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Pairwise comparisons are a well-known method for modelling of the subjectivepreferences of a decision maker. A popular implementation of the method isbased on solving an eigenvalue problem for M - the matrix of pairwisecomparisons. This does not take into account the actual values of preference.The Heuristic Rating Estimation (HRE) approach is a modification of this methodin which allows modelling of the reference values. To determine the relativeorder of preferences is to solve a certain linear equation system defined bythe matrix A and the constant term vector b (both derived from M). The articleexplores the properties of these equation systems. In particular, it is proventhat for some small data inconsistency the A matrix is an M-matrix, hence theequation proposed by the HRE approach has a unique strictly positive solution.

Notes on the existence of solutions in the pairwise comparisons method using the Heuristic Rating Estimation approach