Vector (L2) Normalisation

This card describes the second entry for an operation also separately recorded in the DecisionMind library as "Vector Normalisation (Scaling by the Euclidean Column Norm)." Both do exactly the same computation: they divide every value in a column by the number that is the square root of the sum of the squares of that column's values. This operation brings criteria measured in different units (Mach, million dollars, a 1-9 score) onto the same scale, and is the standard pre-processing step for methods that rely on Euclidean distance, such as TOPSIS. Hwang and Yoon published this step in 1981 as part of TOPSIS's own definition.

DecisionMind's own audit record states that this manifest is "the same canonical operation as NORM-VECTOR" and flags this as "a duplication awaiting Block A taxonomy review." That is, these two records entered DecisionMind at different times historically, under different catalogue codes (this record 1.7.1/3.2.1, the other 1.1.3/3.2.1) and with different maturity labels (this record "canonical," the other "verified"), and have not yet been merged. The practical consequence for the user: whichever record is used, the calculation is identical; it simply appears as two separate pages in the library.

The philosophy of this operation is to define a criterion's value not "relative to the largest possible value" but "relative to the total magnitude within this set of alternatives." The values in a column are scaled not against a single point (such as the maximum value), but against the length of the vector all of them together form. This is a geometric outlook that treats every alternative as a point in a multi-dimensional space and fits that space onto a unit sphere. At the normalisation stage, no decision is made about which criterion is "lower is better" and which is "higher is better"; that decision is left to a later step, such as TOPSIS. The method is therefore a neutral scaler that carries no preference of its own.

The operation consists of two steps.

First, finding the column length. For every criterion column, the square root of the sum of the squares of its values is computed.

Second, division. Every value in the column is divided by this length. As a result, the sum of the squares of the normalised values in that column comes out to exactly 1.

These two steps form TOPSIS's first step and do not on their own establish a criterion's direction. The formulas behind each step are given on the DecisionMind method page; this card carries no formulas.

The output of this operation is not a decision but an intermediate table feeding into the next step. A normalised value (say, 0.57) does not on its own mean "good" or "bad"; it carries meaning only when compared against the other values in the same column. A column's normalised values having squares that sum to 1 is a check confirming the calculation ran correctly, not a measure of success. The normalised table cannot be compared against another set of alternatives, because column length is computed from that set's own values.

Thus instead of writing:

"The normalised value is 0.57, meaning this alternative is 57 per cent successful"

the report should read:

"This value is a scaled figure carrying the criterion's relative share within this set of alternatives; it is not a percentage in its own right"

It works with crisp data: one number per cell, no empty cells. If all of a column's values are zero, the division becomes undefined; that criterion should be removed from the table or re-measured. It works mathematically with negative values, but whether the next step is compatible with negative input should be checked.

This is a building-block card and has no extension of its own (a fuzzy, grey or intuitionistic version) in DecisionMind. Five other normalisation options do the same job: linear scaling by the maximum, linear scaling by the sum, logarithmic normalisation, min-max normalisation and z-score normalisation; each is described on its own building-block card. A minimum of two alternatives and one criterion is sufficient.

This operation is suitable where you do not want a single extreme alternative to dominate a column on its own, and where you are preparing data for a method that relies on Euclidean distance, such as TOPSIS, because the divisor is the joint contribution of all alternatives, not a single extreme value.

It should not be used where the normalised value is wanted to be read intuitively, in a way such as "let the best alternative receive 1.0"; that reading is a property of linear scaling by the maximum, not of this operation. If the data is already in the same unit and there is no difference in magnitude between criteria, this step is unnecessary.

Numerical criteria in different units, feeding into a Euclidean-based method → this operation

An intuitive reading such as "let the best alternative receive 1.0" is wanted → linear scaling by the maximum

The data is sensitive to extreme values and relative position matters → z-score normalisation

Not normalisation but a ranking directly is needed → full methods such as TOPSIS, VIKOR

Its greatest strength is that the divisor is not a single extreme value but the joint magnitude of all alternatives; this gives a result less sensitive to a single extreme alternative than linear scaling by the maximum does. Its computational burden is small, it works without difficulty with negative values, and it is, alongside TOPSIS, the most extensively tested form of normalisation.

Because a normalised value carries no intuitive reference point (such as 1.0), it is hard to interpret in reporting (Jahan and Edwards, 2015). When the set of alternatives changes (a new alternative is added or removed), column length, and with it the entire normalised table, changes. Which form of normalisation is chosen can change the final ranking; this operation and linear scaling by the maximum can give the same data a different order (Vafaei, Ribeiro and Camarinha-Matos, 2018; Milani et al., 2005). It disregards the shape of the distribution within a column, such as skewness or extreme values.

The most common mistake is reading a normalised value directly as a percentage; this value is meaningful only within that set of alternatives. A second mistake is choosing a form of normalisation without justification and assuming the result is unaffected by this choice. A third mistake is feeding a column whose values are all zero into this operation unnoticed. A fourth mistake is adding an alternative after the analysis is finished and being surprised that the normalised table has changed.

The governing principle is this:

This operation makes criteria comparable, but which form of normalisation is chosen is itself a decision, and this decision can change the final ranking; the report must show this sensitivity.