Intuitionistic fuzzy WASPAS

Four things change. The logic of blending the sum with the product stays the same.

Cells. In crisp WASPAS every cell is a single number. Here every cell is a judgement's degree of support and degree of rejection (μ, ν); their sum cannot exceed 1, and the gap between them is the degree of hesitancy. Weights remain crisp; intuitionistic fuzzy data is used only in the decision-matrix cells.

Complementing instead of scale equalisation. Crisp WASPAS equalises every column by dividing it by its own best value; this division is not defined for (μ, ν) pairs. Instead, for cost criteria every cell's degree of support and degree of rejection swap places (μ and ν exchange positions, called taking the complement): on a "lower is better" criterion, a high value is now read through low support. Benefit criteria are left as they are. This is the counterpart of crisp WASPAS's direction correction, but it is a swap, not a division.

IFWA and IFWG instead of sum and product. Crisp WASPAS's weighted sum (WSM) is met here by the Intuitionistic Fuzzy Weighted Averaging operator (IFWA). Degrees of support are combined across criteria in an optimistic manner; strong support on a single criterion can pull the overall result upward. This resembles the compensatory character of summation. Crisp WASPAS's weighted product (WPM) is met by the Intuitionistic Fuzzy Weighted Geometric operator (IFWG). Here degrees of rejection dominate; strong rejection on a single criterion pulls the overall result downward. This resembles the punitive character of multiplication. Both aggregations produce their own (μ, ν) pair; these are reduced to a single number by the Chen-Tan (1994) score function, that is, support minus rejection.

Result and combination. The two scores, one from IFWA and one from IFWG, are blended with λ. DecisionMind does not compute λ from the data as Fuzzy WASPAS does; it uses crisp WASPAS's default value of 0.5. This value is not an interface parameter like the weights; it can be changed as an additional input between 0 and 1.

DecisionMind fixes, for IF-WASPAS, the complementing rule, the IFWA/IFWG combination and the Chen-Tan score function; 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; 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 here. The score is no longer a single measurement but a blend of two distinct views: one optimistic (IFWA), one cautious (IFWG). Each answers "how much is it supported, how much is it rejected" from its own angle, and λ=0.5 gives both views equal say. If the two components agree on the ranking, the result is robust. If they suggest a different ranking, the combined score depends on the choice of λ, and this dependency must be shown in the report. Furthermore, the size of the score alone does not say "how reliable." The gap between an alternative scoring 0.41 and one scoring 0.33 can close if the experts become one notch more supportive, or rejecting, on a single criterion.

Thus instead of writing:

"According to IF-WASPAS, A2 is the most reliable alternative"

the report should read:

"When the IFWA and IFWG components are blended with λ=0.5, A2 has the highest combined score; however, this lead reverses if A3's degree of support on the heaviest criterion rises by one notch, so the gap is sensitive to a single notch of expert judgement"

Use this extension when a criterion assessment comes from a judgement, and the opposing view (rejection) carries as much information as support does. It suits committee votes, board decisions, and expert opinions gathered in the form of "how much do you agree, how much do you disagree." If the degree of rejection is computed as 1 minus support (with no separate source), this extension is unnecessary; the crisp or fuzzy structure carries the same information. If the value is measured, switch to the base method.

WASPAS's exit condition applies exactly as it does there: the IFWA component is compensatory, so it is unsuitable if no compromise is acceptable on one criterion; if weights rather than a ranking are needed, methods such as AHP, BWM, CRITIC or Entropy should be used.

Allowing the sum of support and rejection to exceed 1. A pair with μ + ν > 1 does not fit this structure; complementing and the IFWA/IFWG combination give an undefined or misleading result once this constraint is broken.

Writing rejection as 1 minus support. The core warning on the intuitionistic fuzzy data-type card applies here too; doing so zeroes the degree of hesitancy and erases the extension's own contribution, namely that rejection has its own separate source.

Skipping the complement on a cost criterion. The direction correction here is a swap of (μ, ν), not a division; skip this step and a high degree of support on a "lower is better" criterion is rewarded by mistake.

Confusing IFWA with IFWG, or reporting only one. The two have different characters, one optimistic, one cautious; showing only one and omitting the other hides the combined score's sensitivity to λ.

Changing the score function. The Chen-Tan (1994) score (support minus rejection) is fixed here; a different score function (support alone, for instance) can produce a different ranking and must not be changed without stating so in the report.

The governing principle is this:

Degrees of support and rejection must come from separate sources, and their sum must not exceed 1; if IFWA and IFWG suggest different rankings, the report must clearly show that the combined score depends on the choice of λ.

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