Methods · Normalisation
Vector Normalisation (Scaling by the Euclidean Column Norm)
Vector normalisation is a preparatory step that divides every column of a decision table by its own Euclidean length, making all criteria unit-free and comparable.
Base method's data type: Classical
What Is the Method?
If your decision table has speed in Mach, price in millions of dollars and reliability as an expert score from 1 to 9, you cannot simply sum these three columns or take their weighted average; the column with large figures overwhelms the one with small figures. Vector normalisation solves this. It divides the values in each column by the square root of the sum of their squares, so that every column ends up between 0 and 1 and unit-free. Its output is not an order or a score but a scaled table that feeds into the next step, be that weighting, building an ideal point, or measuring distance. Vector normalisation is therefore not a decision method in its own right, but a building block used inside methods such as TOPSIS. Hwang and Yoon defined it in 1981 as TOPSIS's standard pre-processing step.
DecisionMind holds this operation as two separate entries: NORM-VECTOR and its sibling card, Vector (L2) Normalisation. Both define the same formula from the same source (Hwang and Yoon, 1981); the only difference between them is their record history, their catalogue code and maturity label. DecisionMind's own audit note has flagged these two entries as "the same canonical operation, awaiting a taxonomy review to be merged." The practical upshot for a reader of this card is that whichever entry is chosen, the calculation is the same and the result does not change.
The Philosophy Behind It
The question behind vector normalisation is this: "is a score of 5 on a criterion meaningful relative to that criterion's largest possible value, or relative to the criterion's overall magnitude?" Vector normalisation chooses the second path. It scales a column not against a single largest value but against the length of the vector formed by all of that column's values. This is a geometric viewpoint: each alternative is a point in a multidimensional space, each criterion an axis, and normalisation fits this space onto a unit sphere. A philosophical property follows from this: normalisation makes no direction decision (whether a criterion is "higher is better" or "lower is better") during the process; that decision is left to the next step, the ranking method. Vector normalisation is thus a neutral scaler, carrying no preference of its own.
How It Works
The operation consists of two steps.
First, finding the column length. For every criterion column, the square root of the sum of the squared values is taken. This figure is the column's geometric "length"; the larger the values in the column, the larger this length grows.
Second, division. Every value in the column is divided by that column's length. The result is that the sum of the squares of every column's values comes to exactly 1; the columns are now unit-free and comparable with one another.
These two steps form the first step in TOPSIS, before weighting. Vector normalisation on its own does not determine a criterion's direction (whether lower or higher is better for price, say); that information is used in the next step. The formulas behind each step are given on the DecisionMind method page; this card carries no formulas.
How to Read the Output
The output of vector normalisation is not a decision but an intermediate table. A normalised value on its own (say, 0.71) does not mean "good" or "bad"; it gains meaning only when compared with the other values in the same column. Once a column's values are normalised, the sum of their squares is always 1; this is a check that the calculation was done correctly, not a measure of success. A normalised table cannot be compared with one from a different alternative set, because the column length is computed from that set's own values; change the alternative set and the whole column is rescaled.
Therefore, instead of writing:
"The normalised score is 0.71, meaning this alternative is 71 per cent successful on this criterion"
the report should read:
"This value is a scaled number carrying this criterion's relative weight within this alternative set; on its own it is not a percentage or a success rate"
Data Type and Inputs
Vector normalisation works with crisp data: one number per cell, no empty cells. Negative values cause no mathematical problem, but whether the next step, usually a ranking method, works with negative input must be checked. If every value in a column is zero, the division becomes undefined; in that case the criterion must be removed from the table or measured afresh.
This is a building-block card; DecisionMind holds no extension of it in its own right (fuzzy, grey, intuitionistic version). Alongside this operation DecisionMind holds five other normalisation options: linear division by the maximum, linear division by the sum, logarithmic normalisation, min–max normalisation and z-score normalisation. Each offers a different solution to the same problem, making criteria comparable, and each is described in its own building-block card. A minimum of two alternatives and one criterion is sufficient; there is no upper limit on the number of columns.
When to Use It, When Not To
Vector normalisation is suitable where you do not want a single alternative's extreme value to dominate an entire column, since the divisor is not one largest value but the combined contribution of every alternative. It fits naturally with Euclidean-distance-based methods such as TOPSIS, since both share the same geometric logic, a unit sphere and Euclidean distance.
The case where it should not be used arises when the decision-maker wants to interpret the normalised value directly, since, unlike linear division-by-maximum normalisation, it does not give an intuitive reading such as "the best alternative takes 1.0." Also, if the data is already in the same unit and there is no scale difference among the criteria, the normalisation step adds unnecessary complexity.
Numerical criteria in different units, feeding into a Euclidean-based method (TOPSIS) → vector normalisation
An intuitive reading such as "the best alternative takes 1.0" is wanted → linear division-by-maximum normalisation
The data is sensitive to extreme values and relative position matters → z-score normalisation
Not normalisation but a full ranking method is needed → complete methods such as TOPSIS, VIKOR
Strengths
Vector normalisation's most important advantage is that it keeps every alternative's contribution in balance; the divisor is not a single extreme value but a magnitude computed from the whole column, making it less sensitive to a single extreme alternative than linear division-by-maximum normalisation. The computational load is small and runs at the same speed for large tables too. It works without trouble on negative and positive values alike. It is the most tested and most applied form of normalisation used alongside TOPSIS.
Weaknesses
It has limitations. First, the normalised value has no intuitive reading; it carries no clear reference point such as "1.0" or "0," which makes reporting harder (Jahan and Edwards, 2015). Second, when the alternative set changes (a new alternative added or removed), the column length changes, affecting the entire normalised table and so the result. Third, which normalisation form is chosen can change the final ranking; vector normalisation and linear division-by-maximum can give the same data a different order (Milani et al., 2005; Vafaei, Ribeiro and Camarinha-Matos, 2018). Fourth, dividing the whole column by a single number disregards the shape of the distribution within the column, such as skewness or outliers.
Common Mistakes
The most common mistake is reading the normalised value directly as a percentage or success rate; this value is meaningful only within that alternative set. A second mistake is choosing a normalisation form (vector, linear maximum, min–max) without justification and assuming the result is unaffected by this choice; the cases below show this assumption is often wrong. A third mistake is trying to normalise a column whose values are all zero without noticing; this operation is undefined and the engine should raise an error. A fourth mistake is adding an alternative once the analysis is finished and being surprised the normalised table has changed; because the column length is computed from every alternative, each new alternative changes the table from the ground up.
The governing principle is this:
Vector normalisation makes criteria comparable, but which normalisation is chosen is also a decision, and this decision can change the final order; the report must show this sensitivity.
Cases
Each case opens with a decision table, describes in words what the method does to it, and shows how the result changes when the choice of normalisation form changes.
1. Engineering: Verifying division by column length
This case describes not a real institution's decision but DecisionMind engine's own validation example; three alternatives are scored on three criteria, all taken as "higher is better."
| Alternative | C1 | C2 | C3 |
|---|---|---|---|
| A1 | 3.0 | 5.0 | 4.0 |
| A2 | 5.0 | 3.0 | 2.0 |
| A3 | 4.0 | 4.0 | 3.0 |
| Direction | higher is better | higher is better | higher is better |
The method first finds each column's length: for C1, √(3²+5²+4²)=√50≈7.071; for C2, the same, ≈7.071; for C3, √(4²+2²+3²)=√29≈5.385. It then divides every value by its own column's length.
| Alternative | C1 (normalised) | C2 (normalised) | C3 (normalised) |
|---|---|---|---|
| A1 | 0.4243 | 0.7071 | 0.7428 |
| A2 | 0.7071 | 0.4243 | 0.3714 |
| A3 | 0.5657 | 0.5657 | 0.5571 |
The result reads as follows: the sum of the squares of every column's normalised values is exactly 1 (0.4243²+0.7071²+0.5657²=1.000); this is the check that the calculation ran correctly. Because A2's value on C1 (5.0) is the largest, its normalised C1 value is also the largest (0.7071); but because A2's value on C3 (2.0) is the smallest, its normalised C3 value is also the smallest (0.3714). Normalisation does not change the order, it only equalises the scale.
There is no hesitation here, since this is a scaling step, not a ranking; how it converts into a ranking is seen in the following cases.
In the report: "All three columns have been divided by their own Euclidean length; the sum of each column's normalised squares comes to exactly 1.000, confirming the calculation."
Source: This example is DecisionMind's validation example for the vector-normalisation engine; the formula rests on Hwang and Yoon's (1981) definition of TOPSIS, but this particular 3×3 table is not taken from the book, it was built to test the engine.
2. Logistics: Choosing a cargo vehicle for the fleet
A cargo company is comparing three vehicles to add to its fleet on three criteria: fleet-capacity score (weight 0.50), reliability score (weight 0.30) and fuel-efficiency score (weight 0.20). All three have been scored by expert assessment from 1 to 10, and all are "higher is better."
| Vehicle | Capacity (C1) | Reliability (C2) | Efficiency (C3) |
|---|---|---|---|
| Vehicle 1 | 4 | 3 | 8 |
| Vehicle 2 | 1 | 4 | 1 |
| Vehicle 3 | 4 | 7 | 2 |
| Weight | 0.50 | 0.30 | 0.20 |
The company first tries vector normalisation: each column is divided by its own Euclidean length (5.745 for C1; 8.602 for C2; 8.307 for C3), then multiplied by the weights and summed. The weighted total comes to 0.6454 for Vehicle 1, 0.6404 for Vehicle 3, 0.2506 for Vehicle 2; the order is Vehicle 1, Vehicle 3, Vehicle 2, but the gap between Vehicle 1 and Vehicle 3 is only 0.0050.
The fleet manager hesitates over how small this gap is and has the same data recalculated with linear division-by-maximum normalisation (dividing every value by the column's largest). This time the figures come to 0.8286 for Vehicle 1, 0.8500 for Vehicle 3, 0.3214 for Vehicle 2; the order reverses to Vehicle 3, Vehicle 1, Vehicle 2, and the gap widens to 0.0214. The same data, the same weights, only the normalisation form has changed, and the top place has changed hands.
In the report: "With vector normalisation, Vehicle 1 (0.6454) comes out ahead; but its gap to Vehicle 3 (0.6404) is as small as 0.0050, and the order turns in Vehicle 3's favour once linear division-by-maximum normalisation is used. The decision is sensitive to the choice of normalisation."
3. Museum Curation: Choosing a conservation system for artefacts
A museum's technical team will choose among three climate-control and humidity-management systems. The criteria are humidity-control precision, energy-consumption efficiency and ease-of-maintenance score; all are scored as "higher is better." The team computes a weighted total with vector normalisation, and two systems' scores come out very close to one another.
The team hesitates here: with the scores this close, they ask whether changing the normalisation form would change the order. As in the previous case, switching to linear division-by-maximum normalisation could plausibly swap the two systems' places, because which system is "best" on a given criterion (what the divisor looks at) and which system's overall magnitude stands out (what the vector norm looks at) are different questions. Rather than presenting a thin gap produced by a single normalisation form as a definite advantage, the team decides to run the calculation with both normalisation forms and add whether the result changes to its report.
In the report: "The scores of the two candidate systems are very close; this gap can change direction when the normalisation form is changed, and so the two systems have been placed on the shortlist together."
4. What Not to Do
The first error is reporting Case 2's score of 0.6454 as "Vehicle 1 is 65 per cent suitable"; this score only ranks these three vehicles relative to one another, it is not a percentage. The second error is declaring the 0.0050 gap between Vehicle 1 and Vehicle 3 a definite advantage without ever questioning the normalisation form, when the same data reverses the order under a different normalisation. The third error is adding a fourth vehicle to the fleet after the analysis is finished; because the column length is computed from the values of every vehicle, adding a new one changes the normalised values of all three original vehicles, and even their own relative order among themselves can shift.
Sources
For the formulas behind each step and the intermediate tables, see the DecisionMind method page: decisionmind.app/library/norm-vector
Hwang, C. L., & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications, A State-of-the-Art Survey. Lecture Notes in Economics and Mathematical Systems, Vol. 186. Springer-Verlag. DOI: 10.1007/978-3-642-48318-9
Jahan, A., & Edwards, K. L. (2015). A state-of-the-art survey on the influence of normalization techniques in ranking: Improving the materials selection process in engineering design. Materials & Design, 65, 335-342. DOI: 10.1016/j.matdes.2014.09.022
Milani, A. S., Shanian, A., Madoliat, R., & Nemes, J. A. (2005). The effect of normalization norms in multiple attribute decision making models: a case study in gear material selection. Structural and Multidisciplinary Optimization, 29, 312-318. DOI: 10.1007/s00158-004-0473-1
Vafaei, N., Ribeiro, R. A., & Camarinha-Matos, L. M. (2018). Selection of Normalization Technique for Weighted Average Multi-criteria Decision Making. In Technological Innovation for Resilient Systems. DOI: 10.1007/978-3-319-78574-5_4
Roszkowska, E. (2011). Multi-criteria decision making models by applying the TOPSIS method to crisp and interval data. Multiple Criteria Decision Making '10-11, University of Economics in Katowice, 200-230. (no DOI)