Min-Max Normalization - linear rescaling of each criterion column to [0, 1]

Overview

Min-max normalisation maps every criterion to [0,1] with 1 = best performer and 0 = worst performer in the current dataset. The result is dataset-dependent: adding or removing alternatives changes all normalised values. Use before methods that require [0,1] inputs (e.g. EDAS, CODAS, MARCOS).

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

   For each criterion j, compute the column minimum x_j^min and maximum x_j^max across all m alternatives.

   $$x_j^{\min }=\underset{i=1}{\overset{m}{\min }}{x}_{ij},x_j^{\max }=\underset{i=1}{\overset{m}{\max }}{x}_{ij}$$

   Hwang & Yoon 1981, p.130 (range computation; pending PDF page verification)

2. 2

   Apply direction-aware linear rescaling. Benefit criteria: r_ij = (x_ij − x_j^min)/(x_j^max − x_j^min). Cost criteria: r_ij = (x_j^max − x_ij)/(x_j^max − x_j^min). Result: r_ij ∈ [0,1], with 1 always assigned to the best performance and 0 to the worst.

   $$r_{ij}=\{\begin{array}{ll}\frac{x_{ij}-x_j^{\min }}{x_j^{\max }-x_j^{\min }}& j\in J^{+}\text{ (benefit)}\\ \frac{x_j^{\max }-x_{ij}}{x_j^{\max }-x_j^{\min }}& j\in J^{-}\text{ (cost)}\end{array}$$

   Hwang & Yoon 1981, p.130 Eqs.(4.1a)-(4.1b) (pending PDF page verification)

Edge cases and pitfalls

• Constant column (x_j^max = x_j^min): division by zero - either remove the criterion or add a small ε to the range.

• Rank reversal risk: adding a new extreme alternative shifts all normalised values even when relative performance among existing alternatives is unchanged.

Works with

How to cite