Linear Max Normalization - division by column maximum (benefit) or column minimum over value (cost)

Overview

r_ij ∈ (0,1] for positive data. The best alternative always scores 1. Unlike min-max, the worst alternative does not necessarily score 0. Used in ARAS, COPRAS, WPM.

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

   Apply: benefit criterion j → r_ij = x_ij / x_j^max. Cost criterion j → r_ij = x_j^min / x_ij. Result: best alternative scores 1 in both cases.

   $$r_{ij}=\{\begin{array}{ll}{x}_{ij}/x_j^{\max }& j\in J^{+}\\ x_j^{\min }/x_{ij}& j\in J^{-}\end{array}$$

   Fishburn 1967, p.538 (ratio normalisation; pending PDF page verification)

Edge cases and pitfalls

• Cost variant requires x_ij > 0; zero or negative cost values are undefined.

Works with

How to cite