Finally Consistency Ratio CR is a result of the division of our consistency index by the random index (consistency index of the totally random matrix – those values have been calculated up to 15th dimension): Where 4 stands for the dimension of the comparison matrix. First step in calculation of the consistency ratio is the calculation of the consistency index (which according to Saaty is the max eigenvalue of the comparison matrix) which is done as follows: The final step in obtaining the preference of choices is to calculate the consistency ratio of the selection, which according to Saaty should be less or equal to 10% (according to his famous rule of thumb the consistency or uncertainty of the decision should be order of magnitude less than decision). So the resulting priority matrix tells us that with respect to the criterion 3 (sightseeing) Rome has the highest priority. In the next step we normalize each matrix element by the sum of elements in each column and we calculate the sum for each row:Īfter that we normalize the sum of the rows, which yields: Our preferences are summarized in the matrix below: In this example we judge Rome to be more preferable that Barcelona, much more preferable than NY and we find that SF is as equally less preferable to Rome as Barcelona. In the matrix below A=Barcelona, B=Rome, C=NY, D=SF. Note that number of judgements that are required diminishes with every selection: user starts with 3 unique comparisons for Barcelona, then only 2 for Rome (because the Rome to Barcelona is inverse of Barcelona to Rome) and only one for SF. We perform a pairwise comparison between Barcelona and the rest of spots, then Rome and finally NYC. We want evaluate our vacation spots with respect to sightseeing. Consistency of the selection and the judgements involved is derived from the estimation of the eigenvalue of the decision matrix. The choice is then calculated using linear algebra transformation of the decision matrix. After carrying out pair-wise comparisons on criteria with respect to the goal and then choices with respect to each criterion one obtains a matrix representation of the model (aka the decision matrix). Scale ranges from 1 (equal weight) to 9 (where one option is extremely more preferable than the other).
Saaty proposed 9 grade value scale to compare choices in the model. Selection of preferences for both criteria and choices is done via pair-wise comparison. According to AHP to make a rational decision you should build a model as shown on the figure below in which the goal rests on selection criteria which depend on choices. Let me illustrate the AHP with the following example: Imagine you are deciding on a vacation spot out of 4 choices A,B,C,D (for instance SF, NY, Rome and Barcelona) and using three criteria 1,2,3 (for example cost, night life and sightseeing). The AHP was proposed by Thomas Saaty who developed it while at the US Disarment Agency and the Wharton School, UPenn.
The Analytic Hierarchy Process is a solution / theory that is part of the Operations Research - a branch of economics that attempts to apply modeling and statistics to decision selection and execution of decisions within the enterprise.