q-Rung orthopair WASPAS (DecisionMind derivation)

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

Cells. In crisp WASPAS every cell is a single number. Here every cell is a pair: μ support degree, ν rejection degree. These two values must satisfy μ^q + ν^q ≤ 1; as q grows, the range of accepted pairs widens. In DecisionMind, q is not entered by the user; for this extension q = 3, and it comes bundled with the method's definition, unchangeable cell by cell.

Scale equalisation. Crisp WASPAS equalises every column by scaling it against its own best value. There is no such division step here, because support-rejection pairs are already between 0 and 1. Instead, μ and ν are swapped for a cost criterion. This operation does not equalise scale, it reverses direction.

Sum and product components. Crisp WASPAS's weighted sum (WSM) is matched here by the q-rung orthopair weighted average (q-ROFWA). Each alternative's support-rejection pair across the criteria is first reduced to a single pair. This pair is then turned, by the score function (s = μ^q − ν^q), into a single number called Q1. Crisp WASPAS's weighted product (WPM), in turn, is matched by the q-rung orthopair weighted geometric mean (q-ROFWG); the same score turns it into Q2. q-ROFWA magnifies support degrees optimistically, while q-ROFWG brings rejection degrees to the fore; this corresponds to the compensatory character of the sum and the penalising character of the product, respectively.

Result and combination. In crisp WASPAS, λ is a parameter adjustable in the interface between 0 and 1, defaulting to 0.5. In this extension λ is not offered as an interface field; it arrives as an additional input the engine accepts, with a default of 0.5, and can be changed through this additional input when needed.

DecisionMind fixes this direction correction, the q-ROFWA/q-ROFWG combination and the score s = μ^q − ν^q in QR-WASPAS; q comes from the method's definition, weights are taken from outside as fixed numbers, and the method does not generate weights.

The output is a combined score and a rank, as in crisp WASPAS; 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); with which weight (λ) this blend is made does not show in the interface, and is 0.5 unless changed by the additional input. Beneath the score there is also a support-rejection pair, and this pair, under the constraint μ^q + ν^q ≤ 1 at q = 3, covers a wider area than Pythagorean fuzzy (q = 2). The gap between two scores depends on how much support and rejection was given on which criterion.

Thus instead of writing:

"According to QR-WASPAS, E2 is the best alternative"

the report should read:

"This score comes from combining the support-rejection pairs separately with q-ROFWA and q-ROFWG and blending them with λ=0.5; this lead depends on how much support and rejection was scored on which criterion, and can shift once the weights change"

Use this extension when experts give a judgement both very strong support and a very strong reservation at once, and these pairs also exceed the Pythagorean constraint (μ² + ν² ≤ 1). If expert pairs already fit within the intuitionistic or Pythagorean constraint, raising q brings no benefit; it only reduces discriminating power. It should be kept in mind that the whole matrix runs on a single q; the q a given cell requires is the q required by the most extreme pair in the matrix. The exit condition of crisp WASPAS holds here too: if no compromise is acceptable on one criterion, this extension too is fully compensatory.

Domain violation. Every cell must satisfy μ ∈ [0,1], ν ∈ [0,1] and μ^q + ν^q ≤ 1; q = 1 corresponds to intuitionistic fuzzy, q = 2 to Pythagorean fuzzy. Entering the calculation without checking this invalidates the method.

Choosing a large q without regard to the data. In this extension q is fixed and comes with the method's definition; asking for a different q "for flexibility" is not what this extension is for. If expert pairs exceed the range that q = 3 requires, this extension falls short and a method with a larger q is needed.

Writing ν as 1 − μ. In that case the sum is always exactly 1, and the extra acceptance range that q = 3 provides is never used at all.

Looking for λ in the interface. λ here is not an interface field like the weights; it arrives as an additional input with a default of 0.5. This distinction must be stated in the report; otherwise the reader may assume λ never changed at all.

The governing principle is this:

In QR-WASPAS, the support-rejection pair is combined and scored separately with q-ROFWA and q-ROFWG; q comes from the method's definition, it is not chosen cell by cell, and it is single for the whole matrix.