第二十八讲层次分析法简介

yunjiang · 发表于 2004-7-22 10:48:49

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>　　层次分析法是对复杂问题作出决策的一种简明有效的新方法。随着科学技术的发展，对以前在社会、经济、生物、心理、组织管理等领域只能定性描述的因素、事物和概念等，现在迫切需要作出量化研究。层次分析法在一定程度上满足了这种需要。它把定性分析与定量分析相结合，根据问题的总目标，以系统的观点，把问题分解成若干因素，并按其支配关系构成递阶层次结构模型，然后应用两两比较的方法确定决策方案之间的相对重要性，从而获得满意的决策。 
　　一、层次分析法的基本步骤<

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　　层次分析法的基本思路与人对一个复杂问题的思维、判断过程大体上是一样的。举一个简单的例子：“五一”长假期间你准备去旅游，是去风光秀丽的苏杭二州，还是去迷人的海南三亚，或者是去长春净月旅游村就近一游？假期不过七天，三地均游不可能，因此你必须对此作出选择与决策。不妨设上面三个旅游方案为

1，P2和P3，你会根据诸如景色、费用、居住和旅途条件等一些准则去反复比较那三个侯选方案。
　　首先，你会确定这些准则在你的心目中各占多大比重，如果你经济宽绰，醉心旅游，便会特别看重景色条件，而平素俭朴或手头拮据的人则会优先考虑费用，中老年旅游者还会对居住、饮食等条件寄以较大关注。其次，你会就每一个准则将三个方案进行对比，例如，P2景色最好，P1次之；P3费用最，P1次之；P1居住条件较好，P2次之等等。最后。你要将这几个层次的比较进行综合，在P1、P2、P3中确定哪个作为最佳地点。
　　上面的思维过程可以加工整理成以下几个步骤：
　　1．将决策问题分解为三个层次，最上层为总目标层，即选择旅游地；最下层为方案层，分P1、P2、P3三个供选择方案；中间层为准则层，有景色、费用、居住与饮食、旅途四个准则。各层间的联系用相连的直线段表示（如图6-1）
　　2．通过相互比较确定各准则对于目标的权重，各方案对于每个准则的权重。注意，这些权重在通常人的思维过程中一般是定性的，但在层次分析中则要给出权重的定量描述方法。
　　3．将方案层对准则层的权重及准则层对目标层的权重进行综合，最终确定方案层对目标层的权重，从而给出最终的结果。层次分析法将给出进行综合的计算方法。<a href="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/01.htm#" target="_blank" >图6-1</A>
<IMG src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/50.gif">

yunjiang · 发表于 2004-7-22 10:49:09

二、实例分析:公司利润的合理使用<

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　　问题某工厂有一笔企业留存利润，要由厂领导和职代会决定如何使用。可供选择的方案有：作为奖金发给职工；扩建集体福利设施；引进新技术、新设备等。为进一步促进企业发展，该如何合理使用这笔利润？<

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　　1．建立层次结构模型<

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　　上述各项方案的最终目的都是为了促进企业更大发展，因此该问题的递阶层次结构模型如图6—2所示。<TABLE cellSpacing=0 cellPadding=0 align=center><TR><TD ="53"></TD></TR><TR><TD></TD><TD><img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image007.gif"></TD></TR></TABLE>　　图6—2

yunjiang · 发表于 2004-7-22 10:49:22

2．构造判断矩阵<

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　　构造判断矩阵是进行层次分析的关键。<

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　　设有n个因素（准则）C1，C2，…，Cn对总目标G有影响，要确定它们在G中的比重。采用成对比较法，即每次取两个因素ci和cj，用aij表示ci与cj对G影响之比。譬如本例中 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image009.gif"> 与 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image011.gif"> 之比设定为1：5，与 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image013.gif"> 之比为1：3， <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image015.gif"> =3：1。将全部比较的结果用矩阵<

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　　A=(aij)n×n (aij>0)
　　表示。则 A是正互反阵，因为
　　aii=1,aji=1/aij (i,j=1,2,…,n)。
　　如何确定aij的值，人们常取数字1～9及其倒数作为aij的取值范围。这是因为在进行定性成对比较时，人们头脑中的5个明显等级可数量化为如表6-1所示。在每两个等级之间各有一个中间状态，依次用2，4，6，8将其量化。
　　表6-1 因素等级<TABLE medium none; BORDER-TOP: medium none; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt" cellSpacing=0 cellPadding=0 align=center border=1><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: windowtext 0.5pt solid; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 52: " ="70">ci比cj</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 52: " ="70">相同</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 52: " ="70">稍强</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 52: " ="70">强</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 52: " ="70">很强</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 52: " ="70">绝对强</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: windowtext 0.5pt solid; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-top-alt: solid windowtext .5pt; 52: " ="70">aij</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 52: " ="70">1</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 52: " ="70">3</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 52: " ="70">5</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 52: " ="70">7</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 52: " ="70">9</TD></TR></TABLE>　　于是本问题的判断矩阵G-C构造为
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image017.gif"> 
　　再分别构造判断矩阵C1-P，C2-P，C3-P，不妨设定为　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image019.gif"> ， <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image021.gif"> ， <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image023.gif"> 。
　　仔细分析上述成对比较矩阵 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image025.gif"> 可以发现，既然
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image027.gif"> 应该是 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image029.gif"> 而不是 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image031.gif"> ，才能说明这个成对比较是一致的。使用数学语言即，对三个因素xi,xj,xk进行两两比较：由xi与xj相比得到aij，由xj与xk相比得到ajk，再由xi与xk相比得到aik，则应有aijajk=aik，但完全可能出现aijajk≠aik。
　　由于人们要做众多因素比较，全部一致的要求是太苛刻了。因此，我们给出在成对比较不一致的情况下计算各因素对上一因素的权重的方法，并且确定这种不一致的容许范围。自然，此前进行相应矩阵的一致性检验是必须的。
　　所谓一个正互反矩阵A为一致性的，是指其元素满足aijajk=aik，i,j,k =1,2,…n。我们还有
　　引理 正互反阵的最大特征根是单根且是正实数，对应着正的特征向量。
　　定理 n阶正互反矩阵A=(aij)n×n是一致阵当且仅当 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image033.gif"> =n，其中 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image035.gif"> 是A的最大特征值

yunjiang · 发表于 2004-7-22 10:49:34

<

>　　3．层次单排序与一致性检验<

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　　在构造判断矩阵之后，求出判断矩阵的最大特征值 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image036.gif"> 和对应的特征向量W，W经过标准化后，即为同一层次中相应元素对于上一层次中的某个因素相对重要性的排序权值，这一过程称为层次单排序。<

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　　首先我们观察两个判断矩阵的一致性。例如
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image038.gif"> 
　　求得 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image040.gif"> =3，W1=（0.6,0.3,0.1）T，所以A1是一致阵；再看
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image042.gif"> 
　　求得 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image044.gif"> =3.01,W2=(0.588,0.322,0.09)T，所以A2不是一致阵。而且 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image046.gif"> 越大于n，A不一致的程度将越大，反之亦然。
　　所谓一致性指标，是指用来衡量判断矩阵不一致程度的数量指标。记作CI，定义为
　　 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image048.gif"> （6.1）
　　显然，CI=0 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image050.gif"> 判断矩阵是一致阵。CI的值越大，判断矩阵不一致程度越严重。
　　那么，判断矩阵不一致程度在什么范围内，层次分析法仍然可以使用呢？为此，我们引入随机一致性指标。

yunjiang · 发表于 2004-7-22 10:49:51

对于固定的n，随机地从 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image052.gif"> 中任取一数作为aij，并构成正互反阵<

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　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image054.gif"> <

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　　这样的 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image056.gif"> 是最不一致的，设其最大特征值为 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image058.gif"> ，则定义<

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　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image060.gif"> 
　　为随机一致性指标。当随机一致性比例
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image062.gif"> （6.2）
　　时，A的不一致性仍可接受，否则，必须调整判断矩阵。
　　另外，为了检验方便，我们给出一致性指标RI对各阶矩阵的相应数值如表6-2。
　　表6-2 随机一致性指标
<TABLE medium none; BORDER-TOP: medium none; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt" cellSpacing=0 cellPadding=0 align=center border=1><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 26: " vAlign=top ="35">n</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">1</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">2</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">3</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">4</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">5</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">6</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">7</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">8</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">9</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">10</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 0.5pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 26: " vAlign=top ="35">11</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">RI</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">0</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">0</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">0.58</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">0.9</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">1.12</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">1.24</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">1.32</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">1.41</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">1.45</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">1.49</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 26: " vAlign=top ="35">1.51</TD></TR></TABLE>
　　对于矩阵A利用（6.1）、（6.2）式和表6-2进行的检验称为一致性检验。

yunjiang · 发表于 2004-7-22 10:50:04

前面我们已经求出问题的各判断矩阵，下面求它们的最大特征值、特征向量、一致性指标和随机一致性比例。<

>
　　判断矩阵G-C：<TABLE medium none; BORDER-TOP: medium none; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt" cellSpacing=0 cellPadding=0 align=center border=1><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 60: " ="80"><H1>G<

></H1></TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 60: " ="80"><

>C1</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 60: " ="80">C2</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 60: " ="80">C3</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 60: " ="80"><H1>W</H1></TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-top-alt: solid windowtext .5pt; 60: " ="80">C1C2C3</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 60: " ="80">153</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 60: " ="80">1/511/3</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 60: " ="80">1/331</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 60: " ="80">0.1050.6370.258</TD></TR></TABLE>　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image064.gif"> <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image066.gif"> 
　　故通过一致性检验，从而上述W可作为权向量。又判断矩阵 C1-P：<DIV align=left></DIV><TABLE medium none; BORDER-TOP: medium none; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt" cellSpacing=0 cellPadding=0 align=center border=1><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 35: " ="48">C2</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 37: " ="50">P1</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 40: " ="53">P2</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 79: " ="105">W</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-top-alt: solid windowtext .5pt; 35: " ="48">P1</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 37: " ="50">1</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-top-alt: solid windowtext .5pt; 40: " ="53">3</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 79: " ="105">0.75</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; 35: " ="48">P2</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; 37: " ="50">1/3</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; 40: " ="53">1</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; 79: " ="105">0.25</TD></TR></TABLE>　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image068.gif"> 
　　判断矩阵C2-P：<DIV align=left></DIV><TABLE medium none; BORDER-TOP: medium none; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt" cellSpacing=0 cellPadding=0 align=center border=1><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 35: " ="48">C2</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 37: " ="50">P2</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 40: " ="53">P3</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 79: " ="105">W</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-top-alt: solid windowtext .5pt; 35: " ="48">P2</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 37: " ="50">1</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-top-alt: solid windowtext .5pt; 40: " ="53">1/5</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 79: " ="105">0.167</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; 35: " ="48">P3</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; 37: " ="50">5</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; 40: " ="53">1</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; 79: " ="105">0.833</TD></TR></TABLE>
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image069.gif"> 
　　判断矩阵C3-P： <TABLE medium none; BORDER-TOP: medium none; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt" cellSpacing=0 cellPadding=0 align=center border=1><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 35: " ="48">C3</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 37: " ="50">P1</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 40: " ="53">P2</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 79: " ="105">W</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-top-alt: solid windowtext .5pt; 35: " ="48">P1</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 37: " ="50">1</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-top-alt: solid windowtext .5pt; 40: " ="53">2</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 79: " ="105">0.667</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; 35: " ="48">P2</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; 37: " ="50">1/2</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; 40: " ="53">1</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; 79: " ="105">0.333</TD></TR></TABLE><DIV align=left></DIV>
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image070.gif">

yunjiang · 发表于 2004-7-22 10:50:18

<

>　　4．层次总排序及其一致性检验<

>
　　层次单排序后，还需要进行总排序，即计算同一层次所有因素对于最高层（总目标）相对重要性的排序权值，称为层次总排序。这一过程是由最高层到最底层逐层进行的。结合本例说明如下。<

>
　　将G-C矩阵的权向量W=（0。105，0。637，0。258）列于表6-3的第一行，再将 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image072.gif"> 矩阵的权向量按列依次列于相应的列上。则层次P的总排序按以下公式计算为
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image074.gif"> 
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image076.gif"> 
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image078.gif"> 
　　即得到表中最后一列。
　　表6-3 层次P的总排序<TABLE medium none; BORDER-TOP: medium none; MARGIN-LEFT: 14.4pt; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt" cellSpacing=0 cellPadding=0 align=center border=1><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 1pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 91: " rowSpan=2 ="122">层次P</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 1pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; 151: " colSpan=3 ="202">层次C</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: windowtext 1pt solid; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 117: " rowSpan=2 ="156">层次P的总排序</TD></TR><TR><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; mso-border-left-alt: solid windowtext .5pt; 52: " ="70">C10。105</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 45: " ="60">C20。637</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 0.5pt solid; 54: " ="72">C30。258</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-top-alt: solid windowtext .5pt; 91: " ="122">P1</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 52: " ="70">0.75</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-top-alt: solid windowtext .5pt; 45: " ="60">0</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; 54: ; mso-border-top-alt: solid windowtext .5pt" ="72">0.667</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; 117: " ="156">0.251</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; 91: " ="122">P2</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; 52: " ="70">0.25</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; 45: " ="60">0.167</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; 54: " ="72">0.333</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; mso-border-left-alt: solid windowtext .5pt; 117: " ="156">0.218</TD></TR><TR><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 1pt solid; 91: " ="122">P3</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 1pt solid; mso-border-left-alt: solid windowtext .5pt; 52: " ="70">0</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 1pt solid; 45: " ="60">0.833</TD><TD windowtext 0.5pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 1pt solid; 54: " ="72">0</TD><TD medium none; PADDING-RIGHT: 5.4pt; BORDER-TOP: medium none; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: medium none; PADDING-TOP: 0cm; BORDER-BOTTOM: windowtext 1pt solid; mso-border-left-alt: solid windowtext .5pt; 117: " ="156">0.531</TD></TR></TABLE>　　层次总排序也要进行一致性检验。检验是从最高层到最底层进行的。设P层中的某因素对Cj的单排序的一致性指标为CI，随机一致性指标是RI，则P层总排序随机一致性比例为
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image080.gif"> ，
　　其中 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image082.gif"> 是W的第 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image084.gif"> 个分量。同样，当CR<0.1时，认为层次总排序结果具有满意的一致性。本例的总排序的一致性检验：
　　CI=0.105×0+0.637×0+0.258×0=0。
　　故　　CR=0 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image086.gif"> 0.1
　　从而对于工厂合理使用企业利润，促进企业发展，所考虑的三种方案相对优先排序为：P3>P1>P2，利润分配比例为：引进新设备应占53.1%，用于发奖金占25.1%，用于改善福利事业应占21.8%。

yunjiang · 发表于 2004-7-22 10:50:34

<

>　　三、关于最大特征值和特征向量的近似计算<

>
　　我们看到，当n较大时，计算 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image088.gif"> 和W很麻烦。因此，在计算判断矩阵的最大特征值和特征向量时可以采取近似计算的方法。现介绍几个方法供选用。<

>
　　1．方根法
　　计算的主要步骤：
　　（1）计算判断矩阵A的每一行元素乘积
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image090.gif"> 。
　　（2）计算Mi的n次方根
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image092.gif"> 
　　（3）将 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image094.gif"> 标准化为
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image096.gif"> 
　　则 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image098.gif"> 为所求特征向量。
　　（4）计算最大特征值
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image100.gif">
　　例如对本例中的判断矩阵G-C，计算W的近似值过程：
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image102.gif"> 
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image104.gif"> 
　　即 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image106.gif"> 
　　再求最大特征值的近似值
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image108.gif">

yunjiang · 发表于 2004-7-22 10:50:49

2．和法<

>
　　计算的主要步骤：设A是判断矩阵。<

>
　　（1）将A的每一列标准化<

>
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image110.gif"> 
　　（2）将 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image112.gif"> 按行求和得
　　<img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image114.gif"> 
　　（3）将 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image116.gif"> 标准化为 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image118.gif"> ，则
　　W=（W1,W2,…，Wn） <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image120.gif"> 
　　为近似特征向量。
　　（4）计算 <img src="http://202.205.160.49:8080/media_file/rm/ip3/zhangxh/2004_03_01/sxjm_28/htm/sxjm_28.files/image122.gif"> 
　　在本讲的最后，将层次分析法的基本步骤归纳如下：
　　1．建立层次结构模型 在深入分析实际问题的基础上，将有关的各个因素按照不同属性自上而下地分解成若干层次。同一层的诸因素从属于上一层的因素或对上层因素有影响，同时又支配下一层的因素或受到下层因素的作用。最上层为目标层，通常只有1个因素，最下层通常为方案或对象层，中间可以有1个或几个层次，通常为准则或指标层。当准则过多时（譬如多于9个）应进一步分解出子准则层。
　　2．构造成对比较阵 从层次结构模型的第2层开始，对于从属于（或影响及）上一层每个因素的同一层诸因素，用成对比较法和1-9比较尺度构造成对比较阵，直到最下层。
　　3．计算权向量并做一致性检验 对于每一个成对比较阵计算最大特征根及对应特征向量，利用一致性指标，随机一致性指标和一致性比率做一致性检验。若检验通过，特征向量（归一化后）即为权向量；若不通过，需重新构造成对比较阵。
　　4．计算组合权向量并做组合一致性检验 计算最下层对目标的组合权向量，并做组合一致性检验。若检验通过，则可按照组合权向量表示的结果进行决策，否则需重新考虑模型或重新构造那些一致性比率CR较大的成对比较阵。

coeeoc · 发表于 2004-8-10 03:31:32

谢谢

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第二十八讲 层次分析法简介

第二十八讲层次分析法简介