Linear Sum Normalization

Linear Sum Normalization is a scale-equalisation tool that divides each column of a decision table into shares of a whole. It can feed into ranking methods such as TOPSIS, and COPRAS carries a step very close to this same logic inside its own calculation. Its output is not a ranking; it is a table rescaled so that each cell shows its share of its own column total, with every column summing to 1. Zavadskas, Turskis, Peldschus and Kaklauskas used this form of normalisation in 1994, in a book comparing contractors' bids in the construction sector, and it went on to underpin the normalisation step of COPRAS and similar proportional assessment methods.

The question this method asks differs from Linear Max's. It is not "how do you compare with the best", but "how much of the total performance falls to this alternative". The reference point is not the best value in the column but the column's total. This makes the results readable as a share of a whole: like slices of a pie, the shares of every alternative in a column always add up to 1. The philosophical consequence follows: even the best alternative never scores 1, because the other alternatives also take a share of the total. As the number of alternatives grows, everyone's share shrinks; values are therefore comparable only among the alternatives within one analysis, never across different analyses.

The method proceeds through two steps.

First, finding the reference value. For a benefit criterion, the sum of every value in the column is taken. For a cost criterion, the reciprocal (1 divided by the value) of every cell is calculated first, and then these inverted values are summed.

Second, dividing by the total. For a benefit criterion, each cell is divided by the column total. For a cost criterion, each cell's reciprocal is divided by the sum of the inverted values. For example, take a cost column with values 39.1, 25.9 and 68.3: the reciprocals are first taken, 1/39.1 = 0.0256, 1/25.9 = 0.0386 and 1/68.3 = 0.0146. These three numbers sum to 0.0788. The first alternative's share is 0.0256/0.0788 = 0.3245; the cheapest alternative takes the largest share, because its reciprocal is the largest number.

Zero and negative values are undefined in this method. If a cell in a cost column is 0, the division by 1 is undefined, the result runs to infinity, and this alternative swallows the total on its own. If a cell is negative, its reciprocal is negative too, the sign of the total can be upset, and the share can come out negative or greater than 1; this entirely invalidates the "share summing to 1" reading. The method is therefore used only with strictly positive crisp data.

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

The normalised value shows the share of a criterion's total performance that falls to one alternative, and nothing more. A value of 0.47 does not mean "47 per cent good"; it means "47 per cent of the column total belongs to this alternative". All the shares in a column always sum to 1, so any one alternative's share depends on the number and magnitude of the other alternatives. The same alternative receives a smaller share once a new alternative is added to the table, even though its own performance has not changed. Thus instead of writing:

"This alternative scored 0.47, so it is half as good"

the report should read:

"Among these three alternatives, this one takes roughly 47 per cent of the total performance; adding a fourth alternative will shrink this share"

The method requires crisp data: a single positive number per cell. DecisionMind holds no separate fuzzy, grey or intuitionistic extension of this building block; every data type uses its own host method's own normalisation step. 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 weights nor requires them. A minimum of two alternatives is needed; as the number of alternatives grows, the size of the shares shrinks, but the method's logic stays the same.

If your data is crisp and positive, and you want the result read as a share or proportion, this method is suitable. It gives a natural reading in methods that carry a proportional assessment logic, such as COPRAS, and in resource-allocation contexts. It should not be used when the data contains a zero or negative value; switch to min-max normalisation instead. It is also unsuited to cases where the number of alternatives is expected to change during the analysis, because every new alternative shrinks the others' shares and the comparison grows inconsistent over time.

Data crisp, positive, result to be read as a share/proportion → Linear Sum Normalization

Data contains zero or negative values → Min-Max Normalization

A ratio to the best is wanted, no need for share logic → Linear Max Normalization

Values carry a very large scale difference → Logarithmic Normalization

The alternative set will not stay fixed during the analysis → lock the alternative set before choosing a host method that fixes the normalisation

The method's strength is that it presents its result as a directly interpretable share; it gives the decision-maker an intuitive answer to "who takes how much of the total performance". Its calculation is simple and can be checked by hand. Criterion direction is folded into the same step and needs no separate handling afterwards. It sits naturally alongside proportion-based methods such as COPRAS.

Its weakness is that the shares are sensitive to the number of alternatives. Vafaei, Ribeiro and Camarinha-Matos (2018) showed that the results of sum-based normalisation forms change with the size of the alternative set. Çelen (2014) found that different normalisation methods gave different rankings on the same banking data, with sum-based methods among those that diverged. The inversion applied to cost criteria is non-linear; a small cost value can inflate its reciprocal to the point of dominating the total. The method works only with positive data and cannot be used on tables containing zero or negative values.

The most common mistake is reading the share directly as a percentage measure of success; the share only shows relative standing within this particular alternative set. A second mistake is dividing by the column total without inverting a cost criterion first; the most expensive alternative then takes the largest share and the ranking is reversed. A third is adding a new alternative once the analysis is finished and being surprised that the shares shrink; the share distribution depends on the alternative set and must be recalculated whenever the set changes. A fourth is continuing to apply the method when a cost value in the data sits very close to zero; that value's reciprocal grows excessively large and dominates the total on its own.

The governing principle is this:

Linear Sum Normalization measures an alternative by the share it takes of the total performance; this share is meaningful only relative to the other alternatives in the same analysis, and must be recalculated whenever the alternative set changes.