2. Assumptions
We assume that the competition rules of each sport do not change.
Although sports are developing, we do not take into account of time in the competition rules in order to compare the coaches of different years more fairly. We neglect tied competitions since they have the same effect on the two compared
teams. We only take the Division I into consideration.
mo.crotha.mww w模数
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Competitions are divided into three parts: Division I, II and III according to the level of
sport strengths of different colleges. Since Division I always concludes top coaches, we only take Division I into consideration. The selected data are valid.
Additional assumptions are made to simplify analysis for individual sections. These
assumptions will be discussed at the appropriate sections.
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3. Model
3.1. Additional assumptions
The evaluation system includes two parts: Objective Metrics(OM) and Subjective
Metrics (SM). We assume that OM include four specific indexes: the total number of wins, the
winning-percentage, the number of final fours and the number of champions.
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Tenure and media popularity are considered in SM.
In the subjective metrics of ranking coaches, some factors are hard to investigate qualitatively and quantitatively due to lacking data, such as, his or her influence to players, range of knowledge, studying ability, team spirits, searching talents, acting in competitions, salary and so on. Therefore, we neglect these indexes in SM.
Time only makes a difference in the total number of wins, and the winning
percentage.
In fact, the numbers of final fours and champions have no effect on the other two in OM, since the number of teams which are able to enter into final fours and even achieve champions is fixed. And we neglect the influence of time on media popularity in order to simplify the model.
3.2. Notations
Notations
Si xj n m x, x’, x’’, x*
t pi , qi W(t) s(t) Mj mj
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Table 1: Notations and Descriptions
Descriptions
Evaluation object Evaluation index
The number of evaluation objects The number of evaluation indexes Evaluation index matrix Time
Influence coefficients of time
The total number of competitions in t
The standard deviation of all winning-percentage in t Maximum of xij Minimum of xij
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Notations f(x) A λ w CI RI CR B
Descriptions
Subordinate function
Pairwise comparison matrix The largest eigenvalue Weight vector Consistency index
Random consistency index Consistency ratio
Evaluation vector of AHP
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rxi 0 ,xi j
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Grey relational coefficient Absolute difference
Minimum difference Maximum difference Relation degree vector
Evaluation vector of Grey Relation Degree Partial coefficient
Ultimate evaluation vector
3.3. Evaluation System
We define n as the number of evaluation objects, and S1, S2, , Sn (n>1) are the evaluation objects. m is the number of evaluation indexes, and x1, x2, , xm are the evaluation indexes. Evaluation index vector is
The total evaluation indexes include OM: the total number of wins, the winning- percentage(pct.), the number of final fours and the number of champions and SM: tenure and media popularity. Som 6,
Where:
x1 — the total number of wins vector. x2 — the winning-percentage vector. x3 — the number of final fours vector. x4 — the number of champions vector. x5 — tenure vector.
x6 — media popularity vector.
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Δmin Δmax r C α , β U
x x1,x2, ,xm m 1 .
T
x x1,x2,x3,x4,x5,x6
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pi
1
Wtmi
Figure 1: Flow chart of model I
Undoubtedly, time plays an important role in evaluating top coaches. According to the assumptions, time only makes a difference in the total number of wins, the winning- percentage.
3.3.1. The influence of time on the total number of wins
With the development of sports, the competition is getting relatively fiercer than ever, which means the disparity between teams become wider. The total number of games also increases with time going on. Therefore, when evaluating coaches in the previous century, the later certain coach begin his coaching career, the more likely he will get more wins. So we should put less weight on the coaches active in a later time period. And we can get a fairer evaluation of coaches within different time periods.
In order to compensate the influence of t, we establish Influence Coefficients of Time (ICT)
pi i 1,2, ,n . We assume that the total number of competitions in t isW t .W t can
be obtained by statistical regression and simulating and curve fitting of selected data. So we define: