2014建模美赛B题

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2014 数学建模美赛B题

PROBLEM B: College Coaching LegendsSports Illustrated, a magazine for sports enthusiasts, is looking for the “best all time college coach”male or female for the previous century. Build a mathematical model to choosethe best college coach or coaches (past or present) from among either male or female coaches in such sports as college hockey or field hockey, football, baseball or softball, basketball, or soccer. Does it make a difference which time line horizon that you use in your analysis, i.e., does coaching in 1913 differ from coaching in 2013? Clearly articulate your metrics for assessment. Discuss how your model can be applied in general across both genders and all possible sports. Present your model’s top 5 coaches in each of 3 different sports.In addition to the MCM format and requirements, prepare a 1-2 page article for Sports Illustrated that explains your results and includes a non-technical explanation of your mathematical model that sports fans will understand.问题B：大学教练的故事体育画报，为运动爱好者杂志，正在寻找上个世纪堪称“史上最优秀大学教练”的男性或女性。

2014美赛B数学建模美赛B题数据

2009 2011 1987 1945 1980 1997 1955 1939 1916 1905 1928 2012 1983 1975 1971 1920 1910 1925 1937 1923 1922 1978 1990 1907 1896 1926 1946 1924 1943 1922 1902 1992 1926 1918 1918 1917 1946 1944 1935 1904 1926 1923 1912 1945 1914 1912 1901
Brown Hawaii Fresno State Fordham Towson Stephen F. Austin Murray State Maine Tulsa Oregon State Rochester (NY) Lipscomb Oral Roberts Saint Louis Oklahoma State Georgia Tech St. John's (NY) Kent State Louisville Navy Kentucky Marshall Tennessee State Kansas State Yale DePaul Saint Mary's (CA) Rice Chicago Santa Clara Bloomsburg Sacramento State Utah Virginia Military Institute Muhlenburg Boston University South Carolina Denison Wichita State Brown Valparaiso Rice Roanoke Nevada North Dakota Arizona State Mount Union
names

美国大学生数学建模比赛2014年B题

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

2014年AMC_12真题 (B)

400 11
(C)
75 2
(D) 40
(E)
300 7
4. Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is how many times as expensive as a banana?
15. When p 6 , the number e p is an integer. What is the largest power of 2 that is k 1k ln k a factor of e p ? (A) 212 (B) 214 (C) 216 (D) 218 (E) 220
r and s such that the line through Q with slope m does not intersect P if and only if r < m < s. What is r + s? (A)1 (B)26 (C)40 (D)52 (E)80
18. The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every n from 1 to 15 one can find a subset of the numbers that appear consecutively on the circle that sum to n. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

2014高教社杯全国大学生数学建模竞赛(B,C,D)题目

2014高教社杯全国大学生数学建模竞赛题目（请先阅读“全国大学生数学建模竞赛论文格式规范”）B题创意平板折叠桌某公司生产一种可折叠的桌子，桌面呈圆形，桌腿随着铰链的活动可以平摊成一张平板（如图1-2所示）。

桌腿由若干根木条组成，分成两组，每组各用一根钢筋将木条连接，钢筋两端分别固定在桌腿各组最外侧的两根木条上，并且沿木条有空槽以保证滑动的自由度（见图3）。

桌子外形由直纹曲面构成，造型美观。

附件视频展示了折叠桌的动态变化过程。

试建立数学模型讨论下列问题：1. 给定长方形平板尺寸为120 cm × 50 cm × 3 cm，每根木条宽2.5 cm，连接桌腿木条的钢筋固定在桌腿最外侧木条的中心位置，折叠后桌子的高度为53 cm。

试建立模型描述此折叠桌的动态变化过程，在此基础上给出此折叠桌的设计加工参数（例如，桌腿木条开槽的长度等）和桌脚边缘线（图4中红色曲线）的数学描述。

2. 折叠桌的设计应做到产品稳固性好、加工方便、用材最少。

对于任意给定的折叠桌高度和圆形桌面直径的设计要求，讨论长方形平板材料和折叠桌的最优设计加工参数，例如，平板尺寸、钢筋位置、开槽长度等。

对于桌高70 cm，桌面直径80 cm的情形，确定最优设计加工参数。

3. 公司计划开发一种折叠桌设计软件，根据客户任意设定的折叠桌高度、桌面边缘线的形状大小和桌脚边缘线的大致形状，给出所需平板材料的形状尺寸和切实可行的最优设计加工参数，使得生产的折叠桌尽可能接近客户所期望的形状。

你们团队的任务是帮助给出这一软件设计的数学模型，并根据所建立的模型给出几个你们自己设计的创意平板折叠桌。

要求给出相应的设计加工参数，画出至少8张动态变化过程的示意图。

图1图2图3图4附件：视频2014高教社杯全国大学生数学建模竞赛题目（请先阅读“全国大学生数学建模竞赛论文格式规范”）C题生猪养殖场的经营管理某养猪场最多能养10000头猪，该养猪场利用自己的种猪进行繁育。

2014全国数学建模大赛B题

2014高教社杯全国大学生数学建模竞赛承诺书我们仔细阅读了《全国大学生数学建模竞赛章程》和《全国大学生数学建模竞赛参赛规则》（以下简称为“竞赛章程和参赛规则”，可从全国大学生数学建模竞赛网站下载）。

我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。

我们知道，抄袭别人的成果是违反竞赛章程和参赛规则的，如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。

我们郑重承诺，严格遵守竞赛章程和参赛规则，以保证竞赛的公正、公平性。

如有违反竞赛章程和参赛规则的行为，我们将受到严肃处理。

我们授权全国大学生数学建模竞赛组委会，可将我们的论文以任何形式进行公开展示（包括进行网上公示，在书籍、期刊和其他媒体进行正式或非正式发表等）。

我们参赛选择的题号是（从A/B/C/D中选择一项填写）：我们的参赛报名号为（如果赛区设置报名号的话）：所属学校（请填写完整的全名）：参赛队员(打印并签名)：1.2.指导教师或指导教师组负责人(打印并签名)：（论文纸质版与电子版中的以上信息必须一致，只是电子版中无需签名。

以上内容请仔细核对，提交后将不再允许做任何修改。

如填写错误，论文可能被取消评奖资格。

）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）：创意平板折叠桌摘要折叠与伸展也已成为家具设计行业普遍应用的一个基本设计理念，占用空间面积小而且家具的功能又更加多样化自然会受到人们的欢迎，着看创意桌子把一整块板分成若干木条，组合在一起，也可以变成很有创意的桌子，就像是变魔术一样，真的是创意无法想象。

2014数学建模b题

对创意平板折叠桌的最优化设计摘要本文主要研究了创意平板折叠桌的相关问题。

对于问题一，首先，我们根据所提供的已知尺寸的长方形平板和桌面形状，桌高的要求，以圆桌面中心作为原点建立了相应的空间直角坐标系，分别求出了各个桌腿的长度，根据在折叠过程中，钢筋穿过的每个点距离桌面的高度相同这一性质，利用MATLAB程序计算出了每根木棒卡槽的长度和桌脚底端每个点的坐标，其中卡槽长度依次为（从最外侧开始，单位：cm）：0、 4.3564、7.663、10.3684、12.5926、14.393、15.8031、16.8445、17.5314、17.8728，并且根据底端坐标拟合出了桌脚边缘线的方程并进行了检验。

另外，我们通过桌脚边缘线的变化图像来描述折叠桌的折叠过程。

对于问题二，我们以用材最少为目标函数，以稳固性好为约束条件，通过对桌腿进行力学分析和几何分析得到了使得用材最少且稳固性好的圆桌需要满足的条件是钢筋穿过最长腿的位置满足一个不等式。

并且，当平板的长为163.4702cm,宽为80cm,厚度为3cm,最外侧桌腿钢筋处到桌腿底端的距离与桌腿的长度之比为0.4186时，木板的用材最小，其对应的体积V为392330cm3。

对于问题三，为了满足客户需求，使得生产的折叠桌尽可能接近客户所期望的形状，我们给出了软件设计的基本算法。

我们考虑了“操场形”桌面和“双曲线形”桌面，得到了“操场形”桌面的的创意平板折叠桌槽长为（从最外侧开始，单位：cm）：0、4.3564、7.6637、10.3684、12.5926、14.3930、15.8031、16.8445、17.5314、17.8728; “曲线形”桌面的创意平板折叠桌槽长为（从最外侧开始，单位：cm）：0、1.5756、2.8917、3.9886、4.9005、5.6532、6.2641、6.7397、7.0741、7.2501。

最后，给出了两种桌面的动态变化图。

关键字：曲线拟合最优化设计几何模型折叠桌桌脚边缘线一、问题重述问题背景某公司生产一种可折叠的桌子，桌面呈圆形，桌腿随着铰链的活动可以平摊成一张平板。

14美国数学建模B题

For office use only T1________________ T2________________ T3________________ T4________________Team Control Number30213Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________The Keep-Right-Except-To-Pass RuleAbstractIn this paper, five mathematical models are proposed, respectively around the performance of the keep-right-except-to-pass rule on multi-lane freeways in light and heavy traffic.Three mathematical models are established to analyze tradeoffs between multi-line traffic flow and safety, traffic speed, and other factors under the keep-right-except-to-pass rule. At first, by taking overtaking and mixed vehicle types as viscous resistance, and according to the mass conservation law, we develop continuous and discrete fluid dynamic traffic flow model. Then by taking three variables (space and temporal distance between vehicles, vehicle speed) into account, we establish traffic safety control model. Finally, by considering synthetically vehicle operation efficiency, security, comfort, fuel economy and so on, we develop a multi-objective programming model to limit the highest speed.In this paper we give several different traffic rules, and present some improvement for the keep-right-except-to-pass rule, such as vehicles of different types on different lanes, and different lanes with different speed limits. In a segment of a multi-lane freeway, new measures can improve 15.11% of traffic flow.We can apply our mathematical models to the keep-left-except-to-pass rule with a simple change of orientation. For example, the drivers need drive at the right of the car for more safety.Two mathematical models are established to analyze the traffic flow of multi-lane freeways under the control of an intelligent system. At first, we present five security operating patterns basing on seven indices in security operation about person, vehicle and surroundings. We thus propose a microscopic safety control model basing on RBF neural network, and conduct a simulation test of recognition of security operating patterns. Finally, we study the vehicle’s speed change, total service flow of the freeway, the vehicle’s travel time. We thus propose three different macroscopic optimal control models for freeway sections with low traffic density, medium traffic density and high traffic density, respectively. The three models are dynamic nonlinear programming models. The result shows that the effect of the intelligent system promotes the traffic flow effectively.1 Introduction (1)2 Definitions and Variables (1)3 Assumptions (2)4 Establishment and Solution of Three Models for Problem 1 (2)4.1 Model Constraints (2)4.2 Fluid Dynamic Traffic Flow Model (3)4.2.1 Impact of mixed-type vehicles and overtaking (3)4.2.2 Continuous fluid dynamic traffic flow model (4)4.2.3 Discrete fluid dynamic traffic flow model (5)4.3 Traffic Flow Safety Control Model (6)4.3.1 Determination of the vehicle operating safety (6)4.3.2 Traffic flow safety control model [4] (7)4.4 Multi-objective Programming Model of the Maximum Speed Limit [9] (8)4.4.1 Performance indicators of multi-objective programming model [5] (8)4.4.2 Constraints about security and comfort property (8)4.4.3 Establishment of a multi-objective programming model to limit maximum speed (9)4.5 Improvement and Evaluation of the Traffic Rules (10)4.5.1 Improvement of the traffic rules (10)4.5.2 Evaluation of the traffic rules (10)5 Analysis of Problem 2 (11)6 Establishment and Solution of Two Models for Problem 3 (11)6.1 Microscopic Safety Control Model Basing on RBF Neural Network (12)6.1.1 Indices of security operation about person, vehicle and surroundings (12)6.1.2 Safe operation patterns (12)6.1.3 Microscopic security control model based on neural network (12)6.2 Macroscopic Optimal Control Model of the Traffic Flow (14)6.2.1 Control model with variable speed limitation (15)6.2.2 Constraints of the optimal control model (15)6.2.3 Performance index of the macroscopic optimal control model (15)6.2.4 Three macroscopic optimal control models of the traffic flow (16)7 Analysis and Evaluation of Models (17)7.1 Evaluation of fluid dynamic traffic flow models (17)7.2 Evaluation of the safety control model to limit speed (17)7.3 Evaluation of the microscopic safety control model with RBF neural network (18)7.4 Evaluation of the macroscopic optimal control model (18)7.5 Improvements of the Models (18)References (18)1 IntroductionSome countries, such as USA, China and so on, have the keep-right-except-to-pass rule on multi-lane freeways. This rule requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane. But there are also many countries providing the keep-left-except-to-pass rule. The traffic rule relies upon human judgment for compliance.To analyze the performance of the traffic rule in light andheavy traffic, we study some mathematical models to solve thefollowing problems.Problem 1: Examine tradeoffs between traffic flow and safety,the role of under- or over-posted speed limits, and other factorsfor the keep-right-except-to-pass rule. Judge the efficiency of therule in promoting traffic flow. Suggest alternatives that mightpromote greater traffic flow, safety, or other factors if the rule isnot effective.Problem 2: Argue whether or not above analysis can be carriedover to the keep-left-except-to-pass rule.Problem 3: Stuffy the performance of the traffic rule under thecontrol of an intelligent system-either part of the road networkor imbedded in the design of all vehicles using the roadway.Show the changing extent of the results of earlier analysis.In order to reach Problem 1, we first take overtaking and mixed vehicle types as viscous resistance, and develop continuous and discrete fluid dynamic traffic flow models according to the mass conservation law. Then we take space and temporal distance between vehicles and vehicle speed into account to establish traffic safety control model. Finally, we consider synthetically vehicle operation efficiency, security, comfort property and fuel economy and develop a two-objective programming model to limit the highest speed. We also look for other traffic rules and make comparison.For Problem 2, we give a simple change of orientation in order to apply our mathematical models to the keep-left-except-to-pass rule. For example, the drivers need drive at the right of the vehicles with different types on different lanes on different lanes with different speed limit for more safety and more traffic flow.In order to reach Problem 3, we study security operating patterns basing on indices in security operation about person, vehicle and surroundings. By applying RBF neural network, we propose a microscopic safety control model, and conduct a corresponding simulation test. Next, we research the vehicle’s speed change, total service flow of the freeway, the vehicle’s travel time and freeway sections with low traffic density, medium traffic density and high traffic density. Then according to the thinking of dynamic nonlinear programming, we propose three different macroscopic optimal control models. At last, we test the efficiency of the intelligent traffic system.2 Definitions and VariablesThe following definitions and variables will be used in our discussion of the traffic rule problem.·A rule is drivers drive in the right-most lane on multi-lane freeways.·u is the speed of the car in the freeways.·k is the density of the traffic flow .·q is the traffic flow on the freeways·w τis viscous drag ，mains micelles produced fluid occurs between the relative tangential slip resistance·μ is viscosity coefficient·η is the proportion of small cars·T is the driver's reaction time·A T is safe distance from the vehicle·A K is safety traffic density.·K is the density of the·L is safe distance between vehicles·()t k i is the average density of the i vehicle·g P is gasoline prices3 Assumptions(1) All factors that cause damage to the vehicle speed are defined as the viscous resistance w ι, including every factor that overtaking and mixed vehicle types and so on.(2) There are three types of vehicles in freeway: small car, midsize car, and super-huge type. Because the super-huge vehicle occupies a least proportion, we regard super-huge vehicle as midsize car.(3) Traffic flow is stable traffic flow which is continuous, uninterrupted, evenly distributed.(4)Traffic speed, flow, traffic density are continuous functions with respect to special and temporal variables.(5) There are five kinds of vehicle operation patterns: following pattern, accelerate pattern, retard pattern, overtake pattern, break pattern.(6) In low traffic density area, the mutual interference between vehicles is very small.(7) Under the intelligent control systems, pilots, vehicles, environment and other factors have some influence on the traffic safety.4 Establishment and Solution of Three Models for Problem 14.1 Model ConstraintsTo study the performance of the keep-right-except-to-pass rule, we consider the three elements of traffic flow: traffic flow, safety and speed limits. In China, we have a traffic speed limit rule on the roadway for safety, as follows:On the basis of security, we set up a model to analyze traffic flow, speed and vehicle density. First, speed u , density k and flow q satisfy thatku q = (1)For multi-lane freeway with small k , the driver will drive free at the speed of f u . According to Mass Conservation Law, we set up the following single-lane model without lane change0=∂∂+∂∂xq t k (2) For multi-lane with lane change, we should add a generation item of traffic flow to above equation. Denote the flow in and out of the lane line by r . We establish the following equationdxdr x q t k =∂∂+∂∂ (3) 4.2.1 Impact of mixed-type vehicles and overtaking In fluid mechanics, the fluid viscosity is defined as tangential resistance generating when fluid micelle occurs relative slippage. The size of resistance is in proportion to contact area and velocity gradient. There we assume that all factors that causes damage to the vehicle speed are defined as the viscous resistance w ι, including every factor that overtaking and mixed vehicle types and so on. The higher the degree of mixed vehicle types (Viscosity coefficient of performance μ) is, the more influence traffic stream has; the greater the mixed degree is, the greater the viscosity coefficient μ is; a single model of the traffic has the minimum mutual interference between them. There are three types of vehicles in freeway: small car, midsize car, and super-huge type. Because the super-huge vehicle occupies a least proportion, we regard super-huge vehicle as midsize car. According to [1], we assume that the ratio η of μand small cars to meetηημ++=312 (4) The viscous resistance which date from mixed models is also proportional to the difference between the free-flow speed f μμ-and density gradient between lanes k .drivers are inpursuit of the free-flow speed, and the vehicle speed, difference between the free stream velocity and the size of the density joint determines the possible of overtaking. The smaller the difference, the greater the density is, the more impossible of overtaking. Moreover, inter-lane overtaking has a great impact on the upstream traffic flows. The greater ratio of overtaking is, the greater impact on productivity of downstream, the greater the resistance. This is because the viscous resistance which date from mixed models is proportional to the difference between the free-flow speed and density gradient between lanes .Through the foregoing analysis, the viscous resistance w ιcan be represented by the follow formula:x k n u u f w ∂∂+-=])([μι (5) Corresponding model equations is:w e t x u k u Tdt du ι+-=)],()([1 (6) where T means that the reaction time of the driver and()⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧<<⎪⎪⎭⎫ ⎝⎛-≤≤----≤≤⎥⎥⎦⎤⎢⎢⎣⎡-+=n j n j q r j f n j f n j qr j f n f n j j q r f f e e k k e k e k k m u e k k k m m e k e k k m u e k m m e k k k e k k m m u u 21121141411411 (7) 4.2.2 Continuous fluid dynamic traffic flow modelBecause speed is a function with distance and time, hence we can getxu u t u dt dx x u t u dt t x du dt du a ∂∂+∂∂=⋅∂∂+∂∂===),( (8) We take the second equation and the fifth equation into the third equation, gaining another mixed equation.xk n u u u k u T x u u t u f e ∂∂+-+-=∂∂+∂∂])([])([1μ (9) Therefore, we combine the third equation with the eighth equation for obtaining a new kinetic model as follows:()[]()[]x k n u u t t u k u Tx u u t u dxdr x q t k f w w e ∂∂+-=+-=∂∂+∂∂=∂∂+∂∂μ1 (10) This is the hydrodynamic model which corresponding with the mixed traffic flow that has the phenomenon of overtaking. The model is continuous model . Dispersing the density andvelocity by using numerical differential methods to ninth equation then we can create Discrete Models.4.2.3 Discrete fluid dynamic traffic flow modelt r u k u k xt k k n i n i n i n i n i n i n i ∆+-∆∆+=--+)(111 (11) ()()()()]()⎩⎨⎧⎭⎬⎫-+-+-+-⨯-+∆∆+=---+n i n i n i f n i n i e n i n i e n i n i n i n i k k n u u u u T u u u u x t u u 1111[1μμ (12) where ,3,2,1=i …, ,3,2,1,0=n …. The finite difference numerical format only keeps stability when it meets the following V on Neumann conditions, that ismuu k k x t f j ∆∆≤∆ (13) We use MA TLAB software to simulate the above established mathematical model and then get three-dimensional map which includes changes in the density and the corresponding density and in the speed and corresponding speed when other factors remain unchanged. Drawn diagram below:Figuer1 Density - time curve Figuer2 Three-dimensional map of the changes of densityFiguer3 Velocity- time curve Figuer4 Three-dimensional map of the changes of velocityThrough simulating the change map of speed and density, we can clear observe the relationship between the velocity, density and traffic flow.Utilizing the shifty relation diagram with speed, density and flow, we work out the vehicle flow ’s shifty diagram under low load and high load with the rule. As follow:Figure 5 Low load and high load ’s shifty vehicle flowBy observing the shifty diagram of vehicle under low load and high load, we can study clearly the affect of the right rule. On the low load condition, changes in volatility of vehicle are small. There is a rush hour under high load. Letter on, it would tend to smooth and steady.4.3 Traffic Flow Safety Control ModelFluid dynamic traffic flow model study the relationship between the flow, speed and density and the driver driving safety [4]. In the traffic, we should also consider security problems. Then we improve the fluid dynamic traffic flow model, establishing a new model on flow, speed and security.4.3.1 Determination of the vehicle operating safetyThere are many factors affecting the safety of vehicle running. We study spacing, distance and speed.A T is the shortest length time of don’t crash when the ahead vehicle stop suddenly. Then:321t t t T A ++= (14)In above equation, 1t is the driver perception reaction time, it depends on the driver’s visual function space, motion perception sensitive, other physiological condition and weather condition. 2t denotes the time from life to the breaking, it depends on the driver’s reaction capacity, operational experience, vehicle performance. 3t is the time from breaking start to the end of the break, it depends vehicle performance, road conditions, meteorological condition.Safety clearance A L is the shortest distance between cars of don’t crash when the ahead vehicle stop suddenly. Suppose V is the speed when the vehicle break, safety distance A L can be expressed when vehicle operating.()()212254A K V L L t t V h g =+++- (15)In above equation, K L is a constant,h is adhesion coefficient on tire and road, g is corrected parameter for h with special meteorological conditions such as rain, snow, ice and so on4.3.2 Traffic flow safety control model [4]Suppose f V is pass impeded speed,j K is jam density, X is the length on a road, traffic flow is continuous, uninterrupted, evenly distributed, consisting of small cars. m L is the length of the car, q is a traffic flow into the road, references[4], Select the security control model for the road traffic flowVK V K K qV V V V VKq j f j f f f -=⎥⎥⎦⎤⎢⎢⎣⎡-±==4212 ()[]TL V X L L X K m =+= (16) ()()AAK A A V V K K i g h V V t t L L L t t t T T ≤≤±-+++=≥++=≥254221321In this formula, K is vehicle density,L is vehicle spacing,T is vehicle interval,A Vis safety speed, A K is safety traffic density.Figure 6 Three-dimensional map of speed-density-flowUnder the condition of abiding by the rules of right of traffic, When other factors unchanged,different cars have a higher requirement for the driver.4.4 Multi-objective Programming Model of the Maximum Speed Limit [9]Application of multi-objective optimization ideas proposed maximum speed limit method based on the operational efficiency, safety, economy and comfort of the highway. The time and fuel consumption costs of highway speed to a minimum as the goal , accident mortality and comfort to satisfy the corresponding allowable range as constraints, and we have established a multi-objective programming model of a maximum speed of about costs, fuel costs and other factors limit.4.4.1 Performance indicators of multi-objective programming model [5]4.4.1.1 Time cost function for operating efficiencyPassengers’ time in transit can not to reach its ideal vehicle speed which resulting in the decreased value created that due to the increased time in transit as the cost of highway travelers. The fee of highway passenger time in transit expressed as follows⎪⎪⎭⎫ ⎝⎛-⨯⨯⨯⨯=i t v L v L q E G C 8365 (17) where t C represents the cost of highway travelers’ time in transit:, G represents the GDP per capita, E represents the average load factor, q represents the number of traffic ，L represents the length of the highway ，v represents the running speed ，i v represents the ideal vehicle speed of drivers.4.4.1.2 Fuel cost function about fuel consumptionWhen the vehicle is traveling on the highway, the increasing of speed causes the fuel consumption. Therefore, we have established a highway fuel cost function as follows:())(100497.222518.00013.02e d g g v v Q P L v v C >⨯⨯⨯+-= where g C represents d riving on the highway fuel consumption costs ，g P represents market price of gasoline ，d Q represents the number of daily traffic ，e v represents economic speed .4.4.2 Constraints about security and comfort property4.4.2.1 Constraints about securityDue to maximum speed limit on the highway accident mortality should ensure that the value is less than the acceptable tolerance, and We constructed the constraints that based on security for highest highway speed limits as follows:()[]Death h h h h Death I v v E v v E v v E v v E I <--+-----+--=)(13314)(176)(076234 (18)where h v represents the maximum speed limit ，v represents average operating speed ，[]Death I represents the tolerability of traffic accident mortality law.4.4.2.2 Constraints about comfort propertyIn the course of traveling, everyone has a comfortable speed, here we choose t he highest threshold with more comfortable as constraints of in highway speed limits maximum comfort as follows:h km v h /132≤. The following table shows the relationship between speed and comfortable of the driver's.4.4.3 4.4.3.1 Single goal programming model of the maximum operating efficiencyWe take the maximum operating efficiency (the minimum cost)as the goal that can limit the maximum speed of the highway. we establish efficient single target programming model of the highest speed limits, as follows ：⎥⎦⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛-⨯⨯⨯⨯=i t v L v L q E G C 8365min )min( (19) 4.4.3.2 Single goal programming model of the maximum economic limitWe take the minimum fuel consumption (the minimum cost of the fuel consumption)as the goal that can limit the maximum speed of the highway. we establish economic target programming model of the highest speed limits, as follows ：())(100497.222518.00013.0min )min(2e d g g v v Q P L v v C >⎥⎦⎤⎢⎣⎡⨯⨯⨯+-= (20) 4.4.3.3 Multi-objective programming model to limit the maximum speedConsidering both operating efficiency and fuel consumption performance indicators, we establish a multi-objective programming model of the maximum speed limit, as follows:()[]hb g hb t v C v C +)(min ⎥⎥⎦⎤⎪⎪⎭⎫ ⎝⎛⨯⨯-⨯⎢⎢⎢⎢⎣⎡+⨯⨯+⨯-⨯=t g g g v GE GL P v E G QL v P QL v P QL 183********.2283652518.01000013.0100min 2 (21) []⎪⎩⎪⎨⎧≤≤<<132.hbDeath Death i hb e v I I v v v t s (22)where hb v represents small cars on the highway the maximum speed limit reference value.4.4.3.4 Solution of the multi-objective programming modelAfter calculation, we obtain that the multi-objective programming model has only one real solution. Represented by the following formula: ()()()()()2920,)0351.2(,2056.262736546273654613132322313232E G f P E e P E d de ef d f d e f d d e e f d f d e f d d vg g ⨯=⨯-=⨯-=++++++++=- (23) According to the solution of our model, we can get the maximum speed limit under the different design in the following table.and heavy traffic.4.5 Improvement and Evaluation of the Traffic Rules4.5.1 Improvement of the traffic rulesThere are many kinds of safe driving of the traffic rules, for example:(1)Driving on the right, allowing the overtaking(2)Driving on the right, and each lane no overhead(3)The lane with the same speed limit(4)Each lane has different speed limitIn these traffic rules, (1) and (4) combined is most efficient. Besides, in order to improve the traffic flow, we put forward some improvement measures. i.e.:(5)constant driving rules, increase the number of lanes(6)In a case of a safe, flexible choice about overtaking lane(7)Large and medium-sized car in the right lane, small cars can be arbitrary choice(8)Each lane have different speed limit. For example, in the case of four lanes, left lane speed limit for 90km/h-120km/h, right lane speed limit for 60km/h-90km/h .4.5.2 Evaluation of the traffic rulesWe evaluate above the traffic rules, in the case of guarantee the safe operation of the vehicle, we study traffic flow on the combination of (1),（7）and (8),then we compare the results with previous model.Under the condition of without passing phenomenon, we use MA TLAB software to solve the traffic flow, speed and density. Results of different speed limits are in the following tables.Table 5 Results of different speed limitMaxspeed(km/h)The initial density The final density The original speed The final speed The initial flow The final flow 12052.1 12.42 44.9 93.35 2339.3 1160.4 90 52.1 23.75 44.9 83.28 2339.3 1978.260 52.1 30.15 44.9 88.51 2339.3 2682.7Table6 The traffic condition with or without overtakingTypes Maxspeed(km/h)TheinitialdensityTheFinaldensityTheOriginalspeedThefinalspeedTheinitialflowThefinalflowDon’t overtake 120 52.1 12.42 44.9 93.35 2339.3 1160.4 90 52.1 23.75 44.9 83.28 2339.3 1978.2 60 52.1 30.15 44.9 88.51 2339.3 2682.7overtake 120 52.1 44.83 44.9 87.66 2339.3 3930.3 According to table 5 and table 6, we work out the traffic flow, the traffic flow is 4660.9 when we not improve the traffic rules. The traffic flow is 5365.4 when we improve the traffic rules. Increased by 15.11% between the two traffic rules. We obtain the most of the traffic follow the combination of (1),（7）and(8),and it have high degree of safety.5 Analysis of Problem 2In the countries with Left-hand drive and right lines，drivers require driving on the left lane on the highway. When they want to overtake other cars, they first travel to the right lane, then return to the original lane. This is right in principle to the rule that driving on the right . Nowadays, the number of left-hand drive and right line is third more than the number of the right-hand and left line. However, it not mean the rule, left-hand drive and right line, is superior the rule, right-hand drive and left line. This lies on the driving habits of the drive’s own, geographic position, environmental factors, vehicle driving position and so on. For instants, in countries in the northern hemisphere, vehicles produce a slight shift to the right in the direction of running because of the earth rotation. At the same time, driving on the right is more beneficial to traffic safety. Vehicle driving position of the right of vehicle is on the left side. In this way, drivers can obtain a broader vision for traffic safety.To apply the model fluid dynamic traffic flow model built in the rule of to left-hand drive and right line to the rule of right-hand drive and left line, we should modify the modified model based on relative requirements, for instants:(1) Converting the driving position of vehicles of left-hand drive and right line into right side;(2) Driving on the left line and allowing overtaking;(3) Large and medium-sized cars drive on the left line, cars run on any freeway;(4) Limit speed in every freeway is different.6 Establishment and Solution of Two Models for Problem 3We studied the traffic rules, and the improved depends on the people follow the traffic rules and other factors on the road，under the supervision of intelligent system，The car will be increased relative sensitivity，conduct to the vehicle safe and efficient operation，At the same time greatly enhance the utilization of freeway lanes. We have established hydrodynamic traffic flow model can be modified, create a new mathematical model for the intelligent system control，to follow traffic rules traffic situation. Under the intelligent system control, there is the impact of macro and micro effects of the impact on the vehicle runs. Macroeconomic impact of the shift, highway traffic services, vehicle travel time and other factors that together determine the vehicle, the microscopic effects are determined by personal factors, vehicle performance, weather and other factors. In this regard, we have established asecure micro-and macro-control network model optimal control model.6.1 Microscopic Safety Control Model Basing on RBF Neural Network On the highway, there are 5 types of security operating patterns on the multi-lane traffic, including following pattern, accelerating pattern, decelerating pattern, overtaking pattern and breaking pattern. For simplicity, we study the security operating patterns on two-way four-lane freeways.6.1.1 Indices of security operation about person, vehicle and surroundingsIn the driving process, considering the impact of various factors about the drivers, vehicle and traffic environment, security operating pattern is determined by the following seven indices:（1）absolute speed of vehicle 1v ;（2）the proportion of the vehicle speed 1v and speed 2v of the vehicle in front ；（3）the distance of the car and the vehicle in front ；（4）the distance of the car and the vehicle in latter on the overtaking freeway ；（5）overtaking signal of the car in latter （0 means no overtaking ，1 means overtaking ）；（6）driver's fatigue condition （0 means relaxation ，1 means fatigue ）；（7）the weather condition （0 means sunny ，1 means snow ）.6.1.2 Safe operation patternsThe above seven characteristics determine the following five indicators in the safe operation mode highway vehicles [3]：（1）accelerating pattern ：Measured in run mode is under the distance the car and the vehicle in front ，and no vehicle enters the ultra-lane carriageway circumstances taken a run mode.（2）overtaking pattern ：Measured in run mode is under the vehicle speed is up to speed with the vehicle in front ，and distance before the car reaches a range of overtaking, nor behind the vehicle taken by a passing mode of operation.（3）following pattern ：Measured in run mode is ready to overtake ， but after overtaking car overtaking signal taken a run mode ；（4）decelerating pattern: there is a world run mode is the speed of the vehicle and the vehicle in front of a large vehicle, and the close proximity of the vehicle in front or behind the vehicle overtaking under taken by one mode of operation;（5）breaking pattern ：This model is in front of unexpected events occur, impassable taken the safe operation mode.6.1.3 Microscopic security control model based on neural networkCars run on the highway, accidents often occur in a relatively short period of time. On the background of highway operation decision-making research, real time is the most critical point. The BP neural network is a kind of approximation network, training quickly and meeting the real-time requirements. The BP network simulation diagram is figure 5. Input layer choose 7 neurons, i.e. the input vector is 17 feature index vector of the network. The radial base (implicit strata) choose 5 neurons, the target output layer choose a neuron, the target vector T =[1 2 3 4 5] represent the five kinds of safe operation of the model。

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。

2014年美赛B题主成分分析排名

基于主成份分析的排名方法模型建立采用y(执教年份)，G （参赛场数），WL （胜率）和sum(特殊得分)作为衡量教练排名得分的指标。

由于有4个指标，每个指标都与排名是正相关的，每个指标的增加都对排名增加了得分。

因此，将每个指标对排名增加的得分累加，即为排名得分。

考虑到各个指标之间可能是相关的，导致得分重复计算，因此需要对指标进行坐标变换，使其正交旋转到互不相关的新坐标轴上，去除冗余信息。

此外，考虑到各指标数量级不一样，还应将各变量进行标准化处理。

处理过程如下：（1）假设有n 个教练，记录第i 个教训4个指标的观测值分别为：421,...,,i i i x x x ，则所有n 个教练4个指标的观测值可以表示成以下矩阵：⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=42124222112114n n n x x x x x x x x x X（2）考虑到每个变量的数量级与及标准差不一样，对数据进行标准化处理：，处理方法：∑∑==--====-=n i k ik kni ik k k k ik ik x x n s n x x k n i s x x x 1221'][11,/4,3,2,1,....2,1,/][式中，标准化处理后，指标的方差为1，均值为0; (3)设观察值构成的相关系数矩阵为：⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=444241242221141211r r r r r r r r r R经标准化处理后的数据的相关系数为：)4,3,2,1,(,111''=-=∑=j i x x n r n k kj ki ij（4）对应于相关系数矩阵R ，求特征方程=-I R λ的p 个非负的特征值4321,,,λλλλ。

对应于特征值i λ的特征向量为：4,3,2,1,),,,('4321==i c c c c C i i i i i（5）求主成分。

由特征向量组成的4个主成分为：42211....X c X c X c F pi i i i +++=主成分4321,,,F F F F 之间相互无关，且它们的方差是递减的。

2014年美赛B题翻译

一本为体育爱好者的杂志《体育画报》正在为上世纪寻找“最佳全职大学教练”。

建立一个数学模型，从大学曲棍球和陆上曲棍球,足球,棒球或垒球、篮球、足球的男教练或者女教练中，选择现在或者过去的最佳大学教练。

当你在分析中采用不同的时间点时，会有不同吗？例如：1913年的教练是否与2013年的不同？
清晰的表达你的评价指标。

讨论你的模型怎样用于男、女教练和所有可能的运动。

介绍一下你模型中，在3种不同运动中的前五名的教练。

除了MCM的格式和要求，为《体育画报》准备一份1-2页的文章来解释你的结果，这篇文章需要包括你的数学模型的非技术性解释，从而使体育迷们理解。

2010-2014MCMProblems建模竞赛美赛题目重点

2010-2014MCMProblems建模竞赛美赛题目重点2010 MCM ProblemsPROBLEM A: The Sweet SpotExplain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of t he bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe that “corking” a bat (h ollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash or metal (usually aluminum bats? Is this why Major League Baseball prohibits metal bats?MCM 2010 A题：解释棒球棒上的“最佳击球点”每一个棒球手都知道在棒球棒比较粗的部分有一个击球点，这里可以把打击球的力量最大程度地转移到球上。

2014数学建模B题

2014高教社杯全国大学生数学建模竞赛承诺书我们仔细阅读了《全国大学生数学建模竞赛章程》和《全国大学生数学建模竞赛参赛规则》(以下简称为“竞赛章程和参赛规则”，可从全国大学生数学建模竞赛网站下载)。

我们完全明白，在竞赛开始后参赛队员不能以任何方式(包括电话、电子邮件、网上咨询等)与队外的任何人(包括指导教师)研究、讨论与赛题有关的问题。

我们知道，抄袭别人的成果是违反竞赛章程和参赛规则的，如果引用别人的成果或其他公开的资料(包括网上查到的资料)，必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。

我们郑重承诺，严格遵守竞赛章程和参赛规则，以保证竞赛的公正、公平性。

如有违反竞赛章程和参赛规则的行为，我们将受到严肃处理。

我们授权全国大学生数学建模竞赛组委会，可将我们的论文以任何形式进行公开展示(包括进行网上公示，在书籍、期刊和其他媒体进行正式或非正式发表等)。

我们参赛选择的题号是(从A/B/C/D中选择一项填写)：B我们的报名参赛队号为(8位数字组成的编号)：所属学校(请填写完整的全名)：参赛队员(打印并签名)：1.2.3.指导教师或指导教师组负责人(打印并签名)：(论文纸质版与电子版中的以上信息必须一致，只是电子版中无需签名。

以上内容请仔细核对，提交后将不再允许做任何修改。

如填写错误，论文可能被取消评奖资格。

)日期：年月日赛区评阅编号(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)：全国评阅编号(由全国组委会评阅前进行编号)：创意平板折叠桌的设计摘要随着人类思维的不断进步，极具创意的作品也层出不穷。

本文对创意平板折叠桌进行分析，运用三维坐标对不同平板折叠桌的结构进行描述。

桌子外形由直纹曲面构成，桌面近似圆形，桌腿分成两组，每组各用一根钢筋将木条连接，钢筋两端分别固定在桌腿各组最外侧的两根木条上。

随着铰链的活动，折叠桌可以平摊成一张平板，折叠时，沿木条有空槽以保证滑动的自由度。

(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

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年美国大学生数学建模大赛赛题

2014 MCM ProblemsPROBLEM A: The Keep-Right-Except-To-Pass RuleIn countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employ a rule that requires drivers to drive in the right-most lane unless they arepassing another vehicle, in which case they move one lane to the left,pass, and return to their former travel lane.Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wish to examine tradeoffs between traffic flow and safety, the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement. Is this rule effective in promoting better traffic flow? If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flow, safety, and/or other factors that you deem important.In countries where driving automobiles on the left is the norm, argue whether or not your solution can be carried over with a simple change of orientation, or would additional requirements be needed.Lastly, the rule as stated above relies upon human judgment for compliance. If vehicle transportation on the same roadway was fully under the control of an intelligent system – either part of the road network or imbedded in the design of all vehicles using the roadway – to what extent would this change the results of your earlier analysis?PROBLEM B: College Coaching LegendsSports Illustrated, a magazine for sports enthusiasts, is looking for the “best all time college coach” male or female for the previous century. Build a mathematical model to choose the best college coach or coaches (past or present) from among either male or female coaches in such sports as college hockey or field hockey, football, baseball or softball, basketball, or soccer. Does it make a difference which time line horizon that you use in your analysis, i.e., does coaching in 1913 differ from coaching in 2013? Clearly articulate yourmetrics for assessment. Discuss how your model can be applied in general across both gen ders and all possible sports. Present your model’s top 5 coaches in each of 3 different sports.In addition to the MCM format and requirements, prepare a 1-2 page article for Sports Illustrated that explains your results and includes a non-technical explanation of your mathematical model that sports fans will understand.。

2014美国数学建模B题数据大学教练-曲棍球

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2015数学建模美赛试题

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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。