Deriving priorities from pairwise comparison matrices with a novel consistency index
Abstract It is important to measure the inconsistency level of a pairwise comparison matrix (PCM) and derive the priority vector in the analytic hierarchy process (AHP). In the present study, a new consistency index is proposed by using the cosine similarity measures of two row/column vectors in a PCM. It is called as the double cosine similarity consistency index (DCSCI) since the row and column vectors are all considered. Some interesting properties of DCSCI are investigated and the thresholds for inconsistency tolerance level are discussed in detail. Then following the idea of DCSCI, we provide a new method for obtaining the priority vector from a PCM. Through maximizing the sum of the cosine similarity measures between the priority vector and the row/column vectors, the priority vector is derived by solving the constructed optimization problem. By analyzing the proposed double cosine similarity maximization (DCSM) method, it is found that the existing cosine maximization (CM) method can be retrieved. By carrying out numerical examples, some comparisons with the existing methods show that the proposed index and method are effective and flexible.