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Pairwise comparison matrices (PCMs) are inevitable tools in some important multiple-criteria decision-making methods, for example AHP/ANP, TOPSIS, PROMETHEE and others. In this paper, we investigate some important properties of PCMs which influence the generated priority vectors for the final ranking of the given alternatives. The main subproblem of the Analytic Hierarchy Process (AHP) is to calculate the priority vectors, that is, the weights assigned to the elements of the hierarchy (criteria, sub-criteria, and/or alternatives or variants), by using the information provided in the form of a pairwise comparison matrix. Given a set of elements, and a corresponding pairwise comparison matrix, whose entries evaluate the relative importance of the elements with respect to a given criterion, the final ranking of the given alternatives is evaluated. We investigate some important and natural properties of PCMs called the desirable properties, particularly, the non-dominance, consistency, intensity and coherence, which influence the generated priority vectors. Usually, the priority vector is calculated based on some well-known method, for example, the Eigenvector Method, the Arithmetic Mean Method, the Geometric Mean Method, the Least Square Method, and so forth. The novelty of our approach is that the priority vector is calculated as the solution of an optimization problem where an error objective function is minimised with respect to constraints given by the desirable properties. The properties of the optimal solution are discussed and some illustrating examples are presented. The corresponding software tool has been developed and demonstrated in some examples.