Interval-valued intuitionistic fuzzy WASPAS (a DecisionMind derivation)

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

Cells. In crisp WASPAS every cell is a single number. Here every cell consists of four numbers: the lower and upper bound of the degree of support, and the lower and upper bound of the degree of rejection. A judgement such as "my confidence in this application is between 0.65 and 0.75, my reservation between 0.15 and 0.20" corresponds to these four numbers. Criterion weights are taken from outside as crisp numbers.

Scale equalisation. Crisp WASPAS equalises every column by ratioing it to its own best value. There is no such division step here, because support-rejection intervals already lie between 0 and 1. Instead, on a cost criterion, the support and rejection intervals swap places: ([μL,μU], [νL,νU]) → ([νL,νU], [μL,μU]). This is this family's counterpart to crisp WASPAS's ratio-to-best step; it reverses direction, not scale.

Sum and product components. Crisp WASPAS's weighted sum (WSM) is met here by interval-valued intuitionistic fuzzy weighted averaging (IIFWA). Every alternative's support-rejection interval pair across the criteria is first reduced to a single interval pair, and this pair then descends to a single number, Q1, through a score (subtracting the sum of the rejection bounds from the sum of the support bounds and dividing by two). Crisp WASPAS's weighted product (WPM) is met by interval-valued intuitionistic fuzzy weighted geometric aggregation (IIFWG), which descends to Q2 with the same score. IIFWA grows support in an optimistic way, whereas IIFWG brings rejection to the fore; these correspond respectively to the compensatory character of summation and the punitive character of the product.

Result and combination. In crisp 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 additional input as in intuitionistic fuzzy WASPAS. The combined score is always the equally weighted average of Q1 and Q2.

DecisionMind keeps this complementing rule, the IIFWA/IIFWG aggregation, and the λ=0.5 constant unchangeable in IV-WASPAS; weights are taken from outside as crisp numbers, and the method does not generate weights.

The output is a combined score and a ranking, as in crisp WASPAS, and 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 total view (Q1) with a cautious product view (Q2) in equal measure, and this blend can never be changed. Beneath the score also lies a four-number support-rejection interval; two alternatives' score gap may come from one narrow and one wide interval. Interval width shows how much the assessors agree on this judgement, and it does not appear in the score.

Thus instead of writing:

"According to IV-WASPAS, G2 is the best alternative"

the report should read:

"This score is a value that emerges from combining the support-rejection intervals separately with IIFWA and IIFWG and blending them in equal measure; which alternative's interval is wider, that is, less agreed upon among assessors, must be reported separately"

This extension suits situations where a criterion assessment rests on a judgement, and that judgement's own degree of support and rejection is itself uncertain. This is natural where several assessors' support-rejection scores are known only by a lower and an upper bound with no single consensus point. If support and rejection already sit at a single point (interval width zero), this extra width contributes nothing, and intuitionistic fuzzy WASPAS suffices. Crisp WASPAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is likewise fully compensatory.

If control over λ is wanted (if the balance between WSM and WPM needs to be adjusted by hand), this extension is not suitable; λ is fixed here. In that case, crisp WASPAS, or intuitionistic fuzzy WASPAS, which offers λ as an additional input, should be considered instead.

Allowing the lower bound to exceed the upper bound. In every interval the lower bound must be less than or equal to the upper bound; if this is not checked, both the support-rejection constraint (μU+νU≤1) and the IIFWA/IIFWG aggregation give a meaningless result.

Assuming λ can be changed. In crisp WASPAS, and in some members of the family, λ changes 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 incorrect for this extension; λ never changes in this family.

Fabricating ν as 1 − μ. This zeroes the interval's width without justification and silently collapses the structure back into single-point intuitionistic fuzzy data.

Reading the score as "what percentage probability." The score is a combination of support and rejection bounds, not a probability. The score used here is a mid-point definition; other score definitions also exist in the literature, and which one was used must be stated in the report.

The governing principle is this:

In IV-WASPAS, the support-rejection interval is aggregated and scored separately with IIFWA and IIFWG, and these two scores are then blended, unchangeably, in equal measure; interval width shows the degree of agreement among assessors, and the report must state this separately.