Computing the Consistency Matrix

In the Analytic Hierarchy Process (AHP), computing the consistency matrix is crucial for verifying whether decision-makers maintain logical consistency when constructing judgment matrices. After experts or decision-makers derive the judgment matrix through pairwise comparisons, calculating the consistency matrix becomes essential for validating its reasonableness.

First, compute the maximum eigenvalue and its corresponding eigenvector of the judgment matrix, where the eigenvector represents the weight vector of various factors. This can be implemented using eigenvalue decomposition algorithms like the power iteration method or built-in functions such as numpy.linalg.eig() in Python. The mathematical formulation involves solving: [ CI = \frac{\lambda_{max} - n}{n - 1} ] where ( \lambda_{max} ) is the maximum eigenvalue and ( n ) is the matrix order.

Next, introduce the random consistency index (RI), typically obtained from predefined tables (e.g., RI values for n=3 to 10: [0.58, 0.90, 1.12, 1.24, 1.32, 1.41, 1.45, 1.49]). Finally, calculate the consistency ratio (CR): [ CR = \frac{CI}{RI} ] In code implementation, this requires conditional checking where if CR < 0.1, the judgment matrix is considered acceptably consistent; otherwise, the matrix must be revised through iterative pairwise comparisons until consistency requirements are met.

The computation of the consistency matrix ensures the scientific rigor and reliability of AHP, preventing analytical biases caused by inconsistent subjective judgments. Algorithmically, this process can be automated using while-loops with convergence checks for practical applications.