FUCOM (Full Consistency Method)

FUCOM is a method that derives criterion weights from an importance ranking set by the expert and the importance ratios between successive criteria. Its output is a crisp weight vector summing to one; it does not rank, it does not assess alternatives. Pamučar, Stević and Sremac (2018) proposed it as an answer to the problems that AHP requires a large number of comparisons and that BWM, in some situations, allows more than one solution; it has found a wide range of applications, from supplier selection to sustainability assessment.

FUCOM's underlying idea is to guarantee consistency from the outset rather than checking it afterwards. In AHP the expert compares every pair of criteria, and consistency is then measured afterwards with a ratio; if it is inconsistent, the expert is asked to compare again. FUCOM instead asks the expert only for the ratio between successive criteria, and finds the weights through a mathematical optimisation that fits them as closely as possible both to these successive ratios and to the transitive ratios that derive from them (the ratio of the first criterion to the third must equal the product of the successive ratios). A "deviation value" results from this; if this value is zero, the expert's ratios are already fully consistent, and if it is greater than zero, the weights are reconciled with these ratios at the point of smallest deviation.

This carries a philosophical consequence: in FUCOM, consistency is not an "after-the-fact check" but part of how the weights are calculated. This places the method in a position that requires less data than AHP, yet is more precise (having a single optimal solution) than BWM. The price, as with DIBR, is that the expert is asked only for the ratio between successive criteria; the ratio between criteria that are far apart in the ranking is not asked directly, it is determined implicitly through the consistency optimisation.

The method proceeds through three steps.

First, ranking. The expert arranges the criteria in a complete order from most important to least important.

Second, the successive importance ratio. For every pair of successive criteria only, the expert gives a ratio in the form "how many times more important the preceding criterion is than the next." All of these ratios must be greater than or equal to one, because the next criterion in the ranking can never be considered more important than the one before it.

Third, weighting with minimum deviation. The weights are found through an optimisation, subject to the condition that they sum to one, that fits them as closely as possible both to the given successive ratios and to the transitive ratios that derive from them (for example, the rule that the ratio of the first criterion to the third must equal the product of the two successive ratios). This optimisation aims to minimise the largest deviation among the ratios; the resulting minimum deviation value is itself a result that shows how consistent the expert's given ratios are.

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

The weight reflects, as consistently as possible, the ranking and successive importance ratios set by the expert. If the deviation value is zero (or very close to zero), the weights are fully consistent with the ratios the expert gave; if the deviation value is large, the successive ratios the expert gave are internally contradictory, and the weights are a compromise point that minimises this contradiction. The deviation value must always be shown in the report; giving only the weights and hiding the deviation value conceals how solid a foundation the weights stand on.

Thus instead of writing:

"FUCOM produced the most consistent weights"

the report should read:

"These weights are consistent with the expert's successive ratios at a deviation close to zero; the deviation value is this much, and it is a measure of the ratios' internal consistency"

FUCOM works with crisp (numerical) data: a ranking and importance ratios for successive criterion pairs. In DecisionMind, a fuzzy extension of FUCOM (FUCOM-F) is registered as a separate method; when the data is linguistic or fuzzy, that extension should be used instead. You need: a complete importance ranking of the criteria, and, for every successive pair, a ratio greater than or equal to one. FUCOM produces weights, it does not ask for weights from outside. Three to twelve criteria work comfortably; as the number of criteria grows, both setting the ranking correctly and giving consistent successive ratios becomes harder.

If the expert can easily rank the criteria, can give the ratio between successive criteria, and structural guarantee of consistency is wanted, FUCOM is a suitable choice. If the expert also wants to report their genuine perception of criteria that are far apart in the ranking directly, a method that asks about every pair, such as AHP, should be preferred. If the data consists of linguistic or fuzzy expressions, FUCOM's crisp form is not sufficient, and its fuzzy extension (FUCOM-F) should be used.

Criteria can be ranked, consistency should be guaranteed from the start → FUCOM

All pairs should be compared, the ratio between far-apart criteria should also be asked directly → AHP

Only a comparison between the best and worst criterion is possible → BWM

Data is linguistic/fuzzy → FUCOM-F

FUCOM's greatest strength is that it works with a small number of comparisons (only n-1 successive ratios) and makes consistency part of the weight calculation rather than checking it afterwards. Compared with BWM, FUCOM's optimisation formulation generally reaches a single solution; BWM's allowing more than one optimal solution in some inputs is seen less often in FUCOM (Grošelj and Dolinar, 2025). The method has become widespread in application because it combines easily with different multi-criteria decision methods (such as MAIRCA, CoCoSo and TOPSIS) (Đalić and Durmić, 2019).

The method's fundamental limitation, as with DIBR, is that it asks directly only for information between successive criteria; the genuine perceived relationship between two criteria that are far apart in the ranking is never asked directly, and is determined implicitly during the optimisation. Second, if two criteria are placed in the wrong order in the ranking, the entire subsequent calculation is wrong from the start, and FUCOM cannot catch this error through the deviation value, because the deviation measures only the internal consistency of the ratios, not the correctness of the ranking. Third, Akbari and colleagues' (2021) comparative study shows that the weights FUCOM produces can differ systematically from those of AHP and BWM; this difference is not an error, but which method gives the "correct" weight remains an open debate in the literature.

The most common mistake is giving a successive ratio smaller than one; by definition, the next criterion in the ranking is considered less than or equal in importance to the one before it, and a ratio smaller than one contradicts the ranking. A second mistake is ignoring the deviation value that results from the optimisation and reporting only the weights; a large deviation shows that the ratios the expert gave are internally contradictory, and this information must not be hidden. A third mistake is setting the ranking through the analyst's own assumption without consulting the expert; the ranking is the foundation of the entire calculation and must come from the expert. A fourth mistake is directly comparing the weights FUCOM produces with those of another method (for example, AHP) and asking "which one is correct"; the two methods work with different assumptions, and both are valid within their own assumptions.

The governing principle is this:

A FUCOM weight matches the ranking and successive ratios the expert gave as consistently as possible; the deviation value, which shows how tight this match is, must be reported alongside the weights.