Bonferroni Mean (BM)

1. 1. Verify n ≥ 2, p ≥ 0, q ≥ 0. If weights supplied, verify Σ w_j = 1.

   $$n\ge 2,p\ge 0,q\ge 0$$

2. 2. Compute the unweighted Bonferroni Mean BM^{p,q}: average all ordered pairs (i,j) with i≠j of a_i^p × a_j^q, then take the (p+q)-th root.

   $$B{M}^{p,q}(a_1,\dots ,a_n)={\left(\frac{1}{n(n-1)}\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{a}_i^p{a}_j^q\right)}^{\frac{1}{p+q}}$$

3. 3. Weighted Bonferroni Mean WBM^{p,q}: replace the uniform average by a weighted double sum. Weights w=(w_1,...,w_n) with Σw_j=1.

   $$WB{M}^{p,q}(a_1,\dots ,a_n)={\left(\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n\frac{w_i{w}_j}{1-w_i}{a}_i^p{a}_j^q\right)}^{\frac{1}{p+q}}$$

4. 4. Rank alternatives by BM (or WBM) score in descending order.

   $$\text{rank}(i)\propto -B{M}^{p,q}(a_{i1},\dots ,a_{in})$$