Min-Max Normalisation

Min-Max Normalization is a scale-equalisation tool used in the normalisation step of many ranking methods, TOPSIS foremost among them. It rescales every column of the decision table according to the range between the worst and best value observed in that column. Its output is not a ranking; it is a table compressed to between 0 and 1, showing where each cell sits within its own column's range. Hwang and Yoon defined this form in their 1981 book bringing together multi-criteria decision-making methods; today it is one of the most widely used normalisation building blocks.

The question the method asks is this: where does this alternative sit between the worst and best in this set? The reference point is not a single extreme but the range itself. This sets it apart from Linear Max Normalization, which scales only against the best value, so that the worst value need not automatically come out at 0, whereas Min-Max fixes both ends. The philosophical consequence is that the column's best always takes exactly 1, its worst always takes exactly 0, and every alternative in between sits at a linear position between these two extremes. The method makes full use of a chosen set's own relative spread; because of this, two analyses run on different alternative sets can never be compared directly.

The method proceeds through two steps.

First, finding the range. For every criterion, the smallest and largest value in the column are identified.

Second, placing values within the range. For a benefit criterion, the column's smallest value is subtracted from every cell, and the result is divided by the width of the range (largest minus smallest). For a cost criterion, every cell is subtracted from the column's largest value, again divided by the width of the range. This operation rests on subtraction and division, not inversion; unlike the other three normalisation building blocks (Linear Max, Linear Sum, Logarithmic), it therefore works without trouble on zero and negative raw data. If, for instance, a criterion's value ranges between a loss and a profit, say minus 5 and plus 10, the method places this range too neatly between 0 and 1.

The one real risk is that every value in a column is equal. In this case the largest and smallest value coincide, the width of the range comes out at zero, and the division is undefined. In this case, either the criterion in question must be dropped from the table or a small constant must be added to the denominator. Also, if a very extreme alternative is added to the table afterwards, the column's smallest or largest value changes; in this case the normalised values of all the previous alternatives must be recalculated too, because the reference range has shifted.

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

The normalised value shows where an alternative sits, on that criterion, between the worst and best observed; it says nothing more. A value of 1 does not mean "perfect" but "the best in this set," a value of 0 does not mean "a failure" but "the worst in this set." Values of 0.70 from two different analyses cannot be compared, because each analysis's own range is built from that analysis's own alternative set. Thus instead of writing:

"This alternative scored 0, so it is of no use at all"

the report should read:

"This alternative sits worst on this criterion among these three alternatives; the same raw value could score differently in a different alternative set"

The method requires crisp data: one number per cell. Unlike the other three normalisation building blocks, this number may be zero or negative; the method only requires the range to be non-zero, meaning not every value in the column may be identical. DecisionMind holds no separate extension of this building block for fuzzy, grey or intuitionistic data types; every data type uses its own host method's own normalisation step. You need alternatives in rows, criteria in columns, one number per cell, and direction information for each criterion. The method produces no weights and requires none. A minimum of two alternatives is required; with two alternatives, the range is built at its narrowest, one taking 0 and the other 1.

This method is suitable if your data is crisp, contains zero or negative values, or if you want the result read as position within this alternative set's range. It is one of the default choices in TOPSIS, SAW and many ranking methods, because it fixes both ends exactly and carries the fewest restrictions on data type. It should not be used where the alternative set is expected to change, during an analysis or between analyses; adding a new extreme alternative rescales the entire table, and even the previous alternatives' order may change. Where all the values in a column already sit very close together, meaning the range is close to zero, small measurement errors can also appear as large differences in the normalised table.

Data contains zero or negative values → Min-Max Normalization

Alternative set fixed during analysis, ratio against the best is wanted → Linear Max Normalization

Result to be read as a share/proportion → Linear Sum Normalization

Values carry a very large difference in scale → Logarithmic Normalization

New candidates expected to join the alternative set → the risk of rank reversal must be stated in the report from the outset

The method's greatest strength is its flexibility across data types. Because it rests on subtraction and division, it also works with zero and negative raw data, which is not possible with any of the Linear Max, Linear Sum or Logarithmic normalisations. Because it fixes both ends exactly at 0 and 1, it makes full use of the column's entire range, giving a decision-maker reading the table a clear visual reference. Its calculation is simple and can be verified by hand.

Its weakness is that the result is only meaningful relative to the observed alternative set. As Hwang and Yoon (1981) themselves noted in their own book, the range shifts whenever the alternative set changes. Jahan and Edwards (2015) have shown that the choice of normalisation form can materially affect a ranking result, and that range-based methods are included in this effect. Chakraborty and Yeh (2009) demonstrated by simulation that range-based normalisation in TOPSIS can produce a different order compared to other normalisation forms. The worst alternative in a column always takes exactly 0, however small its actual gap, which can make it look far weaker than an alternative that is in reality only slightly worse.

The most common mistake is reading a normalised score of 0 as an absolute failure; the score only shows the worst position within this set. A second mistake is adding a new alternative once the analysis has finished and being surprised that the previous alternatives' scores change; this is an expected result, because the range has been rebuilt. A third mistake is running the method as it stands when every value in a column is equal; the width of the range is then zero and the division is undefined. A fourth mistake is comparing min-max scores from different analyses directly; each analysis's range is built from its own alternative set and carries no meaning relative to another's.

The governing principle is this:

Min-Max Normalization measures an alternative only by its position between the worst and best of the observed set; when this set changes, the range, and with it the entire normalised table, is rebuilt.