例：在旅游问题中，求目标层到准则层的成对比较矩阵为A的特征向量和最大特征根： 选择旅游地
准则层：
景费居饮旅
色 用 住 食 途
方案层： P1 P2 P3
??1?12433???21755??10433??755???1111??.5??21A?1?4723?＝??0.250.10.50.333?
?11?35211?143??0.3330.2211?????0.3330311??11?35311???.2???W1?利用“和法”求A的特征向量W???????和特征根?max
?Wn??（S1）将A??Wij?nxn的元素按列归一化得：
??0.2650.2450.2350.2860.29??0.5100.4890.4110.4760.484??A?W~ij?nxn???0.0640.0700.0590.0480.032???0.0850.0980.1180.0950.097?
??0.0850.0980.1760.0950.097???2?1?2?0.25?0.333?0.333?3.917?2?0.5?1?0.143?0.2?0.2?2.043?3?4?7?1?2?3?17
?4?3?5?0.5?1?1?10.5?5?3?5?0.333?1?1?10.333
n（S2）将A?W~?~按行求和得各行元素之和：W~~ijnxn中元素Wiji??Wij
j?1 16
??1.312???2.?AW~?37??~i??0.273??W
??0.493???0.511??（S3）再将上述矩阵向量归一化得到特征向量近似值，
?1.312??0~?.262??2.37????0.474??W?Wi1??n?0.273???0.055?W4.999?i??0.493????0.099? 特征向量
?i?1?0.511????0.102??5其中
?W~i?(1.312?2.37?0.273?0.493?0.51)1?4.999 1（S4）计算与特征向量相对应最大特征根（的近似值）
??1n?AW?imaxn?i?1Wi?nnnnn??a?1?1jWi?a2jWi?a3jWi??i?j?1i?j?1i?j?1i?a4jWi?j?1i?a5jWi?j?1?5??W????1W??2W3W4W5???? 17
??0.262??0.262??0.262????????0.4740.4740.474?????????1 0.5 4 33??0.055??21 7 5 5??0.055??0.25 0.1431 0.5 0.333 ??0.055?????????0.099??0.099??0.099???0.102??0.102??0.102?1????????5??0.262?0.474?0.055?????????0.262???0.474??0.262????0.337 0.2 2 11 ???0.055???0.474????0.333 0.2 3 11 ????0.055???0.099??????0.102???0.099?? ??0.102???0.099?0.102??????????1?0.263?0.237?0.22?0.297?0.3060.524?0.474?0.385?0.495?0.55??0.262?0.474? 0.066?0.068?0.055?0.0495?0.0340.087?0.095?0.11?0.099?0.1020.055?0.099? 0.087?0.095?0.165?0.099?0.102?0.102???1?5?1.323?0.262?2.3880.474?0.2730.4930.548?0.055?0.099?0.102???15?5.05?5.038?4.960?4.98?5.373??15?25.401?5.0802??0.262??0.474??故有最大特征根?max?5.0802 , W???0.055?
??0.099???0.102??
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对A一致性检验指标：CI??max?nn?1?5.0802?50.0802??0.02
44RI?1.12 0.02CR??0.018?0.11.12故通过检验。
十、应用实例
对前面旅游问题进行决策
目标层： A 选择旅游地点
0.262 0.474 0.099 0.102 0.055
景费居饮
色 用 住 食 准则层：
B2 B3 B4 B1
0.595 0.129 0.129 0.277
决策层： P1 P2 P3
已知：①目标A对准则Bi i?1, 2, 3, 4, 5的权重向量为：
T，并已通过一致性检W??0.262 0.474 0.055 0.099 0.102?（由前面已算出）
旅途 B5 验。
②准则B1, B2, B3, B4, B5相对于P1, P2, P3的成对比较矩阵为
B1对P1, P2, P3作用的成对比较矩阵为：
?b11?B1??b21?b?31b12b22b32b13??12??b23???Y21?1b33?Y???525??2? ?1??同样B2对P1, P2, P3作用的成对比较矩阵为：
19
?1?B1??3??8?13131??118??1? B3??113??YY1?3?3?3??3? 1???134??11???B4??111? B5??11?3??1?44?11???4?解：
1?4?1? 4?1??对以上每个比较矩阵都可计算出最大特征根?max及对象的特征向量W（即权重向量），并进行一致性检验：CI?RI CR
以B1为例用“和法”求出B1的特征根?max及对立的特征向量W1
25??1???B1??0.512?
?0.20.51????0.5880.5710.625???~（S1）对B1按列归一化得：B1Wij??0.2940.2860.25?
?0.1180.1430.125??????1.784???~（S2）对按列归一化反向量再按行求和：W??Wij??0.83?
j?1?0.386???n~Wi~（S3）对W按行归一化得到特征向量W W?n
~W?ii?1?1.784??0.595??1.784?0.83?0.386?????0.83??W???0.277? ?1.784?0.83?0.386????0.386??0.129???????1.784?0.83?0.386??1（S4）计算特征根?max
(B)?max?1???BW1?? B??0.5ini?1Wi15??12?
?0.20.51???2 20