Implementing the Analytic Hierarchy Process (AHP) in Python: Step-by-Step Guide

The Analytic Hierarchy Process (AHP) is a decision‑making method that derives priority weights from a pairwise comparison matrix. The following Python function implements the core AHP calculations, including eigenvalue extraction, consistency index (CI) evaluation, and weight normalization.

<code>def ahp(judge_matrix):
    """ Compute AHP model
    Parameters:
        judge_matrix: pairwise comparison matrix
    Returns:
        ahp_weights: weight vector
    """
    judge_matrix = np.array(judge_matrix, dtype=np.float64)  # convert to numpy array
    n = judge_matrix.shape[0]  # matrix dimension
    eigenvalues, eigenvectors = np.linalg.eig(judge_matrix)  # eigenvalues and eigenvectors
    value_max = eigenvalues.max()  # maximum eigenvalue
    value_max_index = eigenvalues.argmax()  # index of max eigenvalue
    CI = (value_max - n) / (n - 1)  # consistency index
    RI_list = [0,0,0,0.58,0.9,1.12,1.24,1.32,1.41,1.45,1.49]  # random index list
    if CI / RI_list[n] > 0.1:
        return 'Judgement Matrix is NOT consistent'
    else:
        print('The Max Eigenvalue is ', value_max)
        ahp_weights = eigenvectors[:, value_max_index]
        ahp_weights = ahp_weights / ahp_weights.sum()  # normalize
        return ahp_weights
</code>

Example 1 uses an inconsistent matrix:

<code>weights = ahp(np.array([[1,2,3],[1/2,1,9],[1/3,1/9,1]]))
</code>

The function returns:

'Judgement Matrix is NOT consistent'

indicating that the pairwise comparisons violate the consistency threshold.

After adjusting the matrix to improve consistency:

<code>weights = ahp(np.array([[1,2,3],[1/2,1,1],[1/3,1,1]]))
</code>

The output shows an acceptable consistency:

The Max Eigenvalue is  (3.0182947072896287+0j)

and the resulting weight vector is:

array([0.54994561+0.j, 0.24021087+0.j, 0.20984352+0.j])

This demonstrates how the AHP implementation can be used to assess decision criteria, verify matrix consistency, and obtain normalized priority weights.