Intuitionistic fuzzy Entropy (Hung & Chen, 2010)

Three things change; the entropy logic (a flat distribution carries little information, a spread-out one carries much) is preserved, but the formula itself is different.

Cells. In crisp entropy 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. This extension supports group decisions: more than one decision-maker's pairs are combined into a single collective matrix using the intuitionistic fuzzy weighted average (IFWA) operator, with a weight for each decision-maker; with a single decision-maker, this step is an identity transformation, that is, nothing changes.

Scale equalisation. Crisp entropy converts cost criteria and then divides by the column sum to obtain probabilities, then uses Shannon's two-term formula (−p·ln p). This extension never converts to probability at all: the support-rejection-indecision triple (μ, ν, π = 1−μ−ν) is used directly in a four-term entropy expression. For every alternative, μ·ln μ + ν·ln ν − (1−π)·ln(1−π) − π·ln 2 is summed, then scaled by the number of alternatives and ln 2. The benefit/cost direction does not affect this formula, because the measure is symmetric under swapping support and rejection; exchanging μ and ν leaves entropy unchanged. This is the biggest difference from crisp entropy: direction information is unnecessary here.

Entropy and weight. The remaining steps follow the same logic as crisp entropy: divergence (1 − entropy) is computed, and dividing each divergence by the sum of all divergences gives a weight vector summing to 1. The only difference is how entropy is computed (a four-term intuitionistic fuzzy measure, in place of a two-term probabilistic measure).

DecisionMind fixes four things in this extension: the IFWA aggregation on group input, the Vlachos-Sergiadis (2007) four-term entropy measure, the 0·ln(0) = 0 convergence rule, and keeping the divergence-then-normalise steps in the same form as crisp entropy.

The output is a weight vector summing to 1, as in crisp entropy, and is read by the same principle: the weight shows not the criterion's importance, but how much it discriminates between alternatives in this particular table.

The difference lies here. "Discrimination" here looks not only at variation in the degree of support, but also at variation in the share of indecision. If all alternatives on a criterion have high support and low rejection, but their shares of indecision differ substantially, that criterion can still receive low entropy, that is, high weight. This is because the measure treats indecision itself as a source of divergence. Crisp entropy cannot see this at all, because it has no concept of indecision.

Thus instead of writing:

"This criterion's weight came out low because there was little difference in support among the alternatives"

the report should read:

"This criterion's weight came out low because support, rejection and the share of indecision were all close to one another across the alternatives; intuitionistic fuzzy entropy assesses all three together"

Use this extension when an assessment is given as support for and rejection of a judgement, and the opposing view comes from a separate source; the rule given on the intuitionistic fuzzy data-type card applies here in exactly the same way. The exit condition of crisp entropy applies here as well: if a criterion is identical across all alternatives, its weight comes out at zero and it should be removed from the table. In addition, if all alternatives share exactly the same support-rejection pair on one criterion, that criterion's entropy comes out at exactly 1, its divergence at exactly 0; if every criterion is like this, the weight denominator goes to zero and the method becomes undefined.

Value-space violation. Every cell must satisfy μ ∈ [0,1], ν ∈ [0,1], μ + ν ≤ 1; otherwise the share of indecision comes out negative and the entropy measure becomes meaningless.

Confusing it with crisp ENTROPY. This is the mistake the manifest itself warns about most strongly: intuitionistic fuzzy entropy (four-term, the Vlachos-Sergiadis 2007 measure) and crisp entropy (two-term, Shannon 1948, on probability-normalised data) are different formulas. Treating the degree of support (μ) directly as a probability and running the classic Shannon formula, that is, ignoring the rejection and indecision information, can produce a completely different, even reversed, weight distribution.

Failing to apply the special rule at the boundary. If the x·ln(x) = 0 convergence rule is not applied when μ = 0 or ν = 0, the calculation comes out undefined (ln 0).

Failing to notice a table where every criterion is degenerate (identical). If every alternative's support-rejection pair is very close to every other's on every criterion, total divergence approaches zero and weights become undefined; DecisionMind raises an error in this case, rather than silently filling the result with "equal weight."

The governing principle is this:

Intuitionistic fuzzy entropy processes the support-rejection-indecision triple with a four-term measure, without converting it to probability; confusing this with crisp entropy's two-term probabilistic measure produces a different, even reversed, weight distribution.