Fuzzy MARCOS (Stanković et al., 2020)

Four things change; the ideal/anti-ideal-referenced ratio logic does not.

Cells. In crisp MARCOS every cell is a single number. Here every cell is three numbers: lowest, most likely, highest. Criterion weights are likewise triangular; if a crisp weight is to be used, its three components are written identical and its width is zero. The founding paper (Stanković, Stević, Das, Subotić & Pamučar, 2020) proposed the method for road traffic risk analysis. DecisionMind's Fuzzy MARCOS does not pool the scores of more than one decision-maker; it works on a single fuzzy matrix.

Scale equalisation. Crisp MARCOS builds the ideal (the best observed column value) and anti-ideal (the worst observed column value) reference points and ratios every cell to the ideal's value: for a benefit criterion the cell is divided by the ideal, for a cost criterion the ideal is divided by the cell. Fuzzy MARCOS applies this same rule separately to the triangle's three components; the ideal and anti-ideal are also added to the table as triangles, with all three corners equal and zero width. The rule stays the same, because the step that defines MARCOS is ideal/anti-ideal-referenced normalisation; fuzziness only requires this step to be repeated separately in each component.

Distance, score and combination. In crisp MARCOS the weighted sum is a single number. Here the weighted sum is a triangle; the three components are summed separately. An alternative's "utility ratio" to the ideal and the anti-ideal (K+ and K−) is found by division between triangles. This division is carried out with cross-matched components: one triangle's lowest end is divided by the other's highest end. This is because the worst case must also be matched with the worst case in the division.

Result and defuzzification. The utility-ratio triangles are reduced to a single number by the centre of gravity (the average of the three components); this defuzzification happens at the fifth step, AFTER the ratio to the ideal/anti-ideal has been built. The remaining utility function and final utility degree run with exactly the same formula as crisp MARCOS's sixth and seventh steps.

The choice DecisionMind keeps fixed for Fuzzy MARCOS: centre-of-gravity (COA) defuzzification and corner-based construction of the ideal/anti-ideal (F2, F5). Weights are taken from outside as triangles; the method does not generate weights.

The output is a final utility degree and a ranking, as in crisp MARCOS, and is read the same way. It cannot be compared against another analysis's degree, because the ideal and anti-ideal are built from each analysis's own alternative set every time.

The difference is this: defuzzification happens only at the fifth step, after the utility ratios have been built. This means an uncertainty layer, no longer visible, lies beneath the final degree. The gap between two alternatives may be robust or fragile depending on the width of the inputs.

Thus instead of writing:

"Because Fuzzy MARCOS takes uncertainty into account, the result is more reliable"

the report should read:

"Because the criterion scores are verbal or approximate, the uncertainty has been carried through until the ratio to the ideal and anti-ideal was built, and only then reduced to a single number; which criterion's weight and which score's width the gap between two alternatives is sensitive to should be shown separately"

Use it when criterion scores come from expert judgement or estimation, and reducing this approximation to a single number would create an artificial precision (the criterion given on the fuzzy data-type card). Opening a measured criterion, such as price or time, into a triangle does not model uncertainty, it manufactures it; if the matrix must stay in a single type, the measured value is written as a triangle whose three components are identical.

The exit condition is the same as for crisp MARCOS. If no compromise is acceptable on a criterion, this extension is compensatory too and will not eliminate anything below a threshold. If the alternative set is very small, say only two or three alternatives, the ideal and anti-ideal references stay overly sensitive to the alternatives themselves.

Violating the fuzzy number's l ≤ m ≤ u rule. The three components must sit in this order, and all must be greater than or equal to zero; if the order breaks, the ideal/anti-ideal references break with it.

Changing the defuzzification method without noticing. The centre of gravity (COA, the average of the three components) is the canonical choice, but not the only one; another defuzzification rule, such as taking only the most likely value, can produce a different order. Which rule was used should be stated in the report.

Defuzzifying first and then running crisp MARCOS. Reducing the triangles to a single number at the outset and running the crisp method is not Fuzzy MARCOS. Doing so means the ideal and anti-ideal references are no longer built from fuzzy corners but from a single number. In the illustrative example below, this route shifts A1 from 0.578 to 0.590, A2 from 0.715 to 0.731, and A3 from 0.640 to 0.654. The order happens to stay the same in this table, because the triangles are symmetric and narrow. In an input whose width is skewed upward or downward, whether defuzzification happens at the first step or the fifth step can change the order itself.

Giving crisp weights with fuzzy scores. The method expects the weights to be triangular too; if a crisp weight is used instead, its three components are written identical and this is stated in the report.

The governing principle is this:

Fuzzy MARCOS exists to carry the approximation in criterion scores through until the ratio to the ideal/anti-ideal has been built; any application that hardens the input at the outset, or leaves the scale undefined, destroys the extension's only contribution.

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