Intuitionistic fuzzy MOORA

Cells. In crisp MOORA every cell is a single number. Here every cell is a judgement's pair of support (μ) and rejection (ν) degrees; their sum cannot exceed 1, and what remains is hesitancy. DecisionMind takes criterion weights as crisp, single numbers in this extension; the uncertainty is carried only in the cells of the decision matrix.

Scale equalisation. Crisp MOORA divides a cell's value directly by its column's magnitude. IF-MOORA first reduces every cell's support-rejection pair to a single score: the score is support minus rejection (μ − ν), and because this value can range from −1 to 1, it is shifted by adding 1, putting it between 0 and 2. This is not, however, a "defuzzification" of the kind seen in the fuzzy or neutrosophic extensions. The hesitancy margin plays no part here at all; only the difference between support and rejection is used. The shifted score is then normalised exactly as in crisp MOORA, divided by the root of its own column's magnitude.

Score and ratio system. The normalised, shifted scores are weighted; the sum over the "lower is better" criteria is subtracted from the sum over the "higher is better" criteria. This is exactly the same operation as crisp MOORA's ratio system; the only difference is that the input has first been reduced to a support-rejection score.

Result and defuzzification. The score-shifting step is itself the defuzzification, and it happens at MOORA's first step, before cell normalisation, not at the end. This differs from Fuzzy MOORA and N-MOORA, where defuzzification happens at the last step, by averaging; in IF-MOORA the support-rejection pair descends to a single score as early as the first step, and the rest of the calculation proceeds on this score exactly as in crisp MOORA.

DecisionMind fixes, for classical IF-MOORA, the score function μ − ν (the Chen-Tan score) and its calculation at the first step; the hesitancy margin (π = 1 − μ − ν) is not used separately at any step.

The output is a net score and a rank, as in crisp MOORA, and it is read the same way. The difference is this. The score rests on the difference between a judgement's support and rejection degrees, and never sees the hesitancy margin, the portion the expert neither supported nor rejected. Two assessments with the same score difference (μ − ν) but different hesitancy margins produce the same result in IF-MOORA. This means a judgement with high hesitancy counts exactly as much as one with low hesitancy.

Thus instead of writing:

"IF-MOORA also took hesitancy into account"

the report should read:

"IF-MOORA took into account only the difference between support and rejection; if this difference is the same, the result is the same even where two assessments' hesitancy margins differ, so criteria with high hesitancy must be flagged separately in the report"

Use this extension when criteria come not from a measurement but from a judgement, and the evidence for and against that judgement can be collected separately: board votes, expert support/reservation assessments, the support and dissenting-vote proportions given to a proposition such as "this candidate is suitable". If the expert states only "how suitable" and there is no separate source for the opposing view, that is, if ν would have to be written as 1 − μ, this extension is unnecessary and crisp or fuzzy MOORA is enough. The exit condition is the same as for crisp MOORA: if no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold.

Violating the value space. Every cell must satisfy μ, ν ∈ [0,1] and μ + ν ≤ 1; a pair whose sum exceeds 1 (a judgement given, say, both strong support and a strong reservation) does not fit this structure and requires a different one, such as Pythagorean fuzzy.

Writing ν as 1 − μ. This zeroes the hesitancy margin and reduces the structure to crisp data, erasing IF-MOORA's one contribution, the principle that support and rejection come from separate sources.

Treating the score difference (μ − ν) as the only indicator and never reporting hesitancy. Two criteria can give the same score even though one, such as (0.90; 0.10), is almost fully decided, and the other, such as (0.50; 0.00), is half hesitant; IF-MOORA treats them identically, but the report must tell the reader about this difference.

Turning a measured score directly into a degree of support. Writing a mark of 70 out of 100 as μ = 0.70 and giving ν = 0.30 puts only the score's complement in place of counter-evidence; rejection must be derived from a separate source, such as a dissenting vote or an adverse report.

The governing principle is this:

IF-MOORA carries only the difference between support and rejection; the hesitancy margin neither enters the calculation nor appears in the report of its own accord, so a criterion with high hesitancy must be noted separately.