2014年美赛数模B题-Finalist(2)

Team # 24270 Page 5 of 26 Notations Descriptions

f(x) Subordinate function A Pairwise comparison matrix

λ The largest eigenvalue

w Weight vector CI Consistency index RI Random consistency index CR Consistency ratio B Evaluation vector of AHP

?0??, Grey relational coefficient Absolute difference

??? ???

Δmin Minimum difference

Δmax Maximum difference r Relation degree vector

C Evaluation vector of Grey Relation Degree α , β Partial coefficient U 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

T

?????????????1 2The 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. So m?? 6 ,

T

, x? x ??????? 1 x , , mx . m???

Where:

x x x x x ????????????, , , , ,?x x 1 2 3 4 5 6? 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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Team # 24270 Page 6 of 26

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 is W??t?? . W??t?? can be obtained by statistical regression and simulating and curve fitting of selected data. So we define:

i

p???

1 W??tmi???'

where tmi is the middle year of tenure of Si . And then x1i?? x1i?? pi??i?? 1,2,?, n? .

3.3.2. The influence of time on the winning-percentage

As for the winning-percentage, sports were underdeveloped at an earlier time, and the quality disparity between teams is comparatively narrow. Therefore, the standard deviation of winning-percentage of each coach is closer to zero. Thus we should put less weight on the coaches active in a “mediocre” time period. We define ICT here as qi (i=1,2,…,n), we assume that the standard deviation of all winning-percentage in t is s??t?? . s??t?? can be obtained by statistical regression and simulating and curve fitting of selected data. So we define:

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and x2i?? x2i?? qi??i?? 1,2,?, n? . '

Team # 24270 Page 7 of 26 i

q???

1 s??tmi???

3.3.3. Fuzzy Analysis

As for SM indexes, we assume that they can be divided into five levels: “ Excellent, Very Good, Good, Not Good, Bad”. And we correspond the five levels into 5,4,3,2,1 successively For continuous quantification, we assume:

? As for “Excellent”, we suppose f??5??? 1. ? As for “Very Good”, f??3??? 0.7 .

? As for “Bad”, f??1??? 0.1 . We employ partial large Cauchy distribution and the logarithmic function as the subordinate function[2]:

f (x)?????

??c ln x?? d , 3?? x?? 5

where a, b, c, d stands for undetermined constants. We use the initial conditions above to define their values. And solution of the subordinate function( Figure 2) is:

???? 121 a x b???????????????,1 3x????????????????????

??????? 121 2.8049 0.4417x?????????????????????????????,1 f ( x)???? 3 x ???????????????????????????? (1) ??0.5873ln x?? 0.0548, 3?? x?? 5

Figure 2: Trend of f(x)

Media popularity is measured by the number of search results via Google. The impact of duplication of names can be neglected by means of adding search keywords in order to rule out the redundant information.

We map xj ( j=5,6) into interval [1,5], through function (1),we can obtain:

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Team # 24270 Page 8 of 26 ????4 ji jx m??????????? M 1x f??????12 5,6i n j??? (2)??????????????

j?? m j???where M j?? maxxij , m j?? minxij1????1???i? n

????? j?? 5,6? . As for x3 and x4, we define that x3’= x3, x4’= x4. we use x'j?? j?? 1, 2,?,6? to proceed the following calculation. 3.3.4. Nondimensionalization process

We employ extreme difference method to nondimensionalize the different indexes so that we can compare them[2]on the same level. The method is as follows:

x'ji?? m j

?M j?? m j

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