Pythagorean fuzzy WASPAS (DecisionMind derivation)

Four things change; the idea of blending a sum with a product does not.

Cells. In classical WASPAS every cell is a single number. Here every cell is a pair: μ, the support degree, and ν, the rejection degree. These two values must satisfy μ² + ν² ≤ 1, though their plain sum can exceed 1. Criterion weights remain crisp numbers taken from outside.

Scale equalisation. Classical WASPAS equalises every column by dividing it by its own best value. There is no such division step here, because support-rejection pairs already lie between 0 and 1. Instead, μ and ν swap places on a cost criterion. This operation does not equalise scale; it reverses direction.

Sum and product component. Classical WASPAS's weighted sum (WSM) is met here by the Pythagorean fuzzy weighted average (PFWA). Each alternative's support-rejection pairs across the criteria are first reduced to a single pair. This pair is then turned into a single number, Q1, by the score function (s = μ² − ν²). Classical WASPAS's weighted product (WPM) is met by the Pythagorean fuzzy weighted geometric mean (PFWG), which likewise returns Q2 through the same score. PFWA inflates support optimistically, while PFWG brings rejection to the fore; this mirrors the compensatory character of a sum and the penalising character of a product, respectively.

Result and combination. In classical WASPAS, λ is a parameter that can be changed in the interface, between 0 and 1, with a default of 0.5. In this extension, λ is fixed inside the engine at 0.5, and no input can change it; it is offered neither as an interface field, like the weights, nor as an extra input, as in Intuitionistic Fuzzy WASPAS. The combined score is always the equally weighted average of Q1 and Q2.

DecisionMind keeps this direction correction, the PFWA/PFWG combination, the score s = μ² − ν² and the λ = 0.5 constant unchangeable in PF-WASPAS; weights are taken from outside as crisp numbers, and the method does not generate weights.

The output is a combined score and a rank, as in classical WASPAS, and it is read the same way: it is not a percentage or a probability, and the ranking can shift when the alternative set changes.

The difference is this. The score blends an optimistic sum view (Q1) with a cautious product view (Q2) in equal weight, and this blend can never be altered. Beneath the score also lies a support-rejection pair, and under the constraint μ² + ν² ≤ 1 this pair covers a wider region than an intuitionistic fuzzy one. The score gap between two alternatives depends on how much support and rejection each received on which criterion.

Thus instead of writing:

"PF-WASPAS shows S2 to be the best alternative"

the report should read:

"This score results from combining the support-rejection pairs separately through PFWA and PFWG and blending them in equal weight; this advantage depends on how much support and rejection was given on which criterion, and it can shift if the weights change"

Use this extension when experts give a judgement both strong support and marked reservation at once, and the sum of these two degrees exceeds 1; the intuitionistic fuzzy constraint would force such pairs to shrink. If the sum never exceeds 1, Intuitionistic Fuzzy WASPAS is enough, and there is no need to move to Pythagorean. If the sum of squares also exceeds 1, Pythagorean falls short and a q-rung orthopair form is required. Classical WASPAS's exit condition still applies here: if no compromise is acceptable on one criterion, this extension too is fully compensatory.

If control over λ is wanted, that is, if the balance between WSM and WPM needs manual adjustment, this extension is not suitable; λ is fixed here. Such a need should turn instead to classical WASPAS, or to Intuitionistic Fuzzy WASPAS, which offers λ as an extra input.

Violating the value domain. Every cell must satisfy μ ∈ [0,1], ν ∈ [0,1] and μ² + ν² ≤ 1. Feeding the calculation without checking this invalidates the method.

Assuming λ can be changed. In classical WASPAS, and in some extensions of the family, λ varies as an input; in this extension λ is fixed inside the engine at 0.5. A statement in the report such as "tried with λ = 0.3" is wrong for this extension; λ never changes in this family.

Writing ν as 1 − μ. This makes the sum always exactly 1, and the extra space the Pythagorean structure provides over the intuitionistic one goes entirely unused.

Changing the score function and expecting the same result. s = μ² − ν² is the canonical choice but not the only one. A different score can produce a different Q1 or Q2; whichever score was used must be stated in the report.

The governing principle is this:

In PF-WASPAS, the support-rejection pair is combined and scored separately through PFWA and PFWG; the two scores are then blended in an unchangeable equal weight. Writing ν as 1 − μ, or using this extension for a pair that already fits within the intuitionistic constraint, wastes the extension's one contribution: its wider zone of acceptance.