1993年全国大学生数学建模论文

1、1993年全国大学生数学建模论 文1993年全国大学生数学模型竞赛试题B题足球队排名次下表给出了我国12支足球队在1988-1989年全国足球级联赛中的成绩，要求1 ）设计一个依据这些成绩排出诸队名次的算 法，并给出用该算法排名次的结果。2）把算法推广到任意N个队的情况。3）讨论：数据应具备什么样的条件，用你的方 法才能够排出诸队的名次。T1T2T3T4T5T6T7T8T9T10T11T12022 0011120113311201T1X XX 1124100110001 3101020T2X20121020XX 00 021111 00 01 20 11 3 00 01 01 0T3X4 21

2、 10 02 13 01 01 40 13 11 02 30 12 0XXT4X2 30 10 52 2 11 0 10 0 11 XX33010 11 0 丁5XXXXX01 201 1丁6XXXXXXXT7X1 0200 02 13 01 03 13 02 23 12 0T8X0 11 1 11 3 10 022 00 01T9X3 0100 01 01 0T10X1 02 0T11X1 11 21 1T12X1） 12支球队依次记作 T1,T2,T12.2）符号X表示两队未曾比赛。3）数字表示两队比赛结果，如 T3行与Y8列 交叉处的数字表示：T3与T8比赛了 2场；T 3与T8的进球之

3、比为0 : 1和3 : 1。三个模型模型一假设：1每场比赛中，每支球队都正常发挥。2不考虑进球数，只考虑比赛结果：输或赢。3在每两支队伍的比赛中，多赢 n场，相当于 在基准（1 ）的基础上乘以2n;多输n场，相当于在基准的基础上除以2n;平局则等于基准;没有进行比赛也相当于平局，即等于基准。方法：层次分析法建模： 匚”Z表示足球队；Ai表示在相互的比赛中，第i 支足球队赢的事件；C表示足球队的排名。构造成对比较矩阵：A=1 1 1 8 2 2 1/4 1 4 1 1 1;1 1 1/2 2 1 2 1 1 2 1/2 1 1;1 2 1 2 2 2 1 1 1 1 1 1;1/8 1/2 1/

4、2 1 1/2 1/2 1/4 1 1/2 1/21 1;1/2 1 1/2 2 1 1/2 1 1 1 1 1 1;1/2 1/2 1/2 2 2 1 1 1 1 1 1 1;4 1 1 4 1 1 1 4 8 4 2 2;1 1 1 1 1 1 1/4 1 1/2 1 2 1;1/4 1/2 1 2 1 1 1/8 2 1 4 2 2;1 212111/41 1/4 1 2 2;1 111111/21/2 1/2 1/2 1 1/2;1 111111/21 1/2 1/2 2 1;计算权向量和一致性检验成对比较矩阵A的最大特征值为13.5660该特征值对应的特征向量B= ( 0.35460

5、.25910.12180.21010.22430.20300.27060.24120.2052 ) 归一化向量为:0.29340.60470.17710.07090.1910CI =13.5660 - 1212- 1=0.1424B= ( 0.11200.08190.09270.03850.0664经查随机一致性指标 RI表可得：当n=12 时,RI = 1.54ul00.1424所以CR= -w5T= 0.092 < 0.1 表明A通过一致性检验。所以足球队的排名依次为：T7 T1 T3 T9 T2T10 T6 T5 T12 T8 T11 T4代码如下:syms A;>>

6、A=1 1 1 8 2 2 1/4 1 4 1 1 1;1 1 1/2 2 1 2 1 1 2 1/2 1 1;1 2 1 2 2 2 1 1 1 1 1 1;1/8 1/2 1/2 1 1/2 1/2 1/4 1 1/2 1/21 1;1/2 1 1/2 2 1 1/2 1 1 1 1 1 1;1/2 1/2 1/2 2 2 1 1 1 1 1 1 1;4 114111 484 22;1 111111/411/2121;1/4 1/2 1 2 1 1 1/8 2 1 4 2 2;1 212111/411/4122;1 111111/21/2 1/21/2 1 1/21 111111/211/

7、2 1/2 2 1;A =1至11列1.00008.00000.25001.00001.00002.00001.00000.50001.00001.00002.00001.00001.00001.00001.00001.00001.00002.00002.00001.00002.00004.00000.50002.00002.00001.00002.00002.00001.00001.00001.00001.00001.00000.12500.50000.50001.00000.50000.50000.25001.00000.50000.50001.00000.50001.00000.5000

8、2.00001.00000.50001.00001.00001.00001.00001.00000.50000.50000.50002.00002.00001.00001.00001.00001.00001.00001.00004.00001.00001.00004.00001.00001.00001.00004.00008.00004.00002.00001.00001.00001.00001.00001.00001.00000.25001.00000.50001.00002.00000.25000.50001.00002.00001.00001.00000.12502.00001.0000

9、4.00002.00001.00002.00001.00002.00001.00001.00000.25001.00000.25001.00002.00001.00001.00001.00001.00001.00001.00000.50000.50000.50000.50001.00001.00001.00001.00001.00001.00001.00000.50001.00000.50000.50002.000012歹U1.00001.00001.00001.00001.00001.00002.00001.00002.00002.00000.50001.0000>> x,y=e

10、ig(A)x =1至6歹U0.3546+0.0000i0.28920.1213i0.2892- 0.1213i-0.1199+0.5389i-0.1199-0.5389i0.1379 + 0.3456i0.2591+0.0000i0.07440.1607i0.0744 + 0.1607i-0.02040.0305i-0.0204+0.0305i-0.1975 + 0.1511i0.2934+0.0000i-0.11010.1609i-0.1101 + 0.1609i-0.1026+0.0041i-0.1026-0.0041i0.0432 - 0.2452i0.1218+0.0000i-0.03

11、70+0.0529i-0.0370- 0.0529i-0.0714- 0.0384i-0.0714+0.0384i- 0.1053 + 0.2051i0.2101+0.0000i0.0117-0.1124i0.0117 + 0.1124i-0.0735- 0.2027i-0.0735+0.2027i0.3720 + 0.0000i0.2243+0.0000i-0.0118-0.1047i-0.0118 + 0.1047i-0.1692- 0.1217i-0.1692+0.1217i0.0827 - 0.3554i0.6047+0.0000i0.7875+0.0000i0.7875 + 0.00

12、00i0.7473+0.0000i0.7473+0.0000i0.0174 + 0.3707i0.2030+0.0000i-0.10140.0288i-0.1014- 0.0288i0.0828+0.0227i0.08280.0227i-0.1784 - 0.0810i0.2706+0.0000i-0.02620.3383i-0.0262- 0.3383i-0.0802- 0.0141i-0.0802+0.0141i- 0.0990 - 0.2834i0.2412+0.0000i-0.1683+0.0426i-0.1683- 0.0426i0.0346+0.0345i0.0346-0.0345

13、i0.2400 + 0.1807i0.1771+0.0000i-0.0731-0.0958i-0.0731 + 0.0958i0.0733+0.0142i0.0733-0.0142i- 0.0227 - 0.1776i0.2052+0.0000i-0.1041-0.0486i-0.1041 + 0.0486i0.0634- 0.0486i0.0634+0.0486i -0.1560 + 0.0090i7至12列0.13790.3456i0.48250.0000i-0.4780- 0.1446i-0.4780+0.1446i0.4606+0.0000i0.4606 + 0.0000i-0.197

14、50.1511i-0.15510.0000i0.2273 + 0.2785i0.22730.2785i-0.2686+0.0168i-0.2686 - 0.0168i0.0432+0.2452i-0.28740.0000i0.5545 + 0.0000i0.5545+0.0000i0.09090.0238i0.0909 + 0.0238i-0.10530.2051i0.20690.0000i-0.10280.1069i-0.1028+0.1069i0.17410.0480i0.1741 + 0.0480i0.3720+0.0000i-0.15530.0000i0.0169- 0.0849i0.

15、0169+0.0849i-0.3258+0.1821i-0.3258 - 0.1821i0.0827+0.3554i0.14370.0000i-0.2031 + 0.1752i-0.20310.1752i0.14560.0242i0.1456 + 0.0242i0.01740.3707i0.02130.0000i0.0168 + 0.0365i0.01680.0365i0.31950.2739i0.3195 + 0.2739i-0.1784+ 0.0810i0.54190.0000i-0.0967- 0.0531i-0.0967+0.0531i0.2035+0.0984i0.2035 - 0.

16、0984i-0.0990+0.2834i-0.46540.0000i0.2953 + 0.0704i0.29530.0704i-0.4009+0.0546i-0.4009 - 0.0546i0.24000.1807i-0.01730.0000i-0.0150 + 0.1750i-0.01500.1750i0.06960.0971i0.0696 + 0.0971i-0.0227+0.1776i-0.19320.0000i0.0905- 0.1373i0.0905+0.1373i-0.2974-0.0706i-0.2974 + 0.0706i-0.1560- 0.0090i-0.15700.000

17、0i-0.2054- 0.0805i-0.2054+0.0805i0.0777+0.0846i0.00000.0000+0.0000i0.14930.0000+0.0000i0.0777 - 0.0846i13.5660+ 0.0000i0.0000i0.0000 + 0.0000i+0.0000i0.00000.0000 + 0.0000i0.0000+0.0000i3.6332i0.0000 + 0.0000i+0.0000i0.00000.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.1493- 3.6332i0.0000+0.0000i0.000

18、0+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i-0.3782+2.2734i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i-0.37822.2734i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i-0.2768 + 1.2155i0.0000+0.000

19、0i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.0000+0

20、.0000i0.00000.0000+0.0000i0.00000.0000+0.0000i+0.0000i0.00000.0000 + 0.0000i0.0000+0.0000i0.0000i0.0000 + 0.0000i+0.0000i0.00000.0000 + 0.0000i0.0000+0.0000i0.0000i0.0000 + 0.0000i+0.0000i0.00000.0000 + 0.0000i7至12列0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.000

21、0i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 +

22、0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i-0.27681.2155i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i-0.02000.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0

23、000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0699 + 0.7032i0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.06990.7032i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000i0.0000 + 0.0000i0.0000+0.0000i-0.3372+0.1954i0.0000 + 0.0000i0.0000+0.0000i0.00000.0000

24、i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i-0.3372 - 0.1954i>> syms B;>> B=x(:,1)B =0.35460.25910.29340.12180.21010.22430.60470.20300.27060.24120.17710.205201120340.0818620.0926980.038482006638:一二0.1120 0.0819 0.0927 0.0385 0.0664 0.0709，。北立。臼 0.1910 0.0641 0.0855 0.0762 0.0560 0.06480

25、.0641370.0354950.0762060.0559540.064832模型二假设：1每场比赛中，每支球队都正常发挥。2不考虑输赢，只考虑每支队伍总的进球数建模:Z表示足球队；Bi表示每支足球队总进球数；C 表示足球队排名。构造成对比较矩阵B=1.00001.71430.85711.55564.66671.33334.00001.86674.00002.00001.41184.80000.58330.93331.16673.11111.86670.50000.80000.37501.00000.60000.12500.33332.66671.60001.00000.93332.00009

26、.33330.85710.80000.64293.00000.60000.21431.00008.00001.60000.50000.35002.00001.00000.70000.42860.30001.71430.32140.22501.28570.10710.07500.60003.42861.16670.82352.80002.33331.64715.60001.00000.70592.40000.75000.52941.80000.25000.17650.20000.20000.42860.60001.66674.44442.66670.70831.88891.13330.62501

27、.66671.00000.62501.66671.00000.29170.77780.46670.20830.55560.33332.85711.428613.3333 1.000022.66671.21435.66671.13331.07145.00001.00001.07145.00001.00000.50002.33330.46670.35711.66670.33335.71430.60710.42502.42860.53570.37502.14290.53570.37502.14290.25000.17501.00000.17860.12500.71433.3333.35298.000

28、01.41671.00003.40001.25000.88243.00001.25000.88243.00000.58330.41181.40000.41670.29411.0000计算权向量和一致性检验成对比较矩阵A的最大特征值为12 ;该特征值对应的特征向量D= ( 0.37020.21590.43190.18510.13880.04630.61700.26220.23140.23140.10800.0771 )归化向量为 D= (0.1270 0.0741 0.1481 0.0635 0.0476 0.0159 0.2116 0.0899 0.0794 0.0794 0.0370 0.0

29、264)因为B为一致性矩阵，所以不用进行一致性检 验。所以足球队的排名为：T7 T3 T1 T8 T9=T10 T2 T4 T5 T11 T12 T6代码如下syms B;>> B=1.00001.71430.85718.00001.60000.60003.42862.00002.66671.41181.60004.80000.58331.00000.50001.16671.55564.66670.35000.82350.93330.93332.00002.80001.16672.00001.00002.33333.11119.33330.70001.64711.86671.866

30、74.00005.60000.50000.85710.42861.00001.33334.00000.30000.70590.80000.80001.71432.40000.37500.64290.32140.75001.00003.00000.22500.52940.60000.60001.28571.80000.12500.21430.10710.25000.33331.00000.07500.17650.20000.20000.42860.60001.66672.85711.42863.33334.444413.3333 1.00002.35292.66672.66675.71438.0

31、0000.70831.21430.60711.41671.88895.66670.42501.00001.13331.13332.42863.40000.62501.07140.53571.25001.66675.00000.37501.00001.00002.14290.62501.07140.53571.66675.00000.37501.00001.00002.14290.29170.50000.25000.77782.33330.17500.46670.46671.00000.20830.35710.17860.55561.66670.12500.33330.33330.7143B =

32、1至11列1.00001.71432.00002.66670.60001.41180.88243.00001.25000.88243.00000.58330.41181.40000.41670.29411.00000.85718.00001.60000.58331.00000.50001.60003.42861.16671.55564.66670.35000.82350.93330.93332.00001.16672.00001.00002.33333.11119.33330.70001.64711.86671.86674.00000.50000.85710.42861.00001.33334

33、.00000.30000.70590.80000.80001.71430.37500.64290.32140.75001.00003.00000.22500.52940.60000.60001.28570.12500.21430.10710.25000.33331.00000.07500.17650.20000.20000.42861.66672.85711.42863.33334.444413.33331.00002.35292.66672.66675.71430.70831.21430.60711.41671.88895.66670.42501.00001.13331.13332.4286

34、0.62501.07140.53571.25001.66675.00000.37500.88241.00001.00002.14290.62501.07140.53571.25001.66675.00000.37500.88241.00001.00002.14290.29170.50000.25000.58330.77782.33330.17500.41180.46670.46671.00000.20830.35710.17860.41670.55561.66670.12500.29410.33330.33330.714312列4.80002.80005.60002.40001.80000.6

35、0008.00003.40003.00003.00001.40001.0000 syms a b;>> a,b=eig(B)a =1至6列0.3702+ 0.0000i0.13430.0000i-0.0308 + 0.0000i-0.1732+0.0758i-0.17320.0758i0.0123 + 0.0000i0.2159+0.0000i-0.30660.0000i-0.1981+ 0.0000i-0.07340.2118i-0.0734+0.2118i0.0078 + 0.0000i0.4319+0.0000i0.3301+0.0000i-0.3471+ 0.0000i0.

36、0742+0.1480i0.07420.1480i0.0087 + 0.0000i0.18510.0000i-0.11840.0000i-0.1418 + 0.0000i0.29040.0679i0.2904+0.0679i0.0611 + 0.0000i0.1388+ 0.0000i0.3307+0.0000i-0.0556 + 0.0000i-0.19100.0708i-0.1910+0.0708i-0.0053 + 0.0000i0.0463+ 0.0000i0.3231+0.0000i0.3170 + 0.0000i-0.0746+0.0010i-0.07460.0010i0.0516

37、 + 0.0000i0.6170+ 0.0000i0.2317+0.0000i-0.8109 + 0.0000i0.7240+0.0000i0.7240+0.0000i-0.9933 + 0.0000i0.2622+ 0.0000i0.0165+0.0000i0.1189 + 0.0000i-0.14800.2109i-0.1480+0.2109i0.0594 + 0.0000i0.2314+ 0.0000i-0.19650.0000i0.0248 + 0.0000i0.0178+0.1466i0.01780.1466i0.0176 + 0.0000i0.2314+ 0.0000i-0.196

38、50.0000i0.0248 + 0.0000i0.0178+0.1466i0.01780.1466i0.0176 + 0.0000i0.1080+ 0.0000i0.1177+0.0000i-0.1497 + 0.0000i-0.0252+0.2867i-0.02520.2867i-0.0471 + 0.0000i0.0771+ 0.0000i-0.63730.0000i-0.1425 + 0.0000i0.1176- 0.1410i0.1176+0.1410i0.0132 + 0.0000i7至12列0.3633+ 0.0000i0.15140.0653i0.1514 + 0.0653i0

39、.1840- 0.0098i0.1840+0.0098i- 0.1862 + 0.0000i-0.3543+ 0.0000i-0.03020.0470i-0.0302- 0.0470i0.0126-0.0063i0.0126+0.0063i-0.0987 + 0.0000i-0.3543+ 0.0000i-0.03020.0470i-0.0302- 0.0470i0.0126-0.0063i0.0126+0.0063i-0.0987 + 0.0000i0.4106+ 0.0000i0.02000.1399i0.0200- 0.1399i0.0157-0.0048i0.0157+0.0048i-

40、0.0742 + 0.0000i0.0000+ 0.0000i-0.00000.0000i-0.0000 + 0.0000i-0.0000-0.0000i-0.0000+0.0000i0.0000 + 0.0000i0.0900+ 0.0000i-0.00040.1018i-0.0004 + 0.1018i0.0826+0.0222i0.0826-0.0222i0.0347 + 0.0000i0.5484+ 0.0000i0.95510.0000i0.9551 + 0.0000i-0.9749+0.0000i-0.9749+0.0000i0.2300 + 0.0000i0.0000+ 0.00

41、00i-0.00000.0000i-0.0000 + 0.0000i0.0000+0.0000i0.0000-0.0000i0.0000 + 0.0000i0.0686+ 0.0000i-0.06590.0517i-0.0659- 0.0517i-0.0429+0.0136i-0.0429-0.0136i-0.5833 + 0.0000i0.0686+ 0.0000i-0.06590.0517i-0.06590.0517i-0.0429+0.0136i-0.0429-0.0136i0.7300 + 0.0000i-0.3543+ 0.0000i-0.03020.0470i-0.0302- 0.

42、0470i0.0126-0.0063i0.0126+0.0063i-0.0987 + 0.0000i-0.0683+ 0.0000i-0.07700.0317i-0.0770- 0.0317i-0.0478-0.0341i-0.0478+0.0341i0.0576 + 0.0000ib =0.00000.0000+0.0000i-0.00011至6歹U12.0000+ 0.0000i0.0000i0.0000 + 0.0000i+0.0000i0.00000.0000 + 0.0000i0.0000+ 0.0000i0.0000i0.0000 + 0.0000i0.0000+0.0000i0.

43、0000+0.0000i0.0000 + 0.0000i0.0000+ 0.0000i0.0000+0.0000i0.0000 + 0.0000i+0.0000i0.00000.0000 + 0.0000i0.0000+ 0.0000i0.0000i0.0000 + 0.0000i+0.0001i0.00000.0000 + 0.0000i0.0000+ 0.0000i0.0000i0.0000 + 0.0000i+0.0000i0.00000.0000 + 0.0000i0.0000+ 0.0000i0.0000i0.0000 + 0.0000i+0.0000i0.00000.0000 + 0.0000i0.0000+ 0.0000i0.0000+0.0000i0.0000+0.0000+0.0000i0.0000+0.0000-0.0001i0.0000+0.0000+0.0000i0.0000+0.0000i0.0000 + 0.0000i0.0000+0.0000i0.0000+0.0000i0.0000 +