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 rowcolumn 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 rowcolumn 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.

Negation of Pythagorean Fuzzy Number Based on a New Uncertainty Measure Applied in a Service Supplier Selection System The Pythagorean fuzzy number (PFN) consists of membership and nonmembership as an extension of the intuitionistic fuzzy number. PFN has a larger ambiguity, and it has a stronger ability to express uncertainty. In the multicriteria decisionmaking (MCDM) problem, it is also very difficult to measure the ambiguity degree of a set of PFN. A new entropy of PFN is proposed based on a technique for order of preference by similarity to ideal solution (Topsis) method of revised relative closeness index in this paper. To verify the new entropy with a good performance in uncertainty measure, a new Pythagorean fuzzy number negation approach is proposed. We develop the PFN negation and find the correlation of the uncertainty measure. Existing methods can only evaluate the ambiguity of a single PFN. The newly proposed method is suitable to systematically evaluate the uncertainty of PFN in Topsis. Nowadays, there are no uniform criteria for measuring service quality. It brings challenges to the future development of airlines. Therefore, grasping the future market trends leads to winning with advanced and highquality services. Afterward, the applicability in the service supplier selection system with the new entropy is discussed to evaluate the service quality and measure uncertainty. Finally, the new PFN entropy is verified with a good ability in the last MCDM numerical example.