Intuitionistic fuzzy VIKOR (Devi, 2011)

Three things change; the idea behind the two conditions (acceptable advantage and acceptable stability) does not, although DecisionMind does not calculate either of them in this extension.

Cells. In crisp VIKOR every cell is a single number; here every cell is a degree of support (μ) and a degree of rejection (ν), whose sum does not exceed 1. Criterion weights are crisp. DecisionMind does not combine several decision-makers' votes in the presentation layer for this extension; it works with a single decision-maker's support/rejection pairs.

Scale equalisation (reversing direction). In crisp VIKOR the best and worst value on a cost criterion simply swap places, so that the smallest value counts as good. Here the same idea is applied at the level of the judgement. On a cost criterion, the support and rejection degrees swap places. The support given to the judgement "this alternative is good on this criterion" becomes rejection, and rejection becomes support. The rest of the calculation then proceeds as though every criterion were "higher is better".

Distance / score / aggregation. The method builds the best pair by taking, on each criterion, the highest support and lowest rejection degree, and builds the worst pair the opposite way. It calculates every alternative's distance to this best pair by summing the squares of the support and rejection differences and taking the square root of half that sum. This is a normalised distance that reduces the two degrees to a single number, and the hesitancy margin does not appear separately in this distance. The ratio of this distance to the distance between the best and worst pair is the counterpart of the normalised difference in crisp VIKOR. The method weights and sums this criterion-level ratio (S) and separates out the largest one (R).

Result and defuzzification. The defuzzification here sits between Grey VIKOR and Fuzzy VIKOR. In Grey VIKOR defuzzification happens right at the start, and in Fuzzy VIKOR right at the end; here the support/rejection pair descends to a single number inside the distance calculation itself. From this point on, S, R and Q are entirely crisp numbers. Unlike Fuzzy VIKOR, uncertainty is not carried separately through to the end, but the input is not whitened right from the start either, as in Grey VIKOR. The criterion-level normalised difference is the one genuine trace of uncertainty that remains between the two extremes.

DecisionMind fixes the distance in this form in this extension and takes the compromise coefficient v at a default of 0.5. Unlike Fuzzy VIKOR, it does not calculate the two conditions; it produces only the Q ranking.

A low Q is good. However, the two conditions described on the base VIKOR card (acceptable advantage, acceptable stability) are not tested by DecisionMind in this extension; the user must check this by looking at the S and R columns themselves. In addition, two alternatives' Q values can come out equal, or very close. This is a normal and honest result of VIKOR, not a computational error. Because the data arrives as support/rejection pairs, this closeness may be seen more often than in crisp VIKOR.

Thus instead of writing:

"According to intuitionistic fuzzy VIKOR, the best alternative is A1"

the report should read:

"A1's and A3's compromise index are equal with these weights and at v = 0.5; because DecisionMind does not test the compromise conditions in this extension, which one stands out must be assessed separately by looking at the S and R columns, while A2 is clearly third"

Use this extension when the assessment comes from a judgement and the expert's opposing view also carries measurable information; where several parties' interests conflict and what is sought is not "the best" but "the alternative that draws the least objection", crisp VIKOR's compromise philosophy is preserved here through the support/rejection pair.

If the expert only answers "how suitable" and there is no separate source for the opposing view, there is no need to write ν as 1 − μ and move to this extension; that carries the same information as crisp data. DecisionMind requires a single data type if the table is mixed. Base VIKOR's exit condition applies here too: if no compromise at all is acceptable on one criterion, dominance-based methods should be used instead.

Violating the value space. Both μ and ν must lie between 0 and 1, and their sum must not exceed 1; if it does, DecisionMind rejects the calculation.

Writing ν as 1 − μ. This zeroes the hesitancy margin in every cell and also makes the swap on cost criteria meaningless, because μ and ν become each other's exact complement. In Case 1's table below, every cell's hesitancy margin is a constant 0.10; for that reason, writing ν = 1 − μ does not change the ranking in that particular example. But this is a special symmetry of that table, not a general rule. In a real table where the hesitancy margin varies from cell to cell, the same shortcut can change the order and irreversibly deletes information.

Skipping the swap on a cost criterion. If the support and rejection degrees are left as they are, as though on a benefit criterion, high support on a "lower is better" criterion is wrongly counted as good, and the order reverses.

Deriving support and rejection from a single source. Writing the proportion of experts saying "suitable" as μ and treating the remainder automatically as ν means counting the undecided as opposed.

The governing principle is this:

Support and rejection degrees must come from separate sources, the two must swap together on a cost criterion, and the hesitancy margin must never be zeroed as a number calculated backward from μ.