Abstract
The tolerance analysis on a matrix is highly complicated and has been widely discussed in recent years, especially on a positive reciprocal matrix (PRM) for an evaluation decision. With the requirement of the consistency and reciprocal, tolerance analysis of a PRM can be performed in an eigensystem. The relation between the matrix and the resultant eigenvector that can be regarded as weights of importance of the attributes in an evaluation problem reveals the high complexity yet interesting interpretation in tolerance analysis. Therefore, the 1st issue of our tolerance analysis is given a PRM and the resultant weighting vector, what is the largest tolerance levels of PRM with respective to the vector under the acceptable consistent requirement? Since tolerance analysis on a given PRM is not necessarily significant in relation to the weighting vector, therefore our 2nd issue is given a consistent PRM, what would be the largest tolerance levels such that the required effectiveness will be obtained. After the tolerance levels are acquired, we further propose the possible applications for constructing a consistent PRM with/without the same weight in an eigensystem from the given effective tolerance levels. The procedure has been demonstrated by an illustrative example. The result shows that the tolerance analysis provides the sufficient information to make a better decision.