Achieving matrix consistency in AHP through linearization

Matrices used in the analytic hierarchy process (AHP) compile expert knowledge as pairwise comparisons among various criteria and alternatives in decision-making problems. Many items are usually considered in the same comparison process and so judgment is not completely consistent – and sometimes the level of consistency may be unacceptable. Different methods have been used in the literature to achieve consistency for an inconsistent matrix. In this paper we use a linearization technique that provides the closest consistent matrix to a given inconsistent matrix using orthogonal projection in a linear space. As a result, consistency can be achieved in a closed form. This is simpler and cheaper than for methods relying on optimisation, which are iterative by nature. We apply the process to a real-world decision-making problem in an important industrial context, namely, management of water supply systems regarding leakage policies – an aspect of water management to which great sums of money are devoted every year worldwide.