2014年美国数学建模大赛(MCM)试题译文

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2014年美赛C题翻译

This will take some skilled data extraction and modeling efforts to obtain the correct set of nodes (the Erdös coauthors) and their links (connections with one another as coauthors). 这需要熟练数据提取并在建模上下功夫，以便得到正确的节点和边
Once built, analyze the properties of this network. 建完后分析网络性能(Again, do not include Erdös --- he is the most infnodes in the network. In this case, it’s co-authorship with him that builds the network, but he is not part of the network or the analysis.)
One of the techniques to determine influence of academic research is to build and measure properties of citation or co-author networks.
学术研究的技术来确定影响之一是构建和引文或合著网络的度量属性。
Google Scholar is also a good data tool to use for network influence or impact data collection and analysis.
谷歌学术搜索也是一个好的数据工具用于网络数据收集和分析影响或影响。

美国大学生数学建模比赛2014年B题

Team # 26254
Page 2 oon ............................................................................................................................................................. 3 2. The AHP .................................................................................................................................................................. 3 2.1 The hierarchical structure establishment ....................................................................................................... 4 2.2 Constructing the AHP pair-wise comparison matrix...................................................................................... 4 2.3 Calculate the eigenvalues and eigenvectors and check consistency .............................................................. 5 2.4 Calculate the combination weights vector ..................................................................................................... 6 3. Choosing Best All Time Baseball College Coach via AHP and Fuzzy Comprehensive Evaluation ....................... 6 3.1 Factor analysis and hierarchy relation construction....................................................................................... 7 3.2 Fuzzy comprehensive evaluation ................................................................................................................... 8 3.3 calculating the eigenvectors and eigenvalues ................................................................................................ 9 3.3.1 Construct the pair-wise comparison matrix ........................................................................................ 9 3.3.2 Construct the comparison matrix of the alternatives to the criteria hierarchy .................................. 10 3.4 Ranking the coaches .....................................................................................................................................11 4. Evaluate the performance of other two sports coaches, basketball and football.................................................... 13 5. Discuss the generality of the proposed method for Choosing Best All Time College Coach ................................ 14 6. The strengths and weaknesses of the proposed method to solve the problem ....................................................... 14 7. Conclusions ........................................................................................................................................................... 15

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。

美赛数学建模A题翻译版论文

美赛数学建模A题翻译版论文The document was finally revised on 2021数学建模竞赛(MCM / ICM)汇总表基于细胞的高速公路交通模型自动机和蒙特卡罗方法总结基于元胞自动机和蒙特卡罗方法,我们建立一个模型来讨论“靠右行”规则的影响。

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

然后我们设计规则来模拟车辆的运动模型。

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

我们也设计一个道路的危险指数评价公式。

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

通过计算机和分析数据。

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

我们利用不同的速度限制分析模型的敏感性和看到不同的限速的影响。

左手交通也进行了讨论。

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

1介绍术语假设2模型设计的元胞自动机流入模型跟随模型超车模型超车概率超车条件危险指数两套规则CA模型靠右行无限制行驶规则3补充分析模型加速和减速概率分布的设计设计来避免碰撞4模型实现与计算机5数据分析和模型验证平均速度快车的平均速度密度超车几率危险指数6在不同速度限制下敏感性评价模型7驾驶在左边8交通智能系统智能系统的新规则模型的适应度智能系统结果9结论10优点和缺点优势弱点引用附录。

1 Introduction今天,大约65%的世界人口生活在右手交通的国家和35%在左手交通的国家交通流量。

[worldstandards。

欧盟,2013] 右手交通的国家,比如美国和中国,法规要求驾驶在靠路的右边行走。

多车道高速公路在这些国家经常使用一个规则,要求司机在最右边开车除非他们超过另一辆车,在这种情况下,他们移动到左边的车道、通过，返回到原来的车道。

HIMCM 2014美国中学生数学建模竞赛试题

HIMCM 2014美国中学生数学建模竞赛试题Problem A: Unloading Commuter TrainsTrains arrive often at a central Station, the nexus for many commuter trains from suburbs of larger cities on a “commuter” line. Most trains are long (perhaps 10 or more cars long). The distance a passenger has to walk to exit the train area is quite long. Each train car has only two exits, one near each end so that the cars can carry as many people as possible. Each train car has a center aisle and there are two seats on one side and three seats on the other for each row of seats.To exit a typical station of interest, passengers must exit the car, and then make their way to a stairway to get to the next level to exit the station. Usually these trains are crowded so there is a “fan” of passengers from the train trying to get up the stairway. The stairway could accommodate two columns of people exiting to the top of the stairs.Most commuter train platforms have two tracks adjacent to the platform. In the worst case, if two fully occupied trains arrived at the same time, it might take a long time for all the passengers to get up to the main level of the station.Build a mathematical model to estimate the amount of time for a passenger to reach the street level of the station to exit the complex. Assume there are n cars to a train, each car has length d. The length of the platform is p, and the number of stairs in each staircase is q. Use your model to specifically optimize (minimize) the time traveled to reach street level to exit a station for the following:问题一:通勤列车的负载问题在中央车站,经常有许多的联系从大城市到郊区的通勤列车“通勤”线到达。

2014美国数学建模竞赛赛题翻译

问题A：右行左超规则在美国、中国和大多数除了英国、澳大利亚和一些前英国殖民地的国家，多车道高速公路常常有这样一种规则。

司机必须尽量在最右的车道行使，只有超车时，司机才可以向左移动一个车道来达成目的。

当司机超车完毕后必须回到原车道继续行使。

建立并分析一个数学模型，使得这个模型能够分析这个规则在交通高负荷和低负荷情况下的表现。

你可以从许多角度来思考这个问题，比如车流量和车辆安全之间的权衡，或者一个过快或过慢的车辆限速带来的影响等等。

这个规则可以使我们获得更好的交通流？如果不可以，请提出并分析一个替代方案使得交通流得到优化、安全得到保障、或者其他你认为重要的因素得到实现。

在靠左行使才是规则的国家，论证你的解决方案是否可以通过简单的变换或者通过增加一些新的要求来解决相同的问题。

最后，以上的规则的实行是建立在人们遵守它的基础上的，然而不是所有人都愿意去遵守。

那么现在我们使同一条道（可以只是一段，也可以是全段公路）上的交通车辆都在一个智能系统的严格控制下，这个变化对你之前的分析结果有多大的影响？问题B：体育画刊是一个为体育爱好者们设计的杂志。

这个杂志正在寻找上世纪女性或者男性的“历来最优秀的大学教练”。

建立一个数学模型，从男性或者女性体育教练中选择最好的大学教练（退役或者在役的都可以）。

这些体育教练可以是大学曲棍球、陆上曲棍球、足球、橄榄球、棒球、排球、篮球的教练。

你选择划分的时间会对你的分析有影响吗？也就是说，1913年的教练方式和2013年的会有什么不同吗？清楚的阐述你的评估方式。

讨论你的模型如何通用于两性教练和所有可能的运动项目上。

用你的模型为三项体育项目分别找到五个最佳教练。

再为体育画刊提供一篇1－2页的不涉及技术性问题解释的通俗易懂的文章来解释你们的结果，你们必须保证体育爱好者们能够理解。

2014美赛题目(翻译版)

2014 MCM ProblemsPROBLEM A: The Keep-Right-Except-To-Pass RuleIn countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employ a rule that requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane.在一些以行车靠右为规则的国家中（比如美国、中国以及除了大不列颠、澳大利亚和一些前英属殖民国家以外的其他国家），多行道的高速公路经常采用要求驾驶人在除超车以外时都靠右行驶的交通规则。

Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wish to examine tradeoffs between traffic flow and safety, the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement. Is this rule effective in promoting better traffic flow? If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flow, safety, and/or other factors that you deem important.1、请你建立和分析一个数学模型来分析这个规则在交通畅通和交通堵塞条件下的表现。

2014美赛赛题翻译MCM(非机器翻译)

2014年赛题
MCM 问题
问题 A :除非超车否则靠右行驶的交通规则
在一些汽车靠右行驶的国家(比如美国,中国等等 ,多车道的高速公路常常遵循以下原则:司机必须在最右侧驾驶, 除非他们正在超车, 超车时必须先移到左侧车道在超车后再返回。

建立数学模型来分析这条规则在低负荷和高负荷状态下的交通路况的表现。

你不妨考察一下流量和安全的权衡问题 , 车速过高过低的限制,或者这个问题陈述中可能出现的其他因素。

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

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

最后,以上规则依赖于人的判断,如果相同规则的交通运输完全在智能系统的控制下, 无论是部分网络还是嵌入使用的车辆的设计,在何种程度上会修改你前面的结果?
问题 B :大学传奇教练
体育画报,一个给运动爱好者的杂志,正在寻找“ 最好的大学教练” (男女不限、上个世纪。

建立一个数学模型来选择最佳的高校冰球、曲棍球、足球、棒球、垒球、篮球、足球教练 (们。

你的分析中不同时间的教练水平是否在同一水平线,也就是说, 1913年执教与 2013是否相同?清楚地说明您进行评估的指标。

讨论如何你的模型是否可以推广到一般的跨越男女性别的项目(?和其他的运动。

展示你的模型在每 3个不同的运动项目的前 5名教练。

除了 MCM 的格式和要求,准备一
份 1-2页的文章体育画报,解释你的结果,要让非技术性解释体育迷就明白了你的那个数学模型。

【VIP专享】2010 -2014MCM Problems建模竞赛美赛题目中英文双语翻译版

2010 MCM ProblemsPROBLEM A: The Sweet SpotExplain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe that “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?MCM 2010 A题：解释棒球棒上的“最佳击球点”每一个棒球手都知道在棒球棒比较粗的部分有一个击球点，这里可以把打击球的力量最大程度地转移到球上。

2014美国数学建模竞赛 C题论文【COMAP 31223】

Team # 31223
Page 2 of 19
Contents
1. Introduction 2. Model 2.1Properties of a Successful Model 2.2Assumption 2.3Build Evaluating igist of scoring training ability 2.4.2 the process of scoring training ability 2.5.1 Set the personal quality and management capability system 2.5.2 Build Hierarchical levels 2.5.3build judgment matrix 3.1 the best basketball coach 3.1.1 The Filtering Model 3.1.2 The AHP Model 3.1.3 Consistency Test 3.2the best football coach 3.3the best hockey coach 4. Conclusions 5. Adaptability of the model 5.1 Applicability of the time 5.2 Applicability of the different sports 6.Sensitivity Test 7. Assessment 8. References 9.Advertising shee
Abstract
Considering the coach and the player final destination is to win the games, so we choose the coach’s training ability into our indexes evaluation system. Like that, we continue choose the personal quality and management capability into our indexes evaluation system. Then Among these indexes, we take training ability as the key index. Only the coaches who rank top ten are certificated to enter the next evaluating system, that is the AHP model. Next, the AHP model takes training ability, personal quality and management capability as criteria. So we can find the highest weight coach. In the application of basketball, first, we use the data of win-loss percentage, Conference Champion, NCAA Final Fours and NCAA Championship to measure on the training ability. Then choose 10 coaches from 3513 ones. Then, we make a list of the length of coaching career, life major events, major honors and awards to determine their personal qualities. What is more, to objectively measure the priorities, we apply average weighted sum, Wilcox Rank method to the normalized data. Finally, we obtain the final weights of the 10 coaches. It is shown that the top 5 coaches are John Wooden, Dean Smith，Mike Krzyzewski，Rick Pitino and Roy Williams. In our model, it contains various information of the coach, like whether he has some good or bad behavior, how much is the team improved under the guidance of the coach. So the model can comprehensive evaluation a coach. To test the applicability of the model, we collect 2051 and 351 samples from the internet, and find that the model’s evaluation system can be widely applied to other sports. Through comparison and analysis, we find that although training ability is very important when evaluating coaches, other indexes matters too. For example, Rick Pitino ranks fifth in the end, but his training ability ranks out of top 5. It is his significant management improves his final rank. We adopt Wilcox Rank Test in time horizon problem. Then, the significance level is less than 0.05. So we carefully compare the data and information classified by time. It comes out that more and more regular seasons are held accordingly. But our model does not consider the factor.

2014年美国数学竞赛AMC8题目和答案解析

2.3.What is the value of ?4.Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill?5.Hammie is in the grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?6.The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row?7.Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?8.A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?9.The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?10.What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?11. Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he joggedat a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles perhour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?12. At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pairof sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?13.When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score.By which of the following might her incorrect sum have differed from the correct one?14. Let the two digits be and .The correct score was . Clara misinterpreted it as . The difference between the two is which factors into . Therefore, since the difference is a multiple of 9, the only answer choice that is a multiple of 9 is .15. If , , and , what is the product of , , and ?16. A number of students from Fibonacci Middle School are taking part in a community service project. Theratio of -graders to -graders is , and the the ratio of -graders to -graders is . What is the smallest number of students that could be participating in the project?17. The sum of six consecutive positive integers is 2013. What is the largest of these six integers?18. Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide,and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?--Arpanliku 16:22, 27 November 2013 (EST) Courtesy of Lord.of.AMC19. Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget andCassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?20. A rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of thesemicircle?21.Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blockseast and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?22.Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicksare used altogether?23.Angle of is a right angle. The sides of are the diameters of semicircles as shown. The area of the semicircle on equals , and the arc of the semicircle on has length . What isthe radius of the semicircle on ?24. Squares , , and are equal in area. Points and are the midpoints of sidesand , respectively. What is the ratio of the area of the shaded pentagon to the sum of the areas of the three squares?25. A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprisedof 3 semicircular arcs whose radii are inches, inches, and inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of theball travels over the course from A to B?1.2.The 50% off price of half a pound of fish is $3, so the 100%, or the regular price, of a half pound of fish is $6. Consequently, if half a pound of fish costs $6, then a whole pound of fish is dollars.3.Notice that we can pair up every two numbers to make a sum of 1:Therefore, the answer is .4.Each of her seven friends paid to cover Judi's portion. Therefore, Judi's portion must be . Since Judi was supposed to pay of the total bill, the total bill must be .5.The median here is obviously less than the mean, so option (A) and (B) are out.Lining up the numbers (5, 5, 6, 8, 106), we see that the median weight is 6 pounds.The average weight of the five kids is .Therefore, the average weight is bigger, by pounds, making the answer .6.Solution 1: Working BackwardsLet the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.We see that , making .It follows that , so .Solution 2: Jumping Back to the StartAnother way to do this problem is to realize what makes up the bottommost number. This method doesn't work quite as well for this problem, but in a larger tree, it might be faster. (In this case, Solution 1 would be faster since there's only two missing numbers.)Again, let the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.We can write some equations:Now we can substitute into the first equation using the two others:7.If Trey saw , then he saw .2 minutes and 45 seconds can also be expressed as seconds.Trey's rate of seeing cars, , can be multiplied by on the top and bottom (and preserve the same rate):. It follows that the most likely number of cars is . Solution 2minutes and seconds is equal to .Since Trey probably counts around cars every seconds, there are groups of cars that Trey most likely counts. Since , the closest answer choice is .8.First, there are ways to flip the coins, in order.The ways to get two consecutive heads are HHT and THH.The way to get three consecutive heads is HHH.Therefore, the probability of flipping at least two consecutive heads is .9.This is a geometric sequence in which the common ratio is 2. To find the jump that would be over a 1000 meters, we note that .However, because the first term is and not , the solution to the problem is10. To find either the LCM or the GCF of two numbers, always prime factorize first.The prime factorization of .The prime factorization of .Then, find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is ). Multiply all of these to get 5940.For the GCF of 180 and 594, use the least power of all of the numbers that are in both factorizations and multiply. = 18.Thus the answer = = .We start off with a similar approach as the original solution. From the prime factorizations, the GCF is .It is a well known fact that . So we have,.Dividing by yields .Therefore, .11.We use that fact that . Let d= distance, r= rate or speed, and t=time. In this case, let represent the time.On Monday, he was at a rate of . So, .For Wednesday, he walked at a rate of . Therefore, .On Friday, he walked at a rate of . So, .Adding up the hours yields + + = .We now find the amount of time Grandfather would have taken if he walked at per day. Set up the equation,.To find the amount of time saved, subtract the two amounts: - = . To convert this to minutes, we multiply by .Thus, the solution to this problem is12. First, find the amount of money one will pay for three sandals without the discount. We have.Then, find the amount of money using the discount: .Finding the percentage yields .To find the percent saved, we have13. Let the two digits be and .The correct score was . Clara misinterpreted it as . The difference between the two is which factors into . Therefore, since the difference is a multiple of 9, the only answer choice that is a multiple of 9 is .14. The probability that both show a green bean is . The probability that both show a red bean is . Therefore the probability is15.Therefore, .Therefore, .To most people, it would not be immediately evident that , so we can multiply 6's until we get the desired number:, so .Therefore the answer is .16. Solution 1: AlgebraWe multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:Therefore, the ratio of 8th graders to 7th graders to 6th graders is . Since the ratio is in lowest terms, the smallest number of students participating in the project is .Solution 2: FakesolvingThe number of 8th graders has to be a multiple of 8 and 5, so assume it is 40 (the smallest possibility). Then there are 6th graders and 7th graders. The numbers of students is17. Solution 1The mean of these numbers is . Therefore the numbers are , so the answer isSolution 2Let the number be . Then our desired number is .Our integers are , so we have that.Solution 3Let the first term be . Our integers are . We have,18. Solution 1There are cubes on the base of the box. Then, for each of the 4 layers above the bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are 4 feet left), there arecubes. Hence, the answer is .Solution 2We can just calculate the volume of the prism that was cut out of the original box. Each interior side of the fort will be feet shorter than each side of the outside. Since the floor is foot, the height will be feet. So the volume of the interior box is .The volume of the original box is . Therefore, the number of blocks contained in thefort is .19. If Hannah did better than Cassie, there would be no way she could know for sure that she didn't get thelowest score in the class. Therefore, Hannah did worse than Cassie. Similarly, if Hannah did worse than Bridget, there is no way Bridget could have known that she didn't get the highest in the class. Therefore, Hannah didbetter than Bridget, so our order is .20.A semicircle has symmetry, so the center is exactly at the midpoint of the 2 side on the rectangle, making the radius, by the Pythagorean Theorem, . The area is .21.The number of ways to get from Samantha's house to City Park is , and the number of ways to get from City Park to school is . Since there's one way to go through City Park (just walking straight through), the number of different ways to go from Samantha's house to City Park to school .22.There are vertical columns with a length of toothpicks, and there are horizontal rows with a length of toothpicks. An effective way to verify this is to try a small case, i.e. a grid of toothpicks. Thus, our answer is .23. Solution 1If the semicircle on AB were a full circle, the area would be 16pi. Therefore the diameter of the first circle is 8. The arc of the largest semicircle would normally have a complete diameter of 17. The Pythagorean theoremsays that the other side has length 15, so the radius is .Solution 2We go as in Solution 1, finding the diameter of the circle on AC and AB. Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of the largestis , and the middle one is , so the radius is .24.First let (where is the side length of the squares) for simplicity. We can extend until it hitsthe extension of . Call this point . The area of triangle then is The area of rectangle is . Thus, our desired area is . Now, the ratio of the shaded area to the combinedarea of the three squares is .Solution 2Let the side length of each square be .Let the intersection of and be .Since , . Since and are vertical angles, they are congruent. We also have by definition.So we have by congruence. Therefore, .Since and are midpoints of sides, . This combined with yields.The area of trapezoid is .The area of triangle is .So the area of the pentagon is .The area of the squares is .Therefore, .Solution 3Let the intersection of and be .Now we have and .Because both triangles has a side on congruent squares therefore .Because and are vertical angles .Also both and are right angles so .Therefore by AAS(Angle, Angle, Side) .Then translating/rotating the shaded into the position ofSo the shaded area now completely covers the squareSet the area of a square asTherefore, .25. Solution 1The radius of the ball is 2 inches. If you think about the ball rolling or draw a path for the ball (see figure below), you see that in A and C it loses inches, and it gains inches on B.So, the departure from the length of the track means that the answer is . Solution 2The total length of all of the arcs is . Since we want the path from the center, the actual distance will be shorter. Therefore, the only answer choice less than is . Thissolution may be invalid because the actual distance can be longer if the path the center travels is on the outside of the curve, as it is in the middle bump.。

2014美国数学建模竞赛 MCM A题参考答案元胞自动机

clfclear all%build the GUI%define the plot buttonplotbutton=uicontrol('style','pushbutton',...'string','Run', ...'fontsize',12, ...'position',[100,400,50,20], ...'callback', 'run=1;');%define the stop buttonerasebutton=uicontrol('style','pushbutton',...'string','Stop', ...'fontsize',12, ...'position',[200,400,50,20], ...'callback','freeze=1;');%define the Quit buttonquitbutton=uicontrol('style','pushbutton',...'string','Quit', ...'fontsize',12, ...'position',[300,400,50,20], ...'callback','stop=1;close;');number = uicontrol('style','text', ...'string','1', ...'fontsize',12, ...'position',[20,400,50,20]);%CA setupn=100;%数据初始化z=zeros(1,n);%元胞个数z=roadstart(z,5);%道路状态初始化，路段上随机分布5辆cells=z;vmax=3;%最大速度v=speedstart(cells,vmax);%速度初始化x=1;%记录速度和车辆位置memor_cells=zeros(3600,n);memor_v=zeros(3600,n);imh=imshow(cells);%初始化图像白色有车，黑色空元胞set(imh, 'erasemode', 'none')axis equalaxis tightstop=0; %wait for a quit button pushrun=0; %wait for a drawfreeze=0; %wait for a freeze（冻结）while (stop==0)if(run==1)%边界条件处理，搜素首末车，控制进出，使用开口条件a=searchleadcar(cells);b=searchlastcar(cells);[cells,v]=border_control(cells,a,b,v,vmax); i=searchleadcar(cells);%搜索首车位置for j=1:iif i-j+1==n[z,v]=leadcarupdate(z,v);continue;else%======================================加速、减速、随机慢化 if cells(i-j+1)==0;%判断当前位置是否非空continue;else v(i-j+1)=min(v(i-j+1)+1,vmax);%加速%=================================减速k=searchfrontcar((i-j+1),cells);%搜素前方首个非空元胞位置if k==0;%确定于前车之间的元胞数d=n-(i-j+1);else d=k-(i-j+1)-1;endv(i-j+1)=min(v(i-j+1),d);%==============================%减速%随机慢化v(i-j+1)=randslow(v(i-j+1));new_v=v(i-j+1);%======================================加速、减速、随机慢化%更新车辆位置z(i-j+1)=0;z(i-j+1+new_v)=1;%更新速度v(i-j+1)=0;v(i-j+1+new_v)=new_v;endendendcells=z;memor_cells(x,:)=cells;%记录速度和车辆位置memor_v(x,:)=v;x=x+1;set(imh,'cdata',cells)%更新图像%update the step number diaplaypause(0.2);stepnumber = 1 + str2num(get(number,'string'));set(number,'string',num2str(stepnumber))endif (freeze==1)run = 0;freeze = 0;enddrawnowend///////////////////////////////////////////////////////////////////////Function[new_matrix_cells,new_v]=border_control(matrix_cells,a,b,v,vmax) %边界条件，开口边界，控制车辆出入%出口边界，若头车在道路边界，则以一定该路0.9离去n=length(matrix_cells);if a==nrand('state',sum(100*clock)*rand(1));%¶¨ÒåËæ»úÖÖ×Óp_1=rand(1);%产生随机概率if p_1<=1 %如果随机概率小于0.9，则车辆离开路段，否则不离口matrix_cells(n)=0;v(n)=0;endend%入口边界，泊松分布到达，1s内平均到达车辆数为q，t为1sif b>vmaxt=1;q=0.25;x=1;p=(q*t)^x*exp(-q*t)/prod(x);%1s内有1辆车到达的概率rand('state',sum(100*clock)*rand(1));p_2=rand(1);if p_2<=pm=min(b-vmax,vmax);matrix_cells(m)=1;v(m)=m;endendnew_matrix_cells=matrix_cells;new_v=v;///////////////////////////////////////////////////////////////////////function [new_matrix_cells,new_v]=leadcarupdate(matrix_cells,v)%第一辆车更新规则n=length(matrix_cells);if v(n)~=0matrix_cells(n)=0;v(n)=0;endnew_matrix_cells=matrix_cells;new_v=v;///////////////////////////////////////////////////////////////////////function [new_v]=randslow(v)p=0.3;%慢化概率rand('state',sum(100*clock)*rand(1));%¶¨ÒåËæ»úÖÖ×Óp_rand=rand;%产生随机概率if p_rand<=pv=max(v-1,0);endnew_v=v;///////////////////////////////////////////////////////////////////////function [matrix_cells_start]=roadstart(matrix_cells,n)%道路上的车辆初始化状态，元胞矩阵随机为0或1，matrix_cells初始矩阵，n初始车辆数k=length(matrix_cells);z=round(k*rand(1,n));for i=1:nj=z(i);if j==0matrix_cells(j)=0;elsematrix_cells(j)=1;endendmatrix_cells_start=matrix_cells;///////////////////////////////////////////////////////////////////////function[location_frontcar]=searchfrontcar(current_location,matrix_cells)i=length(matrix_cells);if current_location==ilocation_frontcar=0;elsefor j=current_location+1:iif matrix_cells(j)~=0location_frontcar=j;break;elselocation_frontcar=0;endendend///////////////////////////////////////////////////////////////////////function [location_lastcar]=searchlastcar(matrix_cells)%搜索尾车位置for i=1:length(matrix_cells)if matrix_cells(i)~=0location_lastcar=i;break;else %如果路上无车，则空元胞数设定为道路长度location_lastcar=length(matrix_cells);endend///////////////////////////////////////////////////////////////////////function [location_leadcar]=searchleadcar(matrix_cells)i=length(matrix_cells);for j=1:iif matrix_cells(i-j+1)~=0location_leadcar=i-j+1;break;elselocation_leadcar=0;endend///////////////////////////////////////////////////////////////////////function [v_matixcells]=speedstart(matrix_cells,vmax)%道路初始状态车辆速度初始化v_matixcells=zeros(1,length(matrix_cells));for i=1:length(matrix_cells)if matrix_cells(i)~=0v_matixcells(i)=round(vmax*rand(1));endend。

2014美赛ICM赛题翻译

2014 ICM Problem使用网络来测量的影响和冲击Using Networks to Measure Influence and Impact确定学术研究的影响力的一种方法是构建和测量引文或合著者网络的属性.共同创作的文章通常意味着研究者之间的影响力有了重要的连接。

其中最有名的学术合著者是20世纪的数学家保罗·埃尔德什（Paul Erdös）他拥有超过500共同作者，并发表了超过1400的技术研究论文。

说埃尔德什（Erdös）是具有学科交叉特点的网络科学（science of networks）这一新兴研究的奠基人之一或许具有讽刺意味，或者也不。

特别是1959年，他和阿尔弗雷德的共同撰写的论文“关于随机图”（“On Random Graphs”）的发表，使得埃尔德什的作为合作者的角色在数学领域变得十分重要以至于数学家们经常会通过分析埃尔德什数量惊人的合作者来衡量自己与埃尔德什联系的紧密程度（their closeness to Erdös）。

(see the website http://www.oakla /enp/ )保罗.埃尔德什作为一个天才的数学家，天才的问题解决者，和著名合作者的不寻常和令人着迷的故事公布也在了许多书籍和在线网站上。

(例如,/Biographies/Erdos.html ).也许，他的生活方式就是经常和他的合作者待在一起或者住在一起。

或者把钱给他的学生作为解决问题的奖励，从而使他的合作蓬勃发展，并帮助他在数学的几个领域里建立了具有惊人影响力的网络。

为了衡量诸如埃尔德什等人产生的影响力，已经有了一些基于网络评价的工具（network-based evaluation tools）。

即是利用共同作者和引文数据来确定学者，出版物和学术期刊的影响因子，比如：科学文献索引（Science Citation Index，SCI的），H- factor（一种评价学术成就的新方法）Impact factor （期刊影响因子，SCI），Eigenfactor等等。

2014数学建模美赛题目

PROBLEM A: The Keep-Right-Except-To-Pass RuleIn countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employ a rule that requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane.Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wish to examine tradeoffs between traffic flow and safety, the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement. Is this rule effective in promoting better traffic flow? If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flow, safety, and/or other factors that you deem important.In countries where driving automobiles on the left is the norm, argue whether or not your solution can be carried over with asimple change of orientation, or would additional requirements be needed.Lastly, the rule as stated above relies upon human judgment for compliance. If vehicle transportation on the same roadway was fully under the control of an intelligent system – either part of the road network or imbedded in the design of all vehicles using the roadway – to what extent would this change the results of your earlier analysis?PROBLEM B: College Coaching LegendsSports Illustrated, a magazine for sports enthusiasts, is looking for the “best all time college coach” male or female for the previous century. Build a mathematical model to choose the best college coach or coaches (past or present) from among either male or female coaches in such sports as college hockey or field hockey, football, baseball or softball, basketball, or soccer. 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? Clearly articulate your metrics for assessment. Discuss how your model can be applied in general across both genders and all possible sports. Pre sent your model’s top 5 coaches in each of3 different sports.In addition to the MCM format and requirements, prepare a 1-2 page article for Sports Illustrated that explains your results and includes a non-technical explanation of your mathematical model that sports fans will understand.ICM PROBLEMPROBLEM C: Using Networks to Measure Influence and Impact Click the title below to download a PDF of the 2014 ICM Problem. Your ICM submission should consist of a 1 page Summary Sheet and your solution cannot exceed 20 pages for a maximum of 21 pages.Using Networks to Measure Influence and Impact。

2014年数学建模美赛题目原文及翻译

2014年数学建模美赛题目原文及翻译作者：Ternence Zhang转载注明出处：MCM原题PDF：PROBLEM A: The Keep-Right-Except-To-Pass RuleIn countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employ a rule that requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane.Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wish to examine tradeoffs between traffic flow and safety, the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement. Is this ruleeffective in promoting better traffic flow? If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flow, safety, and/or other factors that you deem important.In countries where driving automobiles on the left is the norm, argue whether or not your solution can be carried over with a simple change of orientation, or would additional requirements be needed.Lastly, the rule as stated above relies upon human judgment for compliance. If vehicle transportation on the same roadway was fully under the control of an intelligent system –either part of the road network or imbedded in the design of all vehicles using the roadway –to what extent would this change the results of your earlier analysis?问题A：车辆右行在一些规定汽车靠右行驶的国家（即美国，中国和其他大多数国家，除了英国，澳大利亚和一些前英国殖民地），多车道的高速公路经常使用这样一条规则：要求司机开车时在最右侧车道行驶，除了在超车的情况下，他们应移动到左侧相邻的车道，超车，然后恢复到原来的行驶车道（即最右车道）。

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2014年美国数学建模大赛（MCM）试题译文
王景璟大连理工大学
问题A：超车之外靠右行原则
在一些开车必须靠右行驶的国家（比如：美国，中国，以及其他除了英国，澳大利亚，和一些前英国殖民地的国家），行驶在多车道高速路必须遵循一个规则，那就是要求驾驶员在超车之外的情况下，必须在最靠右的车道行驶，超车时，他们向左变道，超车，然后再回到之前行驶的车道。

构建一个数学模型来分析该规则在车流量很少和很大的时候的执行情况。

你最好能考察车流量与安全的之间的相互关系，过低或是过量的速度限制的作用（速度设置过低或是过高），以及/或者其他在该问题陈述中没有明确提到的因素。

该原则是否能有效促进更好的车流量？如果无效，请建议和分析其他更有助于提高车流量、安全、以及其他你认为重要的因素的其他方案（可以完全不包括该原则）。

在开车靠左行的国家，讨论一下你的方案在经过对方向的简单修改之后或是添加额外的要求之后是否也适用。

最后，以上原则取决于人们遵循交通规则的判断力。

如果道路上的车流完全在智能系统（要么是道路体系的一部分，要么是包含在使用道路的所有车辆的设计之中）的控制之下，该改变在多大程度上会影响你先前分析的结果？
问题B: 大学教练联盟
《体育画报》，一本体育爱好者的杂志，正在寻找上世纪“最好的大学教练”，包括男性和女性。

建立一个数学模型以从诸如大学曲棍球，曲棍球，橄榄球，棒球，垒球，篮球，或足球等运动的男性或女性教练中选出最好的一个教练或几个教练（过去的或现在的）。

分析中使用的时间分界线是否有影响？即在1913执教和在2013年执教有不同吗？清晰地表达你们模型中的评判标准。

讨论你们的模型如何能广泛地应用于两种性别及所有可能的体育运动。

分别选出你模型中3种不同运动的前5位教练。

除了MCM格式及要求，准备一篇1-2页的文章给《体育画报》以解释你们的结论并包括一份能让体育迷们看懂的对你们数学模型的非技术性解释。

问题C：使用网络模型测量影响力。