)1.2.5Task 4 and 5 Analysis (5)2 Assumptions and Notations (6)2.1 Assumptions (6)2.2 Notations (6)3 Model Establishment and Solution (7)3.1 The effect of single factor on the water supply in a certain area (7)3.1.1Effects of annual average temperature, annual average precipitation andforest coverage on surface water resources in a certain area (7)3.1.2 Effects of annual average temperature and annual precipitation ongroundwater resources in a certain area (8)3.1.3 Influence of total population and per capita water consumption on dailywater consumption in a certain area (8)3.1.4 The influence of average annual rainfall and total population onagricultural water consumption in a certain area (9)3.1.5 Effect of average annual rainfall and population in an area of agriculturalwater use (9)3.2 Function Arrangement (9)3.2.1 Water supply function (9)3.2.2 Water demand function (10)3.2.3 The ability of a region to provide clean water (10)3.3 In order to test the accuracy and usability of the model, this model is selectedas a test area in Shandong Province, China. (10)3.3.1 Total surface water resources (11)3.3.2 Total groundwater resources (14)3.3.3 Total industrial water consumption function (16)3.3.4 Total agricultural water consumption function (17)3.3.5 Assessment of water supply capacity (18)3.4.2 Remediation Measures (19)3.5 Forecast for the next 15 years (20)3.5.1 Forecast of average annual rainfall (20)3.5.2 Prediction of annual temperature (21)3.5.3 Prediction of forest cover (22)3.5.4 Prediction of population (23)3.6 Intervention Program (24)3.6.1 Present ofthe Intervention Program (24)3.6.2 Implement ofthe Intervention Program (25)4 Advantages and Shortcoming of the model (26)4.1Advantages： (26)4.2 Shortcoming (26)5 Improvement of model (26)6 Reference (27)7 Appendices (28)7.1 Data used in task 2 (28)7.2 Matlab Source Code (30)1 Introduction1.1 Problem StatementOn the earth, the water that human beings can use the water directly or indirectly, is an important part of natural resources. At present, the total amount of the earth's water is about billion cubic meters, of which the ocean water is billion cubic meters, accounting for about 96.5% of the total global water. In the remaining water, the surface water accounts for 1.78%, 1.69% of the groundwater. The fresh water that human mainly use of is about billion cubic meters, accounting for only 2.53% in the global total water storage. Few of them is distributed in lakes, rivers, soil and underground water, and most of them are stored in the form of glaciers, permafrost and permafrost, The glacier water storage of about billion cubic meters, accounting for 69% of the world's total water, mostly stored in the Antarctic and Greenland, the available clean water in the dwindlingwith time going by.In order to assess the ability to provide clean water of an area, we set up an assessment model.1.2Problem Analysis1.2.1Task 1 AnalysisTask 1 requires establishing a model to measure the ability of a region to provide clean water. At the same time,we also need to provide a measure standard.This paper make the ratio of water supply and water requirements of a region as the measure standard, by which to measure the ability of a region to provide clean water.A region's main source of water is groundwater, surface water and sewagepurification.The model assumes the volume of groundwater in a region is mainly affected by average annual temperature, annual precipitation;Thevolume of surface water is mainly affected by the average annual temperature, annual precipitation, the forest coverage rate.These factors decide water supply of an area.The waterdemand of an area mainly includes living water, agriculture water and industrial water. We assume living water is affected by the population and per capita consumption decision;Agricultural waterdepends an annual precipitation and population decision;Industrial water is mainly decided by a gross regional product.The above factors decide the water demand in a certain area.1.2.2Task 2 AnalysisAccording to the information provided in the map, in Asia, China's Shandong Province is the region meeting the requirements.Through the data collection of Shandong Province, we can find the annual temperature, annual precipitation, the forest coverage rate, groundwater, surface water, sewage treatment capacity, water, agricultural water, industrial water, population,per capita consumption and GDP data.And then according to the model of Task 1,we analyze the ratio by using multivariate nonlinear regression to make sure that Shandong province is a water-deficient area.After proving that Shandon province is short of water through the model, we analyze the reasons for lack of water from two aspects: physical shortage and economic shortage.1.2.3Task 3 AnalysisBecause we already have the relevant data, we can function to fit the relationship between the variables and the year.Thus it is possible to predict in Shandong Provinc e’s data in the 15 years, then input the data into the model to achieve the purpose of prediction.In addition, it can be combined with the actual situation and the selected areas of the corresponding policy to analysis which factors will have a great change in the15 years.We still can analyze from two aspects of society and the environment.Socialaspects includes the promotion of water conservation, population growth;Environmental aspects includes policy changes to the environment, sewage purification capacity enhancement and so on.1.2.4Task 4 AnalysisFormulating plans for intervention mainly start from the perspective of the main model. According to the content of the model, we can still divide all of factors into two types: the social and environmental factors. The intervention programs can be developed based on two types of factors that affect the supply of water, reducing as much as possible the negative impact of the factors that control factors and intensifying the development of a positive impact. In addition, because Shandong Province is beside the sea, desalination and other measures can be developed to increase clean water supply sources.1.2.5Task 4 and 5 AnalysisTask 4 intervention programs indirectly impact the water supply and demand water through a direct impact on GDP model of forest cover, annual precipitation, annual temperature, water emissions, sewage treatment capacity, population growth and the region.2 Assumptions and Notations2.1 Assumptions●The water resources in a region are derived from the purification of surfacewater, groundwater and sewage, and the demand of water resources comes from domestic water, industrial water and agricultural water.●The surface water supply in a certain area is affected only by the average annualtemperature, annual precipitation and forest coverage. The groundwater supply is affected by the annual average temperature and annual precipitation.●The region's water consumption of a certain region depends on the populationand per capita water consumption; Agricultural water consumption is affected by the average annual precipitation and the numberof people. Industrial water is mainly determined by a regional GDP.● A certain region will not suddenly increase or decrease the population largely.●There will not be a serious natural disasters in a region in the next periodof time.2.2 Notations3 Model Establishment and SolutionThe model established here is a use of a region's water supply and water demand ratio to determine whether the water shortage in the region, the main variables involved.3.1 The effect of single factor on the water supply in a certain area3.1.1Effects of annual average temperature, annual average precipitation and forest coverage on surface water resources in a certain areaDue to the average annual temperature, annual precipitation, the forest coverage rate and surface water of linear or nonlinear relationship exists, so first in order to determine the average annual temperature, annual precipitation, the forest coverage rate and surface water, and then the nonlinear multiple regression analysis method to determine the functional relationship between the three factors and surface water.The surface water content is 1y , Average annual precipitation, annual average temperature and forest coverage rate are 1x ,2x ,3x , Using nonlinear regression statistical methods, the use of MATLAB fitting toolbox were identified 1x ,2x ,3x of the highest regression power(MATLAB fitting toolbox of the highest fitting function is the 9 power, greater than the 9 power function is too complex, not much research value), According to the decision coefficient R 2of the regression equation, the corresponding probability value of the statistic P, the regression coefficients β,0β,1n β,2n β, get the regression equation:35612412312345699999910123123111 (1)n n n n n n n n n n n n n n n y x x x x x x βββββ====++++∑∑∑∑∑∑3.1.2 Effects of annual average temperature and annual precipitation on groundwater resources in a certain areaThere is a linear or nonlinear relationship between the average annual temperature, average annual rainfall and the supply of groundwater, according to the idea of 5.1.1, the relationship between the average annual amount of groundwater supply, the average annual precipitation and the supply of groundwater is calculated. And the regression coefficient is determined, and the function relationship between the average annual temperature, average annual precipitation and the supply of groundwater is based on the regression coefficient:Design of underground water for 2y , with an average annual precipitation, annual average air temperature respectively 1x ,2x , using nonlinear regression statistical methods, according to the regression equation with coefficient of determination R2, F statistic corresponds to the probability value p, to determine the regression coefficients β,0β,1n β,2n β, got the regression equation:312412123499992012121111n n n n n n n n n n y x x x x ββββ=====+++∑∑∑∑ （2）3.1.3 Influence of total population and per capita water consumption on daily water consumption in a certain areaThe total population of a region and the amount of water consumption per capita and the daily use of the product of the relationship between the amount of water = total amount *Water usage per person consumption.Set daily water consumption is 5y , the total population, per capita water consumption were 5x , Q , 5y ,Q ,5x , the function of the relationship between the:55y Qx =（3）3.1.4 The influence of average annual rainfall and total population on agricultural water consumption in a certain areaDue to the annual precipitation and the total population and the area of agriculture of area of a water there is a linear or nonlinear relationship, according to the thought of multivariate nonlinear regression can be calculated average annual precipitation and the total population and the area of agriculture with the function relationship between water and to determine the regression coefficient and regression coefficient write GDP and industrial functional relationship between Gross domestic product GDP and industrial water consumption.Let industrial water consumption of 3y , gross production set 4x , using statistical nonlinear regression, regression equation based on the coefficient of determination 2R , F statistical probability value p corresponding to the amount determined regression coefficients 0β,1n β, the regression equation:11193011n n n y x ββ==+∑（4）3.1.5 Effect of average annual rainfall and population in an area of agricultural water useAgricultural water consumption of 4y , design with an average annual rainfall of 1x , 5x , using statistical nonlinear regression, regression equation based on the coefficient of determination 2R , F statistical probability value p corresponding to the amount determined regression coefficients 0β,1n β, the regression equation:3124121234999940151511n n n n n n n n n n y x x x x ββββ===+++∑∑∑∑（5） 3.2 Function Arrangement3.2.1 Water supply functionThe model takes into account a region's water supply from three aspects: surfacewater resources, groundwater resources and the amount of sewage treatment. The function relation between surface water resources, groundwater resources, sewage treatment and water supply is the function, that is, the amount of water supply = surface water resources + groundwater resources.The amount of water supply isX , and the sewage treatment capacity is *Q , by(2)(1): *12X y y Q =++（6）3.2.2 Water demand functionThe model takes into account the need for a region from three aspects: daily water consumption, industrial water consumption and agricultural water consumption. Daily water consumption, industrial water and agricultural water consumption and water demand is a function of the relationship between the function and the function, that is: water demand = daily water consumption + industrial water + agricultural water consumption.A demand for Y , by (3) (4) (5) to:345Y y y y =++（7）3.2.3 The ability of a region to provide clean waterA region to provide clean water and the area of water supply and water demand about, if water supply is greater than demand, the region provide clean water ability strong; on the contrary, the region provide clean water ability is weak. This model provides that a region to provide clean water capacity by the area of water supply and water demand ratio λ determined by (6) (7) available:(1) 1λ>: the region's ability to provide clean water;(2) 1λ=: the area provides a warning of the ability to provide clean water;(3) 1λ<: the ability of the region to provide clean water is weak;3.3 In order to test the accuracy and usability of the model, this model is selected as a test area in Shandong Province, China.Provide the capacity of water resources in China's Shandong Province, we collected in 2005 to 2014 this decade, Shandong Province, the total water supply, surface water resources amount, quantity of groundwater resources, sewage treatment capacity, agricultural water consumption, industrial water, living water, sewage emissions, forest coverage, total population, per capita water use, annual precipitation, GDP (see Appendix for the specific data).3.3.1 Total surface water resourcesSurface water resources amount 1y , groundwater resources quantity is 2y , industrial water use 3y , agricultural water use for 4y , with an average annual precipitation 1x , with an average annual temperature 2x , the forest coverage rate 3x , GDP for 4x , with a total population of 5x .The factors 1y that is affected by 1x , 2x , 3x , in order to determine the relationship between the 1y , and 1x , 2x , 3x ,first use the data in the appendix table to make 1y with 1x , 2x , 3x scatter plots, such as the figure:Figure 1 Surface water resources and average annual rainfallFigure 2 Surface water resources and annual temperatureFigure 3 Surface water resources and forest coverFigure 1 is obtained by MATLAB fitting curve, the fitting found that 1x and 1y is the 6 power function model (εfor random error).（9）Figure 2 is obtained by MATLAB fitting curve, the fitting found that 2x and 1y isthe 61011n n n y x ββε==++∑8 power function model（10）Figure 3 is obtained by MATLAB fitting curve, the fitting found that 3x and 1y is the 8 power function model（11）Combined with the above analysis, the model (9) (10) (11) established the following regression model（12）Directly using the MATLAB statistics toolbox in the command regress solution, the format is[b,bint,r,rint,stats]=regress(x,y,alpha)Output value of b for the estimation of the regression coefficient β, bint is the confidence intervals for b and r is the residual vector, rint is the confidence interval of r , stats is regression model test statistics, in the first number is a regression equation with coefficient of determination 2R ;The regression coefficients of the model (12) are estimated and their confidence intervals (confidence level α=0.05), test statistic 2R , F , ρ, and the results are shown in table.Table 1 Surface water regression coefficientCan get the regression coefficient from the figure, the estimated value of the regression coefficient into the model (12) forecast equation81021n n n y x ββε==++∑31031n n n y x ββε==++∑33121212312456683999103331231131111n n n n n n n n n n n n n n n y x x x x x x βββββε=======+++++∑∑∑∑∑∑（13）3.3.2 Total groundwater resourcesFactors that affect the 2y include 1x ,2x , in order to determine the relationship between 2y and 1x ,2x , first use the data in the appendix table to make the A3 and A4 and A5 of the scatter diagram, as shown in figure:Figure 4 the amount of groundwater resources Figure 5 the amount of groundwater resources and annual average temperatureand the average annual rainfallFigure 4 is obtained by MATLAB fitting curve, the fitting found that 1x and 2y is the 6 power function model (ε for random error),（14）Figure 5 is obtained by MATLAB fitting curve, the fitting found that 2x and 2y is the 8 power function model.（15）Combined with the above analysis, the model (9) (10) (11) established the following regression model.（16） Directly using the MATLAB statistics toolbox in the command regress solution, the format ^9665444111138273322223724 1.7610 1.0610 1.8910 1.18103.8410 2.610 3.2910 1.0110y x x x x x x x x x --------=+⨯-⨯+⨯+⨯-⨯+⨯-⨯+⨯62011n n n y x ββε==++∑82021n n n y x ββε==++∑121212123468992013121111n n n n n n n n n n y x x x x ββββε=====++++∑∑∑∑is[b,bint,r,rint,stats]=regress(x,y,alpha)Output value of b for the estimation of the regression coefficient β, bint is the confidence intervals for b ,and 2R is the residual vector, rint is the confidence interval of r , stats is regression model test statistics, in the first number is a regression equation with coefficient of determination 2R ;The regression coefficients of the model (12) are estimated and their confidence intervals (confidence level α=0.05), test statistic 2R , F , ρ, and the results are shown in table.Table 2 Regression coefficients of groundwater resourcesCan get the regression coefficient from the figure, the estimated value of the regression coefficient into the model (16) forecast equation（17）Its image is shown in Figure 6^9665445111123821000 1.8210 1.3410 3.37108.49101.3610y x x x x x -----=+⨯-⨯+⨯+⨯-⨯Figure 6 groundwater resources3.3.3 Total industrial water consumption functionFactors that affect 3y is 4x , in order to determine the relationship between 3y and 4x , the first use of the data in the appendix table to make the X and the scatter diagram, as shown in figure:Figure 7 industrial water consumption and GDPFigure 8 industrial water useFigure 7 is obtained by MATLAB fitting curve, the fitting found that 4x and 3y is a function model (εfor random errors),（18）The regression coefficient can be got from the following chartTable 3 Regression coefficient of industrial water consumptionAccording to the above analysis, combined with the model to establish the following regression model, regression coefficient estimation values are substituted into the model (18) to forecast equation.（19）Image as figure 8 3014y x ββε=++^443105.2888410y x -=+⨯3.3.4 Total agricultural water consumption functionFactors that affect the 4y are 1x , 5x , in order to determine the relationship between 4y and 1x , 5x , first using the data in the appendix table to make the 4y and 1x , 5x scatter diagram, as shown in figure:Figure 9 total agricultural water consumption Figure 10 the amount of agricultural water and the average annual rainfalland populationFigure 9 is obtained by MATLAB fitting curve, the fitting found that 4x and 4y is a function model (εfor random errors),（20）Figure 10 is obtained by MATLAB fitting curve, the fitting found that 5x and 4y is a function model (εfor random errors),（21）Combined with the above analysis, the model (20) (21) established the following regression model.（22）Directly using the MATLAB statistics toolbox in the command regress solution, the format is[b,bint,r,rint,stats]=regress(x,y,alpha)Output value of b for the estimation of the regression coefficient β, bint is the confidence intervals for b ,and 2Ris the residual vector, rint is the confidence 3014y x ββε=++74051n n n y x ββε==++∑121212123468994013121111n n n n n n n n n n y x x x x ββββε=====++++∑∑∑∑interval of r , stats is regression model test statistics, in the first number is a regression equation with coefficient of determination 2R ;The regression coefficients of the model (12) are estimated and their confidence intervals (confidence level α=0.05), test statistic 2R , F , ρ, and the results are shown in table.Table 4 regression coefficients of agricultural water useAccording to the above analysis, combined with the model to establish the following regression model, regression coefficient estimation values are substituted into the model (22) to forecast equation.（22）Its image is shown in Figure 11Figure 11 function of agricultural water3.3.5 Assessment of water supply capacityAccording to the data model obtained in 3.2.3, Shandong Province in China, therelevant ^737424155514 1.010 2.110510910 2.56810y x x x x ----=⨯-⨯+⨯-⨯+⨯data and the above function is brought into the model and calculated results:By the conclusion of the model, 1λ< shows that the ability to provide clean water in Shandong province is weak.3.4 Cause Analysis and Treatment Measures Water Shortage.3.4.1 the causes of water shortage in Shandong.（1） Water and soil erosion in hilly areas is serious, and water cannot be brought together into a river（2） Shandong is a temperate monsoon climate. Instability is one of the characters of the monsoon climate. Shandong is located in a part a Plain, and it is short of water. It is a big agricultural province. The water used in industry and agriculture is a lot.（3） Water shortage is the basic situation in the province of Shandong, the contradiction between water supply and demand have become increasingly prominent.（4） Total water resources shortage, average, low mu water resources, less water and more and more people, water resources and population, cultivated land resources serious imbalance, which is the main reason caused by a very prominent contradiction between water supply and demand in Shandong.（5） Have a great relationship with the natural geographical location. Shandong is located at the junction of the north and the south, which is a warm temperate monsoon climate. From the rainfall, the first is the uneven distribution of rainfall during the year.（6） As to rainfall distribution, in the southeast of Shandong Province annual rainfall average is up to 8.5 mm, and northwest region's annual average rainfall is only 550 millimeters, basically showing decreases from the southeast Shandong Province to the northwest of successive trend.（7） East Province is a coastal province, but the sea is not the water for drinking.A lot of rain in the coastal areas is typhoon. The available water in these areas is actually very little.（8） Groundwater levels continue to decline due to over exploitation of underground water in many places. The eastern provinces have formed a number of super mining areas.A series of environmental geological problems, such as groundwater pollution, are formed by the formation of the super mining area.（9）Water must not lack of water in the Yellow River in Shandong province. However, the amount of water in the Yellow River is declining year by year, and the available amount is decreasing.（10） Water conservancy project aging, degradation, water supply reduction3.4.2 Remediation Measures（1）With more rain and floods, water conservation, improvement of water cycle, reserve of groundwater resources, to achieve the use of abundant dry.（2）In strict accordance with the requirements of the state on the implementation of 216.030.741289.69X Y λ===<。

2007美国大学生数学建模竞赛B题特等奖论文

American Airlines' Next Top ModelSara J. BeckSpencer D. K'BurgAlex B. TwistUniversity of Puget SoundTacoma, WAAdvisor: Michael Z. SpiveySummaryWe design a simulation that replicates the behavior of passengers boarding airplanes of different sizes according to procedures currently implemented, as well as a plan not currently in use. Variables in our model are deterministic or stochastic and include walking time, stowage time, and seating time. Boarding delays are measured as the sum of these variables. We physically model and observe common interactions to accurately reflect boarding time.We run 500 simulations for various combinations of airplane sizes and boarding plans. We analyze the sensitivity of each boarding algorithm, as well as the passenger movement algorithm, for a wide range of plane sizes and configurations. We use the simulation results to compare the effectiveness of the boarding plans. We find that for all plane sizes, the novel boarding plan Roller Coaster is the most efficient. The Roller Coaster algorithm essentially modifies the outside-in boarding method. The passengers line up before they board the plane and then board the plane by letter group. This allows most interferences to be avoided. It loads a small plane 67% faster than the next best option, a midsize plane 37% faster than the next best option, and a large plane 35% faster than the next best option.IntroductionThe objectives in our study are:To board (and deboard) various sizes of plane as quickly as possible."* To find a boarding plan that is both efficient (fast) and simple for the passengers.With this in mind:"* We investigate the time for a passenger to stow their luggage and clear the aisle."* We investigate the time for a passenger to clear the aisle when another passenger is seated between them and their seat.* We review the current boarding techniques used by airlines.* We study the floor layout of planes of three different sizes to compare any difference between the efficiency of a given boarding plan as plane size increases and layouts vary."* We construct a simulator that mimics typical passenger behavior during the boarding processes under different techniques."* We realize that there is not very much time savings possible in deboarding while maintaining customer satisfaction."* We calculate the time elapsed for a given plane to load under a given boarding plan by tracking and penalizing the different types of interferences that occur during the simulations."* As an alternative to the boarding techniques currently employed, we suggest an alternative plan andassess it using our simulator."* We make recommendations regarding the algorithms that proved most efficient for small, midsize, and large planes.Interferences and Delays for BoardingThere are two basic causes for interference-someone blocking a passenger,in an aisle and someone blocking a passenger in a row. Aisle interference is caused when the passenger ahead of you has stopped moving and is preventing you from continuing down the aisle towards the row with your seat. Row interference is caused when you have reached the correct row but already-seated passengers between the aisle and your seat are preventing you from immediately taking your seat. A major cause of aisle interference is a passenger experiencing rowinterference.We conducted experiments, using lined-up rows of chairs to simulate rows in an airplane and a team member with outstretched arms to act as an overhead compartment, to estimate parameters for the delays cause by these actions. The times that we found through our experimentation are given in Table 1.We use these times in our simulation to model the speed at which a plane can be boarded. We model separately the delays caused by aisle interference and row interference. Both are simulated using a mixed distribution definedas follows:Y = min{2, X},where X is a normally distributed random variable whose mean and standard deviation are fixed in our experiments. We opt for the distribution being partially normal with a minimum of 2 after reasoning that other alternative and common distributions (such as the exponential) are too prone to throw a small value, which is unrealistic. We find that the average row interference time is approximately 4 s with a standard deviation of 2 s, while the average aisle interference time is approximately 7 s with a standard deviation of 4 s. These values are slightly adjusted based on our team's cumulative experience on airplanes.Typical Plane ConfigurationsEssential to our model are industry standards regarding common layouts of passenger aircraft of varied sizes. We use an Airbus 320 plane to model a small plane (85-210 passengers) and the Boeing 747 for a midsize plane (210-330 passengers). Because of the lack of large planes available on the market, we modify the Boeing 747 by eliminating the first-class section and extending the coach section to fill the entire plane. This puts the Boeing 747 close to its maximum capacity. This modified Boeing 747 has 55 rows, all with the same dimensions as the coach section in the standard Boeing 747. Airbus is in the process of designing planes that can hold up to 800 passengers. The Airbus A380 is a double-decker with occupancy of 555 people in three different classes; but we exclude double-decker models from our simulation because it is the larger, bottom deck that is the limiting factor, not the smaller upper deck.Current Boarding TechniquesWe examine the following industry boarding procedures:* random-order* outside-in* back-to-front (for several group sizes)Additionally, we explore this innovative technique not currently used by airlines:* "Roller Coaster" boarding: Passengers are put in order before they board the plane in a style much like those used by theme parks in filling roller coasters.Passengers are ordered from back of the plane to front, and they board in seatletter groups. This is a modified outside-in technique, the difference being that passengers in the same group are ordered before boarding. Figure 1 shows how this ordering could take place. By doing this, most interferencesare avoided.Current Deboarding TechniquesPlanes are currently deboarded in an aisle-to-window and front-to-back order. This deboarding method comes out of the passengers' desire to be off the plane as quickly as possible. Any modification of this technique could leadto customer dissatisfaction, since passengers may be forced to wait while others seated behind them on theplane are deboarding.Boarding SimulationWe search for the optimal boarding technique by designing a simulation that models the boarding process and running the simulation under different plane configurations and sizes along with different boarding algorithms. We then compare which algorithms yielded the most efficient boarding process.AssumptionsThe environment within a plane during the boarding process is far too unpredictable to be modeled accurately. To make our model more tractable,we make the following simplifying assumptions:"* There is no first-class or special-needs seating. Because the standard industry practice is to board these passengers first, and because they generally make up a small portion of the overall plane capacity, any changes in the overall boarding technique will not apply to these passengers."* All passengers board when their boarding group is called. No passengers arrive late or try to board the plane early."* Passengers do not pass each other in the aisles; the aisles are too narrow."* There are no gaps between boarding groups. Airline staff call a new boarding group before the previous boarding group has finished boarding the plane."* Passengers do not travel in groups. Often, airlines allow passengers boarding with groups, especially with younger children, to board in a manner convenient for them rather than in accordance with the boarding plan. These events are too unpredictable to model precisely."* The plane is full. A full plane would typically cause the most passenger interferences, allowing us to view the worst-case scenario in our model."* Every row contains the same number of seats. In reality, the number of seats in a row varies due to engineering reasons or to accommodate luxury-class passengers.ImplementationWe formulate the boarding process as follows:"* The layout of a plane is represented by a matrix, with the rows representing rows of seats, and each column describing whether a row is next to the window, aisle, etc. The specific dimensions vary with each plane type. Integer parameters track which columns are aisles."* The line of passengers waiting to board is represented by an ordered array of integers that shrinks appropriately as they board the plane."* The boarding technique is modeled in a matrix identical in size to the matrix representing the layout of the plane. This matrix is full of positive integers, one for each passenger, assigned to a specific submatrix, representing each passenger's boarding group location. Within each of these submatrices, seating is assigned randomly torepresent the random order in which passengers line up when their boarding groups are called."* Interferences are counted in every location where they occur within the matrix representing the plane layout. These interferences are then cast into our probability distribution defined above, which gives ameasurement of time delay."* Passengers wait for interferences around them before moving closer to their assigned seats; if an interference is found, the passenger will wait until the time delay has finished counting down to 0."* The simulation ends when all delays caused by interferences have counted down to 0 and all passengers have taken their assigned seats.Strengths and Weaknesses of the ModelStrengths"* It is robust for all plane configurations and sizes. The boarding algorithms that we design can be implemented on a wide variety of planes with minimal effort. Furthermore, the model yields reasonable results as we adjust theparameters of the plane; for example, larger planes require more time to board, while planes with more aisles can load more quickly than similarlysized planes with fewer aisles."* It allows for reasonable amounts of variance in passenger behavior. While with more thorough experimentation a superior stochastic distribution describing the delays associated with interferences could be found, our simulationcan be readily altered to incorporate such advances."* It is simple. We made an effort to minimize the complexity of our simulation, allowing us to run more simulations during a greater time period and mini mizing the risk of exceptions and errors occurring."* It is fairly realistic. Watching the model execute, we can observe passengers boarding the plane, bumping into each other, taking time to load their baggage, and waiting around as passengers in front of them move out of theway. Its ability to incorporate such complex behavior and reduce it are key to completing our objective. Weaknesses"* It does not account for passengers other than economy-class passengers."* It cannot simulate structural differences in the boarding gates which couldpossibly speed up the boarding process. For instance, some airlines in Europeboard planes from two different entrances at once."* It cannot account for people being late to the boarding gate."* It does not account for passenger preferences or satisfaction.Results and Data AnalysisFor each plane layout and boarding algorithm, we ran 500 boarding simulations,calculating mean time and standard deviation. The latter is important because the reliability of plane loading is important for scheduling flights.We simulated the back-to-front method for several possible group sizes.Because of the difference in thenumber of rows in the planes, not all group size possibilities could be implemented on all planes.Small PlaneFor the small plane, Figure 2 shows that all boarding techniques except for the Roller Coaster slowed the boarding process compared to the random boarding process. As more and more structure is added to the boarding process, while passenger seat assignments continue to be random within each of the boarding groups, passenger interference backs up more and more. When passengers board randomly, gaps are created between passengers as some move to the back while others seat themselves immediately upon entering the plane, preventing any more from stepping off of the gate and onto the plane. These gaps prevent passengers who board early and must travel to the back of the plane from causing interference with many passengers behind them. However, when we implement the Roller Coaster algorithm, seat interference is eliminated, with the only passenger causing aisle interference being the very last one to boardfrom each group.Interestingly, the small plane's boarding times for all algorithms are greater than their respective boarding time for the midsize plane! This is because the number of seats per row per aisle is greater in the small plane than in the midsize plane.Midsize PlaneThe results experienced from the simulations of the mid-sized plane areshown in Figure 3 and are comparable to those experienced by the small plane.Again, the Roller Coaster method proved the most effective.Large PlaneFigure 4 shows that the boarding time for a large aircraft, unlike the other plane configurations, drops off when moving from the random boarding algorithm to the outside-in boarding algorithm. Observing the movements by the passengers in the simulation, it is clear that because of the greater number of passengers in this plane, gaps are more likely to form between passengers in the aisles, allowing passengers to move unimpeded by those already on board.However, both instances of back-to-front boarding created too much structure to allow these gaps to form again. Again, because of the elimination of row interference it provides for, Roller Coaster proved to be the most effective boarding method.OverallThe Roller Coaster boarding algorithm is the fastest algorithm for any plane pared to the next fastest boarding procedure, it is 35% faster for a large plane, 37% faster for a midsize plane, and 67% faster for a small plane. The Roller Coaster boarding procedure also has the added benefit of very low standard deviation, thus allowing airlines a more reliable boarding time. The boarding time for the back-to-front algorithms increases with the number of boarding groups and is always slower than a random boarding procedure.The idea behind a back-to-front boarding algorithm is that interference at the front of the plane is avoided until passengers in the back sections are already on the plane. A flaw in this procedure is that having everyone line up in the plane can cause a bottleneck that actually increases the loading time. The outside-in ("Wilma," or window, middle, aisle) algorithm performs better than the random boarding procedure only for the large plane. The benefit of the random procedure is that it evenly distributes interferences throughout theplane, so that they are less likely to impact very many passengers.Validation and Sensitivity AnalysisWe developed a test plane configuration with the sole purpose of implementing our boarding algorithms on planes of all sizes, varying from 24 to 600 passengers with both one or two aisles.We also examined capacities as low as 70%; the trends that we see at full capacity are reflected at these lower capacities. The back-to-front and outside-in algorithms do start to perform better; but this increase inperformance is relatively small, and the Roller Coaster algorithm still substantially outperforms them. Underall circumstances, the algorithms we test are robust. That is, they assign passenger to seats in accordance with the intention of the boarding plans used by airlines and move passengers in a realistic manner.RecommendationsWe recommend that the Roller Coaster boarding plan be implemented for planes of all sizes and configurations for boarding non-luxury-class and nonspecial needs passengers. As planes increase in size, its margin of success in comparison to the next best method decreases; but we are confident that the Roller Coaster method will prove robust. We recommend boarding groups that are traveling together before boarding the rest of the plane, as such groups would cause interferences that slow the boarding. Ideally, such groups would be ordered before boarding.Future WorkIt is inevitable that some passengers will arrive late and not board the plane at their scheduled time. Additionally, we believe that the amount of carry-on baggage permitted would have a larger effect on the boarding time than the specific boarding plan implemented-modeling this would prove insightful.We also recommend modifying the simulation to reflect groups of people traveling (and boarding) together; this is especially important to the Roller Coaster boarding procedure, and why we recommend boarding groups before boarding the rest of the plane.。

2015美国大学生数学建模竞赛一等奖论文

2015 Mathematical Contest in Modeling (MCM) Summary Sheet
Summary
In this paper ,we not only analyze the spread of Ebola, the quantity of the medicine needed, the speed of manufacturing of the vaccine or drug, but also the possible feasible delivery systems and the optimal locations of delivery. Firstly, we analyze the spread of Ebola by using the linear fitting model, and obtain that the trend of development of Ebola increases rapidly before the medicine is used. And then, we build susceptible-infective-removal (SIR) model to predict the trend after the medicine is used, and find that the ratio of patients will decrease. Secondly, we investigate that the quantity of patients equals the quantity of the medicine needed. Via SIR model, the demand of medicine can be calculated and the speed of manufacturing of the vaccine or drug can be gotten by using Calculus (Newton.1671). Thirdly, as for the study of locations of delivery and delivery system, in Guinea, Liberia, and Sierra Leone, we establish the Network graph model and design a kind of arithmetic. Through attaching weights to different points, solving the problem of shortest distance, and taking the optimization mathematical model into consideration, we acquire four optimal locations and the feasible delivery systems on the map. Finally, we consider the other critical factors which may affect the spread of Ebola, such as production capacity, climate, vehicle and terrain, and analyze the extent of every factor. We also analyze the sensitivity of model and give the method that using negative feedback system to improve the accuracy of our models. In addition, we explore our models to apply to other fields such as the H1N1 and the earthquake of Sichuan in China. Via previous analysis, we can predict spread of Ebola and demand of medicine, get the optimal locations. Besides, our model can be applied to many fields.

美国大学生数学建模大赛英文写作

Conference Interpreting and Its Effect Evaluation, Nonlinear Waves in Elastic Rods， Introducing Management into…
写作要求： 1. 简短论文标题一般在10个字内，最多不超过15个词。
多用复合词
如：self-design, cross-sectional, dust-free, water-proof, input-orientation, piece-wiselinear 利用缩略词如：e.g., i.e., vs.（与…相对), ibid.(出处相同）， etc., cit.(在上述引文中), et al.(等人), viz.(即,就是), DEA (data envelopment analysis), OLS(Ordinary least-squares)
“Investigation on …”, “Observation on …”, “The Method of …”, “Some thought on…”, “A research on…”等冗余套语。
4. 少用问题性标题 5. 避免名词与动名词混杂使用如:标题是 “The Treatment of Heating and Eutechticum of Steel” 宜改为 “Heating and Eutechticuming of Steel” 6. 避免使用非标准化的缩略语论文标题要求简洁,但一般不使用缩略语 ,更不能使用非标准化的缩略语。
关键词（Keywords)
基本功能：顾名思义；便于检索语言特点：多用名词；字数有限（4-6）；出处明确写作要求 :论文的关键字一般列在作者与单位之下，论文摘要之上。也有列在论文摘要之下的。关键词除第一个字母大写外，一般不要求大写。关键词间用逗号、分号或大间隔隔开。最末一个关键词一般不加用逗号、分号或句号。

2013年美国大学生数学建模竞赛(MCMICM)参赛规则中英文对照

2013年美国大学生数学建模竞赛（MCMICM）参赛规则中英文对照2 ICM:The InterdisciplinaryContest in ModelingICM：交叉学科建模竞赛ContestRules, Registration and Instructions比赛规则，报名注册和指导(All rules and instructions apply to both ICM and MCM contests, except where otherwisenoted.)（所有MCM的说明和规则除特别说明以外都适用于ICM）To participate in a contest, each team must be sponsored by a faculty advisor fromits institution.参加MCM的每个队伍需有一名在职的高校老师负责指导。

TeamAdvisors: Please read these instructions carefully. It isyour responsibility to make sure that teams are correctly registered and thatall of the following steps required for participation in the contest arecompleted:Pleaseprint a copy of these contest instructions for reference before, during, andafter the contest. Click here for the printer friendly version.指导老师：请认真阅读这些说明事项，确保完成了所有相关的项。

每位指导教师的责任包括确保每个参赛队正确注册并正确完成参加MCM/ICM所要求的相关步骤。

2019年美国大学生数学建模竞赛E题中英文对照

2019年MCM/ICM问题E：环境退化的成本有多大？经济理论经常忽略决策对生物圈（biosphere）的影响、假设资源无限量或生产能力满足其需求。

这个观点有一个缺陷，而且环境现在正面临后果。

生物圈通过许多自然过程为人类生活维持一个健康和可持续的环境，这被称为生态系统服务（ecosystem services）。

例如将废物转变成食物，水的过滤，食物种植，植物授粉，以及二氧化碳转化为氧气的过程。

然而，每当人类改变生态系统时，都会潜在地使这些生态系统服务受到限制或不复存在。

当地小规模土地利用，如建设一些道路、下水道、桥梁、房屋或工厂的影响似乎可以忽略不计。

除了这些小项目之外，还存在有建设或重新安置大型公司总部、在全国范围内建设管道、或扩大或改变水路用于扩展商业用途等大规模工程。

现在思考这些众多项目跨越一个地区、国家以及世界的影响。

虽然这些活动单独而言可能看似对生物圈运作潜力的总体功能无关紧要，但经过累积，它们将直接影响生物多样性（biodiversity）并造成环境退化（environmental degradation）。

一般而言，绝大多数土地利用项目不考虑对生态系统服务的影响，或者无法解释对于生态系统服务的改变。

用于缓解土地利用所造成的各种负面后果（如受污染的河流，不良的空气质量，存在危险的垃圾场，处理不当的废水，气候变更等）的经济成本的变化通常不包括在计划中。

是否可以对土地利用的环境成本进行估价？如何在这些项目成本中增加对于环境退化的考虑？当生态系统服务在一个项目的成本效益比中得到考虑之后，便可以建立起真实和全面的项目价值评估（valuation）。

你的ICM团队已被聘请去创建一个生态服务评估模型从而了解考虑生态系统服务时土地利用项目的真实经济成本。

用你的模型对从小型社区项目直到大型国家项目的不同规模土地利用开发项目进行成本效益分析。

基于你的分析和模型设计来评估你所建立模型的有效性。

你的建模对于土地使用项目规划者和管理者而言有什么含义？你的模型需要如何随时间变化？您的提交应包括：•单页摘要页，•不超过20页的解决方案，加上摘要页合计最多21页。

2016年美国大学生数学建模大赛A题获奖论文A Hot Bath

Part II Finding the best strategy ...................................................................12 VI. Results ........................................................................................................ 12
The first part has five sections: air’s heat radiation, bathtub wall’s heat radiation, person in, hot water in, bubble existed. We discuss some factors that affect water temperature, such as the shape and the volume of the bathtub and person, especially the motions made by the person in the bathtub because the temperature in the bathtub has a great connection with person. Finally, we get the water temperature variation and distribution model.
In this article, we establish two models. One is water temperature variation and distr one is finding best strategy model. We put forward some acceptable hypothesis to simplify the model. What’s more, we clear the meaning of the word “noticeably”.

2016美国大学生数学建模大赛C题特等奖(原版论文)C42939Tsinghua University, China

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

数学建模美赛论文标准格式参考--中英文对照

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MCMTheMathematicalContestinModeling：MCM数学建模竞赛

MCM: The Mathematical Contest in ModelingICM: The Interdisciplinary Contest in ModelingContest Registration and Instructions(All instructions and rules apply to ICM as well as to MCM, except where otherwise noted.)To participate in MCM a team must be sponsored by a faculty advisor from their institution. The registration process must be completed by the advisor.PLEASE NOTE THE CHA NGES TO THE MCM/ICM RULES WHICH ARE HIGHLIGHTED IN RED BELOWThere are several procedures that a team's advisor must go t hrough at various times before, during, and after the contest. Please read these instructions carefully and be sure to complete all the steps involved. It is the advisor's responsibility to make sure that teams are correctly registered and that all steps required for participation in MCM/ICM are completed.We Suggest print ing a copy of these contest instructions for reference before, during, and after the contest.Note that COMAP is in the USA Eastern time zone; all t imes given in these instructions are in terms of Eastern time.If a team has been caught violat ing the rules, the faculty advisor will not be allowed to advise another team for one year and the advisor's instit ution will be put on one year's probation.Should a team from that institut ion be caught violating the rules a second t ime, then that school will not be allowed to compete for a period of at least one year.Before the contest registration deadline at 2pm EST on Thursday February 5, 2009 :Register your team online:The registration process will take you through a series of screens that askyou for your email address and contact information. 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Please allow several weeksafter the results are posted to the contest web site before expecting toreceive your certificate.Prizes∙The Institute for Operations Research and the Management Sciences (INFORMS) will designate an Outstanding team from each of the three problems as anINFORMS winner∙The Society for Industrial and Applied Mathematics (S IAM) will designate one Outstanding team from each problem as a S IAM winner.∙The Mathematical Association of America (MAA) will designate one Outstanding team from each problem for the MCM as a MAA winner.Note: COMAP is the final arbiter of all rules and policies, and may disqualify or refuse to register any team that, in its sole discretion, does not follow these contest regulations and procedures.。

2021年美国大学生数学建模竞赛题目A--真菌范文六篇(含Matlab源代码)

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2021年美国大学生数学建模竞赛题目A--真菌范文六
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美国大学生数学建模一等奖31552

Best all time college coachAbstractIn 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 composite score, namely the order of male basketball coaches, which is shown in Table 7.Adolph Rupp and Mark Few are the last century and this century's "best all time college coach" respectively. It is realistic. The rank of college coaches can be clearly determined through this methods.Next, ANOVA shows that the scores of last century’s coaches and this century’s coaches have significant difference, which demonstrates that time line horizon exerts influence upon the evaluation and gender factor has no significant influence on coaches’ score. The assessment model, therefore, can be applied to both male and female coaches. Nevertheless, based on this, we have drawn coaches’ coaching ability distributing diagram under ideal situation and non-ideal situation according to the data we have found, through which we get that if time line horizon is chosen reasonably, it will not affect the selecting results. In this problem, the time line horizon of the year 2000 will not influence the selecting results.Furthermore, we put the data of the three types of sports, which have been found by us, into the above Model, and get the top 5 coaches of the three sports, which are illustrated in Table10, Table 11, Table12 and Table13 respectively. These results are compared with the results on the Internet[7], so as to examine the reasonableness of our results. We choose the sports randomly which undoubtedly shows that our model can be applied in general across both genders and all possible sports. At the same time, it also shows the practicality and effectiveness of our model.Finally, we have prepared a 1-2 page article for Sports Illustrated that explains our results and includes a non-technical explanation of our mathematical model that sports fans will understand.Key words: TOPSIS Improved Model; Entropy; Analytical Hierarchy Process;Comprehensive Evaluation Model; ANOV AContents Abstract (1)Contents (2)I. Introduction (3)П. The Basic Assumption (4)Ⅲ. Nomenclature (5)Ⅳ. Model (5)4.1 Data Processing (5)4.2 Model analysis (6)4.3 Model building (6)4.3.1 Dominant index weights calculation (7)4.3.2 Hidden index weights calculation (9)4.3.3 Positive and negative ideal solution building (12)4.3.4 Distance calculation (12)4.3.5 Comprehensive evaluation value (13)4.4 Model solution (13)4.4.1 Dominant index weights calculation (13)4.4.2 Hidden factors weights calculation (14)4.4.3 Consolidated score (16)4.5 Judgment of significant differences between the last century’s andthis century’s coaching score. (16)4.5.1 Preliminary investigation of the last century and the coach ofthe century standards (16)4.5.2 Further exploration on the influence of different time linehorizons on the assessment results (18)4.6 Test of model’s applicability to both gender (19)4.7 The selection for the top five college coaches of three sports (20)V. Analysis of our Model (22)5.1 Applications of our models (22)5.2 Strengths (22)5.3 Weaknesses (22)5.4 Future Improvements (22)Ⅵ. Conclusions (23)Ⅶ.A letter to the sports enthusiasts (23)Ⅷ. References (24)I. IntroductionTh e paper is to help "Sports Illustrated" to find the “best all time college coach” male or female.We tackle five main problems:●Build a mathematical model to choose the best college coach or coaches (past orpresent) from among either male or female coaches in such sports as college hockey or field hockey, football, baseball or softball, basketball, or soccer, and clearly articulate our metrics for assessment.●Does it make a difference which time line horizon that you use in your analysis,i.e., does coaching in 1913 differ from coaching in 2013?●Present our model’s top 5 coaches in each of 3 different sports.●Discuss how our model can be applied in general across both genders and allpossible sports.●In addition to the MCM format and requirements, prepare a 1-2 page article forSports Illustrated that explains our results and includes a non-technical explanation of our mathematical model that sports fans will understand.To tackle the first problem, we searched the indicators of Top 600 men’s basketball coaches of the American colleges. Take selecting the best male basketball coach as an example: for the explicit factors that affect assessment standards, we calculate each indicator’s weight by using Entropy method; for those implicit factors, we calculate the weight through experts’evaluation. The determination of each indicator’s score should be given by experts evaluation of each indicator. These indicators are then numericalized, and the importance of each indicator is determined through weight coefficients. Then through the multiplication of the scores of coaches’different ability indicator with corresponding weight coefficients, we get the corresponding scores, and the highest score indicates the best choice.For the second question, we first use ANOV A to determine whether significant difference exists between the scores of coaches in the last century and this century and the gender factor Significance difference shows that the time line horizon, the gender factor has influence on the assessment, whereas insignificant difference shows no influence. And based on this, we have drawn coaches’coaching ability distributing diagram under ideal situation and non-ideal situation according to the data we have found, which help us further research the influence of time line horizon on the assessment.For question 3 and 4, we put the data of the three types of sports, which have been found by us, into the Model , and get the top 5 coaches of the three sports, which are illustrated in Table10, Table 11, Table 12 and Table 13 respectively. These results are compared with the results on the Internet, so as to examine the reasonableness of our results. We choose the sports randomly, which undoubtedly shows that our model can be applied in general across both genders and all possible sports. At the same time, it also shows the practicality and effectiveness of our model.Figure1. The source of the best college coachesП. The Basic Assumption●Experts recessive factors evaluation criteria evaluation is fair and equitable.●Coaches’coaching level will increase with increasing age, but it will decline due to mental declination and the lack of the physical strength.●Assessment experts are fully known on college coaches.●The evaluation criteria only consider the factors enumerated in this paper, without considering other factors.●The evaluation criteria apply equally to men and women coaches.●We used the general data from a reliable website,Website (see Appendix).Ⅲ. NomenclatureXIndex data normalization matrix j w j Index weightsij θ Transformed normalized matrixθ+ "Positive ideal solution"θ- "Negative ideal solution"i φi comprehensive evaluation index values of being evaluated j ej Index entropy j ψ j Index Information utilityFF statisticⅣ. Model4.1 Data ProcessingIn order to better assess the extent of outstanding coaches, we selected a number of indicators to determine the coach for the "best all time college sports coach". We found information on the various indicators of data on the site and get some reliable indicators data of these college coaches. Due to the dimensions of each index inconsistencies exist, so we transformed the data to eliminate the effects of dimensionless. And through poor conversion get a normalized matrix ij m n X x ⨯⎡⎤=⎣⎦ ,1111n X m mn x x x x =, 1,2,;1,2,i m j n == ()41-ij r x =1,2,;1,2,i m j n == ()42-ij x is a dimensionless quantity and ij x []0,1∈, 1,2,;1,2,i m j n ==.4.2 Model analysisIn order to address the problems mentioned above and provide a valid, feasible assessment strategy for Sports Illustrated, we decide to select softball, basketball and football by reviewing the relevant literature. Coaching time, Competition winning rate, Cultural qualities, Athletic ability, Social skills, Ability to withstand, Innovation capacity, Ability to perceive, and so on, which are evaluation indexes. These evaluation indexes are divided into dominant factors and recessive factors. Specific factors of affecting the evaluation criteria are shown in Figure X. These indicators will be quantified and determine the degree of importance of each index by weight coefficient. When selecting coaches, the scores of the indicators multiply corresponding weight coefficient, getting corresponding scores, and the person with the highest score is the best candidate.Multi-level analysis method to determine the weight is more subjective. It is suitable to determine the weights for hidden factors, which are not used widely in both sexes and all possible requirements for sport. We need to build a more reasonable model to determine the weight for the dominant factor and recessive factors. Finally, we determine the “best all time college coach”.4.3 Model buildingWe look for the “best all time college coach” by establishing a mathematical model in Technique for Order Preference by Similarity to Ideal Solution. Take choosing the best college coach or coaches from among male coaches in such sports as basketball as an example. For the dominant factor, we calculate the weight of each indicator in Entropy Method; For the hidden factors, we calculate the weight of each indicator in expert assessment method. According to the situation of the coaches , the scores of all levels should be determined by experts, and these indicators should be quantified. Weighting coefficients represent the importance of each indicator. The scores of the indicators multiply corresponding weight coefficient to obtain the total score, and the person of highest score is the best candidate. This method is more objective, comprehensive, accurate and wide-applicable than the previous evaluation model.Flow chart of looking for the “best all time college coach” is shown in Figure 2.Figure 2.Flow chart of ModelTOPSIS Model (Technique for Order Preference by Similarity to an Ideal Solution ) was firstly introduced by C.L.Hwang and K.Yoon in 1981.TOPSIS Model is based on the proximity of a limited number of evaluation objects and idealistic goals and evaluate the relative merits of existing objects.Meanwhile, TOPSIS Model is an approximation of the ideal solution in order model, the model requires only a monotonically increasing (or decreasing) of each Utility function.Furthermore, TOPSIS multi-objective decision analysis model is a commonly used and effective model, also known as the merits of the solution from the law. The basic principle is evaluated by detecting the distance the optimal solution and the solution of the worst sort, if the evaluation of the optimal solution while the object closest to farthest from the worst solution, the result is optimal; otherwise, is not optimal，ｗhere the value of each index has reached the optimal solution for the optimal value of each index. Each index value solution has reached the worst the worst value of each index."Positive ideal solution" and "negative ideal solution" are two basic concepts TOPSIS Model. "Positive ideal solution" is an envisaged optimal solution (program), it's the individual attribute values to achieve the best value of each option; rather negative ideal solution is a solution envisaged for the worst (program ), each of which have reached the attribute value of each option in the worst value. Program to sort the various alternative rules are the ideal solution and the negative ideal solution for comparison, if one has a solution closest to the ideal solution, while away from the negative ideal solution, the solution is the best alternative solution.4.3.1 Dominant index weights calculationFor the dominant factor, we calculate the entropy method using the weight of each indicator.According to the data we found, we list the dominant influence coaches criteria indicators (see Figure 3). These dominant indicators are intuitive and easy to quantify,due to the weight of these data to calculate the specific rights-based approach, with strong objectivity. Degree of dispersion of data can be seen as the degree of disorder (entropy), the greater the entropy index data, the smaller the proportion of the index.Figure 3. Th e diagram of the entropy and weightsInformation entropy method is a method completely dependent on the data, but it is not affected by subjective factors.Figure 4. The structure of influence the selection criteria for the dominant factorsFormula to calculate the information entropy index for item j :11ln ln j ij mij i e x x m ==-∑，ij x []0,1∈ ()43- Information utility depends on the difference between the value of an index of the index information entropy between A and 1. It directly affects the size of the weight : the greater the utility value of the information , the greater the importance of the evaluation, and the greater the weight .1j je ψ=- ()43- Estimating the weight of each index using entropy method, its essence is to use the value of the coefficient to calculate the index information, the higher the value ofthe coefficient, the greater the importance of the evaluation (or the greater the weight, the bigger contribution to the evaluation results).Right item j index weight is:1j j jmi w ψψ==∑ ()43- 4.3.2 Hidden index weights calculationFor recessive factors, we take expert assessment method to calculate weights. The determination of index score at all levels should be carried out by an expert score for each indicator according to the situation of the coaches.According to the data we found, we cited the impact coach implicit criteria indicators (see Figure 4). These indicators are visually hidden but not easy to quantify. Because of these hidden right index weight calculation method based on highly subjective, we used AHP to accurately calculate the weights of these hidden indicators.AHP is a decision problem in terms of total goals layers of sub-goals, evaluation criteria and specific equipment investment program in order to break down the different hierarchies, then use judgment matrix eigenvector method to obtain the elements of each level of priority on a certain level of heavy elements, and finally re-weighted and hierarchical approach to merge the various alternative solutions to the overall goal of the final weights. "Priorities" is a relative measure, which indicates the alternative criteria for the evaluation of a program or sub-features of the target, which means excellent measure of the relative degree of each sub-target and the target level for the purposes of the relative importance of measure. Specific usage is to judge the matrix, find the maximum eigenvalue, then the corresponding feature vector normalization, finally we can get a level indicator on one level for a related indicators relative importance weights.Features of AHP are based on the nature of complex decision problems, influencing factors and internal factors affecting the relationship between in-depth analysis, and use less quantitative information to make decisions mathematical thinking process, so as to multi-target, multi-standard or non-structural properties of the complex issues simple decision making methods. Especially suitable the occasion for decision-making results difficult to directly and accurately measure.Figure 5. The structure of influence the selection criteria for the hidden factors We can know from the Figure 5 , the hierarchy is divided into one-level indicator, two-level indicators, so it belongs to the multi-level hierarchical structure model.Comparison matrix constructionAccording to the analysis of psychologists, the importance of being divided into nine grades, and secondary indicators for the level indicators can be pairwise comparison of their importance to quantify the value using the following scale.Table 1. Evaluation scaleScale Definition1 i is for j equally important3 i is for j slightly important5 i is more important for j7 i is very important for j9 i is absolutely vital for j2,4,6,8 Two intermediate value corresponding to the scaleReciprocal i compared with j,1ijijcc=or 1ijc=are obtained for the judge valueAccording to the above scale, relative matrix is as follows:By comparison, the comparison matrix of level indicators and secondary indicators are as follows:1111ni n nn c c C c c =()44-Due to the above judgment matrix symmetry, so when filling out, usually the first to fill 1ii c =section, and then judge and triangular or lower triangular() 1/2n n -elements on the form. In exceptional circumstances, the judgment matrix is transitive, that satisfies the equation ：ik kj ij c c c *= .When the formula to determine all the elements of the matrix are established, the consistency of judgment matrix is a matrix.Level single-sorting (Weight vector calculation) and TestFor the judgment of experts to fill in the matrix ，we took advantage of some mathematical methods for sorting. Level single-sorting refers to the various factors of each judgment matrix for weight relative weights of the criteria, so essentially calculating the weight vector. There are many ways to calculate the weight vector, such as the eigen value method, and the method, the root method, power method. Here is a brief overview and method.Principle "and the law", for consistency of judgment matrix, each column after normalization, we can get the corresponding weights. For non-consistency of judgment matrix, each column after normalization, which can be approximated by the corresponding weights, n column vectors and these strike the arithmetic average as the final weight of the weight. Specific formula is:111n iji nj klk c W n c ===∑∑ ()45- It should be noted that, in the layers of the sort , you need to test the consistency of judgment matrix . In exceptional circumstances , determining the matrix has passed and consistency. Under normal circumstances, the judge is not required to meet the strict nature of the matrix . But looking at the human understanding of the law , a right to judge the importance of the matrix there is some sort of logical law . For example, if A is more important than B, and B surpasses C importantly , from a logical perspective , A should be significantly more important than C, if the two a comparison of two important results than C , then the consistency of judgment matrix in violation of norms, logically unreasonable. If pairwise comparisons, C is more important than the result of A, the consistency of judgement matrix in violation of the guidelines, it was logically irrational.Therefore, in practice it is required to meet the general consistency of judgment matrix, which requires consistency checking. Only by testing can it illustrate that the logical judgment matrix is reasonable and to continue to analyze the results. Steps of consistency test are as follows.First , calculate the consistency index ..C I (consistency index).max..1nC I n λ-=- ()46- Second , look-up table to determine the corresponding average random consistency index ..R I （random index ）According to the different order of judgment matrix, we check the table below, and get the average random consistency index ..R I For example, for a 5-order judgment matrix, we can get ..R I = 1.12 easily.0 0 0.52 0.89 1.12 1.26 1.36 1.411.41.491.521.541.561.581.59Third , calculate the proportion of consistency ..C R (consistency ratio) and determine.......C I C R R I = ()47- When ..0.1C R <, the consistency of judgment matrix is considered acceptable and when .. 0.1C R >, it is considered the consistency of judgment matrix does not meet the requirements, we need to re-amend the judgment matrix. 4.3.3 Positive and negative ideal solution buildingWe define ,1,2,;1,2,;ij ij ij w x i m j n θ=•==Determine the positive idealsolution θ+and negative ideal solution θ-；Assuming positive ideal solution θ+Negative ideal solution ：{}min ,1,2,;1,2,;j ij ii m j n θθ-=== Positive ideal solution ：{}min ,1,2,;1,2,;j ij ii m j n θθ+===4.3.4 Distance calculationT he Euclidean distance between being evaluated and Positive ideal solution1,2,i d i m +==⋅⋅⋅ ()48-The Euclidean distance between being evaluated and Negative ideal solution_1,2,i d i m ==⋅⋅⋅ ()49-4.3.5 Comprehensive evaluation valueThe value of comprehensive evaluation index evaluated is,1,2,i i i id i m d d φ-+-==⋅⋅⋅+ ()410- 4.4 Model solution4.4.1 Dominant index weights calculationWe find four dominant indicators for the last century of the impact evaluation criteria through the network, namely, "The time of winning the championship", "The number of races", "Coaching time", "Completion wining rate". Specific data are in Table 3.Table 3. Four indicators for men's basketball coachesHank Iba 29 1085 40 0.693 Ray Meyer 20 1078 42 0.672 Don Haskins 29 1072 38 0.671 Adolph Rupp 71 1066 41 0.822 E.A. Diddle 17 1061 42 0.715 Ralph Miller 17 1044 38 0.646 Slats Gill 13 992 36 0.604 Norm Stewart 30 967 32 0.656 Tony Hinkle 4 952 41 0.586 Norm Sloan 14 917 33 0.609 Jack Friel 3 872 30 0.568 Guy Lewis 26 871 30 0.68 Ned Wulk 17 837 31 0.59 JohnThompson 37835 27 0.714 John Wooden 54 826 29 0.804 Bill E. Foster 7 820 30 0.515 Johnny Orr 13812 29 0.574 … …………We will enter the above data by calculated entropy method to get the dominant index weights as follows:Table 4. Men's basketball coach dominant index weights tableDominant indexThe time ofwinning thechampionshipThe number ofraceCoachingtimeCompletionwinning rateweight0.7481 0.1195 0.1145 0.0178From the Table 4 we can observe "The time of winning the championship" share of the weight is larger than the "Completion wining rate". But the proportion of "The number of race" and "Coaching time", is less. This shows that the dominant indicators, "The time of winning the championship" for the selection of the coach plays a very important role.Figure 6. Men's basketball coach dominant index weights pie From Figure6, we can observe that"The time of winning the championship" significant weightings are larger in the share of other indicators. On the surface this is actually somewhat contradictory, but in fact, as "The time of winning the championship"indicators of the degree of dispersion is larger, therefore,its impact is huge coach rankings, while the smaller degree of dispersion of other indicators,so they rank impact on the coach is smaller.4.4.2 Hidden factors weights calculationUsing the comparison scale of the model we c an go to the comparison matrix level indicators and secondary indicators. Since the pairwise comparison is subjective, the Hidden factors weight is subjective. Using the way of expert reviewing, finding information or questionnaires to get the comparison matrix. Then calculate the weights. Then we examined whether it could through consistency test.We did a series of comparison matrix and then through examination we selected the following comparison matrix.Table 5. The best comparison matrix of the University men's basketball coach indicatorsHidden factors CulturalqualitiesAthleticabilitySocialskillsAbility towithstandInnovationcapacityThe abilityto perceiveCultural qualities 1 1/3 1/3 1/3 1/6 1/7 Athletic ability 3 1 1/3 1/3 1/5 1/5Social skills 3 3 1 1/3 1/5 1/4 Ability to withstand 5 3 3 1 1/3 1/5 Innovation capacity 6 5 5 3 1 1/4 The ability toperceive7 5 4 5 4 1Known by its consistency index, ..0.98700.10C R=<，.. 0.1360C I=, so it can go through consistency test. The maximum value weight , 6.6799λ=,which we calculated are shown in Table 5.The consistency is index..0.98700.10C R=<，.. 0.1360C I=. Through consistency test, the maximum characteristic value is 6.6799λ=. Weights form table below.Table 6. Best college men's basketball coach recessive factor index weights tableHidden factors CulturalqualitiesAthleticabilitySocialskillsAbility towithstandInnovationcapacityThe abilityto perceiveWeights 0.0345 0.0554 0.0836 0.1340 0.2491 0.4435Figure 7. Best college men's basketball coach recessive factor index weights pieWe obtain the weight values though consistency test and Analytic Hierarchy Process, Athletic ability of coaches is great importance of hidden index. Second, the cultural qualities, the innovation capacity is not important. The results are subjective more or less. We can not be generalized, with the development of society, the proportion of innovative indicators may increase.4.4.3 Consolidated scoreweights above put in TOPSIS model can get a score for each coach. Because hidden indicators expert review in our paper is difficult to achieve. Thus weakened expert evaluation index, highlighting calculations dominant indicators.Thus, we get the following scoring table.Table 7. Last century University men's basketball coach total scoreDean Smith 70 1133 36 0.776 0.0181 John Wooden 54 826 29 0.804 0.0139 John Thompson 37 835 27 0.714 0.0097 Norm Stewart 30 967 32 0.656 0.0081 Hank Iba 29 1085 40 0.693 0.0080 Don Haskins 29 1072 38 0.671 0.0080 Guy Lewis 26 871 30 0.68 0.0071 Everett Case 27 511 19 0.738 0.0071 Lou Carnesecca 26 726 24 0.725 0.0070 Gene Bartow 25 744 24 0.66 0.0067 Neil McCarthy 23 681 23 0.665 0.0062 Pete Carril 22 798 30 0.658 0.0061 Frank McGuire 22 785 30 0.699 0.0061 Joe B. Hall 23 463 16 0.721 0.0060 Jack Gardner 22 721 28 0.674 0.0060 Ray Meyer 20 1078 42 0.672 0.0058 Terry Holland 21 634 21 0.659 0.0057 ………………As can be seen from Table 7Adolph Rupp's highest overall score, it is reasonable to judge him in the last century's "best all time college coach". Due to the weakening of the influence of implicit indicators, so here was "best all time college coach" on the hidden indicators may have less. Using the same method to evaluate the coach of the century can be the century of the "best all time college coach" is Mark Few.4.5 Judgment of significant differences between the last century’s and this century’s coaching score.4.5.1 Preliminary investigation of the last century and the coach of the century standards.Taking into account the tremendous changes in the last century and this century,。

美赛：27688---数模英文论文

Team Control NumberFor office use only27688For office use onlyT1 ________________F1 ________________T2 ________________F2 ________________T3 ________________Problem Chosen F3 ________________ T4 ________________C F4 ________________2014Mathematical Contest in Modeling (MCM/ICM) Summary SheetThe research of influence based on the characteristic of a network To find the influential nodes in the network, the key is the definition of “influential”and how to measure the influence. In this paper, we use two kinds of metrics to measure the influence of coauthor network and citation network. In coauthor network, both the Authority and Importance of the researchers are proposed to measure the influential of researcher. And the second one in citation network take the citation times, publication time and the position in the network into account.For the evaluation of coauthor, we first construct a coauthor network with 511 vertices and 18000 edges and it is an undirected graph. Next, we use software UCInet to analyze the degree centrality, eigenvector centrality, closeness centrality and betweenness centrality of the network. Since there is no evident transfer relationship in the coauthor network, we using Authority and Importance to measure the influence of a research. In detail, the Authority is correlated with the coauthoring times with Paul Erdös and the Importance is measured by eigenvector centrality. Finally, we rank the researchers whose authority is larger than 2 according to their importance. And the top 5 most influential researchers are: RODL, VOJTECH; LOVASZ, LASZLO; GRAHAM, RONALD LEWIS; PACH, JANOS; BOLLOBAS, BELA. Finally, we search for some data through websites and verify these people are really influential.For the evaluation of papers, we first compare the difference between the citation network and coauthor network. According to the characteristic of Directed Acyclic Graph(DAG), we define a contribution coefficient and self-contribution coefficient by making an analogy with the energy transfer in the food chain. Considering the less-effectiveness of PageRank Algorithm and Hits Algorithm, we design an algorithm, which is effective in solving the DAG problem, to calculate the contribution coefficient. We find 3 most influential papers: Paper 14, Paper 4 and Paper 2 in the NetSciFoundation.pdf.In the third part, we implement our model to analyze a corporation ownership network. We use the value of the company’s cash, stock, real estate, technical personnel, patent and relationships to define its value. And we use the proportion of stock to measure the control ability of parent company. Applying the model and algorithm of citation network, we find 15 influential companies. Then we find that 9 of them are in the top 20 of authoritative ranking, which verifies the rationality of our result.Finally, we describe how we can utilize these influential models to do some socialized service, to aid in making decision on company acquisition and to carry out strategic attack.Team #27688Page 1 of 18 1. IntroductionNowadays, coauthor network and citation network are built to determine influence of academic research. Paul Erdös, one of the most influential researchers who had over 500 coauthors and published over 1400 technical research papers. There exists a coauthor network among those who had coauthored with Erdös and those who had coauthored with Erdös’s directed coauthors.In this paper, we first analyze this coauthor network and find some researchers who have significant influence. Then, we analyze the citation network of some set of foundational papers in the emerging field of network science. Furthermore, we determine some measures to find some most influential papers. After that, we use the data of US Corporate Ownership to construct a new network and test the applicability of our model and algorithm. Finally, we describe some applications of using the analysis of different networks.In section 3, the coauthor network is an undirected graph. We first analyze four kinds of centrality: Degree Centrality, Eigenvector Centrality, Closeness Centrality and Betweenness Centrality. Additional, the Degree distribution and Clustering coefficient are also the important properties of the network. Then, we define Authority and Importance to measure the influence of a researcher. Authority can be measured by the coauthoring times with Erdös. It is clearly that the researcher who coauthors with more people is more important. Since this is not a problem about “information flow”, we only cons ider the influence of those directed coauthor and neglect the transitivity of influence. That is to say, Importance can be measure by Eigenvector Centrality. Finally, we choose some people with higher authority and rank them according to their Eigenvector Centrality.In section 4, the citation network is different from the coauthor network. As the citation relation is related to publication time, the citation network is a Directed Acyclic Graph(DAG). Traditionally, we calculate the nodes’ importance of a n etwork by using PageRankAlgorithm[17] and HITS Algorithm[18]. However, both of them involve matrix multiplicationand repeated iterative process, which is less-effective. Since the network satisfies theproperty of Directed Acyclic Graph(DAG), we draw on the thought of topological sorting to design a more effective algorithm. In this citation network, there exists transitive relation that does not exist in the coauthor network. We first use software UCInet to calculate thecentrality of each paper. And then we take these metrics, publication time and times cited count into account to develop a new model. In this model, we learn from the energy transfersin the food chain and define an initial contribution coefficient to measure its authority. In addition, we define a self-contribution coefficient to measure the influence from other papers. Finally, we design an algorithm to calculate each paper’s final contribution coefficientto measure the paper’s influence.In section 5, we use nearly 500 US Media Companies to construct an ownership network. Then we set the initial value of each company according to their case, stock, real estate, technical personnel, patent and relationships. And we set a control coefficient to measure the ownership between two companies. Then we can use the algorithm in citation network to find someTeam #27688Page 2 of 18influential companies.In the fourth part, we utilize these influential models to do some social service, aid in making decision on company acquisition and carry out strategic attack.In general, the article is written follows:(1)Build a coauthor network for question 1.(2)Build the evaluation index of the most influential coauthor to estimate the influenceof coauthors in the coauthor network.(3)Build citation network and define the influence criterion of papers to estimate themost influential paper.(4)Implement our model to the US Corporate Ownership network to analyze theimportance and the value of the company.(5)Finally, we discuss about the basic theory, the use and effectiveness of the science ofnetwork.2. Assumptions and Justification(1)We use number 1..16 to represent the paper given in the NetSciFoundation.pdf according totheir sequence. It is worth mentioning that the information of paper 7 given in the file seems to be wrong. Hence, we regard it as an isolated vertex in the network.(2)The researchers’ authority, it is correlated with the coauthoring times with Paul Erdös. Inthe coauthor network, we know that all of them have coauthored with Erdös and Erdös is such an excellent mathematician. So it is suitable for us to assume that more times coauthored with Erdös, more authority the researcher is.(3)We do not consider the influence of the paper’s content and field because the cited times indifferent fields have no comparability. In question 3 we know that 16 papers are in the emerging field of network science, so it is reasonable for us to simplify this problem.(4)When constructing the citation network, we only take those papers citing more than twopapers in 16 given papers and also having been cited by other papers. Absolutely, the citation network is infinite. In this paper, we aim to find influential papers. Hence, we give up those less important papers and restrict the scale of our network.(5)We assume that the citation relation is effective. If a paper cited other papers, we considerthat the author admitted the positively effect of the cited paper. Since the influence of a paper is related to the citation times, our assumption can improve the validity of the result.(6)The data in our paper is effective. Our dataset is searched in Web of Science and GoogleScholar, which are equipped with high authority.Team #27688Page 3 of 18 3. Coauthor Network3.1 Building the modelA coauthor network can be built to help analyze the influence of the researchers whose Erdös Number are 1. Obviously, this is a social network. In the network, each node represents a researcher who has coauthored with Paul Erdös and each link could represent the coauthoring relationship between two researchers. Since the coauthor matrix is symmetrical, we know that there is no different between A coauthors withB and B coauthors with A. Therefore, the coauthor network is an undirected network which has 511 vertices. We use software Gephi to draw the graph and the network diagram is shown in Figure 1.Figure 1: the co-author networkIn this graph, the vertex represents a researcher and the edge represents the coauthoring relation. The size of the vertex represents its coauthoring times with Erdös and the darker the color is, the more people he coauthored with. There are 511 vertices and 18000 edges.In this network, there are many basic measures and metrics, such as Degree, Centrality, Clustering coefficient, Density, Betweenness and so on. In this paper, we first choose several important measures for analyzing this network and show them as follows. [1]Of course, the common property is CENTRALITY. Centrality is a crucial metric to evaluate the influence of a vertex. In the following, we discuss several classic Centralities and analyze theirdifference.⏹DEGREE CENTRALITYThe degree of a vertex in a graph is the number of edges connected to it. We will denote thedegree of vertex i by d i. And the simplest centrality measure, which is called degree centrality ( C d ), is just the degree of a vertex. That means:C d(i ) d iTeam #27688 Page 4 of 18In a social network, for instance, it seems reasonable to suppose that individuals whohave connections to many others might have more influence, more access to information, or more prestige than those who have fewer connections.⏹ EIGENVECTOR CENTRALITYSometimes, all neighbors of a vertex are not equivalent. Hence, Bonacich [2] puts forwardEigenvector centrality to cope with this situation. It assigns relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes.λd i = ∑r ij d j jWhere:r ij represents the elements in the adjacency matrix; d irepresents the degree centrality of vertex iUsually, we choose the eigenvector corresponding to the maximal eigenvalue to be the eihenvector centrality( C e )[3]. ⏹ CLOSENESS CENTRALITYCloseness centrality measures the mean distance from a vertex to other vertices, which canused to analyze the position of a vertex in the network [1]. ∑ D ijC c (i ) = j ( ≠i )-n 1Where: D ij is the distance between vertex i and vertex j ;C c (i ) is the closeness centrality of vertex i ;n is the number of vertices. ⏹ BETWEENNESS CENTRALITYBetweenness centrality measures the extent to which a vertex lies on paths between othervertices [1]. That is to say, a vertex with a higher betweenness centrality plays a moreimportant role in the connection of the network.n nC b ( k ) = ∑∑[ g ij ( k ) / g ij ] i jWhere: g ij ( k ) represents the number of shortest path between i andj through k ; g ij represents the number of shortest path between i andjThen, we use the UCINET to calculate some basic metrics and show them in table 1.Table 1: the basic data of centralitytype Degree Closeness Betweenness EigenvectorAverage 1.292 2.115 0.461 3.055Minimum 0.000 0.196 0.0000.000 Maximum 10.392 2.201 7.508 36.515According to the above table and Figure 1, we can know that about 30 vertices have 3 times more than the average degree. That is to say, these researchers have many coauthors. In addition, since the average value of closeness is close to its maximum, we know that there are few vertices。

美国大学生数学建模竞赛二等奖论文

美国⼤学⽣数学建模竞赛⼆等奖论⽂The P roblem of R epeater C oordination SummaryThis paper mainly focuses on exploring an optimization scheme to serve all the users in a certain area with the least repeaters.The model is optimized better through changing the power of a repeater and distributing PL tones,frequency pairs /doc/d7df31738e9951e79b8927b4.html ing symmetry principle of Graph Theory and maximum coverage principle,we get the most reasonable scheme.This scheme can help us solve the problem that where we should put the repeaters in general cases.It can be suitable for the problem of irrigation,the location of lights in a square and so on.We construct two mathematical models(a basic model and an improve model)to get the scheme based on the relationship between variables.In the basic model,we set a function model to solve the problem under a condition that assumed.There are two variables:‘p’(standing for the power of the signals that a repeater transmits)and‘µ’(standing for the density of users of the area)in the function model.Assume‘p’fixed in the basic one.And in this situation,we change the function model to a geometric one to solve this problem.Based on the basic model,considering the two variables in the improve model is more reasonable to most situations.Then the conclusion can be drawn through calculation and MATLAB programming.We analysis and discuss what we can do if we build repeaters in mountainous areas further.Finally,we discuss strengths and weaknesses of our models and make necessary recommendations.Key words:repeater maximum coverage density PL tones MATLABContents1.Introduction (3)2.The Description of the Problem (3)2.1What problems we are confronting (3)2.2What we do to solve these problems (3)3.Models (4)3.1Basic model (4)3.1.1Terms,Definitions,and Symbols (4)3.1.2Assumptions (4)3.1.3The Foundation of Model (4)3.1.4Solution and Result (5)3.1.5Analysis of the Result (8)3.1.6Strength and Weakness (8)3.1.7Some Improvement (9)3.2Improve Model (9)3.2.1Extra Symbols (10)Assumptions (10)3.2.2AdditionalAdditionalAssumptions3.2.3The Foundation of Model (10)3.2.4Solution and Result (10)3.2.5Analysis of the Result (13)3.2.6Strength and Weakness (14)4.Conclusions (14)4.1Conclusions of the problem (14)4.2Methods used in our models (14)4.3Application of our models (14)5.Future Work (14)6.References (17)7.Appendix (17)Ⅰ.IntroductionIn order to indicate the origin of the repeater coordination problem,the following background is worth mentioning.With the development of technology and society,communications technology has become much more important,more and more people are involved in this.In order to ensure the quality of the signals of communication,we need to build repeaters which pick up weak signals,amplify them,and retransmit them on a different frequency.But the price of a repeater is very high.And the unnecessary repeaters will cause not only the waste of money and resources,but also the difficulty of maintenance.So there comes a problem that how to reduce the number of unnecessary repeaters in a region.We try to explore an optimized model in this paper.Ⅱ.The Description of the Problem2.1What problems we are confrontingThe signals transmit in the way of line-of-sight as a result of reducing the loss of the energy. As a result of the obstacles they meet and the natural attenuation itself,the signals will become unavailable.So a repeater which just picks up weak signals,amplifies them,and retransmits them on a different frequency is needed.However,repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.In addition to geographical separation,the“continuous tone-coded squelch system”(CTCSS),sometimes nicknamed“private line”(PL),technology can be used to mitigate interference.This system associates to each repeater a separate PL tone that is transmitted by all users who wish to communicate through that repeater. The PL tone is like a kind of password.Then determine a user according to the so called password and the specific frequency,in other words a user corresponds a PL tone(password)and a specific frequency.Defects in line-of-sight propagation caused by mountainous areas can also influence the radius.2.2What we do to solve these problemsConsidering the problem we are confronting,the spectrum available is145to148MHz,the transmitter frequency in a repeater is either600kHz above or600kHz below the receiver frequency.That is only5users can communicate with others without interferences when there’s noPL.The situation will be much better once we have PL.However the number of users that a repeater can serve is limited.In addition,in a flat area ,the obstacles such as mountains ,buildings don’t need to be taken into account.Taking the natural attenuation itself is reasonable.Now the most important is the radius that the signals transmit.Reducing the radius is a good way once there are more users.With MATLAB and the method of the coverage in Graph Theory,we solve this problem as follows in this paper.Ⅲ.Models3.1Basic model3.1.1Terms,Definitions,and Symbols3.1.2Assumptions●A user corresponds a PLz tone (password)and a specific frequency.●The users in the area are fixed and they are uniform distribution.●The area that a repeater covers is a regular hexagon.The repeater is in the center of the regular hexagon.●In a flat area ,the obstacles such as mountains ,buildings don’t need to be taken into account.We just take the natural attenuation itself into account.●The power of a repeater is fixed.3.1.3The Foundation of ModelAs the number of PLz tones (password)and frequencies is fixed,and a user corresponds a PLz tone (password)and a specific frequency,we can draw the conclusion that a repeater can serve the limited number of users.Thus it is clear that the number of repeaters we need relates to the density symboldescriptionLfsdfminrpµloss of transmission the distance of transmission operating frequency the number of repeaters that we need the power of the signals that a repeater transmits the density of users of the areaof users of the area.The radius of the area that a repeater covers is also related to the ratio of d and the radius of the circular area.And d is related to the power of a repeater.So we get the model of function()min ,r f p µ=If we ignore the density of users,we can get a Geometric model as follows:In a plane which is extended by regular hexagons whose side length are determined,we move a circle until it covers the least regular hexagons.3.1.4Solution and ResultCalculating the relationship between the radius of the circle and the side length of the regular hexagon.[]()()32.4420lg ()20lg Lfs dB d km f MHz =++In the above formula the unit of ’’is .Lfs dB The unit of ’’is .d Km The unit of ‘‘is .f MHz We can conclude that the loss of transmission of radio is decided by operating frequency and the distance of transmission.When or is as times as its former data,will increase f d 2[]Lfs .6dB Then we will solve the problem by using the formula mentioned above.We have already known the operating frequency is to .According to the 145MHz 148MHz actual situation and some authority material ,we assume a system whose transmit power is and receiver sensitivity is .Thus we can conclude that ()1010dBm mW +106.85dBm ?=.Substituting and to the above formula,we can get the Lfs 106.85dBm ?145MHz 148MHz average distance of transmission .()6.4d km =4mile We can learn the radius of the circle is 40mile .So we can conclude the relationship between the circle and the side length of regular hexagon isR=10d.1)The solution of the modelIn order to cover a certain plane with the least regular hexagons,we connect each regular hexagon as the honeycomb.We use A(standing for a figure)covers B(standing for another figure), only when As don’t overlap each other,the number of As we use is the smallest.Figure1According to the Principle of maximum flow of Graph Theory,the better of the symmetry ofthe honeycomb,the bigger area that it covers(Fig1).When the geometric centers of the circle andthe honeycomb which can extend are at one point,extend the honeycomb.Then we can get Fig2,Fig4:Figure2Fig3demos the evenly distribution of users.Figure4Now prove the circle covers the least regular hexagons.Look at Fig5.If we move the circle slightly as the picture,you can see three more regular hexagons are needed.Figure 52)ResultsThe average distance of transmission of the signals that a repeater transmit is 4miles.1000users can be satisfied with 37repeaters founded.3.1.5Analysis of the Result1)The largest number of users that a repeater can serveA user corresponds a PL and a specific frequency.There are 5wave bands and 54different PL tones available.If we call a code include a PL and a specific frequency,there are 54*5=270codes.However each code in two adjacent regular hexagons shouldn’t be the same in case of interfering with each other.In order to have more code available ,we can distribute every3adjacent regular hexagons 90codes each.And that’s the most optimized,because once any of the three regular hexagons have more codes,it will interfere another one in other regular hexagon.2)Identify the rationality of the basic modelNow we considering the influence of the density of users,according to 1),90*37=3330>1000,so here the number of users have no influence on our model.Our model is rationality.3.1.6Strength and Weakness●Strength:In this paper,we use the model of honeycomb-hexagon structure can maximize the use of resources,avoiding some unnecessary interference effectively.It is much more intuitive once we change the function model to the geometric model.●Weakness:Since each hexagon get too close to another one.Once there are somebuildingsor terrain fluctuations between two repeaters,it can lead to the phenomenon that certain areas will have no signals.In addition,users are distributed evenly is not reasonable.The users are moving,for example some people may get a party.3.1.7Some ImprovementAs we all know,the absolute evenly distribution is not exist.So it is necessary to say something about the normal distribution model.The maximum accommodate number of a repeater is 5*54=270.As for the first model,it is impossible that 270users are communicating in a same repeater.Look at Fig 6.If there are N people in the area 1,the maximum number of the area 2to area 7is 3*(270-N).As 37*90=3330is much larger than 1000,our solution is still reasonable to this model.Figure 63.2Improve Model3.2.1Extra SymbolsSigns and definitions indicated above are still valid.Here are some extra signs and definitions.symboldescription Ra the radius of the circular flat area the side length of a regular hexagon3.2.2Additional AdditionalAssumptionsAssumptions ●The radius that of a repeater covers is adjustable here.●In some limited situations,curved shape is equal to straight line.●Assumptions concerning the anterior process are the same as the Basic Model3.2.3The Foundation of ModelThe same as the Basic Model except that:We only consider one variable(p)in the function model of the basic model ;In this model,we consider two varibles(p and µ)of the function model.3.2.4Solution and Result1)SolutionIf there are 10,000users,the number of regular hexagons that we need is at least ,thus according to the the Principle of maximum flow of Graph Theory,the 10000111.1190=result that we draw needed to be extended further.When the side length of the figure is equal to 7Figure 7regular hexagons,there are 127regular hexagons (Fig 7).Assuming the side length of a regular hexagon is ,then the area of a regular hexagon is a .The area of regular hexagons is equal to a circlewhose radiusis 22a =1000090R.Then according to the formula below:.221000090a R π=We can get.9.5858R a =Mapping with MATLAB as below (Fig 8):Figure 82）Improve the model appropriatelyEnlarge two part of the figure above,we can get two figures below (Fig 9and Fig 10):Figure 9AREAFigure 10Look at the figure above,approximatingAREA a rectangle,then obtaining its area to getthe number of users..The length of the rectangle is approximately equal to the side length of the regular hexagon ,athe width of the rectangle is ,thus the area of AREA is ,then R ?*R awe can get the number of users in AREA is()，2**10000 2.06R a R π=????????9.5858R a =As 2.06<<10,000,2.06can be ignored ,so there is no need to set up a repeater in.There are 6suchareas(92,98,104,110,116,122)that can be ignored.At last,the number of repeaters we should set up is,1276121?=2)Get the side length of the regular hexagon of the improved modelThus we can getmile=km 40 4.1729.5858a == 1.6* 6.675a =3)Calculate the power of a repeaterAccording to the formula[]()()32.4420lg ()20lg Lfs dB d km f MHz =++We get32.4420lg 6.67520lg14592.156Los =++=32.4420lg 6.67520lg14892.334Los =++=So we get106.85-92.156=14.694106.85-92.334=14.516As the result in the basic model,we can get the conclusion the power of a repeater is from 14.694mW to 14.516mW.3.2.5Analysis of the ResultAs 10,000users are much more than 1000users,the distribution of the users is more close toevenly distribution.Thus the model is more reasonable than the basic one.More repeaters are built,the utilization of the outside regular hexagon are higher than the former one.3.2.6Strength and Weakness●Strength:The model is more reasonable than the basic one.●Weakness:Repeaters don’t cover all the area,some places may not receive signals.And thefoundation of this model is based on the evenly distribution of the users in the area,if the situation couldn’t be satisfied,the interference of signals will come out.Ⅳ.Conclusions4.1Conclusions of the problem●Generally speaking,the radius of the area that a repeater covers is4miles in our basic model.●Using the model of honeycomb-hexagon structure can maximize the use of resources,avoiding some unnecessary interference effectively.●The minimum number of repeaters necessary to accommodate1,000simultaneous users is37.The minimum number of repeaters necessary to accommodate10,000simultaneoususers is121.●A repeater's coverage radius relates to external environment such as the density of users andobstacles,and it is also determined by the power of the repeater.4.2Methods used in our models●Analysis the problem with MATLAB●the method of the coverage in Graph Theory4.3Application of our models●Choose the ideal address where we set repeater of the mobile phones.●How to irrigate reasonably in agriculture.●How to distribute the lights and the speakers in squares more reasonably.Ⅴ.Future WorkHow we will do if the area is mountainous?5.1The best position of a repeater is the top of the mountain.As the signals are line-of-sight transmission and reception.We must find a place where the signals can transmit from the repeater to users directly.So the top of the mountain is a good place.5.2In mountainous areas,we must increase the number of repeaters.There are three reasons for this problem.One reason is that there will be more obstacles in the mountainous areas. The signals will be attenuated much more quickly than they transmit in flat area.Another reason is that the signals are line-of-sight transmission and reception,we need more repeaters to satisfy this condition.Then look at Fig11and Fig12,and you will know the third reason.It can be clearly seen that hypotenuse is larger than right-angleFig11edge(R>r).Thus the radius will become smaller.In this case more repeaters are needed.Fig125.3In mountainous areas,people may mainly settle in the flat area,so the distribution of users isn’t uniform.5.4There are different altitudes in the mountainous areas.So in order to increase the rate of resources utilization,we can set up the repeaters in different altitudes.5.5However,if there are more repeaters,and some of them are on mountains,more money will be/doc/d7df31738e9951e79b8927b4.html munication companies will need a lot of money to build them,repair them when they don’t work well and so on.As a result,the communication costs will be high.What’s worse,there are places where there are many mountains but few persons. Communication companies reluctant to build repeaters there.But unexpected things often happen in these places.When people are in trouble,they couldn’t communicate well with the outside.So in my opinion,the government should take some measures to solve this problem.5.6Another new method is described as follows(Fig13):since the repeater on high mountains can beFig13Seen easily by people,so the tower which used to transmit and receive signals can be shorter.That is to say,the tower on flat areas can be a little taller..Ⅵ.References[1]YU Fei,YANG Lv-xi,"Effective cooperative scheme based on relay selection",SoutheastUniversity,Nanjing,210096,China[2]YANG Ming,ZHAO Xiao-bo,DI Wei-guo,NAN Bing-xin,"Call Admission Control Policy based on Microcellular",College of Electical and Electronic Engineering,Shijiazhuang Railway Institute,Shijiazhuang Heibei050043,China[3]TIAN Zhisheng,"Analysis of Mechanism of CTCSS Modulation",Shenzhen HYT Co,Shenzhen,518057,China[4]SHANGGUAN Shi-qing,XIN Hao-ran,"Mathematical Modeling in Bass Station Site Selectionwith Lingo Software",China University of Mining And Technology SRES,Xuzhou;Shandong Finance Institute,Jinan Shandon,250014[5]Leif J.Harcke,Kenneth S.Dueker,and David B.Leeson,"Frequency Coordination in the AmateurRadio Emergency ServiceⅦ.AppendixWe use MATLAB to get these pictures,the code is as follows:1-clc;clear all;2-r=1;3-rc=0.7;4-figure;5-axis square6-hold on;7-A=pi/3*[0:6];8-aa=linspace(0,pi*2,80);9-plot(r*exp(i*A),'k','linewidth',2);10-g1=fill(real(r*exp(i*A)),imag(r*exp(i*A)),'k');11-set(g1,'FaceColor',[1,0.5,0])12-g2=fill(real(rc*exp(i*aa)),imag(rc*exp(i*aa)),'k');13-set(g2,'FaceColor',[1,0.5,0],'edgecolor',[1,0.5,0],'EraseMode','x0r')14-text(0,0,'1','fontsize',10);15-Z=0;16-At=pi/6;17-RA=-pi/2;18-N=1;At=-pi/2-pi/3*[0:6];19-for k=1:2;20-Z=Z+sqrt(3)*r*exp(i*pi/6);21-for pp=1:6;22-for p=1:k;23-N=N+1;24-zp=Z+r*exp(i*A);25-zr=Z+rc*exp(i*aa);26-g1=fill(real(zp),imag(zp),'k');27-set(g1,'FaceColor',[1,0.5,0],'edgecolor',[1,0,0]);28-g2=fill(real(zr),imag(zr),'k');29-set(g2,'FaceColor',[1,0.5,0],'edgecolor',[1,0.5,0],'EraseMode',xor';30-text(real(Z),imag(Z),num2str(N),'fontsize',10);31-Z=Z+sqrt(3)*r*exp(i*At(pp));32-end33-end34-end35-ezplot('x^2+y^2=25',[-5,5]);%This is the circular flat area of radius40miles radius 36-xlim([-6,6]*r) 37-ylim([-6.1,6.1]*r)38-axis off;Then change number19”for k=1:2;”to“for k=1:3;”,then we get another picture:Change the original programme number19“for k=1:2;”to“for k=1:4;”,then we get another picture:。

2012年美国国际大学生数学建模竞赛(MCM+ICM)题目+翻译

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

数学建模竞赛英语

数学建模竞赛英语Mathematical modeling competitions have been widely popular among students around the world. These competitions provide an excellent platform for students to showcase their mathematical abilities and critical thinking skills. One crucial aspect of mathematical modeling competitions is communicating the results in a clear and concise manner, which involves writing a comprehensive report in English. In this article, we will discuss the essential steps in preparing an excellent report for a math modeling competition in English.Step 1: Understanding the Problem StatementThe first step in writing a report for a math modeling competition in English is to understand the problem statement completely. You need to identify the problem statement's core issues and clarify any vague or ambiguous aspects of the problem. It's essential to understand what the competition organizers expect from you and what they're looking for in your report.Step 2: Identify the ObjectivesOnce you have a clear understanding of the problem, the next step is to identify the objectives you need to achieve through your report. It would help if you determined what variables and parameters you need to consider, what data you need to gather, and what questions you need to answer through your report. This step is vital because it helps you stay focused on the actual problem and provides you with a clear outline for your report.Step 3: Data Collection and AnalysisThe next step is to collect and analyze the data required to solve the problem statement. You should verify the validity and reliability of the data sources you use to ensure the accuracy of your results. You should also determine the most suitable data analysis tools for your specific problem.Step 4: Model DevelopmentAfter identifying the objectives and analyzing the data, the next step is to develop a suitable model that solves the problem statement. You should create a comprehensive model that considers every element involved in the problem statement, including any relevant constraints or limitations.Step 5: Report WritingFinally, you need to write a comprehensive report that presents your results, conclusions, and solutions in a clear and concise manner. It is essential to follow specific guidelines for formatting and structuring your report, including an introduction, a detailed description of the model, the results, and a conclusion.In conclusion, writing a report for a math modeling competition in English requires careful planning, attention to detail, and excellent communication skills. To create an effective and well-structured report, you need to have a clear understanding of the problem statement, identify the objectives, collect and analyze data, develop a suitable model, and write the report in a comprehensive and concise manner. By following these steps, you can write a winning report and achieve success in any math modeling competition.。