Picture fuzzy SAW

Cells. In crisp SAW every cell holds a single number. Here every cell holds three degrees: support (yes, μ), abstention (η), and rejection (no, ν); their sum cannot exceed 1. This triple comes from a board's or a survey's vote distribution on the same judgement. Criterion weights are taken from outside, as crisp numbers.

Scale equalisation. Crisp SAW first divides every column by its own magnitude. There is no column-based division in this family; picture fuzzy triples are already a structure between 0 and 1. Instead, in a cost criterion, support and rejection swap places while the abstention degree stays as it is: (μ, η, ν) → (ν, η, μ).

Weighted sum and defuzzification. Crisp SAW weights every column and sums it directly. Here, all the criteria that have been brought to the "higher is better" direction are reduced, in a single operation, directly to one aggregate triple through a picture fuzzy weighted aggregation (PFWA): μ_i = 1−∏(1−μ_ij)^{w_j}, η_i = ∏η_ij^{w_j}, ν_i = ∏ν_ij^{w_j}. The weights are used exponentially inside this aggregation; there is no separate column normalisation as in crisp SAW. Only at this last step is the aggregate triple reduced to a single number with Garg's (2017) score function (s = μ − η − ν); in the event of a tie, the alternative with the lower accuracy measure H = μ + η + ν is placed ahead.

DecisionMind fixes, in classic picture fuzzy SAW, the PFWA aggregation, the Garg score function (μ−η−ν), and the H-based tie-breaking rule. Weights are taken from outside as crisp numbers; the method does not generate weights.

The score is read like crisp SAW's total score; it only ranks this set of alternatives, and is not a percentage or a probability.

The difference is this. The score subtracts all three of μ, η and ν together, giving a value between −1 and 1; the abstention share (η) is a component that pulls the score towards the negative, and it is not as visible as support. Two alternatives can reach the same score, one from low rejection, the other from high abstention.

Thus instead of writing:

"This alternative is definitely the best because its picture fuzzy SAW score comes out highest"

the report should read:

"This score is the support-minus-abstention-minus-rejection difference emerging from the weighted aggregation of the vote distribution; which alternative carries the larger abstention share, that is, how 'open' the decision remains, should be shown separately"

Use this extension when criterion assessments come from a board's, a survey's, or a panel's yes, abstain, no vote distribution on the same judgement; for that, the three shares must be separately countable. Inventing the other two degrees from a single support percentage erases this extension's one contribution; the detail is on the Picture Fuzzy data-type card. There is no need to expand a measured criterion (such as price or time) into a picture fuzzy triple. The exit condition is the same as for crisp SAW: if no concession is acceptable on a criterion, this extension is compensatory too, and will not eliminate anything below a threshold.

Value-space violation. In every cell, the sum μ+η+ν must not exceed 1; this must be checked before aggregation.

Inventing a triple from a single proportion. Saying "there is 60 per cent support, so 40 per cent is opposed" sets the abstention share to zero; every degree must be counted separately.

Computing the score as μ−ν and skipping η. In this engine the score uses all three degrees at once (μ−η−ν); leaving the abstention share out can produce a different, incorrect ranking.

Defuzzifying first and then running crisp SAW. Taking every cell's support ratio alone and applying the crisp method is not this extension; the abstention and opposing-vote information is erased in the first step.

The governing principle is this:

Picture fuzzy SAW exists to carry a board's or a survey's abstention share through to the construction of the aggregate triple. Any application that invents a degree or crispens the input from the outset erases this contribution.