Linear Sum Normalization - column-sum division (probability / stochastic normalisation)

Overview

Each column sums to 1 - r_ij can be interpreted as the fraction of total criterion performance captured by alternative i. Used in MOORA (ratio system) and ENTROPY weighting.

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 = x_ij / Σ_k x_kj. Cost criterion j: first invert (y_ij = 1/x_ij), then r_ij = y_ij / Σ_k y_kj. Each column sums to 1.

   $$r_{ij}=\{\begin{array}{ll}{x}_{ij}/{\sum }_k{x}_{kj}& j\in J^{+}\\ (1/x_{ij})/{\sum }_k(1/x_{kj})& j\in J^{-}\end{array}$$

   Zavadskas et al. 1994 (column-sum normalisation; pending PDF page verification)

Edge cases and pitfalls

• All x_ij must be > 0 for the cost reciprocal variant.

Works with

How to cite