2014年美赛数模B题-Finalist(3)

Team # 24270 Page 10 of 26 r r r r And then we construct the relation degree vector????????????1 2 3 4 5 6r , , , , , rr = , where 1 1r?? . T

We define C as the evaluation vector of AHP, and C can be calculated as follows::

C?? x'?? w (7) In evaluation vector, the greater Ci is, the higher ranking Si is.

? Combination of AHP and GRAP

At first, we employ extreme difference method to nondimensionalize the two evaluation vector B and C. And then, we construct an ultimate evaluation vector:

U???? B???? C (8)

where?? ,?? respectively stands for the weight of AHP and GRAP, which should satisfy the requirements of α + β =1.

Finally, we sort the value of Ui (i=1,2,…,n), and Si that corresponds to the top 5 of Ui are top five coaches.

3.4. Solutions to ModelⅠ

We choose three sports to verify our model and get the results, which include basketball, football and baseball.

3.4.1. Basketball

? Searching and selecting data

We search and select data through the Internet[5][6][7]. For example, first, we search 100 coaches and their evaluation index data. Secondly, we rank them by comprehensively considering the total number of wins and the winning-percentage, so we can get top 40

coaches. And then, we consider other metrics and rank top 20 coaches. Finally, the evaluation system is based on the selected 20 data. Table A1 in Appendix show the selected coaches and their evaluation index data.

? Determining the final evaluation index matrix

At first, we determine vector x’. For x1, via the data in Table A2, we use W(ti)=Num2, where Num represents the total number of teams in ti.

We utilize software MATLAB to plot the graph of W(1)(t) by simulating and curve fitting of data (Figure 3). So we can get pi for each Si, and then we obtain the vector x1’.

Figure 3: Trend of W(1)(x) Figure 4: Trend of s(1)(x)

For x2, via the data in Table A3, we also plot the graph of s(1)(t) by simulating and curve fitting of data (Figure 4). So we can get qi for each Si, and then we obtain the vector x2’.

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Team # 24270 Page 11 of 26 For x5 and x4, from (1)(2), we can obtain x5’ and x6’.

Secondly, from (3), we can obtain x''j?? j?? 1,2,?,6? . Finally, from (4), we can obtain x* . We list the quantitative value of x* in Table A4.

? Obtaining the result via ultimate evaluation vector

At first, from (5), we use AHP and get B . Secondly, we use GRAP and define that ρ=0.3 and wi=0.05(i=1,2,…,n). From (6), we can get the relation degree vector r. And then, from (7), we can obtain C. Finally, from (8), by defining???? 0.6,???? 0.4 , we can obtain the ultimate evaluation vector:

U????0.293,0.288,0.468,0.241,0.422,0.160,0.521,0.868,0.713,0.311,0.481,0.836,0.168,0.998, ?0.130 0.318 0.243 0, , , .482,1.761,0.138 By sorting the value of Ui (i=1,2,…,n), we can obtain the ranking result of Si. And the ranking vector is:

T

????????????????????????????????19,14,8,12,9,7,18,11,3,5,16,10,1,2,17,4,13,6,20,15?1Rank

Table 3: Top 5 Coaches of Basketball

T

Therefore, we list top five coaches of basketball in the previous century in Table 3:

No.1 No.2 No.3 No.4 No.5 S19 S14 S8 S12 S9

John Wooden Dean Smith Mike Krzyzewski Adolph Rupp Bob Knight

This result is largely agreement with the widely accepted result[8][9].

3.4.2. Football

? Searching and selecting data

Like what we do in basketball, we search and select data through the Internet[5][10][11]. However, we calculate that the number of final fours is the sum number of times that teams can enter into Super Bowl.

? Determining the final evaluation index matrix

At first, we determine vector x’. For x1, we use W(ti)=Num2, where Num represents the total number of teams in ti.

We can obtain W(2)(t) by simulating and curve fitting of data. So we can get pi for each Si, and then we obtain the vector x1’.

For x2, we also obtain s(1)(t) by simulating and curve fitting of data. So we can get qi and the vector x2’.

Finally, from (4), we can obtain x* . We list the quantitative value of x* .

? Obtaining the result via ultimate evaluation vector

Like what we do in Basketball, we can obtain the ultimate evaluation vector:

U????0.234,0.879,0.738,0.106,0.359,0.296,0.383,0.193,0.291,0.453,0.494,0.248,0.180,0.615, T

?1.000 0.316 0.151 0, , , .047, 0.392,0.052 By sorting the value of Ui (i=1,2,…,n), we can obtain the ranking result of Si. And the ranking vector is:

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Team # 24270 Page 12 of 26 ?????????????????????????????????,2,3, ,11,10, ,7,5,16, ,9,12,1,8,1315 14 1 ,17,49 6 ,20,18?2Rank Table 4: Top 5 Coaches of Football

T

Therefore, we list top five coaches of basketball in the previous century in Table 4:

No.1 No.2 No.3 No.4 No.5

S15 S2 S3 S14 S11

Joe Paterno Bobby Bowden Bear Bryant Tom Osborne Don James

This result is largely agreement with the widely accepted result[12].

3.4.3. Baseball

? Searching and selecting data

Like what we do in basketball, we search and select data through the Internet[13][14 …… 此处隐藏：8273字，全部文档内容请下载后查看。喜欢就下载吧 ……