Intuitionistic fuzzy WPM

Three things change; the multiplicative logic, and defuzzification coming last, do not.

Cells. In crisp WPM every cell is a single, strictly positive number. Here every cell is a pair of degrees, support (μ) and rejection (ν), given to a judgement; μ + ν cannot exceed 1. Criterion weights are taken from outside as crisp numbers.

Direction handling (cost criteria). In crisp WPM a cost criterion is handled by "divide the smallest value into the cell." In Fuzzy WPM, the value and the weight triple are both inverted together. Here a third route is taken. Every (μ, ν) pair on a cost criterion is converted to its own complement, that is, (μ, ν) is written as (ν, μ). This means the degrees of support and rejection given to the judgement swap places, and it is the intuitionistic fuzzy counterpart of the WPM family's "take the reciprocal" logic.

Aggregation and defuzzification. In crisp WPM, criteria combine directly through a weighted product (Π r_j^{w_j}). Here the same multiplicative logic runs through the intuitionistic fuzzy weighted geometric aggregation operator (IFWG, Xu 2007). The combined support is computed as Π μ_j^{w_j}, and the combined rejection as 1 − Π(1 − ν_j)^{w_j}. Throughout this step the (μ, ν) pair is carried as a whole; it does not descend to a single number.

Defuzzification happens only in the final step, with the Chen-Tan (1994) score function (μ − ν). This order matters. The score cannot be computed first and then fed into crisp WPM's ratio-and-power machine, because μ − ν can come out negative; indeed, this is exactly what happens on the third criterion in the illustrative example below. Crisp WPM, by contrast, requires strictly positive data.

DecisionMind fixes two rules in this extension: the IFWG operator (Xu 2007) as the aggregation rule, and the Chen-Tan (1994) score function (μ − ν) as the defuzzification rule. Weights are taken from outside; the method does not generate weights and does not support group decisions.

As in crisp WPM, the score is only a ranking within this particular alternative set. Crisp WPM's advantage of unit independence does not apply here, because the criteria are no longer measured quantities but degrees of support and rejection given to judgements.

The difference is here. The combined (μ_P, ν_P) pair can be read on its own before it descends to a score. If the combined support is high but the combined rejection is also high, that is, if strong evidence exists both for and against at once, the Chen-Tan score can average this difference out and conceal it. When two alternatives' scores are close, whichever has the lower combined rejection carries fewer reservations. Looking at the combined rejection is therefore more informative than looking at the score gap alone.

Thus instead of writing:

"A2 has the highest score, so A2 is the alternative with the fewest reservations"

the report should read:

"A2 has the highest score; but this score is the difference between the combined support and the combined rejection. Whichever alternative has the lower combined rejection is, from a different angle, the one with fewer reservations, and the two do not always point to the same alternative"

Use this extension when criteria come not from a measurement but from a judgement, and the evidence for and against that judgement can be gathered separately. Computing rejection as 1 minus support zeroes out this extension's contribution and leaves it indistinguishable from crisp WPM, where support alone is used.

The crisp method's exit condition applies here too. A criterion on which every alternative scores identically can make scale equalisation undefined. Pairs with μ + ν > 1 are also invalid and must be screened out before the calculation.

Forgetting to take the complement on a cost criterion. In the illustrative example below, failing to take the complement of the third criterion flips the ranking from A2-A3-A1 to A1-A2-A3; this is not a small shift but a complete reversal.

Descending to the Chen-Tan score (μ − ν) first and only then feeding it into crisp WPM's ratio-and-power machine. In the illustrative example, the μ − ν scores on the third criterion are negative (−0.1; −0.3; −0.5). Since crisp WPM requires strictly positive data, defuzzifying first leads either to an undefined operation (raising a negative base to a power) or to a meaningless shift. The correct order is to combine the (μ, ν) pair as a whole with IFWG first, and only then take the score.

Feeding a pair with μ + ν > 1 into the calculation unnoticed. This is not a valid intuitionistic fuzzy number; the degree of hesitancy comes out negative.

Silently changing the score function. Chen-Tan (μ − ν) is the fixed rule here; using μ alone, or switching to the accuracy function (μ + ν), can produce a different ranking and must not be changed without stating so in the report.

The governing principle is this:

In IF-WPM the (μ, ν) pair is carried as a whole through to the end of the IFWG aggregation, and only in the final step does it descend to a score; reversing this order (score first, aggregation second) produces negative values and breaches crisp WPM's positivity requirement.