This function calculates indicator weights using the Analytic Hierarchy Process (AHP) method based on a pairwise comparison matrix. It computes weights using eigenvalue decomposition and calculates the Consistency Ratio (CR) to assess the logical consistency of the judgments.
Arguments
- pairwise
A square numeric matrix of pairwise comparisons where element [i,j] represents the relative importance of indicator i compared to indicator j. The matrix must be reciprocal (A[i,j] = 1/A[j,i]) with diagonal values equal to 1.
- indicators
Optional character vector of indicator names. If NULL, uses row names from the pairwise matrix or generates default names.
Value
A list with the following components:
- weights
Named numeric vector of normalized weights summing to 1
- CR
Consistency Ratio, a measure of logical consistency (should be < 0.1)
- lambda_max
Maximum eigenvalue of the pairwise matrix
Details
The AHP method uses eigenvalue decomposition to derive weights from pairwise comparisons. The Consistency Ratio (CR) is calculated as: $$CR = CI / RI$$ where CI (Consistency Index) = (lambda_max - n) / (n - 1) and RI is the Random Consistency Index that depends on matrix size.
A CR value less than 0.1 indicates acceptable consistency. Values exceeding 0.1 suggest that the pairwise judgments may be inconsistent and should be revised.
Examples
# Create a 3x3 pairwise comparison matrix
pairwise <- matrix(c(
1, 3, 5,
1/3, 1, 2,
1/5, 1/2, 1
), nrow = 3, byrow = TRUE)
# Calculate weights
result <- ahp_weights(pairwise, indicators = c("pH", "OM", "P"))
print(result$weights)
#> pH OM P
#> 0.6483290 0.2296508 0.1220202
print(result$CR)
#> [1] 0.003184998