Weighted Geometric Mean

1. Verify weight vector: w_j ≥ 0 for all j and Σ_{j=1}^n w_j = 1. Normalize if needed.
$j = 1 \sum n w_{j} = 1, w_{j} \geq 0 \forall j$
2. Compute the Weighted Geometric Mean: multiply each value raised to its weight.
$W GM (x_{1}, \dots, x_{n}; w) = j = 1 \prod n x_{j}^{w_{j}} = x_{1}^{w_{1}} \cdot x_{2}^{w_{2}} \dots x_{n}^{w_{n}}$
3. In group AHP context: aggregate K expert pairwise matrices A^(k) via element-wise WGM to obtain the group matrix Ā.
$\overset{a}{ˉ}_{ij} = W GM (a_{ij}^{(1)}, \dots, a_{ij}^{(K)}; w) = k = 1 \prod K (a_{ij}^{(k)})^{w_{k}}$

1. Verify weight vector: w_j ≥ 0 for all j and Σ_{j=1}^n w_j = 1. Normalize if needed.