Intuitionistic fuzzy MULTIMOORA

Four things change; the dominance logic does not.

Cells. In crisp MULTIMOORA every cell is a single number. Here every cell consists of a degree of support (μ) and a degree of rejection (ν), with μ + ν ≤ 1. Weights are entered as CRISP numbers in this extension; an intuitionistic fuzzy weight is not accepted. This extension is designed for a single decision-maker; combining several experts' pairs must happen outside the card, at the data-preparation stage.

Scale equalisation. Crisp MULTIMOORA divides a column by the root of the sum of its squares. Here, every cell's support-rejection pair is first reduced to a SINGLE number by the Chen-Tan (1994) scoring rule: score = support − rejection. Since this score can range from −1 to 1, it is shifted by adding 1 so that negative values do not corrupt the vector normalisation; it now lies between 0 and 2. AFTER this, however, column-wise vector normalisation, dividing by the root of the sum of squares, is applied exactly as in the crisp method. That is, the uncertainty descends to a single number BEFORE normalisation, at the very first step, and crisp MULTIMOORA's normalisation logic then operates on that single number unchanged.

Distance, score and aggregation. The ratio system and the reference point are calculated on this shifted, normalised score in exactly the same way as crisp MULTIMOORA (subtracting the cost sum from the benefit sum; taking the largest weighted deviation from the best value). The full multiplicative form is DIFFERENT and is the only step that remains genuinely "intuitionistic fuzzy". The complemented matrix is built by swapping, that is complementing, the support-rejection pair on cost criteria. The benefit criteria are then COMBINED among themselves, and the cost criteria among themselves, with the intuitionistic fuzzy weighted geometric (IFWG) operator. So at this step the support-rejection structure is genuinely preserved, and it is only converted to a Chen-Tan score at the end.

Result and defuzzification. The output is a single number (minus the sum of the three sub-ranks). Because the ratio system and the reference point already operate on a score that was defuzzified right at the start, before normalisation, they require no separate defuzzification step; only the full multiplicative form defuzzifies, returning to a Chen-Tan score after the IFWG aggregation.

DecisionMind fixes the following in this extension: the Chen-Tan (1994) scoring rule (support − rejection); the +1 shift when the score is negative; the calculation of the ratio system and reference point on this shifted score; cost complementation plus IFWG aggregation in the full multiplicative form; and the Borda-sum form of dominance theory.

The final rank is the combined summary of three perspectives, as in crisp MULTIMOORA, and it is read the same way.

The difference is this. Two of the three sub-methods, the ratio system and the reference point, reduce the support-rejection pair to a single number as early as the first step, after which they bear no further relation to the intuitionistic fuzzy structure. Only the full multiplicative form carries the support-rejection structure through to the IFWG aggregation. So it is wrong to say that "intuitionistic fuzzy uncertainty was preserved across all three sub-methods".

Thus instead of writing:

"IF-MULTIMOORA is more informative because it carries intuitionistic fuzzy uncertainty throughout the whole calculation"

the report should read:

"Support and rejection degrees are preserved only up to the IFWG aggregation in the full multiplicative form; the ratio system and the reference point operate on a crisp number already reduced to the Chen-Tan score (support minus rejection)"

This extension is suitable when an assessment is given as degrees of support for, and rejection of, a judgement, and the opposing view (rejection) comes from a separate source; the rule stated on the intuitionistic fuzzy data-type card applies here without change. A measured criterion should not be converted into a single judgement degree (μ, 1−μ); that zeroes the hesitancy margin and leaves it indistinguishable from crisp data. Crisp MULTIMOORA's exit condition applies here too: all values must be positive, and this method should not be used where no compromise can ever be accepted on a criterion. In addition, if a criterion has the same Chen-Tan score across all alternatives, so that the shifted score's column norm comes out at zero, the normalisation becomes undefined.

Violating the value space. Every cell must satisfy μ ∈ [0,1], ν ∈ [0,1], and μ + ν ≤ 1; otherwise the hesitancy margin (π = 1 − μ − ν) comes out negative and the whole calculation is invalidated.

Changing the score function. DecisionMind fixes the Chen-Tan (1994) score (support − rejection); a different score function (support alone, or support plus half the hesitancy margin) can produce a different order, and this choice must be stated in the report.

Assuming the ratio system and reference point also preserve the support-rejection structure. These two are reduced to the Chen-Tan score before normalisation; only the full multiplicative form performs genuine intuitionistic fuzzy aggregation, with IFWG.

Skipping the shift step. Because the Chen-Tan score can be negative, feeding it directly into vector normalisation without the +1 shift (since the square of a negative number is positive) corrupts the normalisation and, with it, the choice of the reference point and the ideal. In the illustrative example below, skipping this step COMPLETELY changes the final order.

The governing principle is this:

In IF-MULTIMOORA the support-rejection structure is preserved only in the full multiplicative form; the ratio system and the reference point operate on a crisp number already reduced to the Chen-Tan score, and skipping the shift step renders the normalisation meaningless.