Normalizing criteria

From my doctoral course

This note comes from the doctoral course “Multi-Criteria Decision Making”. I rewrote the explanations in my own words and recomputed the numerical examples. Main source of the course: Asgharpour, Multi-Criteria Decision Making (University of Tehran).

Why normalise?

Price is in millions, battery in hours and beauty a score from 1 to 5. These cannot be added or multiplied directly. Normalisation turns each column into a unitless number, usually between 0 and 1. Non-compensatory methods usually do not need it, since each attribute is measured separately.

Example: one attribute, four alternatives

Values: $x=(50,70,40,90)$.

Method	Formula	Result
Linear (positive attribute)	$x_{ij}/\max x_j$	0.56, 0.78, 0.44, 1.00
Linear (negative attribute)	$\min x_j/x_{ij}$	0.80, 0.57, 1.00, 0.44
Euclidean norm	$x_{ij}/\sqrt{\sum x_{ij}^2}$	0.38, 0.54, 0.31, 0.69 (norm ≈ 130.8)
Ratio (sum)	$x_{ij}/\sum x_{ij}$	0.20, 0.28, 0.16, 0.36
Min-max (positive)	$(x-\min )/(\max -\min )$	0.20, 0.60, 0.00, 1.00
Min-max (negative)	$(\max -x)/(\max -\min )$	0.80, 0.40, 1.00, 0.00
Normal distribution	$(x-\mu )/\sigma$	−0.65, 0.39, −1.17, 1.43

Character of each method

• Linear: the best value is always 1 but the worst is not zero. For a negative attribute there are two ways: the inverse (above) or $1-x/\max$, used when all attributes are negative.
• Norm: all values stay between 0 and 1 and the direction of the attribute (positive or negative) does not change; it is the basis of TOPSIS.
• Ratio: the column sum is always 1; for a negative attribute, take the ratio of the inverses.
• Min-max: best is 1 and worst is 0; it uses the full range but is very sensitive to outliers.
• Normal distribution: with many alternatives we can assume a normal distribution; values lie around zero, usually between −3.5 and +3.5.
• Centre-seeking: when the “most desirable” value is neither the maximum nor the minimum but a midpoint (for example the right level of risk-taking for a job). Each alternative is measured by its distance from that desirable point.

A warning

The choice of normalisation method can change the final ranking. I think a good analysis runs with at least two methods and, if the rankings differ, investigates why.

Related: Introduction: what is multi-attribute decision making? and Weighting criteria.

normalizationmulti_criteria_decision_making