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Evaluation System for College Coaching Legends

Summary
In order to evaluate the performance of a coach, we describe metrics in five
aspects: historical record, game gold content, playoff performance, honors and
contribution to the sports. Moreover, each aspect is subdivided into several secondary
metrics. Take playoff performance as example, we collect postseason result (Sweet
Sixteen, Final Four, etc.) per year from NCAA official website, Wikimedia and so on.
First, Analytic Hierarchy Process (AHP) Model is established to determine the
weight of each metric to coaches’ evaluation grade. All metrics are adequately filled
into the three-hierarchy structure, and then we obtain the metric weight based on
which evaluation grade is calculated. Second, Fuzzy Synthetic Evaluation (FSE) is
built to overcome weakness of excess subjective factors in AHP. This model takes
data processing by membership function to generate fuzzy matrix. After that, entropy
method and linear weighted method are applied to obtain evaluation grade.
To evaluate the accuracy of the two models, hit score is defined. It is supposed to
reflect the difference between our results and standard rankings from several
authorities such as ESPN and Sporting News. Take NCAA basketball as a case study,
AHP receives 78.77 hit score while FSE gets 81.81, which indicates that FSE
performs better than AHP. Afterwards, Aggregation Model (AM) can be developed by
combining the two models based on hit score. The top 5 college basketball coaches,
in turn, are John Wooden, Mike Krzyzewski, Adolph Rupp, Dean Smith and Bob Knight.
Time line horizon does make a difference. According to turning points in NCAA
history, we divide the previous century into six periods with different time weights
which lead to the change of ranking. We apply our model into college women’s
basketball only to find that genders do not matter. Model proves to be efficient in
other sports. The ranking of college football is: Bear Bryant, Knute Rockne, Tom
Osborne, Joe Paterno , Bobby Bowden, and the top 5 coaches in college hockey are
Bob Johnson, Red Berenson, Jack Parker, Jerry York, Ron Mason.
We conduct sensitivity analysis on FSE to find best membership function and
calculation rule. Sensitivity analysis on aggregation weight is also performed. It
proves AM performs better than single model. As a creative use, top 3 presidents
(U.S.) are picked out: Abraham Lincoln, George Washington, Franklin D. Roosevelt.
At last, the strength and weakness of our mode are discussed, non-technical
explanation is presented and the future work is pointed as well.

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I. Introduction ............................................................................................. 3
1.1 Problem Background......................................................................................... 3
1.2 Previous Research ............................................................................................ 3
1.3 Our Work........................................................................................................... 3
II. Symbols, Definitions and Assumptions ................................................ 4
2.1 Symbols and Definitions.................................................................................... 4
2.2 General Assumptions ........................................................................................ 5
III. Articulate our metrics ............................................................................. 5
3.1 Specify evaluation norms .................................................................................. 5
3.2 Collect data ....................................................................................................... 8
3.3 Preprocess data ................................................................................................ 9
IV. Two models for coach ranking............................................................. 10
4.1 Model I: Analytic Hierarchy Process (AHP) ..................................................... 10
4.1.1 The three-hierarchy structure ................................................................... 10
4.1.2 Obtain the index weight ............................................................................ 10
4.1.3 Results & analysis .................................................................................... 11
4.2 Model II: Fuzzy Synthetic Evaluation (FSE) .................................................... 12
4.2.1 Quantify grades in the five aspects........................................................... 12
4.2.2 Determine membership functions ............................................................. 13
4.2.3 Determine the weights using entropy method .......................................... 14
4.2.4 Results & analysis .................................................................................... 15
V. Models Combination ............................................................................. 15
5.1 Evaluation of individual model ......................................................................... 15
5.2 Aggregation Model .......................................................................................... 17
5.3 Results & analysis ........................................................................................... 17
VI. Extend Our Models................................................................................ 18
6.1 Genders do not matter .................................................................................... 18
6.2 Time factor does make a difference ................................................................ 19
6.2.1 Why time factor matters?.......................................................................... 19
6.2.2 How time factor matters?.......................................................................... 19
6.2.3 What is the variation tendency? ............................................................... 23
6.3 Model also works in other sports ..................................................................... 23
VII. Further discussion ................................................................................ 25
7.1 Sensitive Analysis on FSE .............................................................................. 25
7.1.1 Vary Membership function ........................................................................ 25
7.1.2 Vary calculation rule ................................................................................. 27
7.2 Sensitive Analysis on Aggregation weight....................................................... 28
7.3 Explore: Evaluating Best President ................................................................. 29
VIII. Strength and Weakness ................................................................. 30
IX. Non-technical Explanation ................................................................... 30
X. Future work ............................................................................................ 32
XI. References ............................................................................................. 32
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I. Introduction
1.1 Problem Background
Sports Illustrated is an American sports media franchise owned by media
[1]
conglomerate Time Warner . This magazine is looking for the “best all time college
coach” male or female for the previous century. The best college coach or coaches
can be from among either male or female coaches in different fields, such as college
hockey or field hockey, football, baseball or softball, basketball, or soccer.
We face mainly four problems:
 Articulate our own metrics and build a mathematical model;
 Set up the evaluation system for the performance of the model.
 Discuss how our model can be applied with time factor or across both genders
and all possible sports;
 Analyze the influences of the parameters, then discuss whether your model could
be applied into wide fields.

1.2 Previous Research

Some magazines or websites that focus mainly on college sports have ranked the
top college coaches of different sports. For example, rivels.com has made a basket-
ball power rankings [2] which shows the top 25 coaches of college basketball.
Considering the best college coaches is an evaluation problem. There are some
models which can solve such problem. One is the Analytic hierarchy process (AHP),
[3]
which was developed by Thomas L. Saaty in the 1970s. The AHP provides a
comprehensive and rational framework for structuring a decision problem, for
representing and quantifying its elements, for relating those elements to overall goals,
[4]
and for evaluating alternative solutions . Another is the Fuzzy Synthetic Evaluation
Model. Fuzzy mathematics forms a branch of mathematics related to fuzzy set theory
[5]
and fuzzy logic . It started in 1965 after the publication of Lotfi Asker Zadeh's
seminal work Fuzzy sets [6].

1.3 Our Work

In this paper we determine the best college coaches from among either male or
female coaches in different sports. In Section 2, we provide the terminology definitions
and assumptions that will be utilized in the rest of the paper. In Section 3, we give the
definitions of evaluation standard and specific evaluation norms which we used in our
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models, and show some of the data we have collected. In Section 4, we build two
mathematical models to choose the best college coaches, and Section 5 considers
combination of the two models mentioned above. In Section 6 we extend our models
and take time, genders and types of sports into consideration. Section 7 provides
further discuss of our models. In Section 8, we provide an overview of our approach
and give a non-technical explanation of our models that sports fans will understand.
Section 9 shows some work we can do in the future.

II. Symbols, Definitions and Assumptions

2.1 Symbols and Definitions
 Symbols for evaluation norms:
Symbol Definition
𝒂𝒂𝒊𝒊 wins for the 𝑖𝑖𝑡𝑡ℎ year
𝒃𝒃𝒊𝒊 losses for the 𝑖𝑖𝑡𝑡ℎ year
𝐑𝐑 the average SRS
𝐎𝐎 the average SOS
𝒏𝒏𝒌𝒌 the times for each class of playoff
𝓴𝓴𝒊𝒊 the weight of each award
𝓬𝓬𝒊𝒊 point for each aspect of contribution
 Symbols for Analytic Hierarchy Process:
Symbol Definition
𝐀𝐀 the judging matrix
𝝀𝝀𝒎𝒎𝒎𝒎𝒎𝒎 the greatest eigenvalue of matrix A
𝐂𝐂𝐂𝐂 the indicator of consistency check
𝐂𝐂𝐂𝐂 the consistency ratio
𝐑𝐑𝐑𝐑 the random consistency index
𝐂𝐂𝐂𝐂 the weight vector for criteria level
𝐀𝐀𝐀𝐀 the weight vector for alternatives level
𝒀𝒀𝟏𝟏 the evaluation grade for model I
 Symbols for Fuzzy Synthetic Evaluation:
Symbol Definition
𝑿𝑿𝒊𝒊 the grades for each aspect
𝝁𝝁𝒋𝒋 �𝑿𝑿𝒊𝒊𝒊𝒊 � the membership function
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𝑿𝑿𝒇𝒇 the fuzzy matrix

𝒑𝒑𝒊𝒊𝒊𝒊 the characteristic weight
𝒆𝒆𝒋𝒋 the entropy for the 𝑗𝑗𝑡𝑡ℎ evaluation grade
𝑬𝑬𝐖𝐖 the weight vector in entropy method
𝒀𝒀𝟐𝟐 the evaluation grade for model II
 Symbols for Aggregation Model:
Symbol Definition
𝐃𝐃 the average offset distance
𝑾𝑾𝟏𝟏 the weight for model I
𝐘𝐘 the evaluation grade for aggregation model

2.2 General Assumptions

 The elements that we already have taken into consideration play a vital role in the
evaluation.
 The ignored elements of coach do not influence the ranking.
 The data that we have collected is enough and accurate and the quantification is
correct.
 There exists objective and accurate ranking for coaches, and the rankings from
selected media could reflect the accurate ranking to some extent.

III. Articulate our metrics

3.1 Specify evaluation norms
[9]
As for the evaluation standard for players, there are mainly five aspects that
count: strength, speed, skill, defense and attack. Similarly a coach could be evaluated
from following five aspects: historical record, game gold content, play-off performance,
honors and contribution to the sports. What follows in the chapter will hammer at
accounting for the five aspects.
 Historical record: The team’s record undoubtedly accounts for the largest
proportion in the coach evaluation. According to the mainstream statistic indexes
for the team record, wins and losses are most notable. The team’s historical
record could directly reflects the coaching ability.
The total wins a could be calculated as follows:

a = � 𝑎𝑎𝑖𝑖 (3.1)
𝑖𝑖
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-Where 𝑎𝑎𝑖𝑖 denotes wins for the 𝑖𝑖𝑡𝑡ℎ year.

The total loses b could be calculated as follows:

b = � 𝑏𝑏𝑖𝑖 (3.2)
𝑖𝑖

-Where 𝑏𝑏𝑖𝑖 denotes losses for the 𝑖𝑖𝑡𝑡ℎ year.

 Game gold content: If all wins are produced during the fights with weak teams,
apparently the wins could not illustrate the real coaching ability. At the same time,
the average point difference also makes a difference. It reflects the coaching style
that whether a coach is conservative or radical. To illustrate the upon two points,
we choose the following two norms:
 Simple Rating System (SRS) [8]
: The simple rating system works by first
finding how many points, on average, a team wins/loses by. For each game,
the point differential is then weighted based on how much better or worse than
average their opponent's point differential is.
Let R denote the total SRS, then it could be calculated as follows:
∑𝑖𝑖 𝑆𝑆𝑆𝑆𝑆𝑆𝑖𝑖
R= (3.3)
𝑡𝑡
-Where 𝑆𝑆𝑆𝑆𝑆𝑆𝑖𝑖 denotes the SRS value for the 𝑖𝑖𝑡𝑡ℎ year, t denotes the number
of the years.
 Strength of Schedule (SOS) [8]: In sports, strength of schedule (SOS) refers to
the difficulty or ease of a team's/person's opponent as compared to other
teams/persons. This is especially important if teams in a league do not play
each other the same number of times.
Let O denote the total SOS, then it could be calculated as follows:
∑𝑖𝑖 𝑆𝑆𝑆𝑆𝑆𝑆𝑖𝑖
O= (3.4)
𝑡𝑡
-Where 𝑆𝑆𝑆𝑆𝑆𝑆𝑖𝑖 denotes the SOS value for the 𝑖𝑖𝑡𝑡ℎ year, t denotes the number
of the years.
 Playoff performance: Generally, during the regular season, teams play more
games in their division than outside it, but the league's best teams might not play
[11]
against each other in the regular season . Therefore, in the postseason any
group-winning team is eligible to participate thus making the playoff performance
extremely important in the coach evaluating [12]. For college basketball in U.S., the
playoff performance could be divided as follows:
 First round: The team is eliminated in the first round.
 Second round: The team is eliminated in the second round.
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 Sweet sixteen: The last sixteen teams remaining in the playoff tournament
 Final four: The last four teams remaining in the playoff tournament.
 Runner-up: The team loses in the finals.
 Champion: The team wins in the finals.
To quantify the aspect, we count the number of times for each class using the
symbol 𝑛𝑛𝑘𝑘 . Let a binary variable 𝑚𝑚𝑘𝑘𝑘𝑘 denote whether the team get the 𝑘𝑘 𝑡𝑡ℎ (for first
round k = 1, champion k = 7) class in the 𝑖𝑖𝑡𝑡ℎ year. Thus 𝑛𝑛𝑘𝑘 could be calculated as
follows:

𝑛𝑛𝑘𝑘 = � 𝑚𝑚𝑘𝑘𝑘𝑘 (3.5)

𝑖𝑖

 Honors: There are various awards in this field which make up the honors of the
coach, at the same time, the basketball hall of fame and college basketball hall of
fame [7] are also honors. To quantify the honor, we count the times of main award
such as the Naismith College Coach of the Year, Basketball Times National
[9]
Coach of the Year and so on. Different awards have different gold content. To
determine the weight of each award (𝓀𝓀𝑖𝑖 ), we collect its time period, based on
which weights are designated. Let ℋ denote the total weights of all the awards
a coach has got:

ℋ = � 𝓀𝓀𝑖𝑖 (3.6)
𝑖𝑖

Namely ℋ reflects the how much honor a coach have ever obtained.
 Contribution to sports: This concept covers a wide range. In order to quantify
the contribution, we divide the contribution into five parts:
 Star players: Evaluate the number of the star players the coach have trained.
 Coaching age: When the coaching career start and how long does it last.
 Tactical Innovation: Have the coach invented tactical innovation?
 Performance in international competitions: Have the coach ever fight in the
international competitions? Then How many gold or silver medals?
 Popularity: The number of the results when search its name in Google.
We give 𝒸𝒸𝑖𝑖 points for each aspect above: 0 for mediocre, 1 for good, 2 for
excellent. Then add the points up to form the final grade 𝒞𝒞 in this aspect (the full
mark is 10):

𝒞𝒞 = � 𝒸𝒸𝑖𝑖 (3.7)
𝑖𝑖
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A figure is prepared to conclude the evaluation norms above. (See figure)

Figure 3.1 First level evaluation norms

Figure 3.2 Second level evaluation norms

3.2 Collect data

We use men’s college basketball that will be utilized in the following models
discussion as an example, and collect relative data from the Internet. We choose
those 70 coaches who were in the list of the National Collegiate Basketball Hall of
[7]
Fame , because those coaches had gained tremendous glory and are more
competitive to be chosen as the best coaches. What’s more, we select other 5 college
coaches who are not in the Hall of Fame but still made significant contributions.
Searching from the sports-reference.com [8], a website that can provide specific
data about coaches, we can find relative data of our specific evaluation norms.
Combining those data with the statistics we search from the Wikipedia, we finally
conclude the relative statistics of those 75 college coaches and list them in a form.
Here we give statistics of 10 coaches as an example.
In the following table, “FR”, “SR”, “SS”, “EE”, “FF”, “RU”, and “CH” refer to “First
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Round”, “Second Round”, “Sweet Sixteen”, “Elite Eight”, “Final Four”, “Runner-Up”,
and “Campion”, respectively.
Name from to year win lose SRS SOS FR SR SS EE FF RU CH

Jim 1905 1995 48 719 259 15.81 7.27 5 8 11 2 1 2 1

Boeheim
Jim 1972 2001 40 877 382 12.64 4.74 5 5 4 5 1 0 3

Calhoun
Larry 1979 2013 9 210 83 13.08 5.95 0 3 1 0 1 1 1

Brown
…
Mike 1975 2013 39 975 302 20.16 8.78 2 6 6 2 3 4 4

Krzyzewski
Table 3.1 the relevant data of the “best college coaches” candidates
We also collect college basketball coaching record about each season of every
candidate. Here we take Larry Brown as an example.
Season win lose SRS SOS AP Pre AP High AP Final Result
1979-80 22 10 15.67 6.1 8 7 — NCAA Runner-up

1980-81 20 7 14.89 5.26 6 3 10 NCAA Second Round

1983-84 22 10 9.76 5.86 17 17 — NCAA Second Round

1984-85 26 8 11.84 6.27 19 9 13 NCAA Second Round

1985-86 35 4 23.18 10.42 5 2 2 NCAA Final Four

1986-87 25 11 13.36 7.73 8 6 20 NCAA Sweet Sixteen

1987-88 27 11 15.71 10.77 7 7 NCAA Champions

2012-13 15 17 -0.59 -1.33 — — — —

2013-14 18 5 13.88 2.45 — — — —
Table 3.2 the college basketball coaching record about each season of Larry Brown

3.3 Preprocess data

When we collect data from Internet, we notice that some data is missing due to
age. Given the fact, we have to preprocess the data from Internet. As for the data for
college basketball, SRS&SOS are sometimes missing. The solution adopt by us is
filling the data mainly based on interpolation according to the ranking generated by
the other metrics.
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IV. Two models for coach ranking

4.1 Model I: Analytic Hierarchy Process (AHP)
When we try to obtain the weight of mainly five aspects as the first class index and
the weight of several second class index, subjective judgment is ill-considered. So we
[4]
choose the Analytic Hierarchy Process (AHP) as the way to conform the weighting
coefficient of all the indicators in the evaluation system.

4.1.1 The three-hierarchy structure

The three hierarchy structure which contains criteria level and alternatives level is
shown in following table.
Goal Criteria Alternatives
The influence of coach Historical Record Wins
Losses
Game Gold Content SRS
SOS
Playoff Performance First Round
…
Champion
Honors Different Awards
Hall of Fame
Contribution to sports Star Player
Coaching Age
Tactical Innovation
International Games
Popularity
Table 4.1 the three hierarchy structure of our model

4.1.2 Obtain the index weight

 Determine the judging matrix
We use the pairwise comparison method and one-nine method to construct
judging matrix A = (𝑎𝑎𝑖𝑖𝑖𝑖 ).
𝑎𝑎𝑖𝑖𝑖𝑖 ∗ 𝑎𝑎𝑘𝑘𝑘𝑘 = 𝑎𝑎𝑖𝑖𝑖𝑖 (4.1).
Where 𝑎𝑎𝑖𝑖𝑖𝑖 is set according to the one-nine method
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 Calculate the eigenvalues and eigenvectors

The greatest eigenvalue of matrix A is 𝜆𝜆𝑚𝑚𝑚𝑚𝑚𝑚 , and the corresponding eigenvector

is u = (𝑢𝑢1, 𝑢𝑢2, 𝑢𝑢3, … 𝑢𝑢𝑛𝑛 )𝑇𝑇 . Then we normalize the u by the expression:
𝑢𝑢𝑖𝑖
𝑥𝑥𝑖𝑖 = ∑𝑛𝑛 (4.2)
𝑖𝑖=0 𝑢𝑢𝑗𝑗

 Do the consistency check

The indicator of consistency check formula:
𝜆𝜆𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑛𝑛
CI = (4.3)
𝑛𝑛 − 1
Where n denotes the exponent number of matrix.
The expression of consistency ratio:
𝐶𝐶𝐶𝐶
CR = (4.4)
𝑅𝑅𝑅𝑅
As we have confirmed the weighting coefficient of all the indicators in the
evaluation system, now we quantify the importance of coaches.
𝐶𝐶𝐶𝐶𝑖𝑖 denotes the weight of 𝑖𝑖𝑡𝑡ℎ criteria level factor, where 𝐴𝐴𝐴𝐴𝑗𝑗 is the weight of
𝑗𝑗𝑡𝑡ℎ secondary critical level factor, and 𝐹𝐹𝑗𝑗 denotes the 𝑗𝑗𝑡𝑡ℎ secondary critical level
factor.
The evaluation grade 𝑌𝑌1 should be:
5 𝑚𝑚𝑚𝑚

𝑌𝑌1 = � 𝐶𝐶𝐶𝐶𝑖𝑖 ∗ � 𝐴𝐴𝐴𝐴𝑗𝑗 ∗ 𝐹𝐹𝑗𝑗 (4.5)

𝑖𝑖=1 𝑗𝑗=1

4.1.3 Results & analysis

Based on the data we have already collected in section 3.2, we solve the model
and obtain the following results:
 Judging matrix:
1 5 5/9 1 1
⎡ ⎤
1/5 1 1/9 1/5 1/5
⎢ ⎥
A = ⎢7/5 7 1 7/5 9/5⎥
⎢ 1 5 5/7 1 9/5⎥
⎣ 1 5 5/7 1/5 1 ⎦
 Weight vector of criteria level:
CW = [0.1996 0.0399 0.3093 0.2419 0.2092]
𝐶𝐶𝐶𝐶
For this level, CI=0.301, CR=0.0269 satisfying < 0.1.
𝑅𝑅𝑅𝑅

 Weight vector of alternatives level:

 Historical Record:A𝑊𝑊1 = [1.5 −0.5]
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 Game Gold Content:A𝑊𝑊2 = [0.75 0.25]

 Playoff Performance: A𝑊𝑊3 = [0.0079 0.0157 0.0315 0.126 0.252 0.5039]
𝐶𝐶𝐶𝐶
All of these eight vectors satisfy < 0.1.
𝑅𝑅𝑅𝑅

Finally, we can obtain the final rankings of the top ten college basketball coaches
using AHP models.
Rank Name Grade(𝒀𝒀𝟏𝟏 ) Rank Name Grade(𝒀𝒀𝟏𝟏 )
1 Mike Krzyzewski 0.8426 6 Roy Williams 0.5637
2 John Wooden 0.7334 7 Bob Knight 0.5479
3 Adolph Rupp 0.6048 8 Phog Allen 0.4788
4 Jim Boeheim 0.5985 9 Rick Pitino 0.4683
5 Dean Smith 0.5844 10 Lute Olson 0.4132
Table 4.2 the top ten college basketball coaches’ rankings
Conclusion:
 Analyzing the weight vector of criteria level, we can know that the highest value is
the weight of Playoff Performance, so coaches with better game results have
more chance in top ranking.
 SOS plays a less important role than SRS when determining the Game Gold
Content, and the weight of the Game Gold Content is the lowest value in criteria
value.

4.2 Model II: Fuzzy Synthetic Evaluation (FSE)

4.2.1 Quantify grades in the five aspects

[6]
Fuzzy set theory has been developed and extensively applied since 1965
(Zadeh, 1965). It was designed to supplement the interpretation of linguistic or
measured uncertainties for real-world random phenomena.
In section III, we have already articulate our metrics for ranking. Totally, there are
five aspects: historical record, game gold content, playoff performance, honors,
contribution to sports. Before using the fuzzy set theory, we calculate the grades
{𝑋𝑋1 , 𝑋𝑋2 , 𝑋𝑋3 , 𝑋𝑋4 , 𝑋𝑋5 }in each of the 5 aspect using the collected data.
 Calculation rule for historical record:
𝒶𝒶 denotes the number of wins, 𝒷𝒷 denotes the number of loses, 𝜆𝜆𝑤𝑤𝑤𝑤𝑤𝑤 denotes
the weight for single win, and 𝜆𝜆𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 denotes the weight for single lose.
𝑋𝑋1 = 𝜆𝜆𝑤𝑤𝑤𝑤𝑤𝑤 𝒶𝒶 − 𝜆𝜆𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝒷𝒷 (4.6)
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This formula provides a comprehensive assessment for wins and losses,

obviously 𝜆𝜆𝑤𝑤𝑤𝑤𝑤𝑤 > 𝜆𝜆𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 .
 Calculation rule for game gold content:
R denotes the value of SRS, O denotes the value of SOS，𝑂𝑂𝑚𝑚𝑚𝑚𝑚𝑚 denotes the
maximum value of SOS in the Strength of Schedule system.
𝑂𝑂
𝑋𝑋2 = 𝑅𝑅 �1 + � (4.7)
𝑂𝑂𝑚𝑚𝑚𝑚𝑚𝑚
If a coach has higher SRS, it will have higher grade in this aspect because his
team is always far ahead its opponents. At the same time, the higher SOS is, the
harder games are. So we let the SOS be an addition to SRS.
 Calculation rule for playoff performance:
𝑛𝑛𝑘𝑘 denotes the number of times for each class of the playoff results.
7

𝑋𝑋3 = � 2𝑘𝑘 𝑛𝑛𝑘𝑘 (4.8)

𝑘𝑘=1

The number of teams decreases exponentially with power of 2, thus making the
weight increase exponentially with power of 2. Sum up the times by designated weight
then we could finally draw 𝑋𝑋3 .
 Calculation rule for honors:
We have already count up the awards by weight (ℋ) in section III, so the formula:
𝑋𝑋4 = ℋ (4.9)
 Calculation rule for contribution to sports:
We have already give a final grade 𝒞𝒞 for this aspect in section III, so the formula
is:
𝑋𝑋5 = 𝒞𝒞 (4.10)
In conclusion, we form the following quantification rules:
𝑋𝑋1 = 𝜆𝜆𝑤𝑤𝑤𝑤𝑤𝑤 𝒶𝒶 − 𝜆𝜆𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝒷𝒷
⎧ 𝑂𝑂
⎪ 𝑋𝑋2 = 𝑅𝑅 �1 + �
⎪ 𝑂𝑂𝑚𝑚𝑚𝑚𝑚𝑚
7
(4.11)
⎨ 𝑋𝑋3 = � 2𝑘𝑘 𝑛𝑛𝑘𝑘
⎪ 𝑘𝑘=1
⎪ 𝑋𝑋4 = ℋ
⎩ 𝑋𝑋5 = 𝒞𝒞

4.2.2 Determine membership functions

[6]
A fuzzy set is defined in terms of a membership function which maps the
Team#28414 Page 14 of 33

domain of interest, e.g. concentrations, onto the interval [0, 1]. The shape of the
curves shows the membership function for each set. The membership functions
represent the degree, or weighting, that the specified value belongs to the set.
Let 𝑋𝑋𝑖𝑖𝑖𝑖 denote the 𝑋𝑋𝑗𝑗 value for the 𝑖𝑖𝑡𝑡ℎ coach and 𝑋𝑋𝑗𝑗(𝑚𝑚𝑚𝑚𝑚𝑚) denote the maximum
𝑋𝑋𝑗𝑗 value for all the coaches.
Here we use the normalization function as membership function:
𝑋𝑋𝑖𝑖𝑖𝑖
𝜇𝜇𝑗𝑗 �𝑋𝑋𝑖𝑖𝑖𝑖 � = (4.12)
𝑋𝑋𝑗𝑗(𝑚𝑚𝑚𝑚𝑚𝑚)

After calculating 𝜇𝜇𝑗𝑗 �𝑋𝑋𝑖𝑖𝑖𝑖 � for each of the 𝑋𝑋𝑖𝑖𝑖𝑖 , we could concluded the fuzzy
matrix 𝑋𝑋𝑓𝑓 (N denotes the total number of the coaches).
𝜇𝜇1 (𝑋𝑋11 ) … 𝜇𝜇1 (𝑋𝑋15 )
𝑋𝑋𝑓𝑓 = [ … 𝜇𝜇𝑗𝑗 �𝑋𝑋𝑖𝑖𝑖𝑖 � … ]
𝜇𝜇𝑁𝑁 (𝑋𝑋𝑁𝑁1 ) … 𝜇𝜇𝑁𝑁 (𝑋𝑋𝑁𝑁5 )

4.2.3 Determine the weights using entropy method

[13]
The principle of entropy method states that, subject to precisely stated prior
data (such as a proposition that expresses testable information), the probability
distribution which best represents the current state of knowledge is the one with
largest entropy. To use entropy method, there are mainly 5 steps:
 Calculate the characteristic weight 𝑝𝑝𝑖𝑖𝑖𝑖 for the 𝑖𝑖𝑡𝑡ℎ coach’s 𝑗𝑗𝑡𝑡ℎ evaluation grade
(𝑋𝑋𝑖𝑖𝑖𝑖 ) based on the normalized fuzzy matrix 𝑋𝑋𝑓𝑓 :
𝑋𝑋𝑓𝑓(𝑖𝑖,𝑗𝑗)
𝑝𝑝𝑖𝑖𝑖𝑖 = 𝑁𝑁 (4.13)
∑𝑖𝑖=1 𝑋𝑋𝑓𝑓(𝑖𝑖,𝑗𝑗)

 Calculate the entropy for the 𝑗𝑗𝑡𝑡ℎ evaluation grade:

𝑁𝑁
1
𝑒𝑒𝑗𝑗 = − � 𝑝𝑝𝑖𝑖𝑖𝑖 𝑙𝑙𝑙𝑙(𝑝𝑝𝑖𝑖𝑖𝑖 ) (4.14)
ln(𝑁𝑁)
𝑖𝑖=1

 Calculate the diversity factor for the 𝑗𝑗𝑡𝑡ℎ evaluation grade:

𝑔𝑔𝑗𝑗 = 1 − 𝑒𝑒𝑗𝑗 (4.15)
 Determine the weight for each evaluation grade:
𝑔𝑔𝑗𝑗
𝑤𝑤𝑗𝑗 = 5 (4.16)
∑𝑗𝑗=1 𝑔𝑔𝑗𝑗
 Determine the final score Y for each coach；
𝑌𝑌2 = W ∗ 𝑋𝑋𝑓𝑓 (4.17)
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4.2.4 Results & analysis

Characteristic weight, entropy, diversity factor and weight are shown as follows:
𝒑𝒑𝒊𝒊𝒊𝒊 𝑿𝑿𝟏𝟏 𝑿𝑿𝟐𝟐 𝑿𝑿𝟑𝟑 𝑿𝑿𝟑𝟑 𝑿𝑿𝟓𝟓 𝒀𝒀𝟐𝟐

John Wooden 0.04 0.06 0.15 0.13 0.08 0.8708

Mike Krzy 0.07 0.06 0.10 0.13 0.08 0.8629
Adolph Rupp 0.06 0.06 0.08 0.11 0.02 0.675
Dean Smith 0.06 0.06 0.08 0.04 0.05 0.609
Bob Knight 0.05 0.06 0.06 0.07 0.05 0.6052
Roy Williams 0.06 0.05 0.06 0.04 0.08 0.5872
Jim Boeheim 0.06 0.03 0.05 0.03 0.01 0.5864
Phog Allen 0.04 0.04 0.05 0.05 0.05 0.4874
Henry Iba 0.04 0.04 0.04 0.04 0.02 0.4664
Lute Olson 0.06 0.04 0.04 0.06 0.07 0.4538
𝒈𝒈𝒋𝒋 -0.99 -1.00 -1.04 -0.96 -0.93 0.4336
𝒘𝒘𝒋𝒋 0.18 0.16 0.23 0.20 0.22 0.8708
Table 4.3 the results for FSE
 The weights for each aspect is near to each other.
 The playoff performance (𝑋𝑋3 ) plays the most important role (with 0.23 weight) in
FSE evaluating.
 The coaches whose playoff performance is better will enjoy priority to some extent.
At the same time, the coaches who have amazing game gold content (with only
0.16 weight) might not outstand.

V. Models Combination
5.1 Evaluation of individual model
In order to evaluate the accuracy of our two individual models, average offset
distance D is defined.
We collect ranking lists of top 10 NCAA basketball coaches from several
authoritative media such as ESPN, Bleacher Report, Yahoo Sports, and Sporting
News [15]
. Then compare our results to those lists and average offset distance D
reflects the difference.
Here we use the first-order Minkowski distance to denote the average offset
distance of the top 10.
Team#28414 Page 16 of 33

𝑛𝑛 10
1
D= � ��𝑗𝑗 − 𝑟𝑟𝑗𝑗 � (5.1)
10𝑛𝑛
𝑖𝑖=1 𝑗𝑗=1

Where n is the number of the top 10 ranking lists, j is the ranking in the 𝑖𝑖𝑡𝑡ℎ list,

and 𝑟𝑟𝑗𝑗 is the ranking of 𝑗𝑗𝑡𝑡ℎ coach in our results. So �𝑗𝑗 − 𝑟𝑟𝑗𝑗 � denotes the difference

between result of media and ours, and D means the average difference. If our results
are the same as all media selection results, then D is equal to zero.
𝐷𝐷𝛼𝛼 is the average offset distance of top 5
𝑛𝑛 5
1
𝐷𝐷𝛼𝛼 = � ��𝑗𝑗 − 𝑟𝑟𝑗𝑗 � (5.2)
5𝑛𝑛
𝑖𝑖=1 𝑗𝑗=1

𝐷𝐷𝛽𝛽 is the average offset distance of 6th to 10th.

𝑛𝑛 10
1
𝐷𝐷𝛽𝛽 = � ��𝑗𝑗 − 𝑟𝑟𝑗𝑗 � (5.3)
5𝑛𝑛
𝑖𝑖=1 𝑗𝑗=6

Obviously model with smaller average offset distance should get higher score. So
We can define hit score
900
g= (0 < 𝑔𝑔 < 100) (5.4)
9+𝐷𝐷

When D = 0, g= 100, means if there is no average offset distance, this model can
get full marks 100. Here are our results:

AHP FSE
𝑫𝑫𝜶𝜶 1.75 1.15

𝑫𝑫𝜷𝜷 3.1 2.85

𝐃𝐃 2.425 2

𝒈𝒈𝜶𝜶 83.72 88.67

𝒈𝒈𝜷𝜷 73.38 75.94

𝐠𝐠 78.77 81.81
Table 5.1 the results for evaluation
Conclusions:
 Vertical comparison: Either AHP or Fuzzy Synthetic Evaluation 𝐷𝐷𝛼𝛼 is obviously
smaller than 𝐷𝐷𝛽𝛽 . It means that the results are more reasonable in top 5 than in top
10.
Team#28414 Page 17 of 33

 Horizontal comparison: Fuzzy Synthetic Evaluation performs better than AHP in

both top 5 and top 10. It proves that Fuzzy Synthetic Evaluation is more accurate
than AHP. Because AHP depends on artificial scoring which is too subjective.

5.2 Aggregation Model

AHP is a subjective method, it largely depends on artificial scoring; Relatively,
Fuzzy Synthetic Evaluation is an objective method, it all depends on the data. To
comprehensively consider the effect of subjective and objective factors, we adopt
linear weighted method:
𝑊𝑊1 + 𝑊𝑊2 = 1
� (5.5)
Y = 𝑊𝑊1 𝑌𝑌1 + 𝑊𝑊2 𝑌𝑌2

𝑌𝑌1 is the evaluation grade of AHP model , 𝑌𝑌2 is the evaluation grade of Fuzzy
Synthetic Evaluation model. All of them range from 0 to 1.
To determine the weight 𝑊𝑊1 and 𝑊𝑊2 , we take D(the average offset distance) into
consideration. Since smaller average offset distance means the more accuracy
results, we can assign higher weight to the mode with smaller D. Then we get
𝐷𝐷2
⎧𝑊𝑊1 =
𝐷𝐷1 + 𝐷𝐷2
(5.6)
⎨𝑊𝑊 = 𝐷𝐷1
⎩ 2 𝐷𝐷1 + 𝐷𝐷2
In conclusion, our final model can be defined as:
Y = 𝑊𝑊1 𝑌𝑌1 + 𝑊𝑊2 𝑌𝑌2 (5.7)

5.3 Results & analysis

AHP FSE AM
Rank 1 Mike Krzyzewski John Wooden John Wooden
Rank 2 John Wooden Mike Krzyzewski Mike Krzyzewski
Rank 3 Adolph Rupp Adolph Rupp Adolph Rupp
Rank 4 Jim Boeheim Dean Smith Dean Smith
Rank 5 Dean Smith Bob Knight Bob Knight
Rank 6 Roy Williams Roy Williams Jim Boeheim
Rank 7 Bob Knight Jim Boeheim Roy Williams
Rank 8 Phog Allen Phog Allen Phog Allen
Rank 9 Rick Pitino Rick Pitino Rick Pitino
Rank 10 Lute Olson Henry Iba Henry Iba
Team#28414 Page 18 of 33

Top5 Hit score 83.72 88.67 88.67

Top10 Hit score 78.77 81.81 82.57
Table 5.2 the ranking comparison among the models
Conclusion:
 All our models perform better in top 5 than in top 10. It proves that the top 5
coaches in college basketball history are less controversial than top 10.
 The result of AM is very similar to FSE. They have the same hit score 88.67 in top
5; but in top 10, AM have highest hit score 82.57 in these three models. It proves
the combination can improve our model.
 According to our final result, our model’s top 5 coaches in college basketball are
John Wooden, Mike Krzyzewski, Adolph Rupp, Dean Smith and Bob Knight.

VI. Extend Our Models

6.1 Genders do not matter
Now we take genders into consideration. We still use basketball as an example,
and rank the top ten college women’s basketball coaches [20] for the previous century.
Searching from the internet, we collect the relative data about 50 college women’s
[18]
basketball coaches with 600 and other 5 coaches who have established
outstanding traditions, earned many awards and garnered recognition for their
colleges. Then we rank them with our models mentioned above.
Coaches’ ranking with the Aggregation Model:
Rank Name Grade Rank Name Grade
1 Pat Summitt 0.8532 6 Sylvia Hatchell 0.5875
2 Geno Auriemma 0.8434 7 Jody Conradt 0.5673
3 Tara VanDerveer 0.7465 8 Kay Yow 0.5486
4 Leon Barmore 0.7236 9 Sue Gunter 0.4783
5 C. Vivian Stringer 0.6074 10 Gail Goestenkors 0.4379
Table 6.1 the ranking for coaches of women’s basketball
From the sports.yahoo.com [19], we get a list of the all-time top ten NCAA women’s
basketball coaches, and the list is shown in following table.
Rank Name Rank Name
1 Pat Summitt 6 Jody Conradt

2 Geno Auriemma 7 Kay Yow

Team#28414 Page 19 of 33

3 Leon Barmore 8 Sylvia Hatchell

4 C. Vivian Stringer 9 Gail Goestenkors

5 Tara VanDerveer 10 Sue Gunter

Table 6.2 the ranking from Yahoo

Using the average offset distance mentioned in section 5; we can measure the hit
score for our models. All results of our models are in agreement within reasonable
error range (hit score = 87.57), so that we can safely address the conclusion that our
models can be applied in general across both genders.

6.2 Time factor does make a difference

6.2.1 Why time factor matters?

National Collegiate Athletic Association Basketball Tournament [14] started at 1939,
during the 74 years’ development, while the number of teams participating in the
tournament increasing a lot, the competition becomes fiercer. Also in different
historical periods, the NCAA Basketball Tournament gained different popularity, and
this also influences the quality of the evaluation grades.
To quantify the time factor, we attach weight 𝑤𝑤𝑖𝑖 (1-10) to different time periods
mainly based on the turning points that occurred in the period.
The following table shows the critical years in the NCAA history [14]:
Year Turning points 𝒘𝒘𝒊𝒊
1913-1939 There are no national college basketball competition. 5
1939-1951 NCAA Basketball Tournament started, and 8 teams 6
anticipated. There are two college tournament: NIT and
NCAA.
1951-1975 16 teams anticipated, NIT became second class competition. 7
1975-1980 32 teams anticipated, especially in 1979, Magic Johnson fight 8
with Larry Bird in the finals, achieving 24.1% audience rating,
then a golden age came.
1980-1985 48 teams anticipated, 9
1985-2013 64 teams anticipated. 10
Table 6.3 the time weights for each time period

6.2.2 How time factor matters?

The whole metric system will change after introducing the time weight. What
Team#28414 Page 20 of 33

follows in the chapter will be devoted to explaining the changes in detail.

 The evaluation norms will change after introducing the time weight (𝑤𝑤𝑖𝑖 ).

⎧ a = � 𝑤𝑤𝑖𝑖 𝑎𝑎𝑖𝑖
⎪ 𝑖𝑖
⎪
⎪ b = � 𝑤𝑤𝑖𝑖 𝑏𝑏𝑖𝑖
⎪ 𝑖𝑖
⎪ ∑𝑖𝑖 𝑤𝑤𝑖𝑖 𝑆𝑆𝑆𝑆𝑆𝑆𝑖𝑖
⎪R = 𝑡𝑡
⎪ ∑𝑖𝑖 𝑤𝑤𝑖𝑖 𝑆𝑆𝑆𝑆𝑆𝑆𝑖𝑖
O= (6.1)
⎨ 𝑡𝑡
⎪ 𝑛𝑛𝑘𝑘 = � 𝑤𝑤𝑖𝑖 𝑚𝑚𝑘𝑘𝑘𝑘
⎪
𝑖𝑖
⎪
⎪ ℋ = � 𝓀𝓀𝑖𝑖
⎪ 𝑖𝑖
⎪
⎪ 𝒞𝒞 = � 𝒸𝒸𝑖𝑖
⎩ 𝑖𝑖
Where
 a denotes the wins, 𝑎𝑎𝑖𝑖 denotes the wins per year.
 b denotes the loses, 𝑏𝑏𝑖𝑖 denotes the loses per year.
 R denotes the average SRS, 𝑆𝑆𝑆𝑆𝑆𝑆𝑖𝑖 denotes the losses per year.
 O denotes the average SOS, 𝑆𝑆𝑆𝑆𝑆𝑆𝑖𝑖 denotes the losses per year, t denotes the
number of years.
 The binary variable 𝑚𝑚𝑘𝑘𝑘𝑘 denotes whether the team get the 𝑘𝑘 𝑡𝑡ℎ class in the 𝑖𝑖𝑡𝑡ℎ
year. 𝑛𝑛𝑘𝑘 denotes the number of times for each class.
 𝓀𝓀𝑖𝑖 denotes the weight for each award, ℋ denotes the total weights of all the
awards a coach has ever got.
 𝒸𝒸𝑖𝑖 denotes the points for each aspect, 𝒞𝒞 denotes the total points.

 Accordingly, the results for AHP (model I) & FSE (model II) will change.
The following table shows how AHP (model I) will change (The names in bold are the
people whose rank has changed):
AHP(without 𝒘𝒘𝒊𝒊 ) Grades (Top 10) AHP(with 𝒘𝒘𝒊𝒊 ) Grades (Top 10)
Mike Krzyzewski 0.8426 Mike Krzyzewski 0.8894
John Wooden 0.7334 John Wooden 0.7601
Adolph Rupp 0.6048 Jim Boeheim 0.6465
Jim Boeheim 0.5985 Adolph Rupp 0.6322
Dean Smith 0.5844 Dean Smith 0.6251
Roy Williams 0.5637 Roy Williams 0.6137
Bob Knight 0.5479 Bob Knight 0.5922
Team#28414 Page 21 of 33

Phog Allen 0.4788 Rick Pitino 0.5171

Rick Pitino 0.4683 Phog Allen 0.5062
Lute Olson 0.4132 Lute Olson 0.4606
Top10 Hit score 78.77 Top10 Hit score 76.60
Top5 Hit score 83.72 Top5 Hit score 83.56
Table 6.4 what is different in AHP introducing time weight?
Conclusion:
 In model I (AHP), Top5 hit score changes from 83.72 to 83.56, namely, Top5 hit
score nearly remains unchanged.
 Top10 hit score changes from 78.77 to 76.60, namely, Top10 hit score falls to
some extent.
 The changes in rankings occur only locally, not globally.

The following table shows how FSE (model II) will change (The names in bold are
the people whose rank has changed):
FSE (without 𝒘𝒘𝒊𝒊 ) Grades (Top 10) FSE (with 𝒘𝒘𝒊𝒊 ) Grades (Top 10)
John Wooden 0.8708 Mike Krzyzewski 0.9337
Mike Krzyzewski 0.8629 John Wooden 0.7850
Adolph Rupp 0.6750 Roy Williams 0.6556
Dean Smith 0.6090 Jim Boeheim 0.6260
Bob Knight 0.6052 Bob Knight 0.6207
Roy Williams 0.5872 Dean Smith 0.5984
Jim Boeheim 0.5864 Adolph Rupp 0.5445
Phog Allen 0.4874 Rick Pitino 0.5164
Rick Pitino 0.4664 Lute Olson 0.4603
Henry Iba 0.4538 Tom Izzo 0.4292
Top10 Hit score 81.81 Top10 Hit score 75.31
Top5 Hit score 88.67 Top5 Hit score 84.51
Table 6.5 what is different in FSE introducing time weight?
Conclusion:
 In model II (FSE), Top5 hit score changes from 88.67 to 84.51, namely, Top5 hit
score falls to some extent.
 Top10 hit score changes from 81.81 to 75.31, namely, Top10 hit score falls a lot.
The model appears to be more inaccurate.
 The changes in rankings occur globally. The model appears to be easily
influenced by the time weights.

 There is no doubt that the results for Aggregation Model (AM) will also change.
Team#28414 Page 22 of 33

The following table shows how aggregation model (final model) will change (The
names in bold are the people whose rank has changed):
AM (without 𝒘𝒘𝒊𝒊 ) Grades (Top 10) AM (with 𝒘𝒘𝒊𝒊 ) Grades (Top 10)
John Wooden 0.8568 Mike Krzyzewski 0.9204
Mike Krzyzewski 0.8296 John Wooden 0.7775
Adolph Rupp 0.6539 Roy Williams 0.6430
Dean Smith 0.6016 Jim Boeheim 0.6322
Bob Knight 0.5900 Bob Knight 0.6122
Jim Boeheim 0.5880 Dean Smith 0.6064
Roy Williams 0.5802 Adolph Rupp 0.5708
Phog Allen 0.4848 Rick Pitino 0.5166
Rick Pitino 0.4669 Lute Olson 0.4604
Henry Iba 0.4308 Phog Allen 0.4354
Top10 Hit score 82.57 Top10 Hit score 76.6
Top5 Hit score 88.67 Top5 Hit score 85.47
Table 6.6 what is different in AM introducing time weight?
Conclusion:
 The changes in rankings occur globally. The model appears to be easily
influenced by the time weights.
 The performance for AM appears to be easily influenced by FSE because of the
weight distribution for the two models.
Take Bob Knight for example, the following figure shows how evaluation grade
changes in the two different situations:

Figure 6.1 every year’s evaluation of Bob Knight

Team#28414 Page 23 of 33

6.2.3 What is the variation tendency?

From the results in section 6.2.2, variation tendency could be concluded as
follows:
 For coaches in earlier ages, their rankings will fall to some extent.
[21]
Take Phog Allen for example, he is known as the "Father of Basketball
Coaching", but most of his games occurred in 1920-1959, which means that NCAA
had not started or though started the teams were few. The time weight for his age is
relatively low thus making the ranking fall.
 For coaches in recent years, they will enjoy some superiority.
[22] [20]
Take Roy Williams and Adolph Rupp for example, the two coaches’
performance are quiet close to each other, Adolph Rupp was even better in in
historical record, but due to the time weight, the historical record for Adolph Rupp
does not count that much, and Roy Williams is ahead of him.
 Introducing time weights does not necessarily mean higher hit score.
In section V, We have stated that we choose some existed rankings as criterion,
but these rankings generally do not take coaching ages into consideration, moreover,
some authorities hold the viewpoints that the coaches in early ages are of more
authority, leading to the hit score falling.

6.3 Model also works in other sports

It obviously could not live up to our expectations if the model could only be used in
basketball. This chapter we will explain in detail how our models can be applied in
general across all possible sports.
There are mainly 4 steps to apply the model in any sport as you want.
 Step1: Adjust the metrics according to the feature of the sports.
Main differences are in the Playoff Performance, different sports may have
different playoff rules, so the metrics in this aspect should be adapted according to the
rule. Take football as example:
5-aspect norms Metrics for Metrics for
basketball football
Historical record Wins Wins
Losses Losses
Gold content SRS SRS
SOS SOS
Team#28414 Page 24 of 33

First round Fiesta Bowl

Second round Orange Bowl
Sweet sixteen Sugar Bowl
Playoff Elite eight Rose Bowl
Performance Final four National
Runner-up Championship
Game
Champion Non-BSC bowls
Honors Awards Awards
Contribution Star players Star players
to sports Coaching ages Coaching ages
Tactical innovation Tactical innovation
International International
Popularity Popularity
Table 6.7 the metrics for college football
 Step2: Adapt the calculation rules according to the feature of the sports.
For example, the metrics for football in playoff performance have already changed,
each class of performance should be assigned another weight according to the gold
content of different bowls. At the same time, the awards and their weights under
consideration should also be adjusted for the “Honors” aspect.
 Step3: Adjust the time weight according to the history of the sport.
For example, as for football, before 2006, there are no BSC bowls. After 2006, 5
BSC bowls came into being, enjoying extremely high gold content.
 Step4: Solve the Aggregation model again and analyze the results.
The following figure shows the steps of applying the model into all possible sports:
Team#28414 Page 25 of 33

Start

Adjust the metrics

Adjust the calculation rules

Adjust time weights

Solve the model

Figure 6.2 the flow chart of the four steps

Following the 4 steps presented above, we apply the model in other 2 different
sports: football and hockey.
Top5 for football Grades Top5 for hockey Grades
Bear Bryant 0.8874 Bob Johnson 0.8963
Knute Rockne 0.8664 Red Berenson 0.8732
Tom Osborne 0.8538 Jack Parker 0.8525
Joe Paterno 0.7872 Jerry York 0.7756
Bobby Bowden 0.7864 Ron Mason 0.7632
Table 6.8 the rankings for football & hockey

VII. Further discussion

7.1 Sensitivity Analysis on FSE

7.1.1 Vary Membership function

In FSE discussed in section 4.2, we choose (4.12) as the membership function.
But there are also other available membership functions. The following table shows
the membership functions that are taken into consideration in this part.
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Type Membership function

1 𝑋𝑋𝑖𝑖𝑖𝑖
𝜇𝜇𝑗𝑗 �𝑋𝑋𝑖𝑖𝑖𝑖 � = ( )𝑘𝑘
𝑋𝑋𝑗𝑗(𝑚𝑚𝑚𝑚𝑚𝑚)

2 𝑋𝑋𝑖𝑖𝑖𝑖 − 𝑋𝑋𝑗𝑗(𝑚𝑚𝑚𝑚𝑚𝑚) 𝑘𝑘
𝜇𝜇𝑗𝑗 �𝑋𝑋𝑖𝑖𝑖𝑖 � = ( )
𝑋𝑋𝑗𝑗(𝑚𝑚𝑚𝑚𝑚𝑚) − 𝑋𝑋𝑗𝑗(𝑚𝑚𝑚𝑚𝑚𝑚)

3 𝜇𝜇𝑗𝑗 �𝑋𝑋𝑖𝑖𝑖𝑖 � = 1 − 𝑒𝑒 −𝑘𝑘(𝑋𝑋𝑖𝑖𝑖𝑖−𝑋𝑋𝑗𝑗(𝑚𝑚𝑚𝑚𝑚𝑚))

Table 7.1 membership functions

 For type 1, we will analyze how the weight distribution (only for FSE) for different
aspects will change while varying k.

Figure 7.1 the weight of each factor for each aspect

Conclusions:
 Weight distribution is sensitive to the value of k and changes of weight distribution
can be seen clearly in Figure 7.1。
 As the value of k increases, 𝑋𝑋1 and 𝑋𝑋2 tend to be less important but 𝑋𝑋3 and 𝑋𝑋5
tend to be more important.
 When 1 ≤ k ≤ 4, 𝑋𝑋3 and 𝑋𝑋5 have the highest weight, 𝑋𝑋1 and 𝑋𝑋2 have the
lowest weight.
 For all three types, we will analyze how hit score (only for FSE) will change
varying k.
Team#28414 Page 27 of 33

Figure 7.2 sensitivity analysis on k for FSE

Conclusions:
 Membership function 1 and member function 2 are sensitive to the value of k,
while membership function 3 is not sensitive to the value of k.
 Membership function 1 reaches its maximum value at k = 3; membership function
2 reaches its maximum value at k = 2.
 Obviously, membership function1 and membership function 2 perform much
better than membership function 3. And membership function 3 is not suitable for
our model.
 For best results, membership function1 with k = 3 is most appropriate in this
model.

7.1.2 Vary calculation rule

𝜆𝜆𝑤𝑤𝑤𝑤𝑤𝑤
Here we focus on figuring out how hit score will change to the ratio .
𝜆𝜆𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

Figure 7.3 sensitivity analysis to the weight of win and lose

Team#28414 Page 28 of 33

Conclusions:
 Model performs best when the weight of winning a game is two times as much as
losing a game.
 If we attach the same weight to winning a game and losing a game, the model will
have a poor hit score. And if the ratio of the weight of wins and loses is too high, it
will also lead to a bad result.

7.2 Sensitivity Analysis on Aggregation weight

In this part, we will analyze how hit score (For AM) and rankings will change while
varying the weight for AHP (𝑊𝑊1 ).

Figure 7.4 && 7.5

Team#28414 Page 29 of 33

Conclusion:
 As the weight of AHP increases, the rank list will change. The rank of Dean Smith
and Bob knight tend to decline and the rank of Jim Boeheim tend to rise.
 Since AHP is less accurate than FSE, hit score of AM would be optimal when the
weight of AHP is small. But when the weight of AHP is zero, hit score doesn’t
reach the maximum. The maximum hit score is reached when the weight of AHP
is 0.1-0.2.
 It proves that AM can perform well than either AHP or FSE.

7.3 Exploration: Evaluating Best President

Now we use our models to find the top ten presidents of the United States. We
choose the 43 men who have been president of USA, and collect relative data from
the internet [24].
A president can be evaluated also from following five aspects: personal qualities,
presidential achievements, leadership qualities, failures and faults, and popular
opinion:

Figure 7.6 5-aspect norm of president of USA

The personal qualities includes imagination, intelligence and being willing to
take risks, while the presidential achievements can be valued with ability of
domestic accomplishments, executive appointments, foreign policy accomplishments,
and ability of compromise. And the leadership qualities can be measured by party
leadership ability and relations with congress. We also take popular opinion into
consideration, and we gather votes from different polls, such as C-SPAN poll, ABC
News poll, Washington College poll, Gallup poll, Rasmussen poll, and 2012 Gallup
poll. The ranking of presidents of the United States is shown in following table.
Rank Name Rank Name
1 Abraham Lincoln 6 Harry S. Truman

2 George Washington 7 Woodrow Wilson

Team#28414 Page 30 of 33

3 Franklin D. Roosevelt 8 Dwight D. Eisenhower

4 Thomas Jefferson 9 James K. Polk

5 Theodore Roosevelt 10 Andrew Jackson

Table 7.2 the ranking of presidents of the United States

VIII. Strength and Weakness

Strength:
 When we articulate our own metrics for assessment, we try our best to include all
the important elements of a coach to make the ranking more accurate. Time factor,
gender, category are all discussed in the model.
 We evaluate the performance of a coach from 5 specific perspectives.
 We set up two different models to form an aggregation model. AHP includes more
subjective factors while FSE appears to be more objective. The aggregation
model is devoted to make clear the tradeoff between the two sides.
 We states a distinct quantification system which is expected to live up to the
common sense.
Weakness:
 We adopt totally eighteen indicators to evaluate a coach, namely not all elements
is under consideration.
 Weights are everywhere in the model, but some weight assignments might not be
the best scheme.

IX. Non-technical Explanation

For better or worse, coaches are often the faces of college sports programs.
Different from players who stay only for a few years, coaches can exert longer
influence in the college games. Here is a list of the top 5 coaches in the college
basketball, college football, and college hockey.

Rank college basketball college football college hockey

1

John Wooden Bear Bryant Bob Johnson

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Mike Krzyzewski Knute Rockne Red Berenson

3

Adolph Rupp Tom Osborne Jack Parker

4

Dean Smith Joe Paterno Jerry York

5

Bob Knight Bobby Bowden Ron Mason

The rankings proved to be a difficult task and job of a college coach is
multifaceted. Firstly, we choose some coaches, who are in the list of Hall of Fame, or
people who have established outstanding traditions and earned many awards, as our
ranking candidates. Then, searching from the internet or other data sources, we try to
collect relative data as detailed as possible. After choosing proper data, college
coaches’ rankings can be obtained. What’s more, we search the existing rankings
from the internet to serve as the evaluation criterion.
During the procedure of choosing data, we evaluate the coaches in our list of
candidates from five aspects. As is known to all, the best college coaches tend to
have good team’s historical record, such as more wins and high win rate. What’s more,
SRS (Simple Rating System) and SOS (Strength of Schedule) can reflect the
coaching ability. We also examined each coach’s success in the postseason. Taking
basketball as an example, the performance could be valued by counting the number
of times for “Champion”, “Runner-up”, “Final Four”, “Sweet Sixteen”, “Second Round”,
and “First Round”. In many case, we take into account coaches’ contribution to the
sports and honors, such as various awards in their field or putting their name in the
“Hall of Fame”.
After collecting and choosing coaches’ detailed data, we define the importance of
Team#28414 Page 32 of 33

those aspects which can measure coaches’ ability, and use the results to give each
coach a score. The higher the coaches’ scored on the relative aspects, the better their
position on the ranking.
We use the data of the best college basketball coach–John Wooden to give some
example. In his college basketball coach career, his team had won 826 games, and
during his sixteen years NCAA tournament, he won ten championships and twelve
straight trips to Final Fours. John Wooden has been recognized tremendous times for
his achievements and created longer legacies in the college basketball games.
Besides his fantastic and glory record, Wooden was recognized for his impact on
college basketball as a member of the founding class of the National Collegiate
Basketball Hall of Fame and was named The Sporting News "Greatest Coach of All
Time" [23]. With so many honors and awards which can’t be listed in detail there, John
gets the highest score when we rank the coaches and is worthy the title of the best
college basketball coach.

X. Future work
 Consider all possible sports coaches together, and make a college coaches
rankings, regardless of what kind of sports coaches they are.
 Take other relevant coach information into consideration, because research
suggests that characteristics of the coaches, such as the breadth of coaches’
knowledge, authority, the ability of searching and cultivating talents, and attention
to details, play a role in determining the best college coaches.
 Develop a general method to rank everything when there is one way to quantify.

XI. References
[1] http://en.wikipedia.org/wiki/Sports_Illustrated
[2] http://collegebasketball.rivals.com/viewcbse.asp?selposition=9
[3] Saaty, Thomas L.; Peniwati, Kirti (2008). Group Decision Making: Drawing out and
Reconciling Differences. Pittsburgh, Pennsylvania: RWS Publications. ISBN
978-1-888603-08-8.
[4] http://en.wikipedia.org/wiki/Analytic_hierarchy_process
[5] http://en.wikipedia.org/wiki/Fuzzy_mathematics
[6] Zadeh, L. A. (1965) "Fuzzy sets", Information and Control, 8, 338–353
[7] http://en.wikipedia.org/wiki/National_Collegiate_Basketball_Hall_of_Fame
Team#28414 Page 33 of 33

[8] http://www.sports-reference.com/cbb/coaches/
[9] http://www.sportingcharts.com/
[10] http://en.wikipedia.org/wiki/Strength_of_schedule
[11] http://en.wikipedia.org/wiki/Playoffs
[12] http://en.wikipedia.org/wiki/NCAA_Men's_Division_I_Basketball_Championship
[13] Dahiya S, Singh B, Gaur S, et al. Analysis of groundwater quality using fuzzy
synthetic evaluation[J]. Journal of Hazardous Materials, 2007, 147(3): 938-946
[14]http://en.wikipedia.org/wiki/College_basketball.
[15]http://sports.espn.go.com/espn/page2/story?page=list/050304/collegehoopscoac
hes
[16] http://sports.yahoo.com/ncaa/basketball/news?slug=ycn-7791514
[17]http://www.sportingnews.com/ncaa-basketball/story/2009-07-29/sporting-news-50
-greatest-coaches-all-time
[18]http://en.wikipedia.org/wiki/List_of_college_women's_basketball_coaches_with_6
00_wins
[19] http://sports.yahoo.com/ncaa/football/news?slug=ac-7168152
[20] http://en.wikipedia.org/wiki/Adolph_Rupp
[21] http://en.wikipedia.org/wiki/Phog_Allen
[22] http://en.wikipedia.org/wiki/Roy_Williams
[23]http://en.wikipedia.org/wiki/Historical_rankings_of_Presidents_of_the_United_Sta
tes
[24] "Sporting News honors Wooden". ESPN. Associated Press. 30 July 2009.
Retrieved 7 June 2010.