LOPCOW (LOgarithmic Percentage Change-driven Objective Weighting)

LOPCOW, like Entropy and CRITIC, is not a ranking method; it does not rank alternatives, it produces criterion weights. Given a numerical decision table of alternatives and criteria, it scans the table without consulting an expert and derives a weight vector that sums to 1. These weights then feed into a ranking method such as TOPSIS, VIKOR or another.

Ecer and Pamučar proposed the method in 2022, for assessing sustainability performance in the banking sector. Its name summarises how it works: it scales each criterion's column between its smallest and largest values, then measures the percentage change within this scaled column through a logarithmic expression. It has been applied across different fields (logistics, healthcare, banking) in a short space of time and is a current, increasingly widespread objective weighting method.

LOPCOW comes from the same family as Entropy and CRITIC: the more the alternatives differ on a criterion, the more that criterion determines the decision. What sets it apart is how it measures this difference. Entropy looks at the evenness of a probability distribution, CRITIC at standard deviation combined with correlation; LOPCOW instead takes the logarithm of the ratio between the root-mean-square magnitude of the scaled values and their standard deviation. This ratio captures the balance between a criterion's "typical magnitude" and "the spread around that magnitude".

A consequence follows: LOPCOW looks jointly at how large a criterion's values are and how widely they are spread; this is what distinguishes it from Entropy, which looks only at spread. Taking the logarithm stops the weight of a widely spread criterion from growing without limit and gives a scale that is independent of measurement units and therefore comparable. If the view that "both magnitude and spread together determine the decision" is acceptable, LOPCOW is the right tool.

The method proceeds through four steps.

First, building the decision table. Alternatives sit in rows and criteria in columns.

Second, scale equalisation. Every column is scaled to between 0 and 1 across its own smallest and largest value. In a "higher is better" criterion, the largest value becomes 1 and the smallest 0; in a "lower is better" criterion this is reversed. This step removes units and aligns direction at the same time.

Third, the percentage-change value. For every criterion, the ratio between the root-mean-square magnitude of the values in the equalised column and their standard deviation is calculated, its natural logarithm is taken, and this is multiplied by a hundred. This value combines the criterion's typical magnitude and its spread into a single number.

Fourth, the weight. Each criterion's percentage-change value is divided by the sum of these values across all criteria. The result is a weight vector summing to 1.

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

A LOPCOW weight measures not the criterion's importance in the decision-maker's eyes but how much the scaled values on that criterion "speak" in terms of magnitude and spread. A weight of 0.36 does not mean "this criterion is 36 per cent of the decision"; it means "roughly a third of the information separating the alternatives in this table comes from this criterion." The same criterion can take a completely different weight in a different alternative set, because the weight is a property of the table, not of the criterion.

Since the best alternative always takes the value 1 in the scaled column, what really determines the spread is how far behind the best the other alternatives fall; if all alternatives sit close to the best on a criterion, the spread is small and the weight comes out low. Where the number of alternatives is small (three or four), the standard deviation is calculated unreliably, and the weights become sensitive to this.

Thus instead of writing:

"The LOPCOW analysis proved that the intensive-care bed ratio is the most important indicator"

the report should read:

"In this set of provinces, the intensive-care bed ratio is the indicator that most separates the alternatives; the LOPCOW weight of 0.36 reflects this distinguishing power, not the indicator's priority in health policy"

Classical LOPCOW works with crisp data: a single number in every cell. DecisionMind holds a fuzzy extension alongside the base method (two members in total); this extension is DecisionMind's own fuzzy adaptation, used when the data is supplied by expert judgement as triangular fuzzy numbers.

You need alternatives in rows, criteria in columns, a number in every cell, with no empty cells. Direction information ("higher is better" or "lower is better") is needed for every criterion, because scale equalisation is carried out according to this direction. No weight is entered; the method produces the weight. A minimum of two alternatives and two criteria is required; for the standard deviation to be meaningful, more than five alternatives is recommended, as with CRITIC.

Where expert opinion is unavailable or unwanted, and the number of alternatives being compared is large enough (province, institution or company comparisons, for instance), LOPCOW is a reliable weight source. It works well on broad tables of sustainability, performance and efficiency indicators, and in cases where indicators of different units and magnitudes must be brought under one roof.

The situations where it should not be used follow from its philosophy. If the number of alternatives is very small, the standard deviation becomes unreliable and the weights unstable. Where the decision-maker clearly regards one criterion as a priority, LOPCOW cannot see this. If every alternative is nearly identical on a criterion (the spread is close to zero), that criterion's weight also comes out close to zero; this should be anticipated in advance.

No expert opinion, the number of alternatives is large enough → LOPCOW

The relationship between criteria should also enter the calculation → CRITIC

Spread alone is sufficient, relationship is unimportant → Entropy

The decision-maker's priority should show in the result → AHP, BWM, SWARA (subjective)

Few alternatives (fewer than five) → read LOPCOW weights with caution, or switch to a subjective method

LOPCOW's most important strength is that it combines magnitude and spread into a single measure; unlike methods that look only at spread, it also accounts for the general level of a criterion's scaled values. It is objective; the same table gives everyone the same weight. Because of the logarithm, a widely spread criterion's weight does not grow without limit, which makes the weights more balanced. Its calculation is comparatively simple and runs quickly on large indicator sets, which is why it is often chosen for country, province and institution comparisons.

Its limitations stem from the same structure. First, standard deviation is calculated unreliably with few alternatives; in small tables the weights become excessively sensitive to a single alternative's position. Second, the weights depend on the alternative set; adding or removing an alternative changes the whole scaling and hence every weight. Third, "distinguishing power" and "importance" are not the same thing; the decision-maker's values do not show up in the result. Fourth, because the method was proposed in 2022, an independent body of critique is still limited; Zavadskas and Podvezko's (2016) assessment of the general limitations of objective weighting methods also applies here.

The most common mistake is reporting a LOPCOW weight as "importance". The statement "the analysis showed the intensive-care bed ratio is the most important indicator" is wrong; the analysis shows that this indicator most distinguishes the alternatives.

A second mistake is marking criterion direction wrongly; if a "lower is better" criterion is marked "higher is better", the scaling runs in reverse and the weight is built the wrong way round. A third is placing excessive confidence in weights derived from a table with fewer than five alternatives. A fourth is using one study's LOPCOW weights on a different alternative set. A fifth is keeping a criterion in the table on which every alternative takes almost the same value, and mistaking its near-zero weight for "a fault in the method".

The governing principle is this:

A LOPCOW weight is a combination of a criterion's scaled magnitude and its spread; the weight changes when the alternative set changes, and the report must call this "distinguishing power", not "importance".