Linear Max Normalization (Linear Division-by-Maximum Normalisation)

Linear Max Normalization is not, on its own, a decision method. It is a scale-equalisation tool used in the first step of ranking methods such as TOPSIS, SAW and WPM. When the criteria in a decision table are in different units, one in lira, say, another in days and another in points, these values cannot be summed or compared directly. The method produces a unit-free, comparable column by dividing every column by its own best value. Its output is not a score or a rank; it is a normalised table that DecisionMind will weight in the next step. Fishburn defined this form in his 1967 work on additive utility functions; it is one of the oldest normalisation forms used in ranking methods.

The question the method asks is simple: what percentage of the best achievable value on this criterion has this alternative reached? The reference point is not the column's average or range but directly the best performance in the column. This is why the best alternative always receives exactly 1 point in every column, and the other alternatives carry a ratio relative to that peak. The method thus settles into a "percentage relative to the best" logic, showing not an alternative's absolute size but where it stands relative to its strongest rival. Its philosophical consequence is this: if the best value in a column is an extreme outlier, all the remaining alternatives look weaker than they are relative to that outlier. The method is internally consistent, but its reference point depends on a single alternative.

The method proceeds through two steps.

First, finding the reference value. For a benefit criterion, one where more is better, the column's largest value is found. For a cost criterion, one where less is better, the column's smallest value is found, because for cost the good outcome is a small number.

Second, dividing by direction. For a benefit criterion, every cell is divided by the column's largest value. For a cost criterion the operation reverses: the column's smallest value is divided by every cell. This reverse division ensures that the best alternative also receives 1 point on a cost criterion, but it shrinks large cost values disproportionately. When a small number is divided into a large one, the result shrinks quickly. For instance, if the lowest cost is 28.5 and an alternative's cost is 93.7, that alternative receives 28.5/93.7 = 0.304 points; although the cost differs by less than three times, the score drops to about a third.

Zero and negative values are undefined in this method. If a cost-criterion cell is 0, the smallest value is divided by zero and the result runs to infinity. If a cell is negative, the sign of the division changes, the "divide by the smallest" logic breaks down, and an alternative with a negative cost can meaninglessly receive the highest score. The method is therefore used only with crisp, positive data.

The formulas behind each step are given on the DecisionMind method page; this card carries no formulas.

The normalised value shows an alternative's ratio to the column's best on that criterion, and says nothing more. A value of 0.76 does not mean "76 per cent good"; it means "76 per cent of this column's best value." A value of 1 in a column shows that alternative is the best in that column. Values in different columns cannot be compared directly, because each column is scaled separately against its own best value. Thus instead of writing:

"This alternative scored 0.30, so it is a poor alternative"

the report should read:

"This alternative reached 30 per cent of the column's best value on this criterion; it trails the best alternative by roughly three times"

The method requires crisp data: a single positive number in every cell. DecisionMind carries no separate extension of this building block in fuzzy, grey or intuitionistic data types; each data type uses its own host method's normalisation step, fuzzy TOPSIS's, for instance. You need alternatives in rows, criteria in columns, a number greater than zero in every cell, and direction information for every criterion. The method neither produces nor requires weight; weighting is done in the host method's next step, on the normalised table. A minimum of two alternatives and one criterion is enough; as the number of columns grows only the computation grows, the method's logic does not change.

If your data is crisp and positive, and you want a table scaled against each criterion's best value, this method is suitable. It is a common choice in additive methods other than TOPSIS, such as SAW and WPM, because it encodes criterion direction into the table in a single step, avoiding a separate cost/benefit distinction in the next step. It should not be used where the data contains zero or negative values; in that case the cost transformation stays undefined, and min-max normalisation should be used instead. If the best value in a column is a far outlier from the rest, this method makes the remaining alternatives look weaker than they are; in that case, range-based min-max or sum-based linear sum normalisation may be preferable.

Data crisp, positive, a ratio against the best is wanted → Linear Max Normalization

Data contains zero or negative values → Min-Max Normalization

The number of alternatives matters, a share/proportion logic is wanted → Linear Sum Normalization

Values carry a very large scale difference → Logarithmic Normalization

The host method is TOPSIS and its classical form is to be preserved → Vector Normalization

The method's greatest strength is its simplicity. It is calculated with a single division, can be verified easily even by hand, and encodes criterion direction into the table in the same step. The best alternative always scoring 1 gives a decision-maker reading the table an intuitive sense of how far a criterion is filled. Its computational load is small and stays just as simple as the number of alternatives and criteria grows.

Its weakness is that the reference point depends on a single alternative, the column's best. Jahan and Edwards (2015) showed that the choice of normalisation form can seriously affect the ranking result; in methods that divide by the maximum, this effect is especially pronounced when the best value is an outlier. Vafaei, Ribeiro and Camarinha-Matos (2018) showed with numerical examples that different normalisation forms can give the same TOPSIS table a different ranking. The reverse-division operation for a cost criterion is non-linear; large cost differences lead to a disproportionate shrinkage in scores. The method works only with positive data and cannot be used on tables containing zero or negative values.

The most frequent mistake is using the benefit formula on a cost criterion too. In that case the most expensive alternative receives 1 point and the ranking is reversed. A second mistake is reading a normalised value such as 0.30 as poor performance; the value shows only a ratio relative to the best, not an absolute measure of failure. A third mistake is continuing to apply the method when the data contains zero or negative values; the result produces an infinite or sign-broken number that can enter a report unnoticed. A fourth mistake is using different normalisation forms in different host methods and then comparing the results; a score produced with TOPSIS's vector normalisation is not on the same scale as one produced with this method.

The governing principle is this:

Linear Max Normalization measures an alternative only against the best rival in its own column; when that best value changes, or is an outlier, the entire table reshapes itself.