Abstract
Summary: In this study, we consider some consistency indices for a pair-wise comparison matrix. These indices derive from measuring the consistency gaps in the comparison matrix from a idempotent matrix. The consistency of each pair-wise comparison is defined by a cardinal relationship, then this cardinal relationship can be extended to consistency relationships in the comparison matrix. Some consistency indices are shown that those distance functions are a type of norms or a entropy function. By simulations, the idempotent relationship shows more sensitive than the residual relationship for consistency. 1. Consistency of pair-wise comparison There are many types of the consistency of pair-wise comparison matrix, which are defined by a distance between the pair-wise comparison matrix and the model of AHP. Because the distance functions can be generated as matrix norms of the difference between matrices. Basically the consistency is introduced for each element of the comparison matrix as a cardinal relationship. Then the consistency of each element is defined by follows. aij = aik ⋅ akj , for k =1, 2,L , n In this definition, the element must be satisfied above equation for all i and , but it is not required to satisfy the equation for all . Then there is the next theorem for the consistency of comparison matrix. aij k j
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