Logarithmic Normalization - log-ratio column normalisation for multiplicative aggregation contexts

Overview

Logarithmic normalisation is suitable when performance values span several orders of magnitude (multiplicative structure). Each benefit column sums to 1. Requires all x_ij > 0.

Output

normalized matrix, higher is better

Data

Crisp, complete numeric matrix

Size

2+ alternatives, 3-12 criteria works best

Used for

Preprocessing

How it works

1. 1

   Benefit criterion j: r_ij = ln(x_ij) / ln(Π_k x_kj) = ln(x_ij) / Σ_k ln(x_kj). Cost criterion j: first compute benefit-normalised values, then apply r_ij = (1 − r_ij^benefit) / Σ_k (1 − r_kj^benefit).

   $$r_{ij}=\frac{\ln x_{ij}}{\ln {\prod }_{k=1}^m{x}_{kj}}=\frac{\ln x_{ij}}{{\sum }_{k=1}^m\ln x_{kj}},j\in J^{+};r_{ij}=\frac{1-r_{ij}^{+}}{{\sum }_k(1-r_{kj}^{+})},j\in J^{-}$$

   Zavadskas & Turskis 2008, p.306 Eqs.(1)-(2) (pending PDF page verification)

Edge cases and pitfalls

• x_ij ≤ 0 is undefined. If data includes zeros or negatives, shift by a constant before applying.

Works with

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