2014美国数学建模C题ICM附件中Erdos 1 合作者人名---全部大写人名

2014美国建模比赛-详表（2014-2-7）MCM:The Mathematical Contest in ModelingMCM：数学建模竞赛ICM:The InterdisciplinaryContest in ModelingICM：交叉学科建模竞赛ContestRules, Registration and Instructions比赛规则，比赛注册方式和参赛指南(All rules and instructions apply to both ICM and MCMcontests, except whereotherwisenoted.)（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 sponsoredby a faculty advisor fromits institution.每个MCM的参赛队需有一名所在单位的指导教师负责。

Team Advisors: Please read these instructions carefully. It is yourresponsibility to make sure that teams are correctly registered and that all ofthe following steps required for participation in the contest are completed:Please print a copy of these contest instructions forreference before, during, and after the contest. Clickhere for the printer friendly version.指导老师：请认真阅读这些说明，确保完成了所有相关的步骤。

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。

2015 ICM Problem CManaging Human Capital in Organizations组织中的人力资本管理Building an organization filled with good, talented, well-trained people is one of the keys to success. But to do this, an organization needs to do more than recruit and hire the best candidates – they also need to retain good people, keep them properly trained and placed in proper positions, and eventually target new hires to replace those leaving the organization. Individuals play unique roles within their organizations, both formally and informally. Thus, the departure of individuals from an organization leaves important informational and functional components missing that need to be replaced. This is true for sports teams, commercial companies, schools and universities, governments, and almost any formal group or organization of people.构建一个组织填充好，有才华的，训练有素的人是成功的关键之一。

For office use onlyT1 ________________ T2 ________________ T3 ________________ T4 ________________ Team Control Number30303Problem ChosenBFor office use onlyF1 ________________F2 ________________F3 ________________F4 ________________ 2014Mathematical Contest in Modeling (MCM/ICM) Summary Sheet(Attach a copy of this page to your solution paper.)Type a summary of your results on this page. Do not includethe name of your school, advisor, or team members on this page.Sports Illustrated is looking for the "best all time college coach" for the previous century. We synthesize Delphi Method, Analytic Hierarchy Process (AHP) and Principal Component Analysis (PCA) to establish a synthetical evaluation method, DHP.(1) Build a evaluation index system. We improve on Delphi Method to establish a synthetical evaluation index system which includes 4 first-grade indexes (sportsmanship, performance, attitude and abilities) and 16 second-grade indexes(see Table 2) to assess a coach's comprehensive competence and expound the definition of each index and data acquisition.(2) Determine the weight of each index. We adopt AHP and use Y AAHP to calculate the weight of the 4 first-grade indexes:U={U1, U2, U3,U4}={0.1451, 0.4582, 0.1188, 0.2779}. Among those 4 indexes, performance has more influence on comprehensive competence than other three indexes. While abilities comes second and other two indexes has less influence.(3) Establish a synthetical evaluation method. We employ PCA to acquire the synthetical score of each index. Then we determine the comprehensive competence evaluation scores of each coach based on the weight of each index. Considering that performance index is easy to acquire while the other three indexes are difficult to obtain directly (In fact, we can indirectly acquire the data of these three indexes through Expert Decision. Yet there are limits on time and conditions. Also, it is against the rules to seek helps from experts), we might as well assume that each coach has the same indexes of sportsmanship, attitude and abilities. So we use the synthetical evaluation scores of performance index as college coaches' comprehensive competence evaluation scores.(4) The synthetical evaluation of college coaches. We employ PCA only to analyze performance index. We abstract 7 indexes into 4 principle components to acquire comprehensive competence evaluation scores. Taking softball as an example to verify the assumption, we find that the accumulative contribution rate of those 4 principle components accounts for 99.6662%, which means the result is valid. Meanwhile, we rank each coach according to comprehensive competence evaluation score, Z (see Table 7). Also, we assess and rank coaches of basketball and football and list respectively the top 20 college coaches (see Table 8&9). Then we list out respectively the top 5 coaches of those three sports (see Table 10).Looking For the Best College Coaches1.IntroductionSports Illustrated, a magazine for sports enthusiasts, is looking for the "best all time college coach" male or female for the previous century. However, to assess the comprehensive competence of college coaches involves numerous factors, for instance, how a team’s training sessions and games reflect a coach’s comprehensive competence. Therefore, how to make an valid assessment of such competence requires a set of scientific and efficient evaluation methods. In this paper, we are going to build a synthetical evaluation model to assess college coaches' comprehensive competence.Requirement 1: Refer to relevant resources, build a system to evaluate indexes and expound the definition and quantitative approach (sources of data) of each index.Requirement 2: Base on the evaluation index system to establish a synthetical evaluation method which is applicable to male, female and all sports, choose three sports and look for related data, employ the synthetical evaluation method to assess coaches' comprehensive competence and list out the top five college coaches.Requirement 3: Prepare a 1-2 page article for Sports Illustrated that explains the results concluded through the synthetical evaluation method and includes a non-technical explanation of mathematical model that sports fans will understand.2.Symbols and DefinitionsTable 1. Variable DefinitionSymbols DefinitionZ,U Y,U21 G,U22 W,U23comprehensive competence evaluation scoreyears of coachinggameswinsL,U24loses T,U25tiesP,U26 W/Y,U27ijxijxpRwining percentagewins per yearstandardized value of indexvalue of indexthe number of major components correlation coefficient matrixNotes: Those symbols not included here will be given and defined later in this paper.3.Assumptions(1)Neglect the influence of time factor(2)Sex has no effect on the evaluation model(3)The chosen indexes of all sports for evaluation are identical(4)Coaches who have a longer years of coaching are more competent(5)Coaches who coached more games are more competent;(6)Coaches with less losses and ties are more competent;(7)Coaches with higher comprehensive competence evaluation scores are more competent;4. Analysis of the ProblemIt is required to 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. We integrate Delphi, Analytic Hierarchy Process (AHP), Principal Component Analysis (PCA) and establish a synthetical evaluation method, DHP.The following is the detailed methodology of analysis：(1) To improve on Delphi, make it a combination of anonymous questionnaires and group discussion; collect information via consultation and statistics and analyze opinions from experts to establish a synthetical evaluation index system to analyze information.(2) To synthesize experts' judgments to the relative importance of each evaluation criterion and element, adopt AHP to build a index system and determine the weight of each evaluation index.(3) Base on the collected data and employ PCA to calculate the scores of each synthetical evaluation index.(4) Base on scores and weight of each synthetical evaluation index to make a synthetical assessment of each coach.5.Model 1: Determine evaluation index system - Delphi MethodDelphi Method, which relies on personal judgments and panels of experts, now develops as a new visual forecasting method. It is, in essence, an investigation through questionnaires and with feedbacks. In other words, there will be several rounds of questionnaires to a panel of experts. Their answers and questions will be aggregated and sent as anonymous feedbacks to those experts after each round. The experts are allowed to adjust their answers in subsequent rounds. After several rounds of questions and feedbacks, since each member of the panel agrees on what the group thinks as a whole, the Delphi Method could reach arelatively "correct" response through consensus[1].Delphi Method has three notable features：(1)Anonymity. All participants remain anonymous in the process. Therefore, it allows free expression of opinions and encourages open critique.(2) Feedbacks. All participants might make new judgments according to the feedbacks, which prevents or decreases the influence brought by the drawbacks of face-to-face meetings.(3) The statistical feature of predictions. Quantitative analysis is one crucial feature of Delphi Method. By analyzing predictions and assessing answers via statistical approach, the whole panel’s opinions and opinion deviation can be acquired.Since Delphi Method is based on subjective judgments of experts, it is especially applicable to long-term forecast which lacks objective materials or data and technical forecast which difficult to proceed through other methods. Besides, Delphi Method can estimate probability on possible prospect and expected prospect. This provides more options for decision makers to choose while any other methods can not acquire such crucial and probabilistic answers.To improve on Delphi Method, make it a combination of anonymous questionnaires and group discussion; collect information via consultation and statistics and analyze opinions from experts to establish a synthetical evaluation index system to analyze information. The details are as followings[2]:1)Facilitator sends questionnaires to experts on the basis of the issue in question.Meanwhile, experts are allowed to propose new opinions.2)Each expert finishes the questionnaire independently.3)Facilitator aggregates and summaries those answers to the questionnaires.4)Send the result of the first round and a new questionnaire to each experts.5)On the basis of the feedbacks, experts revise their own opinions. The result might lead tonew ideas, or improve the original plan. In the meantime, experts finish the questionnaire of the second round.6)Facilitator synthesizes the experts' opinions and propose several tentative ideas based onthose efforts.7)Model those tentative ideas and analyzes in a quantitative way. Also, the model will berevised time and again according to experts' opinions.8)Compare the quantitative results, integrate experts' opinions and put forward a solutionwith consensus.Index set: U=﹛U1，U2,…,U n﹜According to Delphi Method, together with the findings in article[3], we find that, under the influence of certain environmental elements, college coaches’ comprehensive competence is reflected by four elements, namely sportsmanship, performance, attitude and ability (see Table 2), the analysis is presented as followings:5.1 SportsmanshipSportsmanship is reflected by two indexes, fair competition and respect for referees. In a world with rapid development and diversification of society, politics, culture and economics, excellent professional ethics are the prerequisite of all the other abilities.(1) Fair competitionSports value fair, equal and rational competition. All actions, which involve relying on illegal means such as intentional injury, bias and taking illicit drugs to win game, are by no means moral and fair. Such actions ought to be condemned and punished.(2)Respect for refereesOne should respect for the referee's decisions, not to insult or assault referees, not to interfere with the referee’s ability to make fair decisions.5.2 PerformanceAs competitive sports is getting more and more commercialized, performance becomes a vital index to assess coaches' competence. (Because those evaluation indexes are clear to all, there is no need to further elaborate. ) Indexes are listed as followings:（1）Years of coaching（2）Games coached（3）Wins（4）Loses（5）Ties（6）Wining percentage（7）Wins per year5.3 AttitudeThis is manifested by the respect and love for a job, being responsible, accountable and eager to make progress. Here attitude is reflected by two indexes, enthusiasm and dedication, diligence.(1) Enthusiasm and dedicationIt is the prerequisite for any other jobs. Enthusiasm can promote initiatives and further generate great power and courage to overcome difficulties.(2) DiligenceDiligence is a manifestation of a coach's professional ethics and a fundamental premise for reaching the top level. To realize one’s career dream not only needs enthusiasm, but also needs diligence to learn various relevant knowledges, to lead the whole team to practice hard and to keep exploring the rules and methods of certain sports fixture.5.4 Abilities(1) Knowledge structureA coach's knowledge structure can reflect the degree to which one receives formal education, which is one of the factors that measure one’s knowledge level. Meanwhile, it is also an index to predict a coach’s potentials to lead training sessions and conduct scientific research.(2)Organize training sessionsIt is at the core of the ability structure and is one of the most fundamental ability of a coach.(3) Management abilitySince each athletic team features different characteristics and each athlete varies in psychological, physical and technical conditions, a coach should make full use of the knowledge from managerial theory to conduct comprehensive management in order to promote training and competitiveness.(4) Scientific research and creativityCreativity, which is a synthetical ability based on other abilities, is the core ability of a coach.(5) Ability to command gamesHow to command games well is a must for excellent coaches. A coach's excellent command of the game can help athletes to reach a better condition. Moreover, the command techniques play a decisive role in winning the game.This model adopts improved Delphi Method, integrates the existing results, choose 4 first-grade indexes, 16 second-grade indexes and build an evaluation index system to assess a coach's comprehensive competence.Table 2. An evaluation index system of coaches' comprehensive competence[3]First-grade index Second-grade indexComprehensive competence(U) Sportsmanship(U1)Fair competition (U11)Respect for referees (U12)Performance(U2)Years of coaching(U21)Games coached (U22)Wins (U23)Loses (U24)Ties (U25)Winning percentage (U26)Wins per year (U27) Attitude(U3)Enthusiasm and dedication(U31)Diligence (U32)Abilities(U4)Knowledge structure (U41)Organize training sessions(U42)Management ability (U43)Scientific research andcreativity (U44)Ability to command games(U45)It is easy to quantify performance index on the basis of game statistics which can be acquired via internet. As for other three first-grade indexes and their corresponding second-grade indexes, it is easy to do qualitative analysis instead of quantitative analysis. If we need to quantify those indexes, Expert Decision is usually adopted. Yet this method is time-consuming with certain subjectivity. If time and other conditions are allowed, this method can be adopted to acquire relevant data of those three indexes.6. Model 2: Determine the weight of each index-AHP Method[4]In system analysis of scientific management, people often confront with a complicated system which consists of various correlative and interactive factors and usually lacks quantitative data. While Analytic Hierarchy Process (AHP) provides us with a new, simple and practical way of modeling, especially applicable to those problems that are difficult to be analyzed fully quantitatively.We combine the conclusion from model 1 with the experts' judgments of relative importance of each index, adopt AHP to establish a weight matrix for comparison and judgment, which is the weighted fuzzy subset on U: the following data is all calculated through YAAHP software.In general, there are four steps to employ AHP to model.6.1 Establish hierarchical structure and its featuresWhen adopting AHP to analyze problems and make decisions, firstly we need to methodize the problem and make it stratified in order to establish a structure model with hierarchies. In this model, complicated problem is separated into several elements which form certain hierarchies according to attributes and their relations. Elements in the higher hierarchy serve as the rule and dominate relevant elements in the lower hierarchy.These hierarchies can be classified into three categories:（i）The highest hierarchy: there is only one element in this hierarchy. Generally speaking, it is the predetermined aim or ideal result of problem analysis. It is also called the target hierarchy.（ii）The intermediate hierarchy：this hierarchy includes intermediate links to realize the aim. It consists of several hierarchies, including rules that need attention, sub-rules. Therefore, it is called the rule hierarchy.（iii）The bottom hierarchy：this hierarchy includes alternative measures or options in order to realize one's aim. So it is also called the measure hierarchy or the option hierarchy.The hierarchies in hierarchical structure is associated with the problem's complexity and the detailedness in analysis. Generally, hierarchies are unrestricted in which dominated elements are no more than 9. This is because too many elements will bring difficulties in comparing indexes two by two.Set U as index set and establish a hierarchical structure model, as shown in figure 1.UFigure 1 Hierarchical Structure6.2 Establish a judgment matrixHierarchical structure reflets the relations between elements. Yet the rules in the rule hierarchy account for different proportion in measuring aims. In decision makers' mind, each rule takes up certain proportion.When we determine the proportion of several factors that influence certain element to this element, we frequently confront with difficulties that such proportion is not easy to quantify. Moreover, when a element is influenced by too many factors, if we directly consider the degree to which those factors influence this element, usually we will attend to one thing and lose another. This will lead to discrepancy in importance between the proposed data and what the decision maker actually thinks. The decision maker might even present a set of data with implicit contradiction. In order to see clearly, we could make the following assumptions: First smash a stone which weighs 1 kg into n parts. You can accurately weigh their weight and make their weight as w 1,…,w n . Now, ask someone to estimate the proportion of the weight of the n pieces of stone to the total weight (make sure that he does not know the weight of each stone). This person not only will have difficulties in giving an accurate answer, but also will give self-contradictory data for he might attend to one stone and lose another.Assume that we need to compare the degree to which n factors influence element Z ,{}n x x x X ,,,21⋅⋅⋅=,in what way can the comparison provide us with valid data? Saaty andothers advise one to compare factors two by two and establish a pairwise comparison matrix. That is to say, take two factors x i and x j each time and make a ij equal to the ratio of the degree to which x i influence Z to the degree to which x j influence Z . All the results through comparison are presented by a matrix A =(a ij )n ×n , called A as the pairwise comparison matrix for judgment (judgment matrix for short) between X and Z . Clearly, if the ratio is a ij , than the ratio of the degree to which x j influence Z to the degree to which x i influence Z equalsa a ijji 1=. Definition 1 if a matrix A =(a ij )n ×n meets the following condition,（i ）0>ij a , (ii) a a ij ji 1= ()n j i ,,2,1,⋅⋅⋅=It is called reciprocal matrix (it is easy to find that a ii =1, i =1,2,…,n ).As for how to determine the value of a ij , Saaty advise to use number 1 to 9 and their reciprocal as scales. Table 3 lists out the meaning of scales1 to 9:Table 3. Meaning of scalesU 32 U 3 U 2 U 1 U 31 U 11 U 12 U 22 U 23 U 24 U 25 U 21 U 26 U 27 U 42 U 43 U 44 U 45U 41 U 4ScaleMeaning 1Two factors has the same importance 3The former is a little bit more important than the latter 5The former is obviously more important than the latter 7The former is obviously more important than the latter 9The former is extremely more important than the latter 2,4,6,8Represent the median of the judgments aboveReciprocal If the ratio of the importance of factor i to that of factor j equals j i a , then the ratio of the importance of factor j tothat of factor i equals ij ji a a 1=From a psychological perspective, too many hierarchies are beyond people's judgment ability. It not only adds difficulties in making judgments, but also is prone to provide false data. Saaty and others also tried experimental method to compare the validity of people's judgments under different scales. The experiment showed that adopting scales 1 to 9 is the most appropriate.Based on the theory above, we choose proper scales and establish a judgment matrix of 4 first-grade index:UU 1 U 2 U 3 U 4 U 11 1/62 1/5 U 26 17 5 U 31/2 1/7 1 1/6 U 4 5 1/5 6 16.3 Determine weight and check consistencyWe can decide whether matrix A is consistent or not by examining whether the judgment matrix's largest eigenvalue max λequals to the amount of indexes, n. Since the largest eigenvalue continuously relies on ij a , max λis much bigger than n , and the consistency of A is more serious, so the standardized eigenvector corresponding to max λreflects less truly the proportion of the degree to which {}n x x X ,...,1= influence element Z . Thus, we need to check consistency of the judgment matrix provided by the decision maker to decide whether we can accept it.Steps to check consistency are as following:（i ）calculate the index of consistency CI1max --=n n CI λ（ii ）look up the corresponding mean and random index of consistency RI .Setting 9,,1⋅⋅⋅=n , Saaty gives values to RI which is shown in table 4. Table 4. The value of RIn1 2 3 4 5 6 7 8 9 RI 0 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 The value of RI is calculated in this way. Establish 500 sample matrices in a random way: choose one number randomly from 1 to 9 and their reciprocals to establish reciprocal matrices and calculate the mean of the standardized eigenvector 'max λ, and then define:1'max --=n nRI λ （iii ）Calculate consistency ratio RI CI CR =When 10.0<CR , we consider the consistency of judgment matrix as acceptable, otherwise we will revise the judgment matrix.To calculate the data above in YAAHP software, we can get the largest eigenvalue max λ which equals to 4.0507, then the index of consistency:1440507.41max --=--=n nCI λ=0.0169 In order to make sure whether the judgment matrix has the satisfactory consistency, we need to compare CI with the mean and random index of consistency RI. The mean and random index of consistency of the 4-order matrix is 0.90 (table 4), so:01877.09.00169.0===RI CI CR <0.10 So the judgment matrix has satisfactory consistency. Then we employ AHP and acquire the weight of each first-grade index in this evaluation system.U ={U 1, U 2, U 3, U 4}={0.1451,0.4582,0.1188,0.2779}Which is:U =0.1451U 1+0.4582U 2+0.1188U 3+0.2779U 4We can see from the result above that, in four first-grade indexes, performance index has the biggest proportion, 45.82%. Ability index ranks second with the proportion of 27.79%. While the other two indexes have smaller proportions, which are 14.51% and 11.88% respectively.6. Model 3: Synthetical evaluation-PCA Method[4]Based on the way to acquire data introduced in model 1, we can easily find detailed data of performance index via internet. While there is no direct way to obtain data of the other three indexes. That is to say, no quantified data is available on the internet. (In fact, we can indirectly acquire the data of these three indexes through Expert Decision. Yet there are limits on time and conditions. Also, it is against the rules to seek helps from experts.) Therefore, we might as well first assume that each coach has the same indexes of sportsmanship, attitude and abilities. In other words, we only take the influence of performance index into consideration and set aside the other three indexes. We use the synthetical evaluation scores of performance index as college coaches' comprehensive competence evaluation scores. (Surely, if we can acquire quantified data of the other three indexes, we are also able to get the synthetical scores of each index by employing PCA and then determine thecomprehensive competence evaluation scores according to the weight calculated in Model 2.) Principal Component Analysis (PCA) was first introduced by Pearson in 1901 to deal with non-random variables. Hotelling popularized this method in 1933 to the realm of random variables. PCA, which is based on strict mathematical theory, differs a lot from cluster analysis.The major aim of PCA is to use less variables to explain most variations in the original data. Also, PCA will transform variables with high relevance into variables that are mutually independent or irrelevant. Usually, we choose several new variables which are less than original variables in number and can explain most variations in the original data. These new variables are also called principle components which serve as synthetical indexes to explain data. This shows that PCA is actually a way to reduce dimensions.Steps in Principal Component Analysis 7.1 Standardization of Raw DataSuppose m is the number of the index variable for Principal Component Analysis12,,,m x x x ⋅⋅⋅. The total number of the evaluation objects is n . The value of the j th index of the i th evaluation object is ij x .Standardize all the index values into ij x :,(1,2,,;1,2,,)ij jij jx x x i n j m s -==⋅⋅⋅=⋅⋅⋅in which ()()m j x x n s x n x ni j ij j n i ij j ,,2,111,121121⋅⋅⋅=⎥⎦⎤⎢⎣⎡--==∑∑==That is,j x and j s are the sample mean and sample standard deviation of the j th index.,(1,2,,)i ii jx x x i m s -==⋅⋅⋅is the standardized index variate. 7.2 Standardization of Raw Data Standardization of Raw Data Correlation Coefficient Matrix()ij m mR r ⨯=j1,(,1,2,,)1nkik k ij xx r i j m n =⨯==⋅⋅⋅-∑in which ii r =1,ij r =ji r ,and ij r is the correlation coefficient of the i th index and the j th index.7.3 Standardization of Raw Data Standardization of Raw Data Calculate the Eigenvalue of R , the Correlation Coefficient Matrix120m λλλ≥≥⋅⋅⋅≥≥ , and the corresponding eigenvectors 12,,,m u u u ⋅⋅⋅, in which12(,,,)T j j j nj u u u u =⋅⋅⋅ and the corresponding eigenvectors, in which m , the number of indexvalues composed of the eigenvectors, is11112121212122221122n n n nmm m nm n y u x u x u x y u x u x u x y u x u x u x⎧=++⋅⋅⋅+⎪=++⋅⋅⋅+⎪⎨⋅⋅⋅⋅⋅⋅⎪⎪=++⋅⋅⋅+⎩ in which 1y is the first major component, 2y is the second … and m y is the m th .7.4 Select p (p m ≤) number of major components to calculate the Comprehensive index.1)Calculate the information contribution ratio and accumulated contribution ration of Eigenvalue (1,1,,)j j m λ=⋅⋅⋅. Call1(1,2,,)jj mkk b j m λλ===⋅⋅⋅∑the information contribution ratio of major component i y ；call11pkk p mkk λαλ===∑∑the accumulated contribution ration of major components . When 0.85,0.90,0.95p α= approaches 1, select the first p index variables 12,,p y y y ⋅⋅⋅ as p major components,replacethe original m index variables, and in this way conduct a comprehensive analysis of p components.2)Calculate the Integrate Score.Definition: Z is comprehensive competence evaluation score, which represents college coaches' comprehensive competence and is identical to U 's definition.1pj j j Z b y ==∑Take the data of softball for example. We select 3 or 4 major components, and let p =3 or p =4. A major component analysis is conducted with MATLAB on the m elements, and the first a few characteristic roots and their contribution ratio of the correlation coefficient matrix are listed in Table 5. (Appendix 1 is the program code )Table 5. The Results of Major Components AnalysisNumber Characteristic Roots ContributionRatio Accumulated Contribution Ratio1 4.0577 57.9675 57.96752 1.4338 19.1971 77.16463 0.8969 12.8123 89.97694 0.6782 9.6893 99.66625 0.0121 0.1724 99.8386 6 70.0113 0.00000.1615 0.0000100.0001 100.0001From table 5 we can see that the former four principle components' accumulative contribution is 99.6662%, so we select 4 major components to evaluate comprehensively. Table 6 shows the eigenvectors of the first five characteristic roots. i thTable 6. The Eigenvectors Of the First Four Characteristic Roots of the StandardizedThe 1stEigenvector The 2nd Eigenvector The 3rdEigenvector The 4th Eigenvector 1 -0.4350 0.1311 -0.3719 -0.3400 2 -0.4950 0.0162 -0.0559 0.0364 3-0.4930-0.0706-0.0487-0.0321。

A Networks and Machine Learning Approach toDetermine the Best College Coaches of the20th-21st CenturiesTian-Shun Allan Jiang,Zachary T Polizzi,Christopher Qian YuanMentor:Dr.Dan TeagueThe North Carolina School of Science and Mathematics∗February10,2014Team#30680Page2of18Contents1Problem Statement3 2Planned Approach3 3Assumptions3 4Data Sources and Collection44.1College Football (5)4.2Men’s College Basketball (5)4.3College Baseball (5)5Network-based Model for Team Ranking65.1Building the Network (6)5.2Analyzing the Network (6)5.2.1Degree Centrality (6)5.2.2Betweenness and Closeness Centrality (7)5.2.3Eigenvector Centrality (8)6Separating the Coach Effect106.1When is Coach Skill Important? (11)6.2Margin of Win Probability (12)6.3Optimizing the Probability Function (13)6.3.1Genetic Algorithm (13)6.3.2Nelder-Mead Method (14)6.3.3Powell’s Method (14)7Ranking Coaches157.1Top Coaches of the Last100Years (15)8Testing our Model158.1Sensitivity Analysis (15)8.2Strengths (16)8.3Weaknesses (16)9Conclusions17 10Acknowledgments172Team#30680Page3of181Problem StatementCollege sport coaches often achieve widespread recognition.Coaches like Nick Saban in football and Mike Krzyzewski in basketball repeatedly lead their schools to national championships.Because coaches influence both the per-formance and reputation of the teams they lead,a question of great concern to universities,players,and fans alike is:Who is the best coach in a given sport? Sports Illustrated,a magazine for sports enthusiasts,has asked us tofind the best all-time college coaches for the previous century.We are tasked with creat-ing a model that can be applied in general across both genders and all possible sports at the college-level.The solution proposed within this paper will offer an insight to these problems and will objectively determine the topfive coaches of all time in the sports of baseball,men’s basketball,and football.2Planned ApproachOur objective is to rank the top5coaches in each of3different college-level sports.We need to determine which metrics reflect most accurately the ranking of coaches within the last100years.To determine the most effective ranking system,we will proceed as follows:1.Create a network-based model to visualize all college sports teams,theteams won/lost against,and the margin of win/loss.Each network de-scribes the games of one sport over a single year.2.Analyze various properties of the network in order to calculate the skill ofeach team.3.Develop a means by which to decouple the effect of the coach from theteam performance.4.Create a model that,given the player and coach skills for every team,canpredict the probability of the occurrence of a specific network of a)wins and losses and b)the point margin with which a win or loss occurred.5.Utilize an optimization algorithm to maximize the probability that thecoach skill matrix,once plugged into our model,generates the network of wins/losses and margins described in(1).6.Analyze the results of the optimization algorithm for each year to deter-mine an overall ranking for all coaches across history.3AssumptionsDue to limited data about the coaching habits of all coaches at all teams over the last century in various collegiate sports,we use the following assumptions to3Team#30680Page4of18 complete our model.These simplifying assumptions will be used in our report and can be replaced with more reliable data when it becomes available.•The skill level of a coach is ultimately expressed through his/her team’s wins over another and the margin by which they win.This assumes thata team must win to a certain degree for their coach to be good.Even ifthe coach significantly amplifies the skills of his/her players,he/she still cannot be considered“good”if the team wins no games.•The skills of teams are constant throughout any given year(ex:No players are injured in the middle of a season).This assumption will allow us to compare a team’s games from any point in the season to any other point in the season.In reality,changing player skills throughout the season make it more difficult to determine the effect of the coach on a game.•Winning k games against a good team improves team skill more than winning k games against an average team.This assumption is intuitive and allows us to use the eigenvector centrality metric as a measure of total team skill.•The skill of a team is a function of the skill of the players and the skill of the coach.We assume that the skill of a coach is multiplicative over the skill of the players.That is:T s=C s·P s where T s is the skill of the team,C s is the skill of the coach,and P s is a measure of the skill of the players.Making coach skill multiplicative over player skill assumes that the coach has the same effect on each player.This assumption is important because it simplifies the relationship between player and coach skill to a point where we can easily optimize coach skill vectors.•The effect of coach skill is only large when the difference between player skill is small.For example,if team A has the best players in the conference and team B has the worst,it is likely that even the best coach would not be able to,in the short run,bring about wins over team A.However, if two teams are similarly matched in players,a more-skilled coach will make advantageous plays that lead to his/her team winning more often than not.•When player skills between two teams are similarly matched,coach skill is the only factor that determines the team that wins and the margin by which they win by.By making this assumption,we do not have to account for any other factors.4Data Sources and CollectionSince our model requires as an input the results of all the games played in a season of a particular sport,wefirst set out to collect this data.Since we were unable to identify a single resource that had all of the data that we required,we4Team#30680Page5of18 found a number of different websites,each with a portion of the requisite data. For each of these websites,we created a customized program to scrape the data from the relevant webpages.Once we gathered all the data from our sources,we processed it to standardize the formatting.We then aimed to merge the data gathered from each source into a useable format.For example,we gathered basketball game results from one source,and data identifying team coaches from another.To merge them and show the game data for a specific coach,we attempted to match on commonfields(ex.“Team Name”).Often,however,the data from each source did not match exactly(ex.“Florida State”vs“Florida St.”).In these situations,we had to manually create a matching table that would allow our program to merge the data sources.Although we are seeking to identify the best college coach for each sport of interest for the last century,it should be noted that many current college sports did not exist a century ago.The National Collegiate Athletic Association (NCAA),the current managing body for nearly all college athletics,was only officially established in1906and thefirst NCAA national championship took place in1921,7years short of a century ago.Although some college sports were independently managed before being brought into the NCAA,it is often difficult to gather accurate data for this time.4.1College FootballOne of the earliest college sports,College Football has been popular since its inception in the1800’s.The data that we collected ranges from1869to the present,and includes the results andfinal scores of every game played between Division1men’s college football teams(or the equivalent before the inception of NCAA)[2].Additionally,we have gathered data listing the coach of each team for every year we have collected game data[4],and combined the data in order to match the coach with his/her complete game record for every year that data was available.4.2Men’s College BasketballThe data that we gathered for Men’s College Basketball ranges from the sea-son of thefirst NCAA Men’s Basketball championship in1939to the present. Similarly to College Football,we gathered data on the result andfinal scores of each game in the season and infinals[2].Combining this with another source of coach names for each team and year generated the game record for each coach for each season[4].4.3College BaseballAlthough College Baseball has historically had limited popularity,interest in the sport has grown greatly in the past decades with improved media coverage and collegiate spending on the sport.The game result data that we collected5Team#30680Page6of18 ranges from1949to the present,and was merged with coach data for the same time period.5Network-based Model for Team Ranking Through examination of all games played for a specific year we can accurately rank teams for that year.By creating a network of teams and games played, we can not only analyze the number of wins and losses each team had,but can also break down each win/loss with regard to the opponent’s skill.5.1Building the NetworkWe made use of a weighted digraph to represent all games played in a single year.Each node in the graph represents a single college sports team.If team A wins over team B,a directed edge with a weight of1will be drawn from A pointing towards B.Each additional time A wins over B,the weight of the edge will be increased by1.If B beats A,an edge with the same information is drawn in the opposing direction.Additionally,a list containing the margin of win/loss for each game is associated with the edge.For example,if A beat B twice with score:64−60,55−40,an edge with weight two is constructed and the winning margin list4,15is associated with the edge.Since each graph represents a single season of a specific sport,and we are interested in analyzing a century of data about three different sports,we have created a program to automate the creation of the nearly300graphs used to model this system.The program Gephi was used to visualize and manipulate the generated graphs. 5.2Analyzing the NetworkWe are next interested in calculating the skill of each team based on the graphs generated in the previous section.To do this,we will use the concept of central-ity to investigate the properties of the nodes and their connections.Centrality is a measure of the relative importance of a specific node on a graph based on the connections to and from that node.There are a number of ways to calculate centrality,but the four main measures of centrality are degree,betweenness, closeness,and eigenvector centrality.5.2.1Degree CentralityDegree centrality is the simplest centrality measure,and is simply the total number of edges connecting to a specific node.For a directional graph,indegree is the number of edges directed into the node,while outdegree is the number of edges directed away from the node.Since in our network,edges directed inward are losses and edges directed outwards are wins,indegree represents the total number of losses and outdegree measures the total number of wins.Logically,therefore,outdegreeeindegreee represents the winlossratio of the team.This ratiois often used as a metric of the skill of a team;however,there are several6Team#30680Page7of18Figure1:A complete network for the2009-2010NCAA Div.I basketball season. Each node represents a team,and each edge represents a game between the two teams.Note that,since teams play other teams in their conference most often, many teams have clustered into one of the32NCAA Div.1Conferences. weaknesses to this metric.The most prominent of these weaknesses arises from the fact that,since not every team plays every other team over the course of the season,some teams will naturally play more difficult teams while others will play less difficult teams.This is exaggerated by the fact that many college sports are arranged into conferences,with some conferences containing mostly highly-ranked teams and others containing mostly low-ranked teams.Therefore, win/loss percentage often exaggerates the skill of teams in weaker conferences while failing to highlight teams in more difficult conferences.5.2.2Betweenness and Closeness CentralityBetweenness centrality is defined as a measure of how often a specific node acts as a bridge along the shortest path between two other nodes in the graph. Although a very useful metric in,for example,social networks,betweenness centrality is less relevant in our graphs as the distance between nodes is based on the game schedule and conference layout,and not on team skill.Similarly, closeness centrality is a measure of the average distance of a specific node to7Team#30680Page8of18 another node in the graph-also not particularly relevant in our graphs because distance between nodes is not related to team skills.5.2.3Eigenvector CentralityEigenvector centrality is a measure of the influence of a node in a network based on its connections to other nodes.However,instead of each connection to another node having afixed contribution to the centrality rating(e.g.de-gree centrality),the contribution of each connection in eigenvector centrality is proportional to the eigenvector centrality of the node being connected to. Therefore,connections to high-ranked nodes will have a greater influence on the ranking of a node than connections to low-ranking nodes.When applied to our graph,the metric of eigenvector centrality will assign a higher ranking to teams that win over other high-ranking teams,while winning over lower-ranking nodes has a lesser contribution.This is important because it addresses the main limitation over degree centrality or win/loss percentage,where winning over many low-ranked teams can give a team a high rank.If we let G represent a graph with nodes N,and let A=(a n,t)be an adjacency matrix where a n,t=1if node n is connected to node t and a n,t=0 otherwise.If we define x a as the eigenvector centrality score of node a,then the eigenvector centrality score of node n is given by:x n=1λt∈M(n)x t=1λt∈Ga n,t x t(1)whereλrepresents a constant and M(n)represents the set of neighbors of node n.If we convert this equation into vector notation,wefind that this equation is identical to the eigenvector equation:Ax=λx(2) If we place the restriction that the ranking of each node must be positive, wefind that there is a unique solution for the eigenvector x,where the n th component of x represents the ranking of node n.There are multiple different methods of calculating x;most of them are iterative methods that converge on a final value of x after numerous iterations.One interesting and intuitive method of calculating the eigenvector x is highlighted below.It has been shown that the eigenvector x is proportional to the row sums of a matrix S formed by the following equation[6,9]:S=A+λ−1A2+λ−2A3+...+λn−1A n+ (3)where A is the adjacency matrix of the network andλis a constant(the principle eigenvalue).We know that the powers of an adjacency matrix describe the number of walks of a certain length from node to node.The power of the eigenvalue(x)describes some function of length.Therefore,S and the8Team#30680Page9of18 eigenvector centrality matrix both describe the number of walks of all lengths weighted inversely by the length of the walk.This explanation is an intuitive way to describe the eigenvector centrality metric.We utilized NetworkX(a Python library)to calculate the eigenvector centrality measure for our sports game networks.We can apply eigenvector centrality in the context of this problem because it takes into account both the number of wins and losses and whether those wins and losses were against“good”or“bad”teams.If we have the following graph:A→B→C and know that C is a good team,it follows that A is also a good team because they beat a team who then went on to beat C.This is an example of the kind of interaction that the metric of eigenvector centrality takes into account.Calculating this metric over the entire yearly graph,we can create a list of teams ranked by eigenvector centrality that is quite accurate. Below is a table of top ranks from eigenvector centrality compared to the AP and USA Today polls for a random sample of our data,the2009-2010NCAA Division I Mens Basketball season.It shows that eigenvector centrality creates an accurate ranking of college basketball teams.The italicized entries are ones that appear in the top ten of both eigenvector centrality ranking and one of the AP and USA Today polls.Rank Eigenvector Centrality AP Poll USA Today Poll 1Duke Kansas Kansas2West Virginia Michigan St.Michigan St.3Kansas Texas Texas4Syracuse Kentucky North Carolina5Purdue Villanova Kentucky6Georgetown North Carolina Villanova7Ohio St.Purdue Purdue8Washington West Virginia Duke9Kentucky Duke West Virginia10Kansas St.Tennessee ButlerAs seen in the table above,six out of the top ten teams as determined by eigenvector centrality are also found on the top ten rankings list of popular polls such as AP and USA Today.We can see that the metric we have created using a networks-based model creates results that affirms the results of commonly-accepted rankings.Our team-ranking model has a clear,easy-to-understand basis in networks-based centrality measures and gives reasonably accurate re-sults.It should be noted that we chose this approach to ranking teams over a much simpler approach such as simply gathering the AP rankings for vari-ous reasons,one of which is that there are not reliable sources of college sport ranking data that cover the entire history of the sports we are interested in. Therefore,by calculating the rankings ourselves,we can analyze a wider range of historical data.Below is a graph that visualizes the eigenvector centrality values for all games played in the2010-2011NCAA Division I Mens Football tournament.9Team#30680Page10of18 Larger and darker nodes represent teams that have high eigenvector centrality values,while smaller and lighter nodes represent teams that have low eigenvector centrality values.The large nodes therefore represent the best teams in the 2010-2011season.Figure2:A complete network for the2012-2013NCAA Div.I Men’s Basketball season.The size and darkness of each nodes represents its relative eigenvector centrality value.Again,note the clustering of teams into NCAA conferences. 6Separating the Coach EffectThe model we created in the previous section works well forfinding the relative skills of teams for any given year.However,in order to rank the coaches,it is necessary to decouple the coach skill from the overall team skill.Let us assume that the overall team skill is a function of two main factors,coach skill and player skill.Specifically,if C s is the coach skill,P s is the player skill,and T s is10Team #30680Page 11of 18the team skill,we hypothesize thatT s =C s ·P s ,(4)as C s of any particular team could be thought of as a multiplier on the player skill P s ,which results in team skill T s .Although the relationship between these factors may be more complex in real life,this relationship gives us reasonable results and works well with our model.6.1When is Coach Skill Important?We will now make a key assumption regarding player skill and coach skill.In order to separate the effects of these two factors on the overall team skill,we must define some difference in effect between the two.That is,the player skill will influence the team skill in some fundamentally different way from the coach skill.Think again to a game played between two arbitrary teams A and B .There are two main cases to be considered:Case one:Player skills differ significantly:Without loss of generality,assume that P (A )>>P (B ),where P (x )is a function returning the player skills of any given team x .It is clear that A winning the game is a likely outcome.We can draw a plot approximating the probability of winning by a certain margin,which is shown in Figure 3.Margin of WinProbabilityFigure 3:A has a high chance of winning when its players are more skilled.Because the player skills are very imbalanced,the coach skill will likely not change the outcome of the game.Even if B has an excellent coach,the effect of the coach’s skill will not be enough to make B ’s win likely.Case two:Player skills approximately equal:If the player skills of the two teams are approximately evenly matched,the coach skill has a much higher likelihood of impacting the outcome of the game.When the player skills are11Team #30680Page 12of 18similar for both teams,the Gaussian curve looks like the one shown in Figure 4.In this situation,the coach has a much greater influene on the outcome of the game -crucial calls of time-outs,player substitutions,and strategies can make or break an otherwise evenly matched game.Therefore,if the coach skills are unequal,causing the Gaussian curve is shifted even slightly,one team will have a higher chance of winning (even if the margin of win will likely be small).Margin of WinProbabilityFigure 4:Neither A nor B are more likely to win when player skills are the same (if player skill is the only factor considered).With the assumptions regarding the effect of coach skill given a difference in player skills,we can say that the effect of a coach can be expressed as:(C A −C B )· 11+α|P A −P B |(5)Where C A is the coach skill of team A ,C B is the coach skill of team B ,P A is the player skill of team A ,P B is the player skill of team B ,and αis some scalar constant.With this expression,the coach effect is diminished if the difference in player skills is large,and coach effect is fully present when players have equal skill.6.2Margin of Win ProbabilityNow we wish to use the coach effect expression to create a function giving the probability that team A will beat team B by a margin of x points.A negative value of x means that team B beat team A .The probability that A beats B by x points is:K ·e −1E (C ·player effect +D ·coach effect −margin ) 2(6)where C,D,E are constant weights,player effect is P A −P B ,coach effect is given by Equation 5,and margin is x .12Team#30680Page13of18This probability is maximized whenC·player effect+D·coach effect=margin.This accurately models our situation,as it is more likely that team A wins by a margin equal to their combined coach and team effects over team B.Since team skill is comprised of player skill and coach skill,we may calculate a given team’s player skill using their team skill and coach skill.Thus,the probability that team A beats team B by margin x can be determined solely using the coach skills of the respective teams and their eigenvector centrality measures.6.3Optimizing the Probability FunctionWe want to assign all the coaches various skill levels to maximize the likelihood that the given historical game data occurred.To do this,we maximize the probability function described in Equation6over all games from historical data byfinding an optimal value for the coach skill vectors C A and C B.Formally, the probability that the historical data occurred in a given year isall games K·e−1E(C·player effect+D·coach effect−margin)2.(7)After some algebra,we notice that maximizing this value is equivalent to minimizing the value of the cost function J,whereJ(C s)=all games(C·player effect+D·coach effect−margin)2(8)Because P(A beats B by x)is a nonlinear function of four variables for each edge in our network,and because we must iterate over all edges,calculus and linear algebra techniques are not applicable.We will investigate three techniques (Genetic Algorithm,Nelder-Mead Search,and Powell Search)tofind the global maximum of our probability function.6.3.1Genetic AlgorithmAtfirst,our team set out to implement a Genetic Algorithm to create the coach skill and player skill vectors that would maximize the probability of the win/loss margins occurring.We created a program that would initialize1000random coach skill and player skill vectors.The probability function was calculated for each pair of vectors,and then the steps of the Genetic Algorithm were ran (carry over the“mostfit”solution to the next generation,cross random elements of the coach skill vectors with each other,and mutate a certain percentage of the data randomly).However,our genetic algorithm took a very long time to converge and did not produce the optimal values.Therefore,we decided to forgo optimization with genetic algorithm methods.13Team#30680Page14of186.3.2Nelder-Mead MethodWe wanted to attempt optimization with a technique that would iterate over the function instead of mutating and crossing over.The Nelder-Mead method starts with a randomly initialized coach skills vector C s and uses a simplex to tweak the values of C s to improve the value of a function for the next iteration[7]. However,running Nelder-Mead found local extrema which barely increased the probability of the historical data occurring,so we excluded it from this report.6.3.3Powell’s MethodA more efficient method offinding minima is Powell’s Method.This algorithm works by initializing a random coach skills vector C s,and uses bi-directional search methods along several search vectors tofind the optimal coach skills.A detailed explanation of the mathematical basis for Powell’s method can be found in Powell’s paper on the algorithm[8].We found that Powell’s method was several times faster than the Nelder-Mead Method and produced reasonable results for the minimization of our probability function.Therefore,our team decided to use Powell’s method as the main algorithm to determine the coach skills vector.We implemented this algorithm in Python and ran it across every edge in our network for each year that we had data.It significantly lowered our cost function J over several thousand iterations.Rank1962200020051John Wooden Lute Olson Jim Boeheim2Forrest Twogood John Wooden Roy Williams3LaDell Anderson Jerry Dunn Thad Matta The table above shows the results of running Powell’s method until the probability function shown in Equation6is optimized,for three widely separated arbitrary years.We have chosen to show the top three coaches per year for the purposes of conciseness.We will additionally highlight the performance of our top three three outstanding coaches.John Wooden-UCLA:John Wooden built one of the’greatest dynasties in all of sports at UCLA’,winning10NCAA Division I Basketball tournaments and leading an unmatched streak of seven tournaments in a row from1967to 1973[1].He won88straight games during one stretchJim Boeheim-Syracuse:Boeheim has led Syracuse to the NCAA Tour-nament28of the37years that he has been coaching the team[3].He is second only to Mike Krzyzewsky of Duke in total wins.He consistently performs even when his players vary-he is the only head coach in NCAA history to lead a school to fourfinal four appearances in four separate decades.Roy Williams-North Carolina:Williams is currently the head of the basketball program at North Carolina where he is sixth all-time in the NCAA for winning percentage[5].He performs impressively no matter who his players are-he is one of two coaches in history to have led two different teams to the Final Four at least three times each.14Team#30680Page15of187Ranking CoachesKnowing that we are only concerned withfinding the topfive coaches per sport, we decided to only consider thefive highest-ranked coaches for each year.To calculate the overall ranking of a coach over all possible years,we considered the number of years coached and the frequency which the coach appeared in the yearly topfive list.That is:C v=N aN c(9)Where C v is the overall value assigned to a certain coach,N a is the number of times a coach appears in yearly topfive coach lists,and N c is the number of years that the coach has been active.This method of measuring overall coach skill is especially strong because we can account for instances where coaches change teams.7.1Top Coaches of the Last100YearsAfter optimizing the coach skill vectors for each year,taking the topfive,and ranking the coaches based on the number of times they appeared in the topfive list,we arrived at the following table.This is our definitive ranking of the top five coaches for the last100years,and their associated career-history ranking: Rank Mens Basketball Mens Football Mens Baseball 1John Wooden-0.28Glenn Warner-0.24Mark Marquess-0.27 2Lute Olson-0.26Bobby Bowden-0.23Augie Garrido-0.24 3Jim Boeheim-0.24Jim Grobe-0.18Tom Chandler-0.22 4Gregg Marshall-.23Bob Stoops-0.17Richard Jones-0.19 5Jamie Dixon-.21Bill Peterson-0.16Bill Walkenbach-0.168Testing our Model8.1Sensitivity AnalysisA requirement of any good model is that it must be tolerant to a small amount of error in its inputs.In our model,possible sources of error could include im-properly recorded game results,incorrectfinal scores,or entirely missing games. These sources of error could cause a badly written algorithm to return incorrect results.To test the sensitivity of our model to these sources of error,we decided to create intentional small sources of error in the data and compare the results to the original,unmodified results.Thefirst intentional source of error that we incorporated into our model was the deletion of a game,specifically a regular-season win for Alabama(the team with the top-ranked coach in1975)over Providence with a score of67to 60.We expected that the skill value of the coach of the Alabama team would15。

2014数学建模竞赛承诺书我们仔细阅读了河西学院大学生数学建模竞赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题.我们知道，抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出.我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性.如有违反竞赛规则的行为，我们将受到严肃处理.我们参赛选择的题号是（从A/B/C/D中选择一项填写）： B参赛队员(打印并签名) ：序号姓名(打印) 所在学院签名123指导教师或指导教师组负责人(打印并签名)：日期：年月日评阅编号（由竞赛组委会评阅前进行编号）：2014年数学建模竞赛评阅专用页评阅编号（由竞赛组委会评阅前进行编号）：评阅记录（供竞赛组委会评阅时使用）：评阅人评分备注评阅结果：获奖等级：徽章问题摘要: 在此次会议中，代表们收到不同标记的徽章，与他们的名字有关，所以按照一定的原理，我们建立模型，首先，我们将286名代表按照所收到的徽章标记“+”和“-”进行分类，把收到“+”标记徽章的代表们的名字记作向量ijk X ,把收到“-”标记徽章的代表们的名字记作向量ijk Y ，其中)9,2,1,,( =k j i .再利用编程VC++程序，对对向量ijk X 和向量ijk Y (),,0,,9i j k =中的第)7,6,5,4,3,2,1(=m m 个分量分别进行观察、对比，得出向量ijk X 和向量ijk Y (),,0,,9i j k =中的第()1,3,4,5,6,7n n =个分量都分别有交叉项，则排除以这些分量的特性为分类依据的方法，但是ijk X 和向量ijkY (),,0,,9i j k =中第2个分量没有交叉量，且第二个分量都属于()01,05,09,15,21即(),,,,a e i o u .因此，对于猜想徽章标记于第二个分量有关，进而得出了对徽章的分类给出了方法，即代表们的名字的第二个字母在(),,,,a e i o u 内是，徽章标记为“+”，反之，徽章标记为“-”.对于第二题，我们根据问题一的解答，如果名字的第二个字母为大写时，直接给与”-”标记，则准确率为100%，则错误率为0;如如果名字的第二个字母为大写时，仍然按照问题一的分类方法，则正确率为99.49%，则错误率为0.51%.对于问题三，我们按照以上得出的分类方法，通过编写程序利用VC++程序，对未参加会议的14名代表进行了分类，最后只有Attilio Giordana 的徽章为“-”，其它代表的徽章都为“+”.最后，我们对这个模型评价，本题的优点是数据量有限，有利于建模的假设与解答;不足之处是，这个模型虽然检验结果与实际的相当吻合，但该模型存在着随机性，如果参加会议的代表人数曾多，这个模型还会继续有效吗？因此，我们提出改进，加强分类的指标和条件，例如以前2个字母，前3个字母等其他方法进行分析、讨论，尝试分类，使得分得的两组的人数在理论上相等.关键字：字母；分量；排序;分类一 问题重述在1994年的“机器学习与计算学习理论”的国际会议上，参加会议的280名代表都收到会议组织者发给的一枚徽章，徽章的标记为“＋”或“－”（参加会议的名单及得到的徽章见附表）.会议组织者声明：每位代表得到徽章“＋”或“－”的标记只与他们的姓名有关，并希望代表们能够找出徽章“＋”与“－”的分类方法.问题如下：1．如何对参加会议的代表所得的的徽章找出合理的规律进行分类.2．对自己的分类方法进行分析，如分类的理由、分类的规律、分类的正确率与错误率等.3、由于客观原因，有14名代表（见附表）没能参加此次会议.按照以上找出的方法，如果他们参加会议，他们将得到什么类型的徽章？二 模型假设首先，我们将286名代表按照所收到的徽章标记“+”和“-”进行分类，假设得到徽章只与代表的名字有关系.把收到“+”标记徽章的代表们的名字记作向量ijk X ，其中)9,2,1,,( =k j i ，把收到“-”标记徽章的代表们的名字记作向量ijk Y ，其中)9,2,1,,( =k j i ，把代表们名字中的字母按在名字中的排序分别作为这个向量的第一个分量，第二个分量， ，第七个分量（因为代表们名字中最短的名字只要七个字母，Ken Lang ）,同时在把代表们的名字记为向量时，给26个字母（按字母表的顺序）分别赋值为01,02， ，26.例如第一个收到“+”的代表Naoki Abe 的名字用向量表示为)05,02,01,09,11,15,01,14(001=X ，第一个收到“-”的代表Myrian Abramson 的名字用向量表示为)01,13,01,09,18,25,13(001=Y .然后利用VC++编程，对向量ijk X 和向量ijkY (),,0,,9i j k =中的第)7,6,5,4,3,2,1(=m m 个分量分别进行观察、对比.(1)题中所给的人名是英文名字，由英语知识可知，外国人的姓名顺序与中国人相反，即名在前，姓在后.因此我们猜想，徽章的标记跟向量X i j k 和向量ijkY ()9,,1,0,, =k j i 中的第)7,,1( =l l 个分量有关的概率相对较大.(2)一旦得出的徽章分类方法的正确率较高时（不妨为90%以上），那就采取这种分类方法.三 模型建立模型1利用VC++编程，对向量ijkX和向量ijkY()9,,1,0,, =k j i 中的第)7,6,5,4,3,2,1(=m m 个分量分别进行观察、对比，（程序见附件1）结果如下：由以上结果得到：向量ijk X 和向量ijk Y ()9,,1,0,, =k j i 中的第)7,6,5,4,3,1(=n n 个分量都分别有交叉项，则排除以这些分量的特性为分类依据的方法，但是向量ijk X 和向量ijk Y ()9,,1,0,, =k j i 中第2个分量没有交叉量，且向量ijk X 的第二个分量都属于)21,15,09,05,01(即第二个字母属于()u o i e a ,,,,.因此，我们猜想徽章标记与第二个分量有关.模型2我们在先向量ijk X 后向量ijk Y ()9,,1,0,, =k j i 排序基础上再按向量ijk X 和向量ijkY ()9,,1,0,, =k j i 中的第二个分量的顺序)21,15,09,05,01(进行排序,如下：徽章 符号 向量中的第二个分量 名字中的 第二个字母代表们的名字 代表们的姓氏+ 01 a Javed Aslam + 01 a DavidW. Aha+ 01 a + 05 e Peter Bartlett + 05 e GeorgeBerg+ 05 e + 09 i Timothy P. Barber + 09 i MichaelW. Barley+ 09 i + 15 o Tom Bylander + 15 o JohnCase+ 15 o+ 21 u Susan L. Epstein + 21 u JudyA. Franklin+ 21 u - 02 b R. Bharat Rao - 03 c Scott E. Decatur - 04 dOdedMaron-01至26中除去除a 、e 、i 、o 、u01,05,09,15,21 字母外表（1）由上述表格可以看出：（1）当代表们名字的第二个字母为元音（a、e、i、o、u）并且为小写字母时，他们得到的徽章标记均为“+”号，否则为“-”号.（2）当名字的第二个字母为大写时，则他们得到的徽章均为“-”号，例如：R. Andrew McCallum L. Thorne McCarty 等等.四模型求解（1）徽章的分类方法：参加会议的代表们的名字中的第二个字母为元音字母并且为小写时，他们均得到带“+”号标记的徽章；否则均得到带“-”号标记的徽章. （2）分类的理由：由于代表们所收到的徽章的标记只与他们的名字有关，所以先以徽章标记“+”和“-”分析，得出猜想，再以名字中的第二个字母进行排序、比较、分析，最后得出分类方法，见模型2.分类的正确与错误率：参加会议的代表们的总人数为286，我们根据(1)题的分类方法，如果名字的第二个字母为大写时，直接给”-”标记，则准确率为100%，见表(1),从而错误率为0;如果名字的第二个字母为大写时，仍然按照(1)题的分类方法，则正确率为99.49%，则错误率为0.51%.（3）根据我们的假设(若正确率达到90%，则此方法可行)及题(1)得出的分类方法，于是通过编写程序利用VC++软件，对未参加会议的14名代表进行了分类，程序见（附件2），结果如下：由上边结果知：只有Attilio Giordana的徽章为“-”，其它代表的徽章都为“+”.五模型评价及改进本题的优点是数据量有限，有利于建模的假设与解答;不足之处是，这个模型虽然检验结果与本次会议中给出的数据实际的相当吻合，但该模型存在着随机性，如果参加会议的代表人数曾多，这个模型还会继续有效吗？并且在这些代表中收到”+”号标记的人数为196人，收到”-”号标记的人数为90人，这种分类方法是不好的，因为在做决策，裁定时，这种分类方法使解决问题变的不公平、不平等.因此，我们提出改进，加强分类的指标和条件，例如以前2个字母，前3个字母等其他方法进行分析、讨论，尝试分类，使得分得的两组的人数在理论上相等.参考文献[1] 姜启源,叶俊等.《数学模型》第三版.高等教育出版社，北京，2004[2] 康博创作室.《VC++6.0高级编程》.清华大学出版社，北京，1999[3] 谭浩强著.《C程序设计》(第四版).北京：清华大学出版社，2010.6[4] 袁震东，洪渊，林武忠，蒋鲁敏编著.《数学建模》.华东师范大学出版社，1995.5[5] 高隆昌，杨元著.《数学建模基础理论》.北京：科学出版社，2007附件1#include<iostream>using namespace std;int main(){char letter[30]=" ";int count=0,flag=0;char name[280][30]={"+Naoki Abe", "-Myriam Abramson","+David W. Aha","+Kamal M. Ali ","-Eric Allender ", "+Dana Angluin","-Chidanand Apte","+Minoru Asad","+Lars Asker","+Javed Aslam","+Haralabos Athanassiou","+Jose L. Balcazar","+Timothy P. Barber","+Michael W. Barley","-Cristina Baroglio","+Peter Bartlett","-Eric Baum","+Welton Becket","-Shai Ben-David", "+George Berg","+Neil Berkman","+Malini Bhandaru","+Bir Bhanu","+Reinhard Blasig","-Avrim Blum","-Anselm Blumer ","+Justin Boyan","+Carla E. Brodley","+Nader Bshouty","-Wray Buntine","-Andrey Burago","+Tom Bylander","+Bill Byrne","-Claire Cardie","+Richard A. Caruana","+John Case","+Jason Catlett","+Nicolo Cesa-Bianchi","-Philip Chan","+Mark Changizi ","+Pang-Chieh Chen ","-Zhixiang Chen","+Wan P. Chiang","-Steve A. Chien","+Jeffery Clouse","+William Cohen","+David Cohn ","-Clare Bates Congdon","-Antoine Cornuejols","+Mark W. Craven","+Robert P. Daley","+Lindley Darden","-Chris Darken","-Bhaskar Dasgupta","-Brian D. Davidson","+Michael de la Maza","-Olivier De Vel","-Scott E. Decatur","+Gerald F. DeJong","+Kan Deng","-Thomas G. Dietterich","+Michael J. Donahue","+George A. Drastal","+Harris Drucker","-Chris Drummond","+Hal Duncan","-Thomas Ellman","+Tapio Elomaa","+Susan L. Epstein","+Bob Evans" ,"-Claudio Facchinetti","+Tom Fawcett","-Usama Fayyad","+Aaron Feigelson","+Nicolas Fiechter","+David Finton" ,"+John Fischer","+Paul Fischer","+Seth Flanders","+Lance Fortnow","-Ameur Foued","+Judy A. Franklin","+Yoav Freund","+Johannes Furnkranz","+LeslieGrate","+William A. Greene","+Russell Greiner","+Marko Grobelnik","+Tal Grossman" , "+Margo Guertin","+Tom Hancock","+Earl S. Harris Jr.","+David Haussler","+Matthias Heger", "+Lisa Hellerstein","+David Helmbold","+Daniel Hennessy","+Haym Hirsh","+Jonathan Hodgson","+Robert C. Holte","+Jiarong Hong","-Chun-Nan Hsu","+Kazushi Ikeda","+Masayuki Inaba","-Drago Indjic","+Nitin Indurkhya","+Jeff Jackson","+Sanjay Jain","+Wolfgang Janko","-Klaus P. Jantke","+Nathalie Japkowicz","+George H. John","+Randolph Jones","+Michael I. Jordan","+Leslie Pack Kaelbling","+Bala Kalyanasundaram","-Thomas E. Kammeyer","-Grigoris Karakoulas","+Michael Kearns","+Neela Khan","+Roni Khardon","+Dennis F. Kibler ","+Jorg-Uwe Kietz","-Efim Kinber","-Jyrki Kivinen","-Emanuel Knill","-Craig Knoblock","+Ron Kohavi","+Pascal Koiran","+Moshe Koppel","+Daniel Kortenkamp","+Matevz Kovacic","-Stefan Kramer","+Martinch Krikis","+Martin Kummer","-Eyal Kushilevitz","-Stephen Kwek" ,"+Wai Lam","+Ken Lang","-Steffen Lange", "+Pat Langley","+Mary Soon Lee","+Wee Sun Lee","+Moshe Leshno","+Long-Ji Lin", "-Charles X. Ling", "+Michael Littman","+David Loewenstern","-Phil Long","+Wolfgang Maass","-Bruce A. MacDonald","+Rich Maclin","-Sridhar Mahadevan","-J.Jeffrey Mahoney","+Yishay Mansour","+Mario Marchand","-Shaul Markovitch","-Oded Maron","+Maja Mataric","+David Mathias","+Toshiyasu Matsushima","-Stan Matwin","-Eddy Mayoraz ","-R. Andrew McCallum","-L.Thorne McCarty","-Alexander M. Meystel","+Michael A. Meystel ","-Steven Minton","+Nina Mishra","+Tom M. Mitchell","+Dunja Mladenic","+David Montgomery","-Andrew W. Moore","+Johanne Morin","+Hiroshi Motoda","-Stephen Muggleton ","+Patrick M. Murphy","-Sreerama K. Murthy","+Filippo Neri","-Craig Nevill-Manning","-Andrew Y. Ng","+Nikolay Nikolaev","-Steven W. Norton","+Joseph O'Sullivan","+Dan Oblinger","+Jong-Hoon Oh","-Arlindo Oliveira","+David W. Opitz", "+Sandra Panizza","+Barak A. Pearlmutter","-Ed Pednault","+Jing Peng","+Fernando Pereira","+Aurora Perez","+Bernhard Pfahringer","+David Pierce","-Krishnan Pillaipakkamnatt","+Roberto Piola","+Leonard Pitt","+Lorien Y. Pratt","-Armand Prieditis","+Foster J. Provost","-J. R. Quinlan","+John Rachlin","+Vijay Raghavan","-R. Bharat Rao","-PriscillaRasmussen","+Joel Ratsaby","+Michael Redmond","+Patricia J. Riddle","+Lance Riley","+Ronald L. Rivest","+Huw Roberts","+Dana Ron","+Robert S. Roos","+Justinian Rosca","+John R. Rose","+Dan Roth","+James S. Royer","+Ronitt Rubinfeld","-Stuart Russell","+Lorenza Saitta","+Yoshifumi Sakai","+William Sakas","+Marcos Salganicoff","-Steven Salzberg","-Claude Sammut","+Cullen Schaffer","+Robert Schapire","+Mark Schwabacher","+Michele Sebag","+Gary M. Selzer", "+Sebastian Seung","-Arun Sharma","+Jude Shavlik v ","+Daniel L. Silver","-Glenn Silverstein","+Yoram Singer","+Mona Singh","+Satinder Pal Singh","+Kimmen Sjolander","+David B. Skalak","+Sean Slattery","+Robert Sloan","+Donna Slonim","+Carl H. Smith","+Sonya Snedecor","+Von-Wun Soo","-Thomas G. Spalthoff","+Mark Staley","-Frank Stephan","+Mandayam T. Suraj","+Richard S. Sutton","+Joe Suzuki","-Prasad Tadepalli","+Hiroshi Tanaka","-Irina Tchoumatchenko","-Brian Tester","-Chen K. Tham","+Tatsuo Unemi","-Lyle H. Ungar","+Paul Utgoff","+Karsten Verbeurgt ","+Paul Vitanyi","+Xuemei Wang","+Manfred Warmuth","+Gary Weiss","-Sholom Weiss","-Thomas Wengerek","-Bradley L. Whitehall","-Alma Whitten","+Robert Williamson","+Janusz Wnek","+Kenji Yamanishi","+Takefumi Yamazaki","+Holly Yanco","+John M. Zelle" ,"-Thomas Zeugmann","+Jean-Daniel Zucker","+Darko Zupanic"};for(int m=1;m<8;m++){for(int i=0;i<280;i++){if(name[i][0]=='+'){if(count==0){letter[count]=name[i][m];count++;continue;}//判断读取的字母是否已存入letter数组for(int j=0;j<count;j++){if(name[i][m]==letter[j]){flag++;//如果已存入则给flag加1}}//如果flag等于0，则该字母还未被存入if(flag==0){letter[count]=name[i][m];//存入count++;}flag=0;//初始化flag}}cout<<"发“+”徽章的人名字的第"<<m<<"个字符：";for(int k=0;k<count;k++){if((int)letter[k]>64&&(int)letter[k]<74)cout<<'0'<<(int)letter[k]-64<<" ";else if((int)letter[k]>73&&(int)letter[k]<91) cout<<(int)letter[k]-64<<" ";else if((int)letter[k]>96&&(int)letter[k]<106) cout<<'0'<<(int)letter[k]-96<<" ";else if((int)letter[k]>105&&(int)letter[k]<123) cout<<(int)letter[k]-96<<" ";}for(int l=0;l<30;l++){letter[l]=NULL;}cout<<endl;}/*for(int n=1;n<8;n++){for(int i=0;i<280;i++){if(name[i][0]=='-'){if(count==0){letter[count]=name[i][n];count++;continue;}//判断读取的字母是否已存入letter数组for(int j=0;j<count;j++){if(name[i][n]==letter[j]){flag++;//如果已存入则给flag加1}}//如果flag等于0，则该字母还未被存入if(flag==0){letter[count]=name[i][n];//存入count++;}flag=0;//初始化flag}}cout<<"发“-”徽章的人名字的第"<<n<<"个字符：";for(int k=0;k<count;k++){if((int)letter[k]>64&&(int)letter[k]<74)cout<<'0'<<(int)letter[k]-64<<" ";else if((int)letter[k]>73&&(int)letter[k]<91) cout<<(int)letter[k]-64<<" ";else if((int)letter[k]>96&&(int)letter[k]<106)cout<<'0'<<(int)letter[k]-96<<" ";else if((int)letter[k]>105&&(int)letter[k]<123)cout<<(int)letter[k]-96<<" ";}for(int l=0;l<30;l++){letter[l]=NULL;}cout<<endl;}*/return 0;}附件2#include<iostream>using namespace std;int main(){char name[30];bool flag=true;while(flag){cout<<"Please input Name：";cin>>name;//如果名字的第二个字母是aoeiu里的一个，则给他+徽章，否则给他-徽章if(name[1]=='a'||name[1]=='o'||name[1]=='e'||name[1]=='i'||name[1]=='u')cout<<"Give a '+' Badge!"<<endl;elsecout<<"Give a '-' Badge!"<<endl;cout<<"intput '#' to exit!"<<endl;if(name[0]=='#')flag=false;}return 0;}。

1985 年美国大学生数学建模竞赛MCM 试题1985年MCM:动物种群选择合适的鱼类和哺乳动物数据准确模型。

模型动物的自然表达人口水平与环境相互作用的不同群体的环境的重要参数,然后调整账户获取表单模型符合实际的动物提取的方法。

包括任何食物或限制以外的空间限制,得到数据的支持。

考虑所涉及的各种数量的价值,收获数量和人口规模本身,为了设计一个数字量代表的整体价值收获。

找到一个收集政策的人口规模和时间优化的价值收获在很长一段时间。

检查政策优化价值在现实的环境条件。

1985年MCM B:战略储备管理钴、不产生在美国,许多行业至关重要。

(国防占17%的钴生产。

1979年)钴大部分来自非洲中部,一个政治上不稳定的地区。

1946年的战略和关键材料储备法案需要钴储备,将美国政府通过一项为期三年的战争。

建立了库存在1950年代,出售大部分在1970年代初,然后决定在1970年代末建立起来,与8540万磅。

大约一半的库存目标的储备已经在1982年收购了。

建立一个数学模型来管理储备的战略金属钴。

你需要考虑这样的问题:库存应该有多大?以什么速度应该被收购?一个合理的代价是什么金属?你也要考虑这样的问题:什么时候库存应该画下来吗?以什么速度应该是画下来吗?在金属价格是合理出售什么?它应该如何分配?有用的信息在钴政府计划在2500万年需要2500万磅的钴。

美国大约有1亿磅的钴矿床。

生产变得经济可行当价格达到22美元/磅(如发生在1981年)。

要花四年滚动操作,和thsn六百万英镑每年可以生产。

1980年,120万磅的钴回收,总消费的7%。

1986 年美国大学生数学建模竞赛MCM 试题1986年MCM A:水文数据下表给出了Z的水深度尺表面点的直角坐标X,Y在码(14数据点表省略)。

深度测量在退潮。

你的船有一个五英尺的草案。

你应该避免什么地区内的矩形(75200)X(-50、150)?1986年MCM B:Emergency-Facilities位置迄今为止,力拓的乡牧场没有自己的应急设施。