Interval-valued intuitionistic fuzzy VIKOR (aggregated, a DecisionMind derivation)

Four things change; the idea behind the two conditions (acceptable advantage and acceptable stability) does not, though DecisionMind computes neither of them in this extension.

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 given from outside as crisp numbers. DecisionMind does not combine multiple decision-makers' votes in this extension.

Scale equalisation (direction reversal). In crisp VIKOR the best and worst value swap places on a cost criterion. The same idea is applied here at the judgement level, but through a different route: on a cost criterion, every cell's entire support pair and rejection pair swap places. The support interval given to the judgement "this alternative is good on this criterion" becomes the rejection interval, and the rejection interval becomes the support. This applies the same idea by a different route from IVIF-VIKOR. IVIF-VIKOR only swaps minimum and maximum when picking the best and worst value on a cost criterion, and does not touch the support and rejection intervals themselves. IV-VIKOR instead first reverses the entire cell, and then applies the same best/worst rule to every criterion: the largest value is sought in the support pair, the smallest in the rejection pair.

Distance. Every cell has four components: lower support, upper support, lower rejection, upper rejection. The method takes the largest and the smallest of each of these four components separately, in every criterion, to build the best and worst pair. The distance between these two pairs is computed as the square root of one quarter of the sum of the squared differences of the four components. This is a Euclidean distance. IVIF-VIKOR, by contrast, uses the sum of the absolute differences of the four components; this is a Manhattan distance. The two cards measure what looks like the same cell format with a genuinely different distance, so they can give different Q values on the same table.

Result. Group utility S and individual regret R are, as in crisp VIKOR, the sum and the maximum of the weighted distances. The compromise coefficient v can be changed by the user in IVIF-VIKOR, whereas DecisionMind fixes v at 0.5 in this extension and does not offer it as a parameter.

DecisionMind fixes, in this extension, the whole-cell swap on cost criteria, the Euclidean distance, and the fixed v = 0.5. Most importantly, it does not compute the two compromise conditions (acceptable advantage, acceptable stability) at all; it produces only the Q ranking.

A lower Q is better. However, the two conditions described on the base VIKOR card are not tested by DecisionMind in this extension. The user must check this alone by looking at the S and R columns.

Thus instead of writing:

"According to interval-valued intuitionistic fuzzy VIKOR, the best alternative is A2"

the report should read:

"A2 has the lowest value on both S and R, and this is why it leads clearly on Q. Because DecisionMind does not test the compromise conditions in this extension, the robustness of the second- and third-ranked alternatives must be assessed by looking separately at S and R"

This extension is appropriate when only an interval is known for a judgement's degree of support and degree of rejection, that is, when an expert says "support is not exactly 0.6, but somewhere between 0.5 and 0.7." If support and rejection are each given as a single number (interval width zero), this extra width contributes nothing, and intuitionistic fuzzy VIKOR (IF-VIKOR) suffices. If the criteria are measured, stay with crisp VIKOR. If the matrix is mixed, DecisionMind requires a single data type. Crisp VIKOR's exit condition applies here exactly as it does there: if no compromise at all is acceptable on one criterion, turn to elimination-based methods 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.

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

Assuming the two conditions were tested. DecisionMind does not compute the acceptable-advantage and acceptable-stability conditions in this extension. Declaring the alternative with the lowest Q the "sole compromise solution" is the base VIKOR card's most common mistake, left here uncontrolled by the engine as well.

Skipping the swap on a cost criterion. If the support and rejection intervals are left as they would be for a benefit criterion, high support on a "lower is better" criterion is mistakenly treated as favourable.

The governing principle is this:

The support and rejection intervals must swap together on a cost criterion; it must not be forgotten that the two conditions are not computed by the engine, and this card's Euclidean distance must not be confused with IVIF-VIKOR's Manhattan distance.