Consistency of Pair-wise Comparison Matrix with Fuzzy Elements

 This paper deals with measuring inconsistency and incompatibility of pair-wise comparison matrix with fuzzy elements. Here we deal with some properties of such pair-wise comparisons, particularly, reciprocity consistency and compatibility. Moreover, we show how to measure it defining two new indices. The first index of inconsistency is based on the classical concept of consistency, however, instead of principle eigenvector method used in AHP we apply the logarithmic least squares method. Defining the second index of incompatibility we look for a consistent matrix in the form of the ratio matrix with the maximal membership grade “as close as possible” to the original fuzzy matrix. This leads to solving a nonlinear optimization problem which can be transformed to a sequence of LP ones. We compare properties and application areas of both indices. The first index FI is suitable for non-interactive elements of fuzzy matrices, particularly, when uncertainty of individual elements of the matrix can be reflected/measured. The second index GI is appropriate for interactive elements of fuzzy matrices. By GI a measure of compatibility of fuzzy matrix with the closest consistent matrix is expressed. Illustrating examples and simulations are supplied to characterize the concepts and derived properties.