Interval-valued intuitionistic fuzzy MOORA

Two things change; the vector norm and the ratio-system logic do not.

Cells. In crisp MOORA 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. A judgement such as "my confidence in this supplier is between 0.65 and 0.75, my reservation between 0.15 and 0.20" corresponds to these four numbers. Criterion weights are taken from outside as crisp numbers.

Defuzzification comes first; the vector norm applies to the scores. Every four-number support-rejection interval descends, in the very first step of the calculation, to a single number through a score: the score is found by subtracting the sum of the rejection bounds from the sum of the support bounds and dividing by two. Since the score can range from −1 to 1, 1 is added to shift it into positive territory. Crisp MOORA's vector normalisation is then applied to these shifted scores; there is no separate fuzzy normalisation on the interval itself.

Ratio system. The normalised scores are multiplied by the weights; for every alternative, the sum of the weighted shares on the cost criteria is subtracted from the sum of the weighted shares on the benefit criteria. This is exactly the same operation as crisp MOORA's ratio system.

DecisionMind fixes, in this extension, the support-rejection mid-point score, the +1 shift, and the vector norm applied to these shifted scores. Weights are taken from outside; the method does not generate weights.

The MOORA score is read exactly as in crisp MOORA: an alternative's net advantage in its share of the beneficial criteria over its share of the harmful ones, not comparable with another analysis. The difference is here: this share rests only on the mid-point of the four-number support-rejection interval. How wide that interval is, that is, how precisely the judgement was given, never appears in this score. A score arising from a narrow interval and the same score arising from a wide interval are ultimately indistinguishable.

Thus instead of writing:

"According to interval-valued intuitionistic fuzzy MOORA, the best alternative is this one"

the report should read:

"With these weights and this alternative set, the alternative with the highest net score under the ratio system is this one; this score rests only on the mid-point of the support-rejection interval, and the interval's width is not reflected in it"

If an expert can give degrees of support and rejection as an interval rather than a single number (for instance, "support is between 0.65 and 0.75, rejection between 0.15 and 0.20"), this extension is worth considering; the detail is on the intuitionistic fuzzy data-type card. If support and rejection are each given as a single number, there is no need to open them into intervals; doing so only adds to the computational load without contributing to the score, because the score only looks at the mid-point. Crisp MOORA's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is likewise compensatory.

Domain violation. In every cell, the upper support plus the upper rejection must not exceed 1, and the lower bound cannot exceed its own upper bound.

Reading interval width as a sign of robustness. The score is computed only from the mid-point; a narrow interval and a wide one with the same mid-point give the same score. Interval width, reflecting for instance disagreement among experts, must be reported separately.

Confusing this with fuzzy (triangular) MOORA. Here the cell is not a three-cornered triangle but a four-number support-rejection interval pair; the two structures follow different operational rules.

Marking criterion direction incorrectly. If a cost criterion is marked as a benefit criterion, the alternative with the least favourable support-rejection interval is pulled towards the ideal.

The governing principle is this:

In interval-valued intuitionistic fuzzy MOORA, the one genuine loss of information is that the four-number support-rejection interval descends to a single mid-point score as early as the first step; the interval's width never appears again in the rest of the calculation.