数学建模最佳大学教练.pdf

Team 28586 Page 1 of 24 For office use only T1 T2 T3 T4 Team Control Number 28586 Problem Chosen B For office use only F1 F2 F3 F4 2014 Mathematical 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 include the name of your school advisor or team members on this page Summary To evaluate a coach comprehensively many aspects should be considered especially the performance of his team and personal ability However not every aspect is equally important and the weight of each aspect can be different during different time periods Therefore a comprehensive evaluation model which can be applied to both male and female coaches is created it includes evaluation indexes index weights and evaluation criteria We selected the first level indexes after comparing many scholarly resources and determined the second level indexes and third level indexes with further analysis At the end with the help of experts and the support of statistical evidence we selected indexes which can reflect the coaches performance comprehensively These indexes include the performance of a coach s team the coach s personal ability and so on For index weights based on our research we calculated the ratio of importance of each index and got the judgment matrix Then we obtained the results by using Analytic Hierarchy Process and checked our results at the end to ensure the accuracy Referring to scholarly resources and experts we preset evaluation criteria of rating evaluation remarks and determined membership function to quantify the evaluation criteria of college coaches After obtaining the scores of many excellent coaches and using Fuzzy Analytic Estimation to process the data our model selected top 5 coaches based on their evaluation scores During the process of evaluation we combined qualitative analysis and quantitative analysis to ensure fairness and operability Regarding the influence of time line we calculated by using index weights of corresponding time period in order to do further comparison After calculation we analyzed index weights by using Sensitivity Analysis to confirm the stability of our weights within certain range Our model can apply to various coaches who live in different time periods accurately calculate the evaluation score and ultimately select the best coaches Team 28586 Page 2 of 24 1 Introduction With the rapid development of the modern competitive sports and the intensive competition a comprehensive system has been developed Thus a college coach a strong promoter of the development of competitive sports has a more and more important role to play Generally a coach acts like a guiding light in one athlete s athletic career basically determines the characteristics of the athlete s long term career plan develops training programs and innovates training methods and means It is never an overstatement that a coach holds the grasp of athlete s whole training process and is the key factor influencing the performance of athletes Moreover many experts believe that the abilities of coaches are various Therefore the evaluation of coaches is supposed to be comprehensive and objective and should not just stay in the obtained results of the coached athletes or teams In short developing a comprehensive integrated ability evaluation system is indispensable 1 1 Problem restatement and abstraction Our goal is to find a relatively objective and comprehensive coach evaluation mechanism and discuss the impacts of time and sport event on the model In this question our research range is set at the coaches of Universities To analyze problem deeply we finds that the key is to quantify the overall quality of one university coach achieving the purpose of comparing between two coaches We refer to the existing coach evaluation system and take athlete ranking system as an analogy to turn all aspects of college coaches into quantified statics Then we build models considering the degree of influence of each factor to get the result of one college coach s composite score To calculate the final composite score we must address three key issues a Determining evaluation factors b Finding the influence each factor has on the final outcome c Getting the quantized scores of each coach on each factor Our model is established during the process of searching for optimal solutions of these three issues 1 2 Introduce of model To achieve this goal we referred to a large number of documents for various coach evaluation methods in several decades We found that recent coaches evaluation methods in most countries and among almost all levels generally take the achievements of the athletes or teams that the coach leads unilaterally as the only factor After discussion we believe that the achievements of the athletes or teams that the coach leads is indeed one important aspect to evaluate but the other abilities of coaches cannot be discarded without serious consideration For example sportsmanship is a very important aspect one coach who lacks sportsmanship cheats in games or uses stimulant drug can hardly be considered a good coach no matter how big achievements he or she has accomplished So in our model we include more personal qualities that are directly shown from the coaches including Team 28586 Page 3 of 24 self achievement basic cultural qualities and professional training capabilities etc Thus the innovation of our model breaks the previous evaluation system that only based on the achievements of the leading team of coaches and become more comprehensive scientific fair and credible As seen in Figure1 Figure 1 2 Basic Hypothesis All coaches that are evaluated lead teams that have attended similar levels of competitions In doing so we can use the team s percentage of winning as the factor of evaluation 3 Definitions and Variables F Final evaluation score of one coach F1 Subjective evaluation score of one coach F2 Objective evaluation score of one coach U1 Weight of self achievement U2 Weight of leading team achievement U3 Weight of Basic cultural qualities U4 Professional intelligent quality u11 Weight of self achievement u21 Weight of leading team winning rate u22 Weight of leading team winning games Team 28586 Page 4 of 24 u23 Weight of coaching years u31 Weight of medical knowledge in sports u32 Weight of Knowledge of sport physiology and psychology u33 Weight of basic training theory u34 Weight of knowledge of sports management and Sports Sociology u35 Weight of engineering sport science knowledge u36 Weight of philosophy and social thinking science u41 Weight of training innovation capability u42 Weight of Special training capacity u43 Weight of scientific research capacity max The largest judgment matrix eigenvalue r The rank of judgment matrix CIk CRk RIk Parameters in consistence test V The set of remarks u x Membership function Ri The single factor judgment matrix to evaluate Ai The weight of membership subset of D Fuzzy judgment matrix B1 B2 B3 The first level evaluation vector B1i evaluation vector for coaches from athletes B2i evaluation vector for coaches from managers B3i evaluation vector for coaches from experts S1 S2 S3 Subjective evaluation scores for coaches from athletes managers and experts fB x Borda value of index xPi The weight of index under situation y Total number of cases where is in preference in set A 4 Model for college coach evaluation 4 1 Foundation of evaluation indexes 4 1 1 Preset evaluation indexes Evaluation indexes are specific and behavioral main factors that can reflect the essential characteristics of the evaluated Our primary evaluation indexes are determined by referring to a large number of related materials and are based on Theoretical analysis 4 1 1 1 The first level indexes The first level index is the logic basic point of factor split in evaluation system Team 28586 Page 5 of 24 We can further analyze only after deciding the characteristic and quantity of the first level index according to the purpose of the evaluation Because we want to thoroughly evaluate the coaches abilities we determine ability as the first level index 4 1 1 2 Divide the second level indexes According to the principle of choosing indexes and the usual method of building index system and based on the previously determined first level index we listed indexes that can be used At the end we decided 6 second level index As Seen in Table 1 Level Index First Capability Second Self achievement Leading team achievement Morality Publicity Basic cultural qualities Professional intelligent quality Table 1 4 1 1 3 Preset the third level indexes Developing third level index needs to be based on the second level index reflect the purpose of evaluation further analyze listed indexes and evaluate whether listed indexes can comprehensively reflect the abilities of coaches If not then a change to the list of indexes or alternative names and meanings are needs Based on the principle of choosing indexes theoretical analysis and references we decided 21 third level indexes 4 1 2 Modify evaluation indexes Among the factors analyzed by the determined indexes some can accurately reflect the object that has been analyzed some cannot some are major factors and some are minor factors It is difficult to avoid factors intercross repeat and contradict with each other Therefore we found it necessary to synthesize and filter various indexes in order to reach the purpose of effective evaluation Based on the principle and purpose of evaluation we decided to delete the subjective evaluations indexes whose weights are low in order to reduce the work We chose Borda s Function to attach the weights to the indexes While there are several levels of intensity we can choose and the indexes were decided to the order of first second by the deciders we attached different weights to them The function is following fB x isxPiy 4 1 3Determin evaluation indexes statistics majorization For third level indexes that have been determined by experts we used Factor Analytic Method to optimize the model Our purpose is to improve third level indexes and optimize second level indexes in order to make index system more concise more Team 28586 Page 6 of 24 effective and more comprehensible And we decided 13 third level indexes at last 4 2 Determine the system of index weights We used Analytic Hierarchy Process to calculate the weight of the index system Calculation of the index weights of experts are the same as the calculation of the index weight of athletes We used the 1 9 rating scale method to construct the pairwise comparison matrix 4 2 1 Build the Hierarchical structure model Our model includes three hierarchies Among them first level index is ability and there are 4 second level indexes and 13 third level indexes 4 2 2 Calculate the weight of evaluation indexes 4 2 2 1 Calculation of second level indexes By using Analytic Hierarchy Process AHP to build judgment matrixes of the second level indexes by using the analysis of experienced experts as reference and the 1 9 rating scale method we decided the importance ratio of each two factors The meaning of each scale is seen in Table 2 fij Meaning 1 Element i is no more important than element j 3 Element i is slightly more important than element j 5 Element i is clearly more important than element j 7 Element i is much more important than element j 9 Element i is extremely more important than element j 2 4 6 8 The levels between every two set levels fij 1 fji The ratio of the importance of i and j is reciprocal with the ratio of the importance of j and i Table 2 The comprehensive judgment matrix is seen in Table 3 A U1 U2 U3 U4 U1 1 1 1 2 1 3 U2 7 1 7 3 2 U3 2 1 3 1 1 2 U4 3 1 2 2 1 Table 3 Using the comprehensive judgment matrix we calculated and got the largest eigenvalue and the standard eigenvector Here are the steps of calculating Normalize each column of the judgment matrix A we got Team 28586 Page 7 of 24 1 1 7 2 3 1 1 1 7 1 3 1 2 1 2 1 2 3 1 2 1 3 1 3 2 1 2 1 7 1 7 2 3 1 7 1 1 7 1 3 1 2 3 1 2 3 1 2 2 1 3 2 1 2 1 2 1 7 2 3 1 3 1 1 7 1 3 1 2 1 1 2 3 1 2 1 2 1 3 2 1 2 1 3 1 7 2 3 1 2 1 1 7 1 3 1 2 2 1 2 3 1 2 1 1 3 2 1 2 1 0 077 0 0720 0770 087 0 5380 5060 4620 522 0 1540 1690 1540 130 0 2310 2530 3070 261 After normalize we add all the element of each row 0 313 2 028 0 607 1 052 And normalize the matrix above we can get 0 078 0 507 0 152 0 263 then W 0 0780 5070 1520 263 which is the eigenvector Calculate the largest eigenvalue max 11 1 2 1 3 7 1 7 32 2 1 3 1 1 2 3 1 2 21 0 078 0 507 0 152 0 263 0 749 1 600 0 609 1 055 max 1 4 0 749 0 078 1 600 0 507 0 609 0 152 1 055 0 263 4 194 Consistency check CIk max k 1 r 1 CIk 4 194 4 4 1 0 065 Team 28586 Page 8 of 24 CRk CIk RIk 0 034 The value of RIk we refer to the table of values of coincidence indicator RI The judgment matrix meets CRk CIk RIk 0 1 we believe that matrix A has a satisfactory consistency the eigenvector we got could be used as the indexes weights the table of the value of indexes weights is seen as Table 4 Second level indexes self achievement leading team achievement Basic cultural qualities Professional intelligent quality Weights 0 078 0 507 0 152 0 263 Table 4 4 2 2 2Calculation of third level indexes By using the same method of building third level indexes judgment matrix based on second level indexes calculate the largest eigenvalue and corresponding standard eigenvector will lead to the weight of each third level index Using the same method listed above we can also calculate the index weights of experts and managers After calculating the index weights of three major evaluated targets we need to calculate the weights of three evaluation subsystems also using the same method above The weights are shown in Table 5 Evaluation subjective Athletes Experts Managers Weights 0 60 0 25 0 15 Table 5 4 3 Determine the system of evaluation criteria In evaluation criteria one of the important elements is the ability of a coach It is the basic element of evaluation and the object of reference 4 3 1The formulation of evaluation criteria 4 3 1 1 Quantified criteria Means to divide the third level index into many elements decided a full mark according to the importance of each index and also the highest mark of each index The sum of marks that are acquired by an evaluated target is its final mark 4 3 1 2 Scale criteria Scale standard is a standard that is divided into different categories Scale standard uses the average number as the basic point and standard score as the mark When evaluate we only need to convert each index s original score into standard score and then we can label each index on the standard score scale In Fuzzy Comprehensive Estimation we adopted direct descriptive words as Team 28586 Page 9 of 24 evaluation criteria They are The scores of Excellent Good Middle Poor Very poor are seen in Table 6 The set V of remarks Excellent Good Middle Poor Very poor Score S 95 80 70 60 45 Table 6 4 4 1Build the sets of the evaluation factors Based on previously established index system and the weights of every index assume evaluating a coach of a athletic team In order to briefly explain the process of Fuzzy Comprehensive Estimation in here we only talk about two sets of the evaluation factors U 1 U2 U3 U4 Self achievement Leading team achievement Basic cultural qualities Professional intelligent quality U2 u21 u22 u23 Leading team winning rate Leading team winning games Coaching years U3 u31 u32 u33 u34 u35 u36 Medical knowledge in sports Knowledge of sport physiology and psychology Basic training theory Knowledge of sports management and Sports Sociology Engineering sport science knowledge philosophy and social thinking science U4 u41 u42 u43 Training innovation capability Special training capacity Scientific research capacity Then based on previous calculation determine the set of index weights u11 1 u21 0 297 u22 0 539 u23 0 164 u31 0 230 u32 0 090 u33 0 310 u34 0 070 u35 0 090 u36 0 210 u41 0 350 u42 0 460 u43 0 190 4 4 2 Determine the membership functions Respecting the diversity of evaluated targets we evaluated coaches abilities by the opinions of athletes managers and experts And we used the percent of the Team 28586 Page 10 of 24 number of people of each level to decide the degree of membership But what troubled us is that we just invited few experts so we let the experts to score the indexes to decide the degree of membership of each indexes by using the functions of degree of membership Here are the functions of degree of membership Excellent u x 0 x 80 95 80 1 0 x 80 80 x 90 90 x 100 Good u x x 70 80 70 95 x 95 80 0 70 x 80 80 x 95 others Middle u x x 60 70 60 80 x 80 70 0 60 x 70 70 x 80 others Poor u x x 45 60 15 70 x 70 60 0 45 x 60 60 x 70 others Very poor u x 1 60 x 60 45 0 0 x 45 45 x 60 60 x 100 We can change the scores into the levels by using the functions of degree of membership in order to convenient us to deal with them 5 Testing and Results 5 1 Collection of data We invited 30 athletes 10 managers and 5 experts to evaluate ten college coaches in America who have done outstanding jobs in the fields of college soccer basketball baseball Knowing coaches who are evaluated athletes and managers are required to mark on corresponding level Each element is only allowed to select one level Two experts scored each index then converted it into scales then combine all completed charts Below is the result of one of the coaches Harry Statham His original scores are seen in Table 7 8 9 10 The questionnaires is seen in the appendix Team 28586 Page 11 of 24 Item Evaluation index Score Second level Third level Excellent Good Middle Poor Very poor U1 Self achievement u11 0 333 0 300 0 2 00 0 133 0 034 U3 Basic cultural quality u31 0 200 0 267 0 3 33 0 133 0 067 u32 0 067 0 167 0 4 00 0 200 0 167 u33 0 500 0 233 0 1 33 0 067 0 067 u34 0 067 0 267 0 0 33 0 233 0 100 u35 0 033 0 100 0 1 67 0 533 0 167 u36 0 100 0 167 0 4 00 0 233 0 100 U4 Professional