Score Function Defuzzification for Intuitionistic Fuzzy Numbers

Intuitionistic fuzzy numbers (IFNs) carry, unlike classical fuzzy numbers, two separate degrees: a degree of membership (how well an alternative satisfies a criterion) and a degree of non-membership (how much it fails to satisfy it). The sum of these two degrees need not reach 1; the remaining gap represents the share on which the expert remained undecided or offered no opinion at all. The score function is the defuzzification rule that reduces these two degrees to a single crisp figure. Its output is a single crisp score; it produces no weights and eliminates no alternatives, only converting an intuitionistic fuzzy number into a number. This rule serves as the shared final step for the intuitionistic fuzzy (IFN), Pythagorean fuzzy (PFN) and the more general q-rung orthopair fuzzy (q-ROFS) number families. It is used immediately before the ranking step of methods that work with such data, TOPSIS and VIKOR among them.

The idea behind the score function is "measure the net superiority." An expert may partly rate an alternative as both "good" (membership) and "poor" (non-membership); the score takes the net difference between these two ratings. Looking at the degree of membership alone can mislead, because an alternative with a high degree of membership may simultaneously carry a high degree of non-membership; this means the expert saw both strong positive and strong negative evidence about that alternative, that is, made a contradictory assessment. The score reduces this contradiction to a clear difference.

This philosophy carries a consequence: two alternatives may share the same score while reaching it in different ways, one through "high membership, high non-membership" (contradictory, low accuracy), the other through "moderate membership, moderate non-membership." The score alone does not reveal this distinction; a second measure, the degree of accuracy, therefore comes into play and shows which assessment carries less indeterminacy.

The method proceeds through three steps.

First, calculate the score value for every alternative. The score is found by subtracting the degree of non-membership from the degree of membership, and it measures the alternative's net superiority on that criterion. For the Pythagorean fuzzy and q-rung orthopair fuzzy number families, this difference is taken using an appropriate power of the degrees; the intuitionistic fuzzy number is the simplest special case of this family.

Second, where scores are equal, look at the degree of accuracy. If two alternatives' scores come out equal, the sum of the degrees of membership and non-membership (the degree of accuracy) is compared. The assessment with the higher degree of accuracy is preferred, since it carries less indeterminacy.

Third, rescale the score where required. Some applications do not want the score to come out negative (which happens when non-membership exceeds membership); in that case the score is rescaled to fall between zero and one. This step is optional and used only where the following step expects a non-negative input.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The figure the score function returns is an alternative's net superiority on that criterion; it is not the degree of membership alone. A high degree of membership can turn into a low score if the accompanying degree of non-membership is also high. If two alternatives' scores are equal, this does not mean the alternatives are "equally good"; the degree of accuracy must be examined to see which was assessed with less indeterminacy.

Thus instead of writing:

"The defuzzified score is the alternative's degree of membership (goodness)"

the report should read:

"The defuzzified score is the net difference between the degree of membership and the degree of non-membership; a high degree of membership can turn into a low score when accompanied by a high degree of non-membership, and equal scores are separated by the degree of accuracy"

The score function works with intuitionistic fuzzy data: a degree of membership and a degree of non-membership in every cell, the two summing to no more than 1. DecisionMind holds no independent extension for this building block on its own; it is the shared final step for the intuitionistic, Pythagorean and q-rung orthopair fuzzy families, used within these families' TOPSIS, VIKOR and similar extensions. You need, for every cell, a degree of membership and a degree of non-membership (each between 0 and 1, the two together not exceeding the constraint set by the data family). This data usually comes from two separate judgements an expert provides, "this alternative meets this criterion" and "this alternative fails to meet this criterion"; unlike classical fuzzy numbers, the expert's indeterminacy is here recorded explicitly.

The score function is the correct choice wherever experts can give both "how much I agree" and "how much I disagree" separately, and the difference between them, the indeterminacy, is itself meaningful. Where your data holds only a single "how good" degree, with non-membership never having been asked, it is classical fuzzy data rather than intuitionistic fuzzy data, and rules such as centroid, the bisector or MOM should be used instead. Where two alternatives' scores repeatedly come out equal and the degree of accuracy fails to distinguish them either, asking the experts for a clearer distinction at the data-collection stage is a more reliable solution than merely changing the defuzzification rule.

Membership and non-membership given separately, indeterminacy is meaningful → Score function

Only a single "how good" degree (a triangular/trapezoidal fuzzy number) → Centroid, the bisector or MOM

Scores keep coming out equal → Look to the degree of accuracy; if still equal, review the data collection

Data carrying an upper and lower bound together (Type-2 fuzzy) → Type reduction

The score function's clearest strength is its ability to reduce data to a single figure without losing the expert's indeterminacy; this indeterminacy is simply absent from classical fuzzy numbers before defuzzification, whereas intuitionistic fuzzy data carries it from the outset, and the score takes it into account. Used together with the degree of accuracy, it can distinguish not only the net difference but also how reliably that difference was measured. Because it carries a shared logic across the intuitionistic, Pythagorean and q-rung orthopair families, the same way of reading the result remains valid even when the data family changes.

The score function's limitations arise from its sensitivity to the data family. Hong and Choi (2000) showed that the intuitionistic fuzzy score's power to discriminate can weaken in some cases and needs support from the degree of accuracy. Zhang and Xu (2014) showed that for Pythagorean fuzzy numbers the score formula must be calculated differently from the intuitionistic fuzzy formula (by squaring the degrees), otherwise the wrong formula ends up applied to the wrong family. Yager (2017) showed that this family generalises further still with q-rung orthopair fuzzy numbers. As the value of q rises, the range of permitted membership and non-membership combinations widens, so the formula must be chosen correctly for the data family, that is, the value of q. Finally, scores that remain equal even under the degree of accuracy can still arise; in that case the method cannot produce a decision on its own.

The most common mistake is applying the intuitionistic fuzzy number's simple difference formula (membership minus non-membership) directly to Pythagorean or q-rung orthopair fuzzy data; the correct formula for these families uses an appropriate power of the degrees. A second mistake is concluding two alternatives are "equal" whenever their scores come out equal, without looking at the degree of accuracy; this means disregarding the expert's indeterminacy information. A third is calculating the score without checking whether the sum of membership and non-membership exceeds the constraint set by the data family; if the constraint is exceeded, the data is invalid and the score comes out meaningless.

The governing principle is this:

The score gives the net difference between membership and non-membership; equal scores are separated by the degree of accuracy, and the formula must be chosen correctly for the fuzzy number family the data belongs to, intuitionistic, Pythagorean or q-rung orthopair.