Intuitionistic fuzzy CoCoSo

Two things change; the five-step skeleton and the decision logic do not.

Cells. In crisp CoCoSo every cell is a single number. Here every cell is a pair, a degree of support (μ) for and a degree of rejection (ν) of a judgement; μ + ν cannot exceed 1. Criterion weights are taken from outside as crisp numbers, not as a μ/ν pair. DecisionMind does not support group decisions in this extension; every cell is either a single assessor's judgement or a single already-combined judgement.

Point of defuzzification. Here defuzzification happens at the very start. Every (μ, ν) pair is reduced, in the first step of the calculation, to a single number by the Chen-Tan (1994) score function (s = μ − ν). This score takes on the same role as the "criterion value" number that crisp CoCoSo expects. It is then scaled to the 0-1 range by direction-sensitive min-max normalisation, taken column by column: for a benefit criterion this scaling is anchored on the lowest score, and for a cost criterion it is reversed. The weighted-sum (P-like) and weighted-power-sum (R-like) measures are then computed, and finally the three compromise strategies are applied to produce the final K score; every one of these steps runs on exactly the formulas of crisp CoCoSo. The intuitionistic fuzzy structure constrains only the INPUT, that is, how μ and ν are combined and turned into a score. The OUTPUT mechanics are the same as in the crisp method.

DecisionMind fixes two things in this extension: it uses the Chen-Tan (1994) score function (μ − ν) as the defuzzification rule, and it fixes the balance coefficient in the third compromise strategy at λ=0.5. Weights are taken from outside; the method does not generate weights.

As in crisp CoCoSo, the K score is only a ranking within this particular set of alternatives, and whether the three compromise strategies agree with one another shows its robustness.

The real difference appears here. μ and ν each carry separate information: how strongly something is supported, how strongly it is rejected, how much remains undecided. This information is reduced, at the very first step, to a single difference, μ − ν, and is lost from that point on. The gap between two alternatives can look identical in the K score even when it was reached through very different support/rejection pairs. One alternative might reach the same difference through strong support and strong rejection, another through weak support and weak rejection; the two are indistinguishable in the K score. For this reason, the report should state not only the K score but also which criterion's judgement, support or rejection, is carrying the ranking.

Thus instead of writing:

"The IF-CoCoSo score is more reliable because it also takes the expert's indecision into account"

the report should read:

"The share of indecision stays in the cell only until it enters the μ − ν score; from the moment the score is computed, it is processed exactly as in crisp CoCoSo, so the robustness of the result must still be tested separately (do the three strategies agree, does the ranking break down if one judgement shifts by a single step)"

Use the intuitionistic fuzzy structure when criteria come from a JUDGEMENT rather than a MEASUREMENT, and when evidence for and against this judgement can be gathered separately. This is appropriate, for example, when the support and rejection given to a proposition such as "this supplier is reliable" come from separate sources, say different auditor reports or different pieces of evidence. Computing rejection as 1 − support carries none of this extension's contribution, because the result comes out no different from crisp CoCoSo.

The exit conditions of the crisp method apply here in exactly the same way. The matrix must be of a single type, and the direction of each criterion (more is better or less is better) must be marked correctly. Direction is especially critical here, because it determines which end of the score (μ − ν) is preferred. If direction is marked incorrectly, the ranking is COMPLETELY reversed; see the "What Not to Do" section below for details.

Forgetting the direction marker, or entering it incorrectly. In the illustrative example below, if the third criterion's direction, which is actually "less is better," is marked as "more is better," the ranking flips from A3-A2-A1 to A1-A2-A3; this is a complete reversal, not a small shift.

Feeding a pair with μ + ν > 1 into the calculation without noticing. This is an invalid intuitionistic fuzzy number; the share of indecision comes out negative and the score becomes meaningless.

All alternatives sharing the same μ − ν score in one column. In this case the denominator of the min-max normalisation goes to zero and the calculation becomes undefined; DecisionMind rejects this with a typed error (DEGENERATE_DATA_FOR_METHOD).

Silently changing the score function. Chen-Tan (μ − ν) is the fixed rule here. A different score function, for example using μ alone or adding an accuracy function such as μ + ν, can produce a different ranking. This change should never be made without stating it in the report.

The governing principle is this:

IF-CoCoSo carries the degrees of support and rejection only as far as the Chen-Tan score, at the very start of the calculation. Once the score is computed, it is crisp CoCoSo itself that carries the ranking from then on. What matters most is that the direction marker is entered correctly.