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