Interval-valued intuitionistic fuzzy VIKOR (Park, Cho & Kwun, 2011)

Four things change; the idea of the two conditions (acceptable advantage and acceptable stability) does not, and DecisionMind genuinely computes both here.

Cells. In crisp VIKOR every cell is a single number. Here every cell consists of four numbers: the lower and upper bound of the degree of support, and the lower and upper bound of the degree of rejection. The sum of the upper support and the upper rejection cannot exceed 1. Weights are supplied from outside as crisp numbers. Park, Cho and Kwun's (2011) paper also defines a group mechanism that combines several decision-makers' votes. DecisionMind runs, in this extension, only the single-decision-maker kernel, with weights already given.

Scale equalisation (direction reversal). In crisp VIKOR, the best and worst value on a cost criterion simply swap places. Here the same idea is applied without touching the support and rejection intervals themselves. Every cell has four components: lower support, upper support, lower rejection, upper rejection. On a benefit criterion, the best pair is built from the largest support and smallest rejection values among these four components, taken separately. On a cost criterion this selection reverses. This applies the same idea by a different route from IV-VIKOR: IV-VIKOR reverses the cell wholly, that is, the support and rejection intervals swap places, whereas IVIF-VIKOR leaves the cell untouched and reverses only the best/worst selection.

Distance. The distance between two pairs is computed as the sum of the absolute differences of the four components (lower support, upper support, lower rejection, upper rejection). This is called the Burillo–Bustince distance, a Manhattan distance. Park, Cho and Kwun's (2011) paper also defines two further distances: a modified second distance and the Grzegorzewski distance. The paper shows that the ranking can vary across these three. DecisionMind implements only the first of these, the Burillo–Bustince distance, in this extension; the other two are not in the engine. IV-VIKOR, by contrast, uses a Euclidean distance that takes the square root of the sum of the squares of the four components. The two cards measure what looks like the same cell format with, in fact, a different distance.

Result. Group utility S̃ and individual regret R̃ are, as in crisp VIKOR, the sum and the maximum of the weighted distances. The compromise coefficient v can be changed by the user; its default is 0.5. Unlike IV-VIKOR, DecisionMind computes both compromise conditions here and reports whether the outcome is a single solution or a set of compromise solutions.

DecisionMind fixes the Burillo–Bustince distance in this extension; v is a parameter the user can change.

A lower Q̃ is better, as in classical VIKOR. The outcome of the two conditions, that is, whether the result is a single solution or a set, is also reported here, because DecisionMind genuinely runs this test.

Thus instead of writing:

"Interval-valued intuitionistic fuzzy VIKOR finds A1 to be the best alternative"

the report should read:

"With these weights and v = 0.5, A1 is the single compromise solution; it ranks first on both S̃ and R̃, and the gap to the second alternative exceeds the acceptance threshold"

and if either condition is not met, the compromise set should be reported as it stands.

This extension suits situations where only an interval is known about a judgement's degree of support and rejection, that is, where an expert says "support is not exactly 0.6, but somewhere between 0.5 and 0.7." If support and rejection are given as a single number each, that is, if the interval width is zero, this extra width adds nothing, and intuitionistic fuzzy VIKOR (IF-VIKOR) suffices. Where criteria are measured, crisp VIKOR should be used. If the table is mixed, DecisionMind requires a single data type. Classical VIKOR's exit condition applies unchanged here: if no compromise whatsoever is acceptable on one criterion, dominance-based methods should be used instead.

Value-space violation. In every interval the lower bound must be less than or equal to the upper bound, and the sum of the upper support and the upper rejection must not exceed 1. If two alternatives share exactly the same support–rejection pair on a criterion (making the distance between the ideal and the anti-ideal zero), that criterion's contribution becomes undefined; the engine reports this as an error.

Confusing this card with IV-VIKOR. Both use the same four-number cell format, but they run on different engines. IVIF-VIKOR reverses only the best/worst selection on a cost criterion and uses Manhattan distance; IV-VIKOR reverses the cell wholly and uses Euclidean distance. Feeding the same table to both extensions and expecting the same Q is a mistake.

Forgetting that only one of the distance measures has been implemented. The literature defines three distances (Burillo–Bustince, a modified second distance, Grzegorzewski), and the paper shows these can produce different rankings. DecisionMind runs only the Burillo–Bustince distance; the report contains no claim that "the other distance measures would give the same result."

Skipping the two conditions and declaring the alternative with the smallest Q̃ the "winner." DecisionMind genuinely computes the conditions in this extension. Ignoring them means the most common mistake in classical VIKOR applies here just as much.

The governing principle is this:

In this extension the support–rejection interval is reversed not wholly but only through the best/worst selection. Distance is Manhattan in form, and the two compromise conditions are genuinely computed. These three features distinguish it from IV-VIKOR.