1993年全国大学生数学建模论文设计

wordword/word1993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。X0:11:00:02:21:00:22:03:11:03:11:00:11:30:22:11:04:01:11:1XXX2:00:11:30:02:00:01:12:11:11:10:00:02:01:10:20:0XXX4:21:10:02:13:01:01:40:13:11:02:30:12:0XXX2:30:10:52:32:11:30:10:00:11:1XXX0:1XXXX1:01:20:01:1XXXXXXXX1:02:00:02:13:01:03:13:02:23:12:0X0:11:22:01:11:00:13:10:0X3:01:00:01:01:0X1:02:0X1:11:21:1X1〕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；T3与T8的进球之比为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〕的根底上乘以2n；多输n场，相当于在基准的根底上除以2n；平局如此等于基准；没有进展比赛也相当于平局，即等于基准。方法：层次分析法Z建模：ZA1A2A3A4A5A6A8A7A9A12A11A10A1A2A3A4A5A6A8A7A9A12A11A10CCZ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。构造成比照拟矩阵：A=[1118221/414111;111/22121121/211;121222111111;1/81/21/211/21/21/411/21/211;1/211/2211/2111111;1/21/21/2221111111;411411148422;1111111/411/2121;1/41/212111/821422;1212111/411/4122;1111111/21/21/21/211/2;1111111/211/21/221;]计算权向量和一致性检验成比照拟矩阵A的最大特征值为该特征值对应的特征向量B=〔0.2934〕归一化向量为：B=〔0.11200.08190.09270.03850.06640.07090.19100.06410.08550.07620.05600.0648〕CI经查随机一致性指标RI表可得：当n=12时，RI=所以CR=明确A通过一致性检验。所以足球队的排名依次为：T7T1T3T9T2T10T6T5T12T8T11T4代码如下：symsA;>>A=[1118221/414111;111/22121121/211;121222111111;1/81/21/211/21/21/411/21/211;1/211/2211/2111111;1/21/21/2221111111;411411148422;1111111/411/2121;1/41/212111/821422;1212111/411/4122;1111111/21/21/21/211/2;1111111/211/21/221;]A=1至11列12列>>[x,y]=eig(A)x=1至6列7至12列y=1至6列7至12列>>symsB;>>B=x(:,1)B=0.11200.08190.09270.03850.06640.07090.19100.06410.08550.07620.05600.0648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。Z建模：ZA1A2A3A4A5A6A8A7A9A12A11A10A1A2A3A4A5A6A8A7A9A12A11A10CCZ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。构造成比照拟矩阵B=[1.00001.71430.85712.00002.66678.00000.60001.41181.60001.60003.42864.80000.58331.00000.50001.16671.55564.66670.35000.82350.93330.93332.00002.80001.16672.00001.00002.33333.11119.33330.70001.64711.86671.86674.00005.60000.50000.85710.42861.00001.33334.00000.30000.70590.80000.80001.71432.40000.37500.64290.32140.75001.00003.00000.22500.52940.60000.60001.28571.80000.12500.21430.10710.25000.33331.00000.07500.17650.20000.20000.42860.60001.66672.85711.42863.33334.444413.33331.00002.35292.66672.66675.71438.00000.70831.21430.60711.41671.88895.66670.42501.00001.13331.13332.42863.40000.62501.07140.53571.25001.66675.00000.37500.88241.00001.00002.14293.00000.62501.07140.53571.25001.66675.00000.37500.88241.00001.00002.14293.00000.29170.50000.25000.58330.77782.33330.17500.41180.46670.46671.00001.40000.20830.35710.17860.41670.55561.66670.12500.29410.33330.33330.71431.0000]计算权向量和一致性检验成比照拟矩阵A的最大特征值为12；该特征值对应的特征向量D=〔0.1080〕归一化向量为D=〔〕因为B为一致性矩阵，所以不用进展一致性检验。所以足球队的排名为：T7T3T1T8T9=T10T2T4T5T11T12T6代码如下symsB;>>B=[1.00001.71430.85712.00002.66678.00000.60001.41181.60001.60003.42864.80000.58331.00000.50001.16671.55564.66670.35000.82350.93330.93332.00002.80001.16672.00001.00002.33333.11119.33330.70001.64711.86671.86674.00005.60000.50000.85710.42861.00001.33334.00000.30000.70590.80000.80001.71432.40000.37500.64290.32140.75001.00003.00000.22500.52940.60000.60001.28571.80000.12500.21430.10710.25000.33331.00000.07500.17650.20000.20000.42860.60001.66672.85711.42863.33334.444413.33331.00002.35292.66672.66675.71438.00000.70831.21430.60711.41671.88895.66670.42501.00001.13331.13332.42863.40000.62501.07140.53571.25001.66675.00000.37500.88241.00001.00002.14293.00000.62501.07140.53571.25001.66675.00000.37500.88241.00001.00002.14293.00000.29170.50000.25000.58330.77782.33330.17500.41180.46670.46671.00001.40000.20830.35710.17860.41670.55561.66670.12500.29410.33330.33330.71431.0000]B=