2014 美赛优秀论文1 B题

合集下载

2014美赛b题分析

上一篇文章中我们用题意分解的办法详细解读了2014MCM A题的要点，并总结了6个题意的关键点,大家可以对照自己的文章，看看有没有把这些关键问题解释和说明清楚，如果这些问题全部回答到位,M奖就到手啦，在这些部分中有1~2个有所创新，能够令评委眼前一亮的话,那么就有机会得到更高的奖项哦,大家是不是心里更加有底了呢？接下来，我们继续用上篇文章提到的方法来解析B题的关键点，上题：PROBLEM B: CollegeCoaching Legends （bk1: the topic）Sports Illustrated, amagazine for sports enthusiasts, is looking for the “best all time college coach” male or femalefor the previous century. (bk2: briefintroduction of background）Build a mathematical model （spm1）tochoose the best college coach or coaches （past or present）（rsc1) from among either male or female (rsc2)coaches in such sports as college hockey or fieldhockey, football, baseball or softball，basketball，or soccer (rsc3)。

(These 3 restrictiveconditions ask us the range of our evaluation model should contain thedifference above.)Does it make a difference which time line horizon thatyou use in your analysis, i。

2014年美赛数模B题-Finalist

24270 T4 ________________ F4 ________________Team Control NumberFor office use only For office use only T1 ________________ F1 ________________ T2 ________________ F2 ________________ T3 ________________ F3 ________________ Problem Chosen B2014 Mathematical Contest in Modeling (MCM) Summary SheetSummaryIn order to estimate the excellence of different sports coaches and to give a ranking result, two distinct models are developed. The first model is a comprehensive evaluation method. And the second model is a ranking algorithm analogous to the Journal Influence Algorithm . In the first model, we take into account a variety of metrics, and divide them into twocategories: Objective Metrics and Subjective Metrics . In the Objective Metrics , we consider four factors, the number of wins, winning percentage, champions and final fours. All these factors have contributions to the excellence of a coach. We deem that the total number of games in a year could affect the number of wins, and the unevenness of team quality could affect the winning percentage. By employing statistical regression method to process collected data, we establish two functions of influence coefficient to eliminate thediscrepancy caused by the two kinds of effect. In the Subjective Metrics : we consider two factors, media popularity and tenure. We employ Fuzzy Analysis Method to quantify these two subjective factors. We further incorporate Analytic Hierarchy Process (AHP) and Gray Relational Analysis Grade Method (GRAP) to determine the weight allocation to different metrics. The final ranking gives a comprehensive result by weighing results returned by these two methods. Using data from Sports Reference and other websites, the rankings in basketball, football and baseball accord with previous media commentaries.In the second model, we deem that the excellence of a certain coach can be reflected from the media impact over the span of history and that the interactions between two coaches can reflect the disparity of skill level between them. We use search results returned by Google to quantify the impact of one coach on another. Based on the search results, we build a cross-reference matrix to represent relationships between coaches. In view that the different time periods that two coaches were in may largely affect the interaction between them, and the personal reputation may influence the number of search results, we develop a weight function of two variables to compensate the influence of time and to rule out the redundant information.In consideration of the similarity between personal influence and journal influence, we refer to the Journal Influence Algorithm introduced by Eigenfactor and establish a new ranking algorithm. The basic idea of the algorithm is subtle: using weight function to modify the cross-reference matrix , and taking into consideration of individual influence, the algorithm gives an evaluation vector to rank different coaches. To test the validity of this algorithm, we apply the algorithm into basketball, football and baseball. The algorithm gives a result that is similar to the result obtained in the first model. The ranking also agrees withprevious media commentaries. Furthermore, by slightly adjusting the coefficients, we can apply the algorithm into various sports.“Dream Team” of College Coaches# Team 24270Team # 24270 Page 2 of 26Contents1. Introduction (3)1.1. Restatement of the Problem (3)1.2. Model Overview (3)2. Assumptions (3)3. ModelⅠ (4)3.1. Additional assumptions (4)3.2. Notations (4)3.3. Evaluation System (5)3.3.1. The influence of time on the total number of wins (6)3.3.2. The influence of time on the winning-percentage (6)3.3.3. Fuzzy Analysis (7)3.3.4. Nondimensionalization process (8)3.3.5. Final result (8)3.4. Solutions to ModelⅠ (10)3.4.1. Basketball (10)3.4.2. Football (11)3.4.3. Baseball (12)3.4.4. Sensitivity analysis (13)4. ModelⅡ (13)4.1. Additional assumptions (14)4.2. Notations (14)4.3. The Individual Influence Vector (14)4.3.1. Original data (15)4.3.2. The influence coefficient of time (15)4.3.3. The influence coefficient of reputation (16)4.3.3. The individual influence vector (16)4.4. The Cross-Reference Matrix (16)4.4.1. The weight function (17)4.4.2. The final cross-reference matrix (17)4.5. The Evaluation Vector (18)4.6. Solutions to Model II (18)4.6.1. Basketball (18)4.6.2. Football (19)4.6.3. Baseball (19)4.6.4. Sensitivity analysis (19)5. Applicability (20)6. Strengths and Limitations (21)6.1. ModelⅠ (21)6.2. Model II (21)7. Conclusions (21)8. The Article for Sports Illustrated (22)References (23)Appendix (24)Team # 24270 Page 3 of 261. Introduction1.1. Restatement of the ProblemSports, by definition, is all forms of usually competitive physical activity which aim to usephysical ability while providing entertainment to participants and spectators [1]. No wonder theword “sports” gives us a first impression of fierce competition, agitated spectators, sweating on the running track, combined with a joy of victory. It is the uncertainty that makes the sports game so intriguing. However, where there is competition, there will always be victory, defeat, and ranking. Loyal sport fans could debate day and night over the question who is thebest player or coach. These debates have called forth a need for certain criterion of sports coaches and players. The criterion has to be: (1) all-encompassing to take into consideration a variety of factors; (2) applicable to various sports; (3) robust enough to remain unaffected by fluctuation.1.2. Model Overview● Model Ⅰ The evaluation method in Model Ⅰis based on a comprehensive method sophistically combining Analytic Hierarchy Process(AHP) and Gray Relational Analysis GradeMethod(GRAP). In the evaluation process, we take into consideration the influence of time horizon, and incorporate Fuzzy Analysis Method, which make it feasible to compare diverse factors on the same level. The ranking results in three different sports accord with previous media report, which attest the validity of this method.● Model ⅡIn model II, we assume that the excellence of a certain coach can be reflected from the media impact over the span of history and thus can be gauged by the impact on another coach within or without the same period of time. We use Google search results to quantify the impact of one coach on another. The relationship between coaches can be established as a cross-reference matrix. By further taking into account the influence of time, influence of reputation, and a modification to rule out the redundant information, we obtain a finalevaluation vector. The final ranking result is roughly approximate to the result in model I. To sum up, we only need the search results returned by Google search engine to estimate the excellence of certain coach with high accuracy.2. Assumptions● We assume that the competition rules of each sport do not change.Although sports are developing, we do not take into account of time in the competition rules in order to compare the coaches of different years more fairly.● We neglect tied competitions since they have the same effect on the two comparedteams.● We only take the Division I into consideration.x , x , x ’’, x *Evaluation index matrix Team # 24270 Page 4 of 26Competitions are divided into three parts: Division I, II and III according to the level of sport strengths of different colleges. Since Division I always concludes top coaches, we only take Division I into consideration.● The selected data are valid.● Additional assumptions are made to simplify analysis for individual sections. Theseassumptions will be discussed at the appropriate sections.3. Model Ⅰ3.1. Additional assumptions● The evaluation system includes two parts: Objective Metrics(OM) and SubjectiveMetrics (SM).● We assume that OM include four specific indexes: the total number of wins, the winning-percentage, the number of final fours and the number of champions.● Tenure and media popularity are considered in SM.In the subjective metrics of ranking coaches, some factors are hard to investigate qualitatively and quantitatively due to lacking data, such as, his or her influence to players, range of knowledge, studying ability, team spirits, searching talents, acting in competitions, salary and so on. Therefore, we neglect these indexes in SM.● Time only makes a difference in the total number of wins, and the winningpercentage.In fact, the numbers of final fours and champions have no effect on the other two in OM, since the number of teams which are able to enter into final fours and even achievechampions is fixed. And we neglect the influence of time on media popularity in order to simplify the model.3.2. NotationsTable 1: Notations and DescriptionsNotations DescriptionsS i Evaluation objectx j Evaluation indexn The number of evaluation objectsm The number of evaluation indexes’ t Timep i , q i Influence coefficients of timeW (t ) The total number of competitions in ts (t ) The standard deviation of all winning-percentage in tM j Maximum of x ijm j Minimum of x ij, Grey relational coefficient ∆�� Absolute difference [ ] ( )1 2, , m x x =x . , 1x m >[ ]= 1 2 3 4 5 6x x , , , , ,x x x x x Team # 24270 Page 5 of 26 Notations Descriptionsf (x ) Subordinate functionA Pairwise comparison matrixλ The largest eigenvaluew Weight vectorCI Consistency indexRI Random consistency indexCR Consistency ratioB Evaluation vector of AHP (0)��Δmin Minimum differenceΔmax Maximum differencer Relation degree vectorC Evaluation vector of Grey Relation Degreeα , β Partial coefficientU Ultimate evaluation vector3.3. Evaluation SystemWe define n as the number of evaluation objects, and S 1, S 2,…, S n (n >1) are the evaluation objects. m is the number of evaluation indexes, and x 1, x 2,…, x m are the evaluation indexes. Evaluation index vector isTThe total evaluation indexes include OM: the total number of wins, the winning-percentage(pct.), the number of final fours and the number of champions and SM: tenure and media popularity. So m = 6 ,TWhere: ● x 1 — the total number of wins vector.● x 2 — the winning-percentage vector.● x 3 — the number of final fours vector.● x 4 — the number of champions vector.● x 5 — tenure vector.● x 6 — media popularity vector.i p = ' Team # 24270 Page 6 of 26 Figure 1: Flow chart of model I Undoubtedly, time plays an important role in evaluating top coaches. According to the assumptions, time only makes a difference in the total number of wins, the winning- percentage.3.3.1. The influence of time on the total number of winsWith the development of sports, the competition is getting relatively fiercer than ever, which means the disparity between teams become wider. The total number of games also increases with time going on. Therefore, when evaluating coaches in the previous century, the later certain coach begin his coaching career, the more likely he will get more wins. So we should put less weight on the coaches active in a later time period. And we can get a fairer evaluation of coaches within different time periods.In order to compensate the influence of t , we establish Influence Coefficients of Time (ICT) p i (i = 1,2, , n ) . We assume that the total number of competitions in t is W (t ) . W (t ) canbe obtained by statistical regression and simulating and curve fitting of selected data. So we define:1 W (t mi )where t mi is the middle year of tenure of S i . And then x 1i = x 1i ⋅ p i (i = 1,2, , n ) .3.3.2. The influence of time on the winning-percentageAs for the winning-percentage, sports were underdeveloped at an earlier time, and the quality disparity between teams is comparatively narrow. Therefore, the standard deviation of winning-percentage of each coach is closer to zero. Thus we should put less weight on the coaches active in a “mediocre” time period. We define ICT here as q i (i=1,2,…,n ), we assume that the standard deviation of all winning-percentage in t is s (t ) . s (t ) can be obtained by statistical regression and simulating and curve fitting of selected data. So we define:i q = ,1 3x ⎤ ≤ ≤⎪⎣ ⎦( ) 121 a x b --⎧⎡ + - ,1 3x ⎤ ≤ ≤⎪⎣ ⎦ ( ) 121 2.8049 0.4417x --⎧⎡ + - ' Team # 24270 Page 7 of 261 s (t mi )and x 2i = x 2i ⋅ q i (i = 1,2, , n ) .3.3.3. Fuzzy AnalysisAs for SM indexes, we assume that they can be divided into five levels: “ Excellent, Very Good, Good, Not Good, Bad”. And we correspond the five levels into 5,4,3,2,1 successively For continuous quantification, we assume:As for “Excellent”, we suppose f (5) = 1.As for “Very Good”, f (3) = 0.7 .As for “Bad”, f (1) = 0.1 .We employ partial large Cauchy distribution and the logarithmic function as the subordinate function [2]: f (x ) = ⎨⎩⎪c ln x + d , 3 ≤ x ≤ 5where a , b , c , d stands for undetermined constants. We use the initial conditions above to define their values. And solution of the subordinate function( Figure 2) is:f ( x ) = ⎨ (1) ⎪⎩0.5873ln x + 0.0548, 3 ≤ x ≤ 5Figure 2: Trend of f (x )Media popularity is measured by the number of search results via Google. The impact of duplication of names can be neglected by means of adding search keywords in order to rule out the redundant information.We map x j ( j =5,6) into interval [1,5], through function (1),we can obtain:( )4 ji j x m ⎛ ⎫-1x f ⎪=+()12 5,6i n j == (2)⎝ ⎭ =( )1, 2, ,6j =and ' ' ' ' '1 2 3, , , , j j j j n x x x x ⎡ ⎤= ⎣ ⎦x , , , , ,⎡ ⎤= ⎣ ⎦* '' '' '' '' '' ''1 2 3 4 5 6x x x x x x x(4) 1⎢ 2 ⎥⎢ ⎥1⎢ 5 3 ⎥1 3⎢ 5 3 ⎥3 1 5⎢ 3 ⎥⎥⎢ 2⎢ 3 1 ⎥⎥[ ]0.1248,0.1469,0.4593,0.8125,0.0775,0.2928=w 1≤ ≤1≤i ≤ n 1≤ ≤1≤i ≤ n ⎣ Team # 24270 Page 8 of 26M j - m j ⎪where M j = max {x ij } , m j = min {x ij} ( j = 5,6) .As for x 3 and x 4, we define that x 3’= x 3, x 4’= x 4.we use x 'j ( j = 1, 2, ,6) to proceed the following calculation.3.3.4. Nondimensionalization processWe employ extreme difference method to nondimensionalize the different indexes so that we can compare them [2]on the same level. The method is as follows:x 'ji - m jM j - m jTwhere M j = max {x ij } , m j = min {x ij } ( j = 1, 2, ,6) .and then we obtain the final evaluation index matrix:T3.3.5. Final resultBy using AHP as the subjective evaluation method and GRAP as the objective method, the final represents a comprehensive evaluation combined the merits of these two methods. Analytic Hierarchy Process [3] (AHP)By comparing the effect of two indexes x 'j ,the weights of the two method w ( x 'j )(j =1,2,…,m )are given. Then we construct the pairwise comparison matrix A .1 53 7 1 1⎢ 3 5 ⎥ 1A =⎢ 6 ⎥ ⎢ ⎥15 5 7 1 1 1 ⎢ 3 3 5 ⎦ We can obtain the largest eigenvalue of A :λ=6.0496 and its weight vector :Tn λ - CR = 0.008 0.1= <()(){ }( )1,2, ,x i n = =x( ) ( ) ( )ji i r x x = m m ρ∆ +∆( )0 max ji ρ∆ + ∆(ji i i x ∆ x = - ● —absolute difference.( )min min min i ∆ = ∆ —minimum difference of all indexes data.● ( )max max max i ∆ = ∆ —maximum difference of all indexes data.● 1w =∑(i i i w x = ∑ ( ) (,j r r x x = )j ( ) ,r x )j(6) Team # 24270 Page 9 of 26 After that, we must check the consistency of matrix A . The consistency index is calculated as follows:CI = = 9.92 ⨯10-3n -1From Table 2, the random consistency index RI =1.24Table 2: The Quantitative Values of RI [2]n 1 2 3 4 5 6 7 8 9 10 11 RI 0 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 1.51 Then, we can obtain consistency ratio: CIRI Therefore, we can safely draw the conclusion that the inconsistent degree of matrix A is in a tolerable range, and we can take its eigenvector as weight vector w [3].We define B as the evaluation vector of AHP, and B can be calculated as follows:B = x ' ⋅ w (5)In evaluation vector, the greater B i is, the higher ranking S i is.● Gray Relational Analysis Grade Method [4] (GRAP)We use integral grey relational degree to analyze the metrics data. And we take the total number of wins as the reference sequence:0 0and then we can obtain the gray relational coefficient [4]:, i = 1, 2, , n , j = 1,2, ,6Where:(0) (0)j ) jj i jj i ● ρ —resolution ration.For every coach S i , we determine its weight as w i , which should satisfy the requirements:n0 ≤ w i ≤ 1, ii =1 After determining the weight, we can obtain the relational degree [4]:ni =1And then we construct the relation degree vector [ ]1 2 3 4 5 6r r r = , where 1 1r = . , , , , ,r r r rTWe define C as the evaluation vector of AHP, and C can be calculated as follows::C = x ' ⋅ w (7)In evaluation vector, the greater C i is, the higher ranking S i is.● Combination of AHP and GRAPAt first, we employ extreme difference method to nondimensionalize the two evaluation vector B and C . And then, we construct an ultimate evaluation vector: U = α B + β C (8) where α , β respectively stands for the weight of AHP and GRAP, which should satisfy the requirements of α + β =1. Finally, we sort the value of U i (i =1,2,…,n ), and S i that corresponds to the top 5 of U i are top five coaches. 3.4. Solutions to Model ⅠWe choose three sports to verify our model and get the results, which include basketball, football and baseball.3.4.1. Basketball● Searching and selecting data We search and select data through the Internet [5][6][7]. For example, first, we search 100 coaches and their evaluation index data. Secondly, we rank them by comprehensively considering the total number of wins and the winning-percentage, so we can get top 40coaches. And then, we consider other metrics and rank top 20 coaches. Finally, the evaluation system is based on the selected 20 data. Table A1 in Appendix show the selected coaches and their evaluation index data.● Determining the final evaluation index matrixAt first, we determine vector x ’. For x 1, via the data in Table A2, we use W (t i )=Num 2, where Num represents the total number of teams in t i . We utilize software MATLAB to plot the graph of W (1)(t ) by simulating and curve fitting of data (Figure 3). So we can get p i for each S i , and then we obtain the vector x 1’.Figure 3: Trend of W (1)(x ) Figure 4: Trend of s (1)(x ) For x 2, via the data in Table A3, we also plot the graph of s (1)(t ) by simulating and curve fitting of data (Figure 4). So we can get q i for each S i , and then we obtain the vector x 2’.感谢作者分享]0.130 0.318 0.243 0, , , .482,1.761,0.138( ) [ ]19,14,8,12,9,7,18,11,3,5,16,10,1,2,17,4,13,6,20,15=1Rank ]1.000 0.316 0.151 0, , , .047, 0.392,0.052 For x 5 and x 4, from (1)(2), we can obtain x 5’ and x 6’.Secondly, from (3), we can obtain x ''j ( j = 1,2, ,6) . Finally, from (4), we can obtain x * .We list the quantitative value of x * in Table A4.● Obtaining the result via ultimate evaluation vectorAt first, from (5), we use AHP and get B . Secondly, we use GRAP and define that ρ=0.3and w i =0.05(i =1,2,…,n ). From (6), we can get the relation degree vector r . And then, from (7), we can obtain C . Finally, from (8), by defining α = 0.6, β = 0.4 , we can obtain the ultimate evaluation vector:U = [0.293,0.288,0.468,0.241,0.422,0.160,0.521,0.868,0.713,0.311,0.481,0.836,0.168,0.998, TBy sorting the value of U i (i =1,2,…,n ), we can obtain the ranking result of S i . And the ranking vector is: TTherefore, we list top five coaches of basketball in the previous century in Table 3:Table 3: Top 5 Coaches of Basketball No.1 No.2 No.3 No.4 No.5S 19 S 14 S 8 S 12 S 9John Wooden Dean Smith Mike Krzyzewski Adolph Rupp Bob KnightThis result is largely agreement with the widely accepted result [8][9].3.4.2. Football● Searching and selecting dataLike what we do in basketball, we search and select data through the Internet [5][10][11]. However, we calculate that the number of final fours is the sum number of times that teams can enter into Super Bowl.● Determining the final evaluation index matrixAt first, we determine vector x ’. For x 1, we use W (t i )=Num 2, where Num represents the total number of teams in t i .We can obtain W (2)(t ) by simulating and curve fitting of data. So we can get p i for each S i , and then we obtain the vector x 1’.For x 2, we also obtain s (1)(t ) by simulating and curve fitting of data. So we can get q i and the vector x 2’.Finally, from (4), we can obtain x * . We list the quantitative value of x * .● Obtaining the result via ultimate evaluation vectorLike what we do in Basketball, we can obtain the ultimate evaluation vector:U = [0.234,0.879,0.738,0.106,0.359,0.296,0.383,0.193,0.291,0.453,0.494,0.248,0.180,0.615,TBy sorting the value of U i (i =1,2,…,n ), we can obtain the ranking result of S i . And the ranking vector is: 感谢作者分享( ) [ ],2,3, ,11,10, ,7,5,16, ,9,12,1,8,1315 14 1 ,17,49 6 ,20,18=2Rank ''m t = 5t + ]0.022,0.044,0.232,0.225, 0.138,0.118( ) [ ],3,10,1,9,6, 2,17,18, 4,13,7,5,11,8,19, 20,161 ,12 ,15 4=3RankTTherefore, we list top five coaches of basketball in the previous century in Table 4:Table 4: Top 5 Coaches of Football No.1 No.2 No.3 No.4 No.5S 15 S 2 S 3 S 14 S 11 Joe Paterno Bobby Bowden Bear Bryant Tom Osborne Don JamesThis result is largely agreement with the widely accepted result [12].3.4.3. Baseball● Searching and selecting dataLike what we do in basketball, we search and select data through the Internet [13][14]. But in this sport, we assume that the number of final fours is the number of champions of NCAA competitions that teams can achieve. And we assume that the number of champions is the number of champions of National competitions that teams can get.● Determining the final evaluation index matrixAt first, we determine x ’. For x 1, due to scarcity of the data, we can only search a little information of several years [9]. We use W (t i )=Num 2, where Num represents the total number of competitions of champion in t i .We can obtain W (3)(t ) by simulating and curve fitting of data. So we can get p i for each S i , and then we obtain the vector x 1’.For x 2, due to lacking the standard difference of winning-percentage in every ten year, we choose another approach to get x 2’. Considering the influence of time, first, we employ extreme difference method to nondimensionalize t m into t m’, where t mi is the middle year of tenure of S i . Then, we define1'mand then we define x 2' i = x 2i ⋅ t mi '' . So from (3), we obtain x ''2 .Finally, from (4), we can obtain x *. We list the quantitative value of x *.● Obtaining the result via ultimate evaluation vectorLike what we do in Basketball, we can obtain the ultimate evaluation vector:U = [0.353,0.278,0.587,0.223,0.205,0.302,0.208,0.189,0.314,0.494,0.203,1.000,0.214,0,TBy sorting the value of U i (i =1,2,…,n ), we can obtain the ranking result of S i . And the ranking vector is:TTherefore, we list top five coaches of basketball in the previous century in Table 5:Table 5: Top 5 Coaches of Baseball感谢作者分享Team # 24270 Page 13 of 26No.1 No.2 No.3 No.4 No.5S15S2S3S14S11 John Barry Mike Martin Rod Dedeaux Augie Garrido Jim MorrisThis result is largely agreement with the widely accepted result[15].3.4.4. Sensitivity analysisBy changing the weight of AHP and GRAP in equation (8), we analyze the changing result of basketball. For example, we define α = 0.5, β = 0.5 , and the result is listed in Table 6.The coaches who rank top 5 do not change:Table 6: Top 5 Coaches of BasketballNo.1 No.2 No.3 No.4 No.5S19S12S14S8S9 John Wooden Adolph Rupp Dean Smith Mike Krzyzewski Bob KnightWhen defined α = 0.4, β = 0.6 , the result changes, which is listed in Table 7. The coaches who rank top five change:Table 7: Top 5 Coaches of BasketballNo.1 No.2 No.3 No.4 No.5S19S12S14S7S8 John Wooden Adolph Rupp Dean Smith Hank Iba Mike KrzyzewskiWhen defined α = 0.7, β = 0.3 , the result is listed in Table 8. The coaches who rank top five do not change:Table 8: Top 5 Coaches of BasketballNo.1 No.2 No.3 No.4 No.5S19S14S12S8S9 John Wooden Dean Smith Adolph Rupp Mike Krzyzewski Bob KnightAs can be seen from above, when there is a slight change of weights, the result do not change. But with a relatively greater change, weights have an effect on the result.4. ModelⅡHow could one’s reputation affect another’s? One way is to follow the implication in the saying: “You wouldn’t mention A and B in the same breath.” It means if the differencebetween two people is too wide, it would be unlikely for most of individuals to mention them in a same talk. The same holds true for the sports coaches. That means, if two coaches areabsolutely not on the same level, more likely than not, there will be few reports on these two coaches. On the other hand, if two of them are top coaches, there will be a plethora of reports: such as “The Greatest Coaches Ever” “Basketball Hall of Fame”, on the two coaches.Informed by this natural law, we may find an innovative approach to estimate a coach’s level of excellence and popularity. The working flow is shown as follows:感谢作者分享Team # 24270 Page 14 of 26Figure 5: Flow chart of model II4.1. Additional assumptions● The excellence of a coach and association between two coaches can be accuratelyreflected by the mass media.● The attention that the mass media have on certain coach is related to the search results onGoogle, in terms of number of pages, report orientation and report time.● The media attention is related to time and the excellence of certain coach. The influenceof time and excellence on the media attention remains unchanged to different kinds ofpeople.4.2. NotationsTable 9: Notations and DescriptionsNotations Descriptions��The number of search results of coach ik, b Coefficient of the function through linear regression��Characteristic year of coach iu The number of search results about sports careerICT Influence coefficient of timeICR Influence coefficient of reputationl Individual influence vectorZ Original cross-reference matrixWF Weight functionW Weighted cross-reference matrixα,βPartial coefficient4.3. The Individual Influence VectorBy our hypotheses, the excellence of a coach can be accurately reflected by the mass media.There are several ways to evaluate the media attention on a celebrity. One of the most simple and direct way is to record the number of search results on Google. However, the searchresults can be influenced by a variety of factors, such as time periods, tenure, etc. Bysimulating and curve fitting of sorted data, we evaluate the impact of such factors separately.Finally, we obtain a normalized individual influence vector.感谢作者分享( ) 1, 2, ,i n = (influence coefficient of time)=i ICT Team # 24270 Page 15 of 264.3.1. Original dataHere we define t i as the characteristic year , the average of the year that the coach i start coaching and the year of his or her retirement. (If the coach i is still active, then t is theaverage of the year that the coach i start coaching and this year, that is, 2014)The search results vector a is the original data we use to estimate the individual influence, where a i is the number of search results of coach i . Particularly, the coaches here are sorted bycharacteristic year in a descend order. This can be a great convenience to our later discussionabout time factor.4.3.2. The influence coefficient of timeAccording to the growth law of web information [16], the information aiming at a certainfield is similar to an exponent increase. To test this hypothesis and better apply it to sports, we entered the Google website. Using “1910 basketball”, “1920 basketball” ， and “1930 basketball” as the “exact keywords”[17] respectively. The numbers of search results are shownin Table 10:Table 10: The Numbers of Search ResultsYear 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010Results 2750 5160 7440 11700 16200 26400 40200 25800 27000 67700 326000Assuming that this is an exponential function: y 1 = c ⋅ e dt . We use the least squared methodto obtain the unknown numbers in the function. See Figure 6.Figure 6: Trend of exponential function y 1 Figure 7: Trend of linear function y 2The result gives a satisfying simulation to the numbers of search results. However, thedistinction between 2000s and 1900s is too large. In our observation, the search results of coaches at different period of time is almost of the same magnitude of as each other. So, here we use the natural logarithm of the search results. Again we obtain a linear function y 2 as showed in Figure 7.The difference between maximum and minimum is about half of the minimum value. This is a modest value that we can safely put into use to estimate ICT . Common sense told us that the greater number of total reports is, the more “valuable” the search result is, the greater weight the search result will get. So, we define ICT as1 kt i + b 感谢作者分享。

2014MCM-B优秀论文

2014MCM-B-优秀论文美赛丛书目录（考虑）1. 问题2. 问题背景与问题分析3. 评价指标体系选哪些指标？理由何在？如何度量？4. 排名模型（权重模型）5. 时间因素处理6. 模型检验7. 问题综合分析与进一步研究8. 优秀论文A-26911-东南大学9. 优秀论文B - 30680-美国-北卡26160-重庆大学摘要：灰色与模糊评价模型，另外考虑了性别与时间因素。

AHP筛选特征因子，7个因子，灰色相关模型，模糊综合评价模型，灰色模型略强，时间因素对前十人选影响较小。

26160-重庆大学.pdf评价：除结果图外，乏善可陈，时间因素影响的结论有误。

26636-外经贸大学摘要：灰色相关模型，依据专家意见选择了四个评价指标：NCAA冠军，Pct，胜场数，教练报酬。

模糊相容矩阵确定各个评价指标的权值，结果与ESPN作比较。

最后讨论了时间因素，发现规律：“从前”的教练的胜率要远远高于“现在”的教练，但其他三个指标所受到的影响很小。

引入滑动平均方法，将时间因素纳入胜率计算模型中，这是本文的一个亮点。

Shannon熵用于评价稳定性。

讨论了参数敏感性。

便利与普适是我们模型的最大优点，但存在指标选择的主观性。

26636-外经贸.pdf评价：指标体系以及评价模型一般，有点投机，时间因素讨论、模型结果检验以及敏感性检验是亮点，结果对比表达清晰明了，可信度高。

缺假设与“conclusion”，是硬伤。

26911-东南大学三阶段全面评价模型，指标体系（胜率，稳定性，获得冠军数量，个人报酬，点击率，个人荣誉，职业联赛排名），谷歌趋势统计方法，线性拟合方法，加权和模型，AHP+最大熵模型，灰色相关分析，综合排名26911-东南大学.pdf评价：非常全面，思路很清晰，表达很简洁，值得效仿。

具体说：指标意义讨论充分；指标取值实用、合理；时间因素考虑到位；权重确定有技术含量；结果表达清晰；文章节奏把握好。

如果按更高标准衡量，第二种权重体系中GRA的作用不大显著。

2014建模美赛B题

For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number27820Problem 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.)Research on Choosing the Best College Coaches Based on Data Envelopment AnalysisSummaryIn order to get the rank of coaches in differ ent sports and look for the ―best all time college coach‖ male or female for the previous century, in this paper, we build a comprehensive evaluation model for choosing the best college coaches based on data envelopment analysis. In the established model, we choose the length of coaching career, the number of participation in the NCAA Games, and the number of coaching session as the input indexes, and choose the victory ratio of games, the number of victory session and the number of equivalent champion as the output indexes. In addition, each coach is regarded as a decision making unit (DMU).First of all, with the example of basketball coaches, the relatively excellent basketball coaches are evaluated by the established model. By using LINGO software, the top 5 coaches are obtained as follows: Joe B. Hall, John Wooden, John Calipari, Adolph Rupp and Hank Iba.Secondly, the year 1938 is chosen as a time set apart to divide the time line into two parts. And then, basketball coaches are still taken as an example to evaluate the top 5 coaches used the constructed model in those two parts, respectively. The evaluated results are shown as: Doc Meanwell, Francis Schmidt, Ralph Jones, E.J. Mather, Harry Fisher before 1938, and Joe B. Hall, John Wooden, John Calipari, Adolph Rupp and Hank Iba after 1938. These results are accordant with those best coaches that were universally acknowledged by public. It suggests that the model is valid and effective. As a consequence, it can be applied in general across both genders and all possible sports.Thirdly, just the same as basketball coaches, football and field hockey coaches are also studied by using the model. After the calculation, the top 5 co aches of football’s results are as follows: Phillip Fulmer, Tom Osborne, Dan Devine, Bobby Bowden and Pat Dye, and field hockey’s are Fred Shero, Mike Babcock, Claude Julien, Joel Quenneville and Ken Hitchcock.Finally, although the top 5 coaches in each of 3 different sports have been chosen, the above-mentioned model failed to sort these coaches. Therefore, the super- efficiency DEA model is introduced to solve the problem. This model not only can evaluate the better coaches but also can rank them. As a result, we can choose the ―best all time college coach‖ from all the coaches easily.Type a summary of your results on this page. Do not includethe name of your school, advisor, or team members on this page.Research on Choosing the Best College Coaches Based on DataEnvelopment AnalysisSummaryI n order to get the rank of coaches in different sports and look for the ―best all time college coach‖ male or female for the previous century, in this paper, we build a comprehensive evaluation model for choosing the best college coaches based on data envelopment analysis. In the established model, we choose the length of coaching career, the number of participation in the NCAA Games, and the number of coaching session as the input indexes, and choose the victory ratio of games, the number of victory session and the number of equivalent champion as the output indexes. In addition, each coach is regarded as a decision making unit (DMU).First of all, with the example of basketball coaches, the relatively excellent basketball coaches are evaluated by the established model. By using LINGO software, the top 5 coaches are obtained as follows: Joe B. Hall, John Wooden, John Calipari, Adolph Rupp and Hank Iba.Secondly, the year 1938 is chosen as a time set apart to divide the time line into two parts. And then, basketball coaches are still taken as an example to evaluate the top 5 coaches used the constructed model in those two parts, respectively. The evaluated results are shown as: Doc Meanwell, Francis Schmidt, Ralph Jones, E.J. Mather, Harry Fisher before 1938, and Joe B. Hall, John Wooden, John Calipari, Adolph Rupp and Hank Iba after 1938. These results are accordant with those best coaches that were universally acknowledged by public. It suggests that the model is valid and effective. As a consequence, it can be applied in general across both genders and all possible sports.Thirdly, just the same as basketball coaches, football and field hockey coaches are also studied by using the model. After the calculation, the top 5 coaches of football’s results are as follows: Phillip Fulmer, Tom Osborne, Dan Devine, Bobby Bowden and Pat Dye, and field hockey’s are Fred Shero, Mike Babcock, Claude Julien, Joel Quenneville and Ken Hitchcock.Finally, although the top 5 coaches in each of 3 different sports have been chosen, the above-mentioned model failed to sort these coaches. Therefore, the super- efficiency DEA model is introduced to solve the problem. This model not only can evaluate the better coaches but also can rank them. As a result, we can choose the ―best all time college coach‖ from all the coaches easily.Key words: college coach；data envelopment analysis; decision making unit; comprehensive evaluationContents1. Introduction (4)2. The Description of Problem (4)3. Models (5)3.1Symbols and Definitions (5)3.2 GeneralAssumptions (6)3.3 Analysis of the Problem (6)3.4 The Foundation of Model (6)3.5 Solution and Result (8)3.6 sensitivity analysis (17)3.7 Analysis of the Result (19)3.8 Strength and Weakness (19)4.Improved Model............................................................................................................................ .. (20)4.1super- efficiency DEA model (20)4.2 Solution and Result (21)4.3Strength and Weakness (25)5. Conclusions (25)5.1 Conclusions of the problem...............................................................................,.25 5.2 Methods used in our models (26)5.3 Applications of our models (26)6.The article for Sports Illustrated (26)7.References (28)I. IntroductionAt present, the scientific evaluation index systems related to college coach abilities are limited, and the evaluation of coach abilities are mostly determined by the sports teams’game results, and it lacks of systematic, scientific and accurate evaluation with large subjectivity and one-sidedness, thus it can not objectively reflect the actual training level of coaches. In recent years, there appear many new performance evaluation methods, which mostly consider the integrity of the evaluation system. Thus they overcome a lot of weaknesses that purely based on the evaluation of game results. However, it is followed by the complexity of evaluation process and index system, as well as the great increase of the implementation cost. Data envelopment analysis is a non-parametric technique for evaluating the relative efficiency of a set of homogeneous decision-making units (DMUs) with multiple inputs and multiple outputs by using a ratio of the weighted sum of outputs to the weighted sum of inputs. Therefore, it not only simplifies the number of indexes, but also avoids the interference of subjective consciousness, thus makes the evaluation system more just and scientific.Based on the investigation and research of the US college basketball coach for the previous century, this paper aims at establishing a scientific and objective evaluation index system to assess their coaching abilities comprehensively. It provides reference for the relating sports management department to evaluate coaches and continuously optimize their coaching abilities. For this purpose, the DEA is successfully introduced into this article to establish a comprehensive evaluation model for choosing the best college coaches. It makes the assessment of the coaches in different time line horizon, different gender and different sports to testify the validity and the effectiveness of this approach.II. The Description of the Problem In order to find out the ―best all time college coach‖ for the previous century, a comprehensive evaluation model is needed to set up. Therefore, a set of scientific and objective evaluation index system should be established, which should meet the following principles or requirements:The principle of sufficiency and comprehensivenessThe index system should be sufficiently representative and comprehensivelycover the main contents of the coaches’ coaching abilities.The principle of independenceEach of the index should be clear and comparatively independent.The principle of operabilityThe data of index system comes from the existing statistics data, thus copying the unrealistic index system is not allowed.The principle of comparabilityThe comparative index should be used as far as possible to be convenientlycompared for each coach.After the establishment of evaluation index system, it requires the detailed model to make assessment and analysis for each coach. Currently, the comprehensive assessment is mostly widely used, but most of them need to be gave a weight. It is more subjective and not very scientific and objective. To avoid fixing the weight, the DEA method is adopted, which can figure out coaches’ rank eventually from the coach’s actual data.For the different time line horizon, the coaches’ rank is inevitably influenced by the team’s l evel and the sports, thus it requires discussion in different time line horizon to get the further results.Finally, the DEA model is applied to all coaches (either male or female) and all possible sports to get the rank, and then the model’s whole assessm ent basis and process should be explained to the readers in understandable words.III. Models3.1 Terms Definitions and Symbols Symbol ExplanationDMU k the k th DMU0DMU the target DMU, which is one of the nevaluated DMUs;ik x the i th input variable consumed0i x the i th input variable consumedjk y the j th output variable produced0j y the j th output variable produced1I The length of coaching career2I The number of taking part in NCAAtournament3I coaching session1Ovictory ratio of game3.2 General AssumptionsThe same level game difficulty in different regions and cities is equal for all teams.The value of the champion in different regions and cities is equal (without regard to team’s number in the region, the power and strength of the teams and other factors).The same game’s value is equal in different years (without regard to the team number in the year and other factors).The college’s level has no influence to the coach’s coaching performance.3.3 Analysis of the ProblemFor the current problem, first of all, a comprehensive evaluation model is needed to set up. Therefore, a set of scientific and objective evaluation index system should be established. The evaluation system of the coaches is comparatively mature, but it mainly based on the people’s subjective consciousness, thus the evaluation system we build requires more data to explain the problem, and it tries to assess each coach in a objective and just way without the interference of subjective factors.Secondly, the evaluation system we used is different due to the different games in different time periods. So the influence of different time periods to the evaluation results should be taken into account when we deal with the problem. Furthermore, it should be discussed in different cases.3.4 The Foundation of ModelData Envelopment Analysis (DEA), initially proposed by Charnes, Cooper and Rhodes [3], is a non-parametric technique for evaluating the relative efficiency of a set of homogeneous decision-making units (DMUs) with multiple inputs and multiple outputs by using a ratio of the weighted sum of outputs to the weighted sum of inputs. 2O the number of victory session3O The number of equivalent champion1Q the number of regular games champion2Q the number of league games champion3Qthe number of NCAA league gameschampionOne of the basic DEA models used to evaluate DMUs efficiency is the input-oriented CCR model, which was introduced by Charnes, Cooper and Rhodes [1]. Suppose that there are n comparatively homogenous DMUs (Here, we look upon each coach as a DMU), each of which consumes the same type of m inputs and produces the same type of s outputs. All inputs and outputs are assumed to be nonnegative, but at least one input and one output are positive.DMU k : the k th DMU, 1,2,,=k n ;0DMU : the target DMU, which is one of the n evaluated DMUs;ik x : the i th input variable consumed by DMU k , 1,2,,=i m ; 0i x : the i th input variable consumed by 0DMU , 1,2,,=i m ;jk y : the j th output variable produced by DMU k , 1,2,,=j s ;0j y : the j th output variable produced by 0DMU , 1,2,,=j s ; i u : the i th input weight, 1,2,,=i m ;In DEA model, the efficiency of 0DMU , which is one of the n DMUs, isobtained by using a ratio of the weighted sum of outputs to the weighted sum of inputs under the condition that the ratio of every entity is not larger than 1. The DEA model is formulated by using fractional programming as follows:()()000111112121,1,2,...,..,,,0,,,0max sr rj r m j i ij i sr rj r m i ij i T m T s j n s t v v v v u u u u y u hv x y u v x =====⎧⎪⎪≤=⎪⎪⎨⎪=≥⎪⎪=≥⎪⎩∑∑∑∑ （2）The above model is a fractional programming model, which is equivalent to the following linear programming model:00111010,1,2,...,..1,0,1,2,..;1,2,...,max s j r rj r sm i ij r rj r i m i ij i ir j n s t i m r sy h y w x w x w μμμ=====⎧-≤=⎪⎪⎪=⎨⎪⎪≥==⎪⎩∑∑∑∑ （3） Turned to another form is:101min ..0,1,2,,nj j j n j j j j x x s t j ny y θλθθλλ==⎧≤⎪⎪⎪⎪≥⎨⎪⎪≥=⎪⎪⎩∑∑无约束 3.5 Solution and Result3.5.1 Establishing the input and output index systemIn the DEA model, it requires defining a set of input index and a set of output index, and all the indexes should be the common data for each coach. Regarding the team as an unit, then the contribution that the coach made to the team can be regarded as input, while the achievement that the team made can be regarded as reward. In the following, we take the basketball coaches of NACC as an example to establish the input and output index system. These input indexes could be chosen as follows:1I :The length of coaching careerThe more game seasons a coach takes part in, the more abundant experience he has. This ki nd of coach’s achievement is easily affirmed by others. As the Figure 1 shows, the famous coach mostly experienced the long-time coaching career.Furthermore, the time the coach has contributed to the team is fundamental if they want to have a good result in the game. Thus the length of coaching career can be regarded as an index to evaluate the coach’s contribution to the team.Figure 1 The relationship between the length of coaching career and the number ofchampionsI: The number of taking part in NCAA tournament2Whether the coach takes the team to a higher level game has a direct influence on the team’s performance, and also it can reflect the coach’s coaching abilities, level and other factors.I: coaching session3For the reason of layers of elimination, the coaching session is not necessarily determined by the length of coaching career. It can be shown in the comparison between Figure 2 and Figure 3. Thus the number of coaching session can also be regarded as an index.Figure 2These output indexes could be chosen as follows:O: victory ratio of game1The index reflects the coach’s ability of command and control, and it a ttaches great importance to the evaluation of coach’s coaching abilities.O: the number of victory session2The case that the number of victory session reflected is different from that of victory ratio, only if get the enough number of victory session in a large number of coaching session, the acquired high victory ratio can reflect the coach’s high coaching level. If the victory occurs in a limited games, this kind of high victory ratio can not reflect the rules. It can be shown in the comparison between Figure 3 and Figure 4.Figure 3Figure 43O :The number of equivalent championThe honor that US college basketball teams acquired can be divided into three types: 1Q : the number of regular games champion; 2Q : the number of league games champion; 3Q : the number of NCAA league games champion. The threechampionship honor has different levels, and their importance is increasing in turn according to the reference. The weight 0.2、0.3、0.5 can be given respectively, and the number of equivalent champion can be figured out and used as an output index, as it shown in Table 1.5.0Q 3.0Q 2.0Q O 3213⨯+⨯+⨯=Table 1 Coach names Number of regular games champion (weight 0.2) Number of league games champion (weight 0.3) Number of NCAA league games championNumber of equivalent championAccording to Internet, the data of input and output are given by Table 2.Table 2(weight 0.5)Phog Allen 24 0 1 5.3 Fred Taylor 7 0 1 1.9 Hank Iba 15 0 2 4 Joe B. Hall 8 1 1 2.4 Billy Donovan 7 3 2 3.3 Steve Fisher 3 4 1 2.6 John Calipari 14 11 1 6.6 Tom Izzo 7 3 1 2.8 Nolan Richardso 9 6 1 3.9 John Wooden 16 0 10 8.2 Rick Pitino 9 11 2 6.1 Jerry Tarkanian 18 8 1 6.5 Adolph Rupp 28 13 4 11.5 John Thompson 7 6 1 3.7 Jim Calhoun 16 12 3 8.3 Denny Crum 15 11 2 7.3 Roy Williams 15 6 2 5.8 Dean Smith 17 13 2 8.3 Bob Knight 11 0 3 3.7 Lute Olson 13 4 1 4.3 Mike Krzyzewski 12 13 4 8.3 Jim Boeheim 11 5 1 4.2 Doc Meanwell 10 0 0 2 Ralph Jones 4 0 0 0.8 Francis Schmidt61.2Coach namesInput indexOutput index1I2I3I1O 2O3ONCAA tourament Thelength of coaching careerCoaching session Win-Lose %WinsNumber of equivalent championPhog Allen 4489780.735 719 5.3 Fred Taylor 5 18 455 0.653 297 1.9 Hank Iba84010850.6937524Since the opening of NACC tournament in 1938, thus the year 1938 is chosen as a time set apart. The finishing time point of coaching before 1938 is a period of time, while after 1938 is another period of time.For the time period before 1938, take the length of coaching career 1I , coaching session 3I as input indexes, and then take W-L %1O , victory session 2O , the number of regular games champion 1Q as output indexes. The results is shown in Table 3 after the data statistics of each index.For the time period after 1938, because they all take part in NACC, the input index and output index are just the same as that of all time period. The data statistics is just as shown in Table 3.Table 3Coach namesInput indexOutput index1I3I1O2O1QThe lengthof coachingcareerCoachingsessionW-L % WinsNumber of regular games championJoe B. Hall 10 16 463 0.721 334 2.4 Billy Donovan 13 20 658 0.714 470 3.3 Steve Fisher 13 24 739 0.658 486 2.6 John Calipari 14 22 756 0.774 585 6.6 Tom Izzo 16 19 639 0.717 458 2.8 Nolan Richardson 16 22 716 0.711 509 3.9 John Wooden 16 29 826 0.804 664 8.2 Rick Pitino 18 28 920 0.74 681 6.1 Jerry Tarkanian 18 30 963 0.79 761 6.5 Adolph Rupp 20 41 1066 0.822 876 11.5 John Thompson 20 27 835 0.714 596 3.7 Jim Calhoun 23 40 1259 0.697 877 8.3 Denny Crum 23 30 970 0.696 675 7.3 Roy Williams 23 26 902 0.793 715 5.8 Dean Smith 27 36 1133 0.776 879 8.3 Bob Knight 28 42 1273 0.706 899 3.7 Lute Olson 28 34 1061 0.731 776 4.3 Mike Krzyzewski 29 39 1277 0.764 975 8.3 Jim Boeheim303812560.759424.2Louis Cooke 27 380 0.654 248 5 Zora Clevenger 15 223 0.677 151 2 Harry Fisher 14 249 0.759 189 3 Ralph Jones 17 245 0.792 194 4 Doc Meanwell 22 381 0.735 280 10 Hugh McDermott 17 291 0.636 185 2 E.J. Mather 14 203 0.675 137 3 Craig Ruby 16 278 0.651 181 4 Francis Schmidt 17 330 0.782 258 6 Doc Stewart 15 291 0.663 193 2 James St. Clair162630.58215323.5.2 Solution and ResultIn this section, take Phog Allen as an example and make calculation as follows:Taking Phog Allen as 0DMU , then the input vector is 0x , the output vector is 0y , while the respective input and output weight vector are:From the Figure 2 it can be inferred thatT x )978,48,4(0= T y )3.5,719,735.0(0=After the calculation by LINGO then the efficiency value h 1 of DMU 1 is0.9999992.For other coaches, their efficiency value is figured out by the above calculation process as shown in Table 4.Table 4Coach namesInput indexOutput indexEfficiency value 1I2I3I1O 2O3ONCAA Tourna ment Thelength of coachin g careerCoachin g session W-L % Wins Number ofequivalent championJoe B. Hall 10 16 463 0.721 334 2.4 1John Wooden 16 29 826 0.804 664 8.2 1John Calipari 14 22 756 0.774 585 6.6 1Adolph Rupp 20 41 1066 0.822 876 11.5 1Hank Iba 8 40 1085 0.693 752 4 1Mike Krzyzewski 29 39 1277 0.764 975 8.3 1Roy Williams 23 26 902 0.793 715 5.8 1Jerry Tarkanian 18 30 963 0.79 761 6.5 0.9999997 Fred Taylor 5 18 455 0.653 297 1.9 0.9999996 Phog Allen 4 48 978 0.735 719 5.3 0.9999992 Tom Izzo 16 19 639 0.717 458 2.8 0.9785362 Dean Smith 27 36 1133 0.776 879 8.3 0.9678561 Jim Boeheim 30 38 1256 0.75 942 4.2 0.941031 Billy Donovan 13 20 658 0.714 470 3.3 0.9405422 Rick Pitino 18 28 920 0.74 681 6.1 0.9401934 Nolan Richardson 16 22 716 0.711 509 3.9 0.920282 Lute Olson 28 34 1061 0.731 776 4.3 0.9114966 John Thompson 20 27 835 0.714 596 3.7 0.8967647 Jim Calhoun 23 40 1259 0.697 877 8.3 0.8842854 Denny Crum 23 30 970 0.696 675 7.3 0.8823837 Bob Knight 28 42 1273 0.706 899 3.7 0.8769902 Steve Fisher 13 24 739 0.658 486 2.6 0.8543039For those coaches in the time period after 1938, the efficiency values, which is shown in Table 5, are figured out from the similar calculation process as Phog Allen.Table 5Coach namesInput index Output indexEfficiencyvalue 1I3I1O2O1QThelengthofcoaching careerCoaching sessionW-L % WinsNumberof regularchampionDoc Meanwell 22 381 0.735 280 10 1Francis Schmidt 17 330 0.782 258 6 1Ralph Jones 17 245 0.792 194 4 1E.J. Mather 14 203 0.675 137 3 0.9999999Harry Fisher 14 249 0.759 189 3 0.999999 Zora Clevenger 15 223 0.677 151 2 0.9328517 Doc Stewart 15 291 0.663 193 2 0.8910155 Craig Ruby 16 278 0.651 181 4 0.859112 Louis Cooke 27 380 0.654 248 5 0.8241993 Hugh McDermott 17 291 0.636 185 2 0.8091489 James St. Clair 16 263 0.582 153 2 0.7468237Choose basketball, football and field hockey and make calculationsThe calculation result statistics of basketball is shown in Table 4.The calculation result statistics of football is shown in Table 6.Table 6Coach NamesInput index Output indexEfficiencyvalue Total ofthe BowlThelength ofcoachingcareerCoachingsessionW-L % WinsNumberofchampionPhillip Fulmer 15 17 204 0.743 151 8 1 Tom Osborne 25 25 307 0.836 255 12 1 Dan Devine 10 22 238 0.742 172 7 1 Bobby Bowden 33 40 485 0.74 357 22 1 Pat Dye 10 19 220 0.707 153 7 1 Bobby Dodd 13 22 237 0.713 165 9 1Bo Schembechler 17 27 307 0.775 234 5 1 Woody Hayes 11 28 276 0.761 205 5 1.000001 Joe Paterno 37 46 548 0.749 409 24 1 Nick Saban 14 18 228 0.748 170 8 0.9999993 Darrell Royal 16 23 249 0.749 184 8 0.9653375 John Vaught 18 25 263 0.745 190 10 0.9648578 Steve Spurrier 19 24 300 0.733 219 9 0.9639218 Bear Bryant 29 38 425 0.78 323 15 0.9623039 LaVell Edwards 22 29 361 0.716 257 7 0.9493463 Terry Donahue 13 20 233 0.665 151 8 0.9485608 John Cooper 14 24 282 0.691 192 5 0.94494 Mack Brown 21 29 356 0.67 238 13 0.9399592 Bill Snyder 15 22 269 0.664 178 7 0.9028372 Ken Hatfield 10 27 312 0.545 168 4 0.9014634 Fisher DeBerry 12 23 279 0.608 169 6 0.9008829 Don James 15 22 257 0.687 175 10 0.8961777Bill Mallory 10 27 301 0.561 167 4 0.8960976 Ralph Jordan 12 25 265 0.674 175 5 0.8906943 Frank Beamer 21 27 335 0.672 224 9 0.8827047 Don Nehlen 13 30 338 0.609 202 4 0.8804766 Vince Dooley 20 25 288 0.715 201 8 0.8744984 Jerry Claiborne 11 28 309 0.592 179 3 0.8731701 Lou Holtz 22 33 388 0.651 249 12 0.8682463 Bill Dooley 10 26 293 0.558 161 3 0.8639017 Jackie Sherrill 14 26 304 0.595 179 8 0.8435327 Bill Yeoman 11 25 276 0.594 160 6 0.8328854 George Welsh 15 28 325 0.588 189 5 0.820513 Johnny Majors 16 29 332 0.572 185 9 0.807564 Hayden Fry 17 37 420 0.56 230 7 0.792591The calculation result statistics of field hockey is shown in Table 7.Table 7Coach namesInput index Output indexEfficiencyvalue Total ofthe BowlThelength ofcoachingcareerCoachingsessionW-L % WinsNumberofchampionFred Shero 110 10 734 0.612 390 2 1 Mike Babcock 131 11 842 0.63 470 1 1 Claude Julien 97 11 749 0.61 411 1 1 Joel Quenneville 163 17 1270 0.617 695 2 1 Ken Hitchcock 136 17 1213 0.602 642 1 1 Marc Crawford 83 15 1151 0.556 549 1 0.9999998 Scotty Bowman 353 30 2141 0.657 1244 9 0.9999996 Hap Day 80 10 546 0.549 259 5 0.9999995 Toe Blake 119 13 914 0.634 500 8 0.9999994 Eddie Gerard 21 11 421 0.486 174 1 0.999999 Art Ross 65 18 758 0.545 368 1 0.9854793 Peter Laviolette 82 12 759 0.57 389 1 0.9832692 Bob Hartley 84 11 754 0.56 369 1 0.9725678 Jacques Lemaire 117 17 1262 0.563 617 1 0.9706553 Glen Sather 127 13 932 0.602 497 4 0.9671556 John Tortorella 89 14 912 0.541 437 1 0.9415468 John Muckler 67 10 648 0.493 276 1 0.9379403 Lester Patrick 65 13 604 0.554 281 2 0.9246999 Mike Keenan 173 20 1386 0.551 672 1 0.8921282 Al Arbour 209 23 1607 0.564 782 4 0.8868355 Frank Boucher 27 11 527 0.422 181 1 0.8860263Pat Burns 149 14 1019 0.573 501 1 0.8803399 Punch Imlach 92 14 889 0.537 402 4 0.8795251Darryl Sutter 139 14 1015 0.559 491 1 0.8754745Dick Irvin 190 27 1449 0.557 692 4 0.8688287Jack Adams 105 20 964 0.512 413 3 0.8202896 Jacques Demers 98 14 1007 0.471 409 1 0.7982191 3.6 sensitivity analysisWhen determining the number of equivalent champion, the weight coefficient is artificially determined. During this process, different people has different confirming method.Consequently, we should consider that when the weight coefficient changes in a certain range, what would happen for the evaluation result?For the next step, we will take the basketball coaches as example to illustrate the above-mentioned case.The weight coefficient changes is given by Table 12. The changes of evaluation results is shown in Table 13.Table 12Coach names Number ofregulargameschampion(weight0.2)Number ofleaguegameschampion(weight 0.4)Number ofNCAA leaguegameschampion(weight0.4)Number ofequivalentchampionPhog Allen 24 0 1 5.2 Fred Taylor 7 0 1 1.8 Hank Iba 15 0 2 3.8 Joe B. Hall 8 1 1 2.4 Billy Donovan 7 3 2 3.4 Steve Fisher 3 4 1 2.9 John Calipari 14 11 1 7.6 Tom Izzo 7 3 1 3 NolanRichardso9 6 1 4.4 John Wooden 16 0 10 7.2 Rick Pitino 9 11 2 7 JerryTarkanian18 8 1 7.2 Adolph Rupp 28 13 4 12.4 JohnThompson7 6 1 4.2 Jim Calhoun 16 12 3 9.2 Denny Crum 15 11 2 8.2 Roy Williams 15 6 2 6.2Dean Smith 17 13 2 9.4 Bob Knight 11 0 3 3.4 Lute Olson 13 4 1 4.6 MikeKrzyzewski12 13 4 9.2 Jim Boeheim 11 5 1 4.6Table 13Coach namesI nput index O utput indexEfficiencyvalue 1I2I3I1O2O3ONCAATournamentThelength ofcoachingCareerCoachingsessionW-L %WinsNumber ofequivalentchampionJohn Wooden 16 29 826 0.804 664 7.2 1.395729 John Calipari 14 22 756 0.774 585 7.6 1.318495 Joe B. Hall 10 16 463 0.721 334 2.4 1.220644 Adolph Rupp 20 41 1066 0.822 876 12.4 1.157105Hank Iba 8 40 1085 0.693 752 3.8 1.078677 Roy Williams 23 26 902 0.793 715 6.2 1.034188 Fred Taylor 5 18 455 0.653 297 1.8 1.013679 Jerry Tarkanian 18 30 963 0.79 761 7.2 1.007097 Phog Allen 4 48 978 0.735 719 5.2 0.999999 Jim Boeheim 30 38 1256 0.75 942 4.6 0.984568Tom Izzo 16 19 639 0.717 458 3 0.978536 Dean Smith 27 36 1133 0.776 879 9.4 0.96963Mike Krzyzewski 29 39 1277 0.764 975 9.2 0.960445 Rick Pitino 18 28 920 0.74 681 7 0.940614 Billy Donovan 13 20 658 0.714 470 3.4 0.940542 Nolan Richardson 16 22 716 0.711 509 4.4 0.920282 Lute Olson 28 34 1061 0.731 776 4.6 0.911496 John Thompson 20 27 835 0.714 596 4.2 0.896765 Jim Calhoun 23 40 1259 0.697 877 9.2 0.884227 Denny Crum 23 30 970 0.696 675 8.2 0.882805 Bob Knight 28 42 1273 0.706 899 3.4 0.87699 Steve Fisher 13 24 739 0.658 486 2.9 0.854304From the Table 13, it can been seen that the top 5 coaches are: John Wooden, John Calipari, Joe B. Hall, Adolph Rupp, Hank Iba. The result is in accordance with。

2014年数学建模美赛ABC 题翻译

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是一个为运动爱好者服务的杂志，正在寻找在整个上个世纪的“史上最好的大学教练”。

建立数学模型选择大学中在一下体育项目中最好的教练：曲棍球或场地曲棍球，足球，棒球或垒球，篮球，足球。

时间轴在你的分析中是否会有影响？比如1913年的教练和2013年的教练是否会有所不同？清晰的对你的指标进行评估，讨论一下你的模型应用在跨越性别和所有可能对的体育项目中的效果。

展示你的模型中的在三种不同体育项目中的前五名教练。

除了传统的MCM格式，准备一个1到2页的文章给体育画报，解释你的结果和包括一个体育迷都明白的数学模型的非技术性解释。

使用网络测量的影响和冲击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。

与人合写一手稿通常意味着一个强大的影响力的研究人员之间的联系。

最著名的学术合作者是20世纪的数学家保罗鄂尔多斯曾超过500的合作者和超过1400个技术研究论文发表。

讽刺的是,或者不是,鄂尔多斯也是影响者在构建网络的新兴交叉学科的基础科学,尤其是，尽管他与Alfred Rényi的出版物“随即图标”在1959年。

2014年美国大学生数学建模竞赛心得

2014美赛心得在2014年美国大学生数学建模比赛中，我们小组奋战了几天，最终获得了Ｈ奖的成绩。

这个成绩虽然不是最理想的，但是总体来讲还是十分令人满意的。

而且这也是我们第一次参加美国数学建模比赛，经历这几天的比赛，我们收获的不仅仅是一张奖状，更多的是对数学建模的兴趣和相互合作的进一步认识。

首先，参加这个比赛，使我们对数学建模的认识更进了一步。

我们小组的三个同学都曾修过数学建模课程，对数学建模还是有一定的认识的。

而以前，我们对于数学建模的认识，可能只是停留在课程与考试的阶段。

但是，通过参加本次美国大学生数学建模比赛，我们有了更深刻的认识：数学建模不是去做一个题目，而是真正地分析问题、解决问题、优化问题，在茫茫的数学海洋之中寻觅最合适的模型，甚至是自己创造一个新的模型。

而这也是十分关键的，如果只是生搬硬套公式而不去创新，便很难从众多论文中脱颖而出，也很难真正地解决问题。

例如我们本次参赛所选择的B题，从最基本的意义上来说，这道题目很难有所创新。

初看起来这只是一个评价问题，利用一般的评价模型便可以解决。

但是，几乎所有的参赛组都能够利用评价模型来解决此类问题，没有创新也就不会从成千上万的论文中显现出自己的独特之处，也就不会在比赛中有所斩获。

因此，我们把两种方法加以综合，提出一种新的方法来对大学体育教练进行评价。

而且通过与已有的教练评价对比，我们发现这样的评价方法更具有代表性。

其次，我们对于小组成员之间的合作与分工有了更深的认识。

美国大学生数学建模竞赛要求一个参赛组有很好的资料和数据搜集能力、建模能力、编程能力、论文编写能力等等。

我们小组的三位同学分别在某些方面有着不错的能力。

首先，要恰当地评价大学体育教练，需要大量的数据，包括这些教练的胜场、胜率以及所获荣誉等等，这就要求我们搜集大量的资料。

正是xx同学编写的程序，使得我们能够从网站上搜集到大量的有关这些教练的资料。

在这之后，由于我们都对各类数学模型和算法比较了解，于是我们三人同心，其利断金，顺利地将模型初步建立起来。

2014美国数学建模-B题paper-30680

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。

14年国赛数模B题优秀论文

当考虑稳定性时，重心越低稳定性越高，投影的矩形面积越大，而考虑用材少最少时，最长桌腿长度越短用材越少，投影矩形面积越小。因为桌面高度一定，当矩形为正方形时即 A2 A3 A2 A5 ，达到整体最优，桌子稳定性好且用材少。此时：
h tan r 2 l 2 r 将桌高 70 cm，桌面直径 80 cm 代入上述公式，求得： 27.13 ，最长桌腿的长度 l1 78.65 。则得到此时平板尺寸为 181.3cm 80cm 。根据模型二求得的槽长 Ri 与的函数关系，解出此时各木条的槽长矩阵，考虑到加工方便，本文令所有槽长都等于最长槽长 34.87cm。运用 matlab 软件，运用仿真技术（见附录），画出折叠桌展开的动态过程（图 9）。
五、模型的建立与求解
5.1 问题一 5.1.1 模型一的建立与求解
已知长方形平板尺寸为 120 cm × 50 cm × 3 cm，要将该平板裁剪为桌面呈圆形的折叠桌，由于圆形桌面的对称性和木条的已知宽度，本文假设每组桌腿条数为 19，考虑实际裁剪过程，去掉平板两侧长为 120cm,宽为 1.25cm 的两部分（见图 1 阴影部分）由图 1 将每根木条对应在半圆内的矩形抽象出来，得到图 2，设圆形桌面半径为 r ，已知木条宽 d 为 2.5cm ，那么根据勾股定理： l2 d 2 r2
赛区评阅编号（由赛区组委会评阅前进行编号）：
2014 高教社杯全国大学生数学建模竞赛
编号专用页
赛区评阅编号（由赛区组委会评阅前进行编号）：
赛区评阅记录（可供赛区评阅时使用）：评阅人评分备注
全国统一编号（由赛区组委会送交全国前编号）：
全国评阅编号（由全国组委会评阅前进行编号）：

2014美赛A题论文

For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number27400Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________2014 Mathematical Contest in Modeling (MCM) 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.SummaryThe aim of this paper is to evaluate the performance of the changing lane rule named Keep-Right-Except-To-Pass and compare other rules we’ve constructed with the given one. The performance of changing lane rules mainly manifest in safety and traffic flow. Meanwhile, safety is influenced by posted speed limits and traffic density and traffic flow is influenced by average speed.First we construct Model 1 to describe the relationship between posted speed limits, traffic density and safety, noting that safety has negative correlation with collision times and can be fully expressed by it. Therefore we use Matlab to imitate the changing lane and collision process. Then we construct Model 2 to describe the relationship between traffic flow and average speed. Combining the result of Model 1and 2, we can conclude that the higher the posted speed limits and the lower the traffic density, the higher level safety an traffic flow could reach, and the Keep-To-Right-Except-To-Pass rule has the best performance.Then we construct Model 3 to compare some normal changing lane rules with the given one and with each other. Thus we imitate all the rules using algorithm given by model 1. We introduce a new concept named Standard Time to express traffic flow and still collision times to express safety. The result is that when a freeway road contains 2 lanes, the given rule shows the best performance. And when the number of lanes becomes 3, then the Choose-Speed rule, a very common traffic rule acts the best.As for the left-driving countries, we first choose the best rules and mirror symmetrically modify them. Then use the method of Model 2 to imitate performances of these rules. We have observed that these two best rules can be simply carried over.Finally, we have a short discussion about the intelligent system. Since speed can be fastest and changing lanes can be well planned, we regard this system as absolutely safe traffic which has the largest traffic flow.Team #27400 Page 1 of 161.Introduction1.1 Analysis to the problemThis problem can be divided into 4 requirements that we must meet, listed as follows:(1)Build a mathematical model to demonstrate the essence of Keep-Right-Except-To-Pass rule in light and heavy traffic;(2)Show the effects of other reasonable changing-lane rules or conditions by using themodified model built in the former requirement;(3)Determine whether the left-side-driving countries can use the same rule(s) withsimple mirror symmetric modification, or must add some requirements to guaranteethe safety or traffic flow;(4)Construct an intelligent traffic system which does not depend on humans’ compliance,and then compare the effects with earlier analysis.Therefore, this problem requires us to evaluate the performances of several changing-lane rules, especially the Keep-Right-Except-To-Pass rule while performances mainly manifest in traffic flow and safety. It is obvious that speed limit and the number of cars influent cars’ speed and then the performances of those rules.To solve requirement (2), we will use the same method as above. But all rules weconstructed should be discussed, imitated and compare with each other.To discuss the left-driving countries, we will conclude the best rules from requirement2 and apply to those countries. Then compare results with that of right-driving countries.If the results are totally the same, then those rules can be simply carried over with simple change. If not, additional requirement are needed.As for the intelligent system, we assume no collision will happen. So a simplediscussion is feasible.1.2 Crucial method for the problemTo imitate the process of changing lanes and collision, we will construct an algorithm named Changing-line and Collision. This will show the state of a number of car on the same stretch of freeway road, including going straight, changing lane to pass and traffic collision. The imitation process will be delivered on Matlab;2.Symbols and definitionSymbol Definition UnitσStandard Deviation of Speed miles/hmiles/hV85% Percentile of Normal85Distributionmiles/hV70% Percentile of Normal70DistributionDENS Traffic Density (Note: expressed by thenumber of cars) ST Standard Time \FLO Traffic Flow \CT Collision Times \3.General hypotheses for all models(1) All freeway roads contain either two or three lanes.(2) Overtaking Principle: the latter car must change lane and overtake the car right in front it if its speed is faster.(3) No violation of the rules.(4) The intelligent traffic system has no collision.(5) Road indicates one direction of the road.(6) Each cars remains constant speed no matter it goes straight or change lane.(7) Changing lane doesn’t cost time.(8) Given a traffic density, the average speed of cars is constant.(9) All discussions are confined to one stretch of road that is long enough to allow a series of cars to go across.4.Model establishment4.1 PreparationFirst, we must discuss what “changing-lane” is. For a two-lane road, if one car is going to change its lane to the left, then the right side lane “loses” one car and the left side lane “acquires” one, and vice versa. If the road contains three lanes, we can regard it as two “two-lane road”. This property also holds for an n-lane (n>2) road, which means we could regard it as (n-1) two-lane road. Therefore, changing-lane is the relation between two lanes. We just need to analyze a two-lane road to uncover the principle of changing lane and collision.Second, influence relationship figures are given below. In these figures, factors at lower level influence those at upper level. Safety will be expressed by collision times( ).Figure 1. General influence relationshipThird, we have constructed three additional changing-lane rules and shall describe these five rules in mathematical language. The original Keep-Right-Except-To-Pass rule is denoted Rule 1.Rule 1: Keep-Right-Except-To-Pass. If A B v v >, then car A goes left, overtakes car B and then gets back to the right lane.Figure 2. Rule one: Keep-Right-Except-To-PassRule 2: Free-Overtaking. If A B v v > and A C v v >, then car A changes lane to the left, passing B, and then to the right, passing C. And if there still exists a car D in front of A and A D v v >, car A will change to the left lane, pass D and then go straight. The following figure shows the case when the number of lanes is two, and it is identical for the case of 3 lanes.Figure 3. Rule 2: Free-OvertakingRule 3: Pass-Left. For a three-lane freeway road, if B A C v v v >>, then car A changes to the middle lane, and car B to the left lane, and finally they should get back to their original lane.Figure 4. Rule 3: Pass-LeftRule 4: Choose-Speed. Each lane of the road has a posted speed limit interval. We denote the left lane fast lane, right one slow lane and the middle one mid-lane. Drivers should choose lane according to their cars ’ speed.4.2 Model 1: Changing Lane and Collision Algorithm 4.2.1 Flow Diagram*To check Matlab code, please refer to Appendix 1. Note: we only use collision times to express safety.Figure 5. Flow Diagram of the Changing Lane and Collision Algorithm Comments:(1) The 1000*2 matrix stands for a stretch freeway road that contains two lanes;(2) Speed of cars is a series of random numbers which stands for the units they are to go forward each time;(3) Cars enter the road from the right lane.(4) If two cars collide, their speed will become zero but will not affect other cars. Accordingly, the position of collision will be valuated zero.To show how this algorithm works, we now give an example of a road, length 7 and five carswith different speed:Figure 6. An Example of the Algorithm4.2.2 Relationship between Average Speed, Standard Deviation of Speed and Posted Speed LimitsWe have mentioned in the last section that the speed of cars obeys normal distribution [3]. Now we deduce the relationship between ,μσ and PSL .Figure 7. Speed of cars obey normal distributionWe know from reference [4](page 15) that:857.6570.98V PSL =+⋅①The above foemula can be applied to all types of roads. Equation ② from reference [5](page 53):70-6.541+2.4V σ=⋅②Referring to the table of standard normal distribution, we have70V μσ-=0.52③ 85V μσ-=1.04④Considering formula ②③, we have=1.88-6.541μσ⑤Considering formula ①④, we have7.657+0.98=1.04PSL μσ-⑥Considering formula ⑤⑥, we have14.198+0.98=2.92PSLσ⑦0.98*=13.6012PSL μ-⑧Formula ⑦⑧ explains that the higher the posted speed limits, the greater the variance of speed and same to the average speed.0.9813.6012MU PSL =-⑨4.2.3 Process of ImitationWe now use control-variable method to describe the impact of traffic density and posted speed limits on safety. Therefore, we consider:(1) Traffic density being constant, the impact of variation of posted speed limits on safety; (2) Posted speed limits being constant, the impact of variation of traffic density on safety.When traffic density is constant:Table 1.Variation of Posted Speed LimitsPSL 50 60 70 80 90 100 μ 34.1481 40.4577 46.7673 53.0769 59.3865 65.6961 σ21.6432 24.9993 28.3555 31.7116 35.0678 38.424Figure 8. Variation of μ and σ along with Variation of PSLNote: SIGMA=σ, MU=μWhen posted speed limits is constant:Table 2.Variation of the number of carsThe number of cars 5 10 20 30 40 50Therefore we will acquire 6*6 groups of data when PSL is constant and 6*6 groups of data when traffic density is constant.4.3 Model 2: Relationship between Traffic Flow and Average SpeedIt is obvious that traffic flow is the function of average.We denote:=⋅+>FLO k MU c k(0)4.4 Model 3: Rules and Performances4.4.1 AnalysisIn section 4.1 we have listed four changing lane rules. Now we need to compare their effect and evaluate which rule has the best performance.According to the establishment of model 1, performances are fully expressed by safety and traffic flow, and safety has negative correlation with collision times. We can acquire the related data by imitation like model 1. However, since rules have changed, average speed of the road under different rules is different and is merely decided by PSL.Thus we introduce a new concept, namely, Standard Time(ST). Definition: time cost by each move of cars is called standard time. With this concept, we don’t need to calculate the real time cost by cars when going across a stretch of road, but count how many moves cars should make. Standard time has positive correlation with traffic flow, and we can denote:ST k FLO k=⋅>(0)4.4.2 Imitation ProcessConstruct a matrix. If the road contains two lanes, then the matrix size is 150*2; if it contains three lanes, then the size is 150*3. Arrange 40 random positions for each column vectors, and the numbers, as described before, obey normal distribution. Let other variables be constant, we discuss the impact of different rules on safety and traffic flow.Operations are as follows:(1) Distribute a series positions(40 for each column vector as described) for the matrix.(2) Valuate each position a number that obey normal distribution.(3) Apply one of the rules to the matrix. Those numbers will either “collide ” or “keep moving ” until all the position are valuated zero. (4) Record collision times and the spent standard time.(5) Apply another rule to the matrix, repeat steps (1) through (4).Statistically analyzing the data acquired and comparing the performances of different rules, we can decide advantages and disadvantages of each rule.4.4.3 Use the Method of 4.4.2 to Estimate Left-driving CountriesWe will conclude the best two rules and then apply the methods in section 4.4.2 to the left-driving country and estimate the performance of the two rules. Operation steps:(1) Mirror symmetrically modify the chosen rules; (2) Refer to the steps in section 4.4.2; (3) Compare with the results in 4.4.2.5. Model solution5.1 Model 1During the imitation process, we do the experiment for times for each case and then analyze all the results. We will show part of our statistical data. To view full data, please refer to Appendix 2.(1) Given DENS=30, the result is as follows:Table 3.Relationship between Changing Lane Times, Collision Times and PSL when DENS=30PSL Changing-lane times Average Collision TimesAverag-Line chart:Figure 9. Variation of Changing Lane Times and Collision Times along with Variation ofPSL(2) Given PSL=70, the result is as follows:Table 4.Relationship between Changing Lane Times, Collision Times and Number of Times whenPSL=70NumberChanging Lane Times Average Collision Time Average Line chart:Figure 10. Variation of Changing Lane Times and Collision Times along with Variation ofnumber of cars5.2 Model 2We know from formula ⑨, section 4.2.2 that0.9813.6012MU PSL =-Considering formula ⑨ and formula in 4.3, we have:0.9813.6012FLO k PSL k c =⋅-+5.3 Model 3The result of section 4.4.2 is given below:Table 5.Collision Times and Standard Time of each ruleRule →Rule 1 Rule 2(2Rule 3 Rule 2(3 Rule 4Extract the row of Average ST and Average CTTable 6.Average Collision Times and Average Standard Time of each rule The result of 4.4.3 is given below:Table 7.Collision Times and Standard Time of Rule 1 and Rule 4 Applied to Left-driving Countries6.Conclusion6.1 Conclusion of model 1 and 2We can conclude from model 1 that safety will decrease along with the growth oftraffic density. Safety will also decrease while the posted speed limits(PSL) is raised.However, comparing to PSL, traffic density has greater impact on safety while PSL can hardly influence safety.From model 2, we can conclude that traffic flow has liner growth along with the raise of PSL.The following table has summed up model 1 and 2.Table . 8Evaluation of Rule Performances in Different Road conditions6.2 Conclusion of model 3We can know from table 6 that when the number of lanes is 2, rule 1 is not very different from rule 2 in terms of traffic flow but is much safer than rule 2. When the number of lanes is 3, rule 3 is safest rule, but rule 4 will become both safe and efficient. Meanwhile, Free-Overtaking rule, actually meaning no rule, is most dangerous. Therefore, we have chosen rule 1 and rule 4 as the best rules.Result of section 4.4.3 is showed below. Note that original rules are listed on the left and modified on the right.Table .9Comparison between the Modified Rules and the OriginalFrom the table above, we can conclude that no difference between left-driving and right-driving countries. Therefore, those rules can be simply carried over with simple mirror symmetric modification.6.2 Conclusion of intelligent systemSince all the cars are controlled by a high-tech computer, we assume this system will never have an accident until the computer crashes. So every car reach their fastest speed, the order of overtaking can be calculated to avoid collision. Therefore, safety and traffic flow reach the highest level.7. References[1] Mark M. Meerschaert. Mathematical Modelling (Third Edition). Beijing: China Machine Press, 2008.12[2] Solomon. Accidents on Main Rural Highways Re la ted to Speed, Driver, and Vehicle[R]. Washing ton D. C: Federal Highway Administration, 1964[3] Lu Jian, Sun Xinglong, Daiyue. Regression analysis on speed distribution characteristics of ordinary road. Nanjing: Journal of Southeast University(Natural Science Edition), 2012.2 (in Chinese).[4] Wang Lijin. Research on Speed Limits and Operating Speed for Freeway. Beijing: Beijing University of Technology, 2011.5 (in Chinese).8. AppendixAppendix 1. Matlab Codeclear;clc;highway=zeros(1000,2);%define road%crash_times = 0;%define collition times%change_lane_times=0;%define change lane times%num_of_car=5; %definedefine number of car%MU=84.3898;SIGMA=20.4406;randspe=[];j=1;while j<=num_of_carx=round(normrnd(MU,SIGMA,1,1));if x>0randspe(j,1)=x; %define speed and gualentee %j=j+1;endx=0;endrandpla=[];for i=1:num_of_carx=ceil(rand(1)*1000);if find(randpla==x)x=ceil(rand(1)*1000); %define location and guarantee the car located in different place%endrandpla(i,1)=x;endfor i=1:num_of_carhighway(randpla(i,1),2)=randspe(i,1); %insert every car in different speed at diferent place %speed(i,1)=0;pla(i,1)=0;endinitialr_highway=highway; %record intialr-highway statement%disp(initialr_highway)disp('Aboveis the stateof initialr highway')while sum(sum(highway))~=0i=1;speed(1)=0; pla(1)=0;for j=1:length(highway) %Store the location and speed of each carif highway(j,2)~=0 ....to pla_matrix and speed_matrix separately%speed(i)=highway(j,2);pla(i)=j;i=i+1;endendcount=length(pla);while count>1 %When the differerce of two adjacent car's speed is greater than their distance,thenif speed(count)-speed(count-1)>=abs(pla(count)-pla(count-1)) ... the later car must change lane to the left% highway(pla(count),1)=highway(pla(count),2);highway(pla(count),2)=0;change_lane_times=change_lane_times+1;endcount=count-1;endlane_changed= highway; %record lane_changed highway statement%disp(lane_changed)disp('Aboveis the state of after-changed lane highway ')k=1;pas_spe(1)=0;pas_pla(1)=0;for j=1:length(highway)if highway(j,1)~=0pas_spe(k)=highway(j,1);pas_pla(k)=j;k=k+1;endendcount=length(pas_pla); %delete crashed car%while count>1if pas_spe(count)-pas_spe(count-1)>=abs(pas_pla(count)-pas_pla(count-1))highway(pas_pla(count),1)=0;highway(pas_pla(count-1),1)=0;crash_times=crash_times+1;endcount=count-1;endaft_adjust = highway;disp(aft_adjust)disp('Above is after adjusted highway')for i=1:length(highway)for j=1:2if i-highway(i,j)>0highway(i-highway(i,j),j)=highway(i,j);%move forward%endhighway(i,j)=0;endendfor i=1:length(highway)if highway(i,1)~=0&&highway(i,2)==0%back to right%highway(i,2)=highway(i,1);highway(i,1)=0;endendback_to_right = highway; %record lane_changed highway statement%disp(back_to_right);disp('Aboveis the state of back_to_right lane highway.');pla=0;speed=0;pas_pla=0;pas_spe=0;endfprintf('change_lane_time ','\n');disp(change_lane_times)fprintf('crash_time ','\n');disp(crash_times)Appendix 2. Data of ImitationPSL Change lane times Mean Crash times MeanNumberof cars=5 50 2 2 2 2.0 0 0 0 0.0 60 3 2 3 2.7 0 0 0 0.0 70 1 0 1 0.7 0 0 0 0.0 80 2 0 1 1.0 0 0 0 0.0 90 0 0 3 1.0 0 0 0 0.0 100 1 3 2 2.0 0 0 0 0.0Numberof 50 7 5 11 7.7 0 0 0 0.0 60 8 11 8 9.0 0 1 0 0.3cars=10 70 10 6 7 7.7 1 0 1 0.780 6 7 10 7.7 0 1 1 0.790 4 7 4 5.0 1 2 0 1.0100 6 3 3 4.0 1 1 1 1.0Numberof cars=20 50 47 29 38 38.0 1 2 2 1.7 60 30 34 20 28.0 2 0 1 1.0 70 15 17 22 18.0 0 2 3 1.7 80 18 22 16 18.7 0 2 1 1.0 90 20 10 22 17.3 1 1 2 1.3 100 21 15 13 16.3 2 2 1 1.7Numberof cars=30 50 61 60 48 56.3 5 4 2 3.7 60 46 42 45 44.3 4 1 2 2.3 70 42 38 48 42.7 4 5 3 4.0 80 43 32 47 40.7 2 5 5 4.0 90 32 41 33 35.3 3 4 5 4.0 100 26 27 24 25.7 2 2 4 2.7Numberof cars=40 50 65 100 74 79.7 8 4 8 6.7 60 56 77 75 69.3 5 8 3 5.3 70 53 64 62 59.7 4 5 5 4.7 80 52 60 60 57.3 3 7 3 4.3 90 44 53 48 48.3 5 4 4 4.3 100 43 50 38 43.7 1 7 0 2.7Numberof cars=50 50 82 86 106 91.3 12 12 10 11.3 60 80 68 72 73.3 10 9 13 10.7 70 82 75 81 79.3 8 7 10 8.3 80 61 72 58 63.7 9 7 11 9.0 90 57 54 54 55.0 6 7 11 8.0 100 64 50 51 55.0 7 9 8 8.0PSL=705 1 0 1 0.7 0 0 0 0.0 10 1067 7.7 1 0 1 0.7 20 15 17 22 18.0 0 2 3 1.7 30 42 38 48 42.7 4 5 3 4.0 40 53 64 62 59.7 4 5 5 4.7 50 82 75 81 79.3 8 7 10 8.3PSL Change lanetimescollisiontimesDensityChange lanetimescollisiontimes50 56.33333333 3.666666667 5.0 0.666666667 060 44.33333333 2.333333333 10.0 7.666666667 0.666666667 70 42.66666667 4 20.0 18 1.666666667 80 40.66666667 4 30.0 42.66666667 490 35.33333333 4 40.0 59.66666667 4.666666667 100 25.66666667 2.666666667 50.0 79.33333333 8.333333333。

2014大学英语竞赛b类试题及答案

2014大学英语竞赛b类试题及答案2014年大学英语竞赛B类试题及答案涵盖了听力、阅读、写作和翻译等部分。

以下是该竞赛的部分内容和答案示例：听力部分1. 短对话理解（Short Conversations）- 例题：What does the man mean by saying "I'm all ears"? - 答案：He is showing that he is ready to listen attentively.2. 长对话理解（Long Conversations）- 例题：Why does the woman decide to go to the library?- 答案：She needs to find materials for her term paper.阅读部分1. 阅读理解（Reading Comprehension）- 例题：What is the main idea of the passage?- 答案：The main idea of the passage is to discuss the importance of environmental protection.2. 快速阅读（Fast Reading）- 例题：According to the text, what is the author's attitude towards the new policy?- 答案：The author is skeptical about the effectiveness of the new policy.写作部分1. 写作（Writing）- 题目：Write an essay of 200 words on the topic "The Role of Technology in Modern Education."- 答案示例：Technology has revolutionized the way we approach education. It has made learning more interactive and accessible. With the advent of online courses and educational apps, students can learn at their own pace and from anywherein the world. Moreover, technology has also facilitated personalized learning, allowing educators to tailor their teaching methods to the needs of individual students. However, it is also important to ensure that technology does notreplace the human touch in education, but rather enhances it.翻译部分1. 英译汉（English to Chinese Translation）- 题目：Translate the following sentence into Chinese.- 例题：The rapid development of technology has brought about significant changes in our daily lives.- 答案：技术快速发展已经给我们的日常生活带来了显著的变化。

2014美赛B论文

（4.1）通过上式，根据一九方法，确定 aij 的设置。
8
● 计算特征值和特征向量矩阵 A 的最大特征值是 max ，和对应的特征向量是 u u1 , u2 ,u 3, , un 。然
T
后我们通过表达式规范查的公式指标：
i0 u j
二、符号定义以及假设
1. 符号定义
2
（1）评估标准符号
ai
bi
R
说明第 i 年的赢局第 i 年的输局平均 SRS 平均 SOS 季后赛各等级名额每个奖励的分量每个贡献的分数
O
nk
ki ci
（2）层次分析法符号
A
说明评判矩阵矩阵 A 的最大的特征值连续检查的指示随机连续指标评判水平的权值集合选择水平的权值集合模型 1 的评价分数
Y
2. 假设 1. 假设我们已经考虑到所有对评价起重要作用的因素； 2. 假设没有考虑到的教练因素不影响排名； 3. 假设我们收集到的数据具有多样性和准确性以及定量是准确的； 4. 假设教练存在客观而准确的排名情况，并且媒体所给出的排名能够在一定程度上准确的反映教练的排名。
三、表述我们的指标
3.1 指定评估规范至于球员的评价标准，主要有五个方面[9]：力量，速度，技巧，防御和攻击。类似的一个教练的评价也可以分为五个方面：历史战绩，比赛含金量，季后赛的表现，荣誉以及对体育的贡献。接下来的章节将着重说明这五个方面。 ● 历史战绩：球队的战绩在教练的评价中毫无疑问的占据了最大的比例。我们根据主流的统计指标对球队战绩进行统计，发现输赢是非常显着的。球队的历史战绩可以直接反映执教能力。总得赢局可以按如下计算：
O

i
SOSi t

(完整版)2014大学生数学建模美赛B题数据全

1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956
29 BAA
WSC
30 BAA
WSC
31 BAA
WSC
32 NBA
三
33 NBA
BOS
34 NBA
BOS
35 NBA
BOS
36 NBA
BOS
37 NBA
BOS
38 NBA
BOS
39 NBA
BOS
34 0.585
56
21
35 0.375
82
39
43 0.476
82
26
56 0.317
16
8
8
0.5
70
12
58 0.171
Stan Albeck Stan Albeck Stan Albeck Stan Albeck Bob Bass
1979 1980 1981 1982 1983
K.C. Jones
41 NBA
CAP
46 ABA
SAA
40 NBA
NOJ
42 NBA
WSB
47 ABA
SAA
43 NBA
WSB
42 NBA
NOJ
43 NBA
NOJ
44 NBA
NOJ
51 NBA
SAS
42 NBA
DET
79
5722 0.722806020
0.75
80
58
22 0.725
80
59
21 0.738
80
62
18 0.775
Coach Season

历年美赛数学建模优秀论文大全

2008国际大学生数学建模比赛参赛作品---------WHO所属成员国卫生系统绩效评估作品名称：Less Resources, more outcomes参赛单位：重庆大学参赛时间：2008年2月15日至19日指导老师：何仁斌参赛队员：舒强机械工程学院05级罗双才自动化学院05级黎璨计算机学院05级ContentLess Resources, More Outcomes (4)1. Summary (4)2. Introduction (5)3. Key Terminology (5)4. Choosing output metrics for measuring health care system (5)4.1 Goals of Health Care System (6)4.2 Characteristics of a good health care system (6)4.3 Output metrics for measuring health care system (6)5. Determining the weight of the metrics and data processing (8)5.1 Weights from statistical data (8)5.2 Data processing (9)6. Input and Output of Health Care System (9)6.1 Aspects of Input (10)6.2 Aspects of Output (11)7. Evaluation System I : Absolute Effectiveness of HCS (11)7.1Background (11)7.2Assumptions (11)7.3Two approaches for evaluation (11)1. Approach A : Weighted Average Evaluation Based Model (11)2. Approach B: Fuzzy Comprehensive Evaluation Based Model [19][20] (12)7.4 Applying the Evaluation of Absolute Effectiveness Method (14)8. Evaluation system II: Relative Effectiveness of HCS (16)8.1 Only output doesn’t work (16)8.2 Assumptions (16)8.3 Constructing the Model (16)8.4 Applying the Evaluation of Relative Effectiveness Method (17)9. EAE VS ERE: which is better? (17)9.1 USA VS Norway (18)9.2 USA VS Pakistan (18)10. Less Resources, more outcomes (19)10.1Multiple Logistic Regression Model (19)10.1.1 Output as function of Input (19)10.1.2Assumptions (19)10.1.3Constructing the model (19)10.1.4. Estimation of parameters (20)10.1.5How the six metrics influence the outcomes? (20)10.2 Taking USA into consideration (22)10.2.1Assumptions (22)10.2.2 Allocation Coefficient (22)10.3 Scenario 1: Less expenditure to achieve the same goal (24)10.3.1 Objective function： (24)10.3.2 Constraints (25)10.3.3 Optimization model 1 (25)10.3.4 Solutions of the model (25)10.4. Scenario2: More outcomes with the same expenditure (26)10.4.1Objective function (26)10.4.2Constraints (26)10.4.3 Optimization model 2 (26)10.4.4Solutions to the model (27)15. Strengths and Weaknesses (27)Strengths (27)Weaknesses (27)16. References (28)Less Resources, More Outcomes1. SummaryIn this paper, we regard the health care system (HCS) as a system with input and output, representing total expenditure on health and its goal attainment respectively. Our goal is to minimize the total expenditure on health to archive the same or maximize the attainment under given expenditure.First, five output metrics and six input metrics are specified. Output metrics are overall level of health, distribution of health in the population,etc. Input metrics are physician density per 1000 population, private prepaid plans as % private expenditure on health, etc.Second, to evaluate the effectiveness of HCS, two evaluation systems are employed in this paper:●Evaluation of Absolute Effectiveness(EAE)This evaluation system only deals with the output of HCS，and we define Absolute Total Score (ATS) to quantify the effectiveness. During the evaluation process, weighted average sum of the five output metrics is defined as ATS, and the fuzzy theory is also employed to help assess HCS.●Evaluation of Relative Effectiveness(ERE)This evaluation system deals with the output as well as its input, and also we define Relative Total Score (RTS) to quantify the effectiveness. The measurement to ATS is units of output produced by unit of input.Applying the two kinds of evaluation system to evaluate HCS of 34 countries (USA included), we can find some countries which rank in a higher position in EAE get a relatively lower rank in ERE, such as Norway and USA, indicating that their HCS should have been able to archive more under their current resources .Therefore, taking USA into consideration, we try to explore how the input influences the output and archive the goal: less input, more output. Then three models are constructed to our goal:●Multiple Logistic RegressionWe model the output as function of input by the logistic equation. In more detains, we model ATS (output) as the function of total expenditure on health system. By curve fitting, we estimate the parameters in logistic equation, and statistical test presents us a satisfactory result.●Linear Optimization Model on minimizing the total expenditure on healthWe try to minimize the total expenditure and at the same time archive the same, that is to get a ATS of 0.8116. We employ software to solve the model, and by the analysis of the results. We cut it to 2023.2 billion dollars, compared to the original data 2109.8 billion dollars.●Linear Optimization Model on maximizing the attainment. We try to maximize the attainment (absolute total score) under the same total expenditure in2007.And we optimize the ATS to 0.8823, compared to the original data 0.8116.Finally, we discuss strengths and weaknesses of our models and make necessary recommendations to the policy-makers。

美赛历年赛题及其翻译-推荐下载

2015年：A题一个国际性组织声称他们研发出了一种能够阻止埃博拉，并治愈隐性病毒携带者的新药。

建立一个实际、敏捷、有效的模型，不仅考虑到疾病的传播、药物的需求量、可能的给药措施、给药地点、疫苗或药物的生产速度，而且考虑你们队伍认为重要的、作为模型一部分的其他因素，用于优化埃博拉的根除，或至少缓解目前（治疗）的紧张压力。

除了竞赛需要的建模方案以外，为世界医学协会撰写一封1-2页的非技术性的发言稿，以便其公告使用。

B题回顾马航MH370失事事件。

建立一个通用的数学模型，用以帮助失联飞机的搜救者们规划一个有效的搜索方案。

失联飞机从A地飞往B地，可能坠毁在了大片水域（如大西洋、太平洋、印度洋、南印度洋、北冰洋）中。

假设被淹没的飞机无法发出信号。

你们的模型需要考虑到，有很多种不同型号的可选的飞机，并且有很多种搜救飞机，这些搜救飞机通常使用不同的电子设备和传感器。

此外，为航空公司撰写一份1-2页的文件，以便在其公布未来搜救进展的新闻发布会上发表。

2014美赛A题翻译问题一: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。

大多数火车很长(也许10个或更多的汽车长)。

乘客走到出口的距离也很长，有整个火车区域。

每个火车车厢只有两个出口,一个靠近终端, 因此可以携带尽可能多的人。

每个火车车厢有一个中心过道和过道两边的座椅，一边每排有两个座椅，另一边每排有三个座椅。

走出这样一个典型车站,乘客必须先出火车车厢,然后走入楼梯再到下一个级别的出站口。

通常情况下这些列车都非常拥挤,有大量的火车上的乘客试图挤向楼梯，而楼梯可以容纳两列人退出。

大多数通勤列车站台有两个相邻的轨道平台。

在最坏的情况下,如果两个满载的列车同时到达,所有的乘客可能需要很长时间才能到达主站台。

建立一个数学模型来估计旅客退出这种复杂的状况到达出站口路上的时间。

假设一列火车有n个汽车那么长,每个汽车的长度为d。

站台的长度是p,每个楼梯间的楼梯数量是q。

2014年美国数学建模比赛B题(最佳教练)

F1________________ F2________________ F3________________ F4________________
B
Summary
2014 Mathematical Contest in Modeling (MCM) Summary Sheet
This paper mainly sets a model of selecting five “best all time college coach” in basketball, football and hockey respectively. The model is separated to six sub-problems and its solution in our paper:1)Selecting the best coach in one NCAA basketball season; 2)Screening the college coach at top-20 to top-30 in the past 100 years; 3)Further analysis to these selected coaches, and ranking the top-5; 4)Applying this model to football and hockey; 5)Gender impact analysis and time impact analysis; 6)Analyzing the reasons that some famous coach selected by some magazines and media are not on our ranking list. For sub-problem1:Identify four indicators, and confirm the weight of each indicator by Analytic Hierarchy Process, then transformer the indicator data into scores and add these scores multiplied by the weight of itself to get the final scores. The best coach in one season is the one with highest score. For sub-problem2: Use the method above to score for all of the coaches in the past 100 years , then select the top-20.For purpose of reducing the influence of subjective factors, we use Principal Component Analysis to get another ranking list, also, we select the top-20.And union these two top-20. For sub-problem3:Surveying these coaches in the union to learn how many NBA players they've respectively brought up. And regarding this as an indicator, then dividing all indicators into Experience,Leading ability and Ability of player cultivation three aspects, and using Multilevel Hierarchical Analysis to weight them. Next, score for each person by the Fuzzy Criterion of Composite and rank according to the scores. With the purpose of reducing the influence of subjective factors, using Grey Correlation Analysis to calculate the grey correlation degree between “perfect coach” and these coaches. After that, using t-test to judge whether the two results with significant differences. The conclusion obtained: There are no significant difference. Namely, this method is generality. For sub-problem4:Fine-tuning part of indicators and weights, then apply this model in football and hockey. For sub-problem5:As One-way Analysis of Variance used, and the conclusion obtained: gender doesn’t affect the overall score or the ranking of coaches, but female coach has unique advantages in women's basketball league. And the time has no significant effect on the comprehensive score ,but the most excellent coach were in the 70s. Combined with the history of NCAA development, we think that it is necessary to improve the weight of the indicator---champions that gotten recently, making this model better. .For sub-problem6: Comparing the data of their career with our standards to find the reasons.

2014美国大学生数学建模特等奖优秀论文

Team#31552
Page 1 of 25
Best all time college coach Summary
In order to select the “best all time college coach” in the last century fairly, We take selecting the best male basketball coach as an example, and establish the TOPSIS sort - Comprehensive Evaluation improved model based on entropy and Analytical Hierarchy Process. The model mainly analyzed such indicators as winning rate, coaching time, the time of winning the championship, the number of races and the ability to perceive .Firstly ， Analytical Hierarchy Process and Entropy are integratively utilized to determine the index weights of the selecting indicators Secondly，Standardized matrix and parameter matrix are combined to construct the weighted standardized decision matrix. Finally, we can get the college men's basketball composite score, namely the order of male basketball coaches, which is shown in Table 7. Adolph Rupp and Mark Few are the last century and this century's "best all time college coach" respectively. It is realistic. The rank of college coaches can be clearly determined through this method. Next, ANOVA shows that the scores of last century’s coaches and this century’s coaches have significant difference, which demonstrates that time line horizon exerts influence upon the evaluation and gender factor has no significant influence on coaches’ score. The assessment model, therefore, can be applied to both male and female coaches. Nevertheless, based on this, we have drawn coaches’ coaching ability distributing diagram under ideal situation and non-ideal situation according to the data we have found, through which we get that if time line horizon is chosen reasonably, it will not affect the selecting results. In this problem, the time line horizon of the year 2000 will not influence the selecting results. Furthermore, we put the data of the three types of sports, which have been found by us, into the above Model, and get the top 5 coaches of the three sports, which are illustrated in Table10, Table 11, Table12 and Table13 respectively. These results are compared with the results on the Internet[7], so as to examine the reasonableness of our results. We choose the sports randomly which undoubtedly shows that our model can be applied in general across both genders and all possible sports. At the same time, it also shows the practicality and effectiveness of our model. Finally, we have prepared a 1-2 page article for Sports Illustrated that explains our results and includes a non-technical explanation of our mathematical model that sports fans will understand. Key words: TOPSIS Improved Model; Entropy; Analytical Hierarchy Process; Comprehensive Evaluation Model; ANOVA

2011-2014美赛论文评析与灵敏度分析

2011c题O奖论文题目概要：11325队的解答：通过建立三个微分方程，来预测石油、天然气、电的数量变化。

经济模型主要通过现值原理来确定使用车的成本。

8882队：Logistic回归方程预测三个国家的汽车总量的变化情况；然后建立优化模型（如下）来描述在各方面获得最大的利益，接着用图片表示了每个国家三种车辆数量的变化情况。

对电动汽车和混合动力汽车建立电子需求模型（形式如下）非高峰充电，政府可以鼓励。

最小电力成本模型：确定发电站数量模型：9089队：首先巴斯扩散模型预测电动汽车的销售情况。

然后利用蚁群优化来简化上面的模型，得到预测结果。

10607队：生命周期成本模型分析经济影响。

灵敏度分析通过调整参数的值。

高斯烟羽模型分析环境影响，用来模拟和预测汽车尾气的排放。

然后其他的环境影响就是通过找到的有利的数据进行分析，如下所示的噪音污染。

引入外部和内部影响模型（主要形式都是微分方程），然后在此基础上建立诺顿模型模拟电动汽车的数量和变化趋势（考虑EV和CV两种车）接着估计可以保存的石油量：最后通过AHP分析得到不同类型发电方法的权重。

2012B题O奖论文题目概要模型方法元胞自动机模拟灵敏度或者稳定性分析一般通过微小地改变某些已经设定好的参数的值来分析结果的变化情况，如果结果前后变化不大，那么模型具有低敏感性，满足要求。

反之则不稳定。

2013B题O奖论文题目概要模型方法21185队：给出五个小问题和解决方案：一，预测了中国未来13年水的供给和需求；二，建立模型解决中国水资源在时间和空间上的不均匀分布；三，增加区域水资源总量的方法；四，区域水污染处理和节水；五，水成本分析。

模型一：灰色预测模型：先得到水的消耗和人口之间的关联度，工业生产总值和农业产出的关系，然后得到2025年的水需求量模型二：变异系数、最小生成树（得出最佳的水资源转移计划）模型三：描述每个城市的水资源要求的净化程度，得到一个植物的构建方案模型四：基于以上四种成本战略的成本效益分析，首先用AHP各方面（经济、物理、环境）影响的权重，模型五：一个基于神经网络的合理的战略评价模型。

2014美国数学建模B题数据大学教练-曲棍球

93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139
To Yrs 1976 16 1947 20 2007 1 1969 2 1996 1 1975 1 2010 2 1984 3 2008 23 1989 1 2012 2 2014 11 2002 2 1976 3 1978 1 1980 2 1982 3 1990 10 1994 11 2014 1 1996 1 1981 2 1970 3 1968 13 1976 3 1978 2 1954 11 1950 4 2013 3 2014 7 2002 30 2004 9 2000 7 1989 3 1995 1 1975 2 2004 14 1976 1 2014 6 1998 4 2014 4 2009 3 2014 9 1991 6 1919 2
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92
Frank Carroll Wayne Cashman Bruce Cassidy Dave Chambers Art Chapman Guy Charron Gerry Cheevers* Don Cherry King Clancy* Dit Clapper* Odie Cleghorn Sprague Cleghorn* Cory Clouston Neil Colville* Charlie Conacher Lionel Conacher* Kevin Constantine Bill Cook* Jon Cooper Marc Crawford Pierre Creamer Fred Creighton Terry Crisp Joe Crozier Roger Crozier Randy Cunneyworth John Cunniff Alex Curry Leo Dandurand* Hap Day* Billy Dea Peter DeBoer Alex Delvecchio* Jacques Demers Cy Denneny* Bill Dineen Kevin Dineen Rick Dudley Dick Duff* Jules Dugal Art Duncan Red Dutton* Dallas Eakins Frank Eddolls Phil Esposito* Jack Evans John Ferguson