FUCOM-F (Fuzzy Full Consistency Method)

FUCOM-F extends crisp FUCOM's same logic to situations where the expert expresses the importance ratio between successive criteria not as a single number but as a triangular fuzzy number (a lowest, most likely and highest estimate). Its output is a crisp weight vector summing to one; uncertainty is carried as a fuzzy number through the intermediate calculations and reduced to a single figure only at the last step. Pamucar and Ecer (2020) proposed it for situations where the expert wants to express not a clean ratio such as "exactly 1.5 times" but an uncertainty such as "between 1.5 and 2.5, most likely 2."

Crisp FUCOM asks the expert to give the ratio between successive criteria as a single number; but in real life an expert more often says not "exactly 1.5" but "between 1.5 and 2.5, probably 2." FUCOM-F does not discard this uncertainty; it carries it as a triangular fuzzy number and runs the entire calculation, transitivity between ratios, the deviation value, the weights, with fuzzy arithmetic. Only at the very last step is the fuzzy weight defuzzified (reduced to a single number) to obtain a reportable figure.

This carries a philosophical consequence: in FUCOM-F, the principle holds that "uncertainty is carried from the start of the calculation to the end, and is reduced to a single number only at the very end." This is a compromise that lets the expert say "I am not sure, but within this range," while still handing the decision-maker a single figure. It carries more information than crisp FUCOM (it also records how confident the expert was), but it requires a more complex calculation and more data (three figures for every ratio).

The method proceeds through four steps.

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

Second, the fuzzy successive ratio. For every pair of successive criteria only, the expert states a triangular fuzzy importance ratio (a lowest, most likely and highest estimate) through a label chosen from a nine-level linguistic scale.

Third, fuzzy optimisation. The fuzzy weights and the fuzzy deviation value are found through an optimisation that fits them, separately at each of the triangle's three corners (lower, middle, upper), as closely as possible to both the given successive ratios and the transitive ratios that derive from them, subject to the condition that they sum to one. This step solves three times what crisp FUCOM's single-number optimisation solves once, once for each corner.

Fourth, defuzzification. The fuzzy weights are reduced to a single crisp number through an average that gives more weight to the middle point than to the lower and upper ends. From this step onward, the reported weights and deviation value are crisp numbers.

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

Even though the reported weight is a crisp number, a three-cornered uncertainty lies beneath it; because this uncertainty is defuzzified into a single number at the last step, it does not show up in the report. If the weight gap between two criteria is small, that gap can easily reverse if the input triangles are wide (if the expert gave the ratios over a broad range). The same reading of the deviation value that applies to crisp FUCOM applies here too: a small deviation means the weights are consistent with the expert's fuzzy ratios, a large one means the ratios are internally contradictory; but here the deviation itself starts out as a triangular fuzzy number and is defuzzified before being reported.

Thus instead of writing:

"FUCOM-F found the crisp weights"

the report should read:

"These weights are the single-number reduction of the expert's uncertain fuzzy ratios; if the uncertainty beneath them comes from a wide triangle, small weight gaps do not form a reliable ranking"

FUCOM-F works with fuzzy (triangular fuzzy number) data: a lower-middle-upper triple, drawn from a linguistic label, for every successive criterion pair. In DecisionMind, FUCOM-F is the fuzzy extension of crisp FUCOM; if the data consists of crisp numbers, FUCOM should be used directly. You need: a complete ranking of the criteria, a shared linguistic scale, and a label chosen from that scale for every successive pair. FUCOM-F produces weights, it does not ask for weights from outside. Three to ten criteria and one to seven experts are typical; the method can also work with a single expert, because uncertainty is already carried within the expert's own statement (the width of the triangle).

If the expert wants to express the ratio between successive criteria not as a clean number but as a range of uncertainty, and reflecting this uncertainty in the result matters, FUCOM-F is a suitable choice. If the expert can already give clean, single-number ratios, crisp FUCOM is sufficient and gives the same result with less computational load. If the decision-maker only needs to see a single crisp weight table and does not need the uncertainty shown separately, the fuzzy extension adds unnecessary complexity.

Successive ratios are expressed as a range of uncertainty → FUCOM-F

Successive ratios are clean and can be given as a single number → FUCOM

All criterion pairs should be compared → AHP (Fuzzy AHP for fuzzy data)

Only a best-worst comparison is possible → BWM

FUCOM-F's greatest strength is that it builds the expert's uncertainty, the difference between "I am certain" and "I am estimating," into the model; crisp FUCOM cannot make this distinction, it accepts every ratio with the same certainty. Carrying uncertainty right to the end of the calculation and reducing it to a single number only at the last step delays information loss in the intermediate steps; this is a design principle shared by many fuzzy MCDM methods, much as it is with TOPSIS's benefit degree K.

The method's greatest limitation is that it requires far more data (three figures for every successive ratio) than crisp FUCOM, and a more complex optimisation; this places an extra burden on both the expert and the computing infrastructure. Second, which numerical triangles the nine-level linguistic scale corresponds to (Pamucar and Ecer, 2020) has been adapted from other contexts and may not fit every application exactly. Third, the defuzzification at the final step makes the uncertainty invisible in the report; the reader sees only a single crisp weight and does not know how wide the underlying triangle was, and this information is lost unless presented separately.

The most common mistake is thinking only in terms of the defuzzified single number, never seeing the triangle (lower, middle, upper) behind the label chosen from the linguistic scale, and presenting a small gap between two close weights as a definitive ranking. A second mistake is failing to notice that, with only two criteria, the transitivity constraint never comes into play (there is only a single successive ratio), and generalising a result from this simple case to a result with three or more criteria. A third mistake is giving the lower end of the triangle representing a successive ratio as a value below one; by definition, the next criterion in the ranking is considered less than or equal in importance to the one before it, and the triangle's lower corner must also obey this rule.

The governing principle is this:

Even though the weight FUCOM-F reports is a single crisp number, beneath it lies an uncertainty triangle expressed by the expert; if this triangle is wide, the order of two criteria's weights can easily swap with a small change in the input.