基于奇异值分解的权重计算、一致性检验与改进

利用奇异值分解和可能满意度方法，提出了一种新的层次分析法权重计算、一致性检验与改进过程．根据矩阵最优近似定理和Kullback—Leibler信息法则，指出奇异值分解所得结果是对决策偏好的最优近似，根据可能满意度方法，利用判断矩阵的最大特征值及其Frobenius范数，定义判断矩阵的可能度与满意度，分别考察一致性改进程度和相对原始判断矩阵的信息偏离程度，并将两者并合为一个能够全面衡量一致性改善效果的综合指标：判断矩阵的可能满意度．通过与加权几何平均改进方法相结合，在最大限度保留决策者原始判断信息条件下，逐步达到可接受的一致性．给出了改进算法的收敛性证明，并利用典型算例进行对比分析．

The Weight Calculation, Consistency Test and Improvement Based on Singular Value Decomposition

A new approach for analytical hierarchy process(AHP) weight determination,consistency test and improvement was proposed through singular value decomposition(SVD) and possibility-satisfiability method(PS).According to the theory of low rank approximation of matrix and Kullback-Leibler' information law,the result of SVD is the optimal approximation to decision-maker's preference.With the PS method,the possibility degree and satisfiability degree of the comparison matrix are defined by using the maximum eigenvalue and frobenius-norm of that matrix,which can examine the improvement extent and the degree of deviation of the improved matrix from the original one,respectively. A synthetical index,the possibility-satisfiability degree of the comparison matrix,is obtained by merging the above two measurements,which can evaluate the effect of the improvement generally.The acceptable consistency can be achieved gradually with the original preference information preserved furthest when combining the PS method and the traditional weighted geometric mean algorithm for improvement.Finally,the convergence of the proposed algorithm is proved theoretically,and its practicality is shown by the comparison with other methods through typical examples.