Fuzzy VIKOR (Opricovic, 2011)

Three things change; the two conditions (acceptable advantage and acceptable stability) are still calculated here, exactly as in crisp VIKOR.

Cells. In crisp VIKOR every cell is a single number. Here every cell is a triangular fuzzy number: lowest, most likely, highest. Criterion weights here remain crisp, not fuzzy. This is also the difference from Fuzzy TOPSIS (Chen, 2000): there, weights were triangular too; here only the decision-matrix cells are triangular. DecisionMind offers no mechanism in this extension for combining several decision-makers' scores; it works with a single decision-maker's triangles.

Scale equalisation. As in crisp VIKOR, linear normalisation is applied against the best and worst value on each criterion; the difference is that the best and worst value is chosen by looking at the centroid (the average of the three corners), and normalisation is then applied to all three corners at once. DecisionMind takes this ratio corner by corner: the lowest corner is divided by the lowest corner, the most likely by the most likely, and the highest by the highest. This is not full fuzzy division but a simplified corner-wise ratio, a computational shortcut the engine fixes at this point.

Distance / score / aggregation. Group utility S and individual regret R are, as in crisp VIKOR, the sum and the maximum of the weighted differences; the difference is that this summing and maximum-taking is carried out separately on each of the triangle's three corners. The criterion that determines individual regret is chosen, for each alternative, as the criterion whose weighted difference has the largest centroid; that criterion's entire triangle is then carried forward as R.

Result and defuzzification. Q too is computed as a triangle, formed corner by corner from S and R. Defuzzification happens only at the very end, solely for the ranking comparison: the method takes Q's centroid, and the smaller value is judged better. This is the exact opposite of Grey VIKOR, where whitening happens at the very start and width never enters the calculation at all. Here, by contrast, width is carried corner by corner inside S and R, and only disappears at the final comparison.

In this extension, DecisionMind selects the best/worst values by centroid, fixes the corner-wise ratio, takes weights as crisp, and genuinely computes the two conditions, as in base VIKOR, over a Q that has been reduced to its centroid. This is what distinguishes it from the family's grey, intuitionistic and neutrosophic members: DecisionMind applies the compromise-set test only here.

A smaller Q is better, as in base VIKOR. The outcome of the two conditions, that is, whether the result is a single solution or a set, is reported here as well, because DecisionMind genuinely runs this test. The difference is that beneath Q here lies a three-cornered uncertainty. When the centroid difference between two alternatives is small, this difference may or may not be meaningful, depending on the width of the input triangles.

Thus instead of writing:

"Fuzzy VIKOR is more reliable because it accounts for uncertainty"

the report should read:

"Although the inputs are triangular fuzzy numbers, the final comparison is made on a single Q reduced to the corners' average; the width was carried only inside S and R, and disappears in the comparison"

Use this extension when experts score criteria verbally or approximately and reducing these scores to a single number would create artificial precision; or when several parties' interests conflict and the alternative sought is not "the best" but "the one that draws the least objection" — crisp VIKOR's compromise philosophy is preserved here with fuzzy inputs.

Stay with the base method when criteria are measured: converting a measured price or duration into a triangle is not modelling uncertainty, it is manufacturing it. If the table is mixed and DecisionMind requires a single data type, a measured criterion is also written as a triangle with all three components set to the same number. Base VIKOR's exit condition applies here unchanged: when no compromise at all is acceptable on one criterion, methods based on outranking should be used instead.

Violating the value-space order. In every triangle, the lowest value must be less than or equal to the most likely, and the most likely less than or equal to the highest, and none may be negative; if this order is broken, the corner-wise ratio produces meaningless figures.

Trying to give weights as triangles too. This extension expects weights to be crisp; it should not be confused with Fuzzy TOPSIS (Chen 2000), which accepts triangular weights. If a crisp weight is converted into a triangle and entered, DecisionMind either rejects this as an error or produces a meaningless result.

Defuzzifying first and then running crisp VIKOR. Reducing the triangles to their centroid at the outset and applying the crisp method is not Fuzzy VIKOR; the uncertainty is erased in the very first step.

Skipping the two conditions and declaring the alternative with the smallest Q the "winner." Because DecisionMind genuinely computes the conditions in this extension, ignoring them means that the most common mistake in base VIKOR applies here just as much.

The governing principle is this:

In Fuzzy VIKOR, uncertainty is carried in the corners of S and R and collapses to the centroid only at the final comparison; any implementation that renders the input crisp from the start, or skips the two compromise conditions, destroys the extension's own contribution.