# How are priorities calculated using paired comparisons?

Decision Lens uses the Analytic Hierarchy Process (AHP) methodology for establishing criteria priorities using trade off analysis.   AHP enables the efficient organization and evaluation of complex decision problems for the best outcome. It brings together multiple stakeholders in a decision who may have multiple and competing objectives; uses a hierarchy to relate the objectives; and then uses pairwise comparisons to weight the objectives in relation to one another. The AHP establishes a set of quantified priorities to guide evaluation and selection of alternatives in a rigorous, quantified way. Decision-makers compare the relative importance of elements in a pairwise fashion, each one to each other one. These judgments are entered into a matrix of pairwise comparisons, where each entry of the matrix is a ratio of the relative importance of the row element to the column element.

The calculations under which AHP operates then derive the priorities for all of the elements, with some allowance for inevitable inconsistency. If every judgment is consistent with every other judgment, it is easy to calculate the priorities for each element by simply adding across each row and normalizing. This vector of priorities is called the “Eigenvector”. However, most judgments will have a certain amount of inconsistency. The way to deal with this is to raise the matrix to powers (multiply it against itself) until the priorities implied by adding across each row converge. The Decision Lens algorithm raises the matrix to the 32nd power, which is sufficient to create a convergence of priorities. We then add across each row and normalize as before .

To calculate the inconsistency of the matrix, we first add down each column of the matrix, then multiply each element of that by the corresponding element of the priority vector gained from raising the matrix to powers, then finally sum those values. If we call the result y, The consistent ratio is then: (y - #elements in matrix)/(#elements in matrix – 1). The consistency index is this value divided by the consistency ratio of a perfectly random matrix of the same size, and provides a measure of the consistency or lack thereof in judgments. Perfect consistency is not realistic to expect in the Decision Lens process, but the inconsistency index value generally should not be above 10%.