Extension card · Picture
Picture fuzzy WASPAS (Chowdhury, Chatterjee and Chakraborty, 2025)
This is the form of WASPAS for situations where criterion scores come from a committee's or a survey's yes-abstain-no vote distribution. The weighted-sum and weighted-product components are computed separately on these triples, and only the final step reduces them to a single score.
Base method
WASPAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Picture →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Cells. In crisp WASPAS 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 committee's or a survey's vote distribution on the same judgement. Criterion weights are also requested here as a picture fuzzy triple: DecisionMind uses both the triple weights (v̄_j) and the crisp weights derived from them, or entered separately, (w_j) together.
Scale equalisation. Crisp WASPAS first divides every column by its own magnitude. This family has no column-wise division; picture fuzzy triples already lie between 0 and 1 and share the same structure. Instead, for a cost criterion, support and rejection swap places while the abstention degree stays as it is: (μ, η, ν) → (ν, η, μ).
Sum and product component. Crisp WASPAS's weighted sum (WSM) is matched here by a picture fuzzy combination built with the triple weight (v̄_j): every criterion's triple is first multiplied by its weight, then summed across the criteria; the result is again a triple (Q̃¹). Crisp WASPAS's weighted product (WPM) is matched by the product, across the criteria, of the triples raised to the power of the crisp weight (w_j); the result is again a triple (Q̃²). This is the only WASPAS extension that requires two separate weight inputs, a triple v̄_j and a crisp w_j, rather than a single weight type; if the crisp w_j is not supplied, it is derived from the triple weights using Chowdhury's (2025) formula.
Result and defuzzification. The two triples (Q̃¹, Q̃²) are first blended with λ into a single triple (Q̃); DecisionMind does not present λ as a field in the interface, it arrives as an additional input with a default of 0.5. This combined triple is only defuzzified into a single number at the very last step, with F_i = (μ + (1−η) + (1−ν)) / 3; a larger F_i is better.
DecisionMind fixes, in Picture Fuzzy WASPAS, this dual weight input, the sum-product combination and the defuzzification formula; the method does not generate weights, both the triple and the crisp weight come from outside.
How to Read the Output
F_i is read like the total score in crisp WASPAS; it only ranks this set of alternatives, and it is not a percentage or a probability.
The difference is this. F_i combines the three degrees (μ, η, ν) in a single formula, and the abstention share (η) also enters its share directly; two alternatives with the same F_i can arise from one having low rejection and the other having high abstention. Because this extension also uses two separate weight inputs, which weight (triple or crisp) is driving the result should be tracked separately.
Thus instead of writing:
"According to picture fuzzy WASPAS, this experiment is definitively the best"
the report should read:
"This F_i value is a score arising from combining the vote distribution separately in both sum and product form and then blending the two; which criterion's weight (triple or crisp) is driving the result should also be shown"
When to Prefer This over the Base Method
Use this extension when criterion evaluations come from a committee's, survey's or panel's yes, abstain, no vote distribution, and criterion weights can also be collected from an expert panel as a picture fuzzy triple. Deriving the other two degrees from a single support percentage erases this extension's sole contribution; the detail is on the picture fuzzy data-type card. If only a crisp weight exists and the criterion triple cannot be collected, this extension adds needless complexity; crisp WASPAS, or another extension that requires only one weight type, is sufficient. The exit condition is the same as for crisp WASPAS: if no compromise is acceptable on one criterion, this extension too is compensatory.
Mistakes Specific to This Extension
Violating the value space. In every cell, the sum μ+η+ν must not exceed 1. This also rules out feeding in Pythagorean or intuitionistic fuzzy inputs (which have no η) directly; these do not satisfy the picture fuzzy constraint.
Supplying the dual weight incompletely. This extension requires both a triple weight (v̄_j) and a crisp weight (w_j); supplying only one either halts the calculation or carries it forward with an incompletely derived weight. The relationship between the two is set by Chowdhury's (2025) derivation formula; the crisp weight must not be chosen independently of the triple weight without justification.
Deriving a triple from a single ratio. Saying "there is 60 per cent support, so 40 per cent opposition" zeroes out the abstention share; every degree must be counted separately.
Looking for λ in the interface. λ is not an interface field here the way weights are; it arrives as an additional input with a default of 0.5. This distinction must be stated in the report.
The governing principle is this:
Picture Fuzzy WASPAS works with two separate weight inputs, a triple and a crisp one; if the two are supplied inconsistently or incompletely, or if a triple is fabricated from a single ratio, the method's contribution of preserving vote-distribution information is lost.
Cases
The first case is drawn from a real machining experiment published by Chowdhury, Chatterjee and Chakraborty (2025). The second case is an illustrative construction.
1. Manufacturing: Parameter optimisation for laser-assisted electrochemical grinding (Chowdhury, Chatterjee and Chakraborty, 2025)
A manufacturing study compares 16 different parameter combinations (T1-T16) of a laser-assisted electrochemical grinding (LA-JECM) process on Inconel 718, on three criteria: material removal rate MRR, tool wear rate TAP, and surface roughness SR. The paper's own linguistic-to-numeric conversion turns all three into picture fuzzy triples in the higher-is-better direction; the direction correction was made at this conversion step, and at the algorithm level all three are treated as benefit criteria. Criterion weights are given both as triples (v̄_MRR=⟨0.591; 0.034; 0.031⟩, v̄_TAP=⟨0.644; 0.021; 0.019⟩, v̄_SR=⟨0.391; 0.129; 0.129⟩) and in the crisp form derived from them (MRR=0.355; TAP=0.370; SR=0.275).
| Experiment | MRR | TAP | SR |
|---|---|---|---|
| T1 | (0.060; 0.410; 0.400) | (0.995; 0.000; 0.000) | (0.825; 0.015; 0.015) |
| T2 | (0.060; 0.410; 0.400) | (0.755; 0.043; 0.050) | (0.825; 0.015; 0.015) |
| T3 | (0.060; 0.410; 0.400) | (0.650; 0.131; 0.137) | (0.260; 0.260; 0.260) |
| T4 | (0.060; 0.410; 0.400) | (0.060; 0.410; 0.400) | (0.040; 0.400; 0.400) |
| T5 | (0.150; 0.400; 0.295) | (0.995; 0.000; 0.000) | (0.150; 0.400; 0.295) |
| T6 | (0.825; 0.015; 0.015) | (0.150; 0.400; 0.295) | (0.060; 0.410; 0.400) |
| T7 | (0.040; 0.400; 0.400) | (0.225; 0.390; 0.263) | (0.260; 0.260; 0.260) |
| T8 | (0.150; 0.400; 0.295) | (0.260; 0.260; 0.260) | (0.650; 0.131; 0.137) |
| T9 | (0.260; 0.260; 0.260) | (0.260; 0.260; 0.260) | (0.150; 0.400; 0.295) |
| T10 | (0.060; 0.410; 0.400) | (0.060; 0.410; 0.400) | (0.060; 0.410; 0.400) |
| T11 | (0.825; 0.015; 0.015) | (0.260; 0.260; 0.260) | (0.225; 0.390; 0.263) |
| T12 | (0.650; 0.131; 0.137) | (0.650; 0.131; 0.137) | (0.150; 0.400; 0.295) |
| T13 | (0.060; 0.410; 0.400) | (0.225; 0.390; 0.263) | (0.650; 0.131; 0.137) |
| T14 | (0.225; 0.390; 0.263) | (0.650; 0.131; 0.137) | (0.995; 0.000; 0.000) |
| T15 | (0.825; 0.015; 0.015) | (0.150; 0.400; 0.295) | (0.040; 0.400; 0.400) |
| T16 | (0.995; 0.000; 0.000) | (0.040; 0.400; 0.400) | (0.060; 0.410; 0.400) |
The method builds the sum component with the triple weight, the product component with the crisp weight, blends the two with λ=0.5 and defuzzifies with the F_i formula.
| Experiment | F_i | Rank |
|---|---|---|
| T1 | 0.8601 | 1 |
| T14 | 0.8463 | 2 |
| T5 | 0.8269 | 3 |
| T12 | 0.8238 | 4 |
| T2 | 0.8221 | 5 |
| T11 | 0.8138 | 6 |
| T16 | 0.7780 | 7 |
| T6 | 0.7702 | 8 |
| T15 | 0.7668 | 9 |
| T3 | 0.7379 | 10 |
| T8 | 0.7328 | 11 |
| T9 | 0.6954 | 12 |
| T13 | 0.6947 | 13 |
| T7 | 0.6432 | 14 |
| T10 | 0.5686 | 15 |
| T4 | 0.5671 | 16 |
The result reads as follows. T1 ranks first because it carries almost full support (0.995) on TAP and also strong support (0.825) on SR; T4 ranks last because it carries weak support and high abstention and rejection on all three criteria. T14 ranks second despite holding almost full support (0.995) on SR, because it stays only middling on MRR.
The researcher's hesitation is this: what happens if the triple weights of MRR and SR are swapped, that is, if SR is treated as more important than material removal rate, and the crisp weights are re-derived accordingly? When the kernel is run independently in Python, T1 (0.8881) and T14 (0.8856) keep first and second place, but T2's score rises to 0.8578 and it moves up to third place; T5 (0.8269) drops to fourth and T12 (0.7994) to fifth. This is because T2 carries stronger support on SR than T5 and T12 do (0.825 against 0.150).
In the report: "With the triple and crisp weights given, T1 has the highest F_i value (0.8601); when the SR criterion's weight is raised above that of MRR, the third and subsequent ranks change (T2 rises to third), but the first and second ranks do not change."
Source: Chowdhury, Chatterjee and Chakraborty (2025), §4.1, Table 5 (weights) and Table 8 (decision matrix). The paper's own Table 9 contains a calculation error in the μ column of the product component; the triple reported for T1, ⟨0.690; 0.174; 0.170⟩, sums to 1.034, which violates the picture fuzzy constraint (sum ≤ 1). DecisionMind uses the canonical computation that applies the paper's Cuong (2013) Eq. (6)+Eq. (4) operation rules exactly as stated. This canonical computation exactly reproduces the paper's Table 9 sum (WSM) column and the η/ν components of the product (WPM) column; only the product column's μ component, and the F_i and rank that depend on it, diverge from the paper. The first and last rank (T1 first, T4 last) match the paper; there are small differences in the middle ranks. The detail is recorded in the verification notes. The scores and the weight-swap scenario were independently computed by this card's author by running the kernel directly.
2. Sports facility: Choosing an operator for a municipal swimming pool
A municipality will choose one of three candidate firms to operate a new indoor swimming pool it is opening. There are three criteria: variety of training programmes, hygiene and maintenance standard, and monthly operating fee (this last one lower is better). Members of the sports commission reported yes, abstain, no votes separately for each candidate-criterion pair, and also set the criterion weights through a triple panel vote.
The method swaps support and rejection on the operating-fee criterion, builds the sum and product components with the triple and crisp weights, blends them with λ=0.5 and defuzzifies with F_i. Suppose the firm with the highest support on training-programme variety also has the lowest rejection vote on hygiene standard, and comes first overall.
The commission's hesitation is this: the abstention share on the operating-fee criterion, that is, the proportion of commission members who did not state a clear view on this point, has dissolved inside F_i. Before the contract is signed, the source of this uncertainty, for example an energy-cost estimate, should be clarified.
In the report: "One firm stands out on training-programme variety and hygiene standard; the abstention share within the commission on the operating fee is high and should be clarified before the contract is signed."
3. What Not to Do
Deriving T1's TAP triple, 0.995/0/0, from a single "99 per cent support" ratio and skipping the abstention and rejection shares simplifies the paper's real measurement distribution. The second error is supplying only the crisp weights (0.355; 0.370; 0.275) and never entering the triple weights (v̄_j); this extension requires both, and the WSM component cannot be computed from the crisp weight alone. The third error is reporting T1's F_i value of 0.8601 as "86 per cent quality"; F_i only ranks these 16 experiments relative to one another.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pif-waspas
Chowdhury, D., Chatterjee, P., & Chakraborty, S. (2025). Optimization of hybrid non-traditional machining processes using multi-criteria decision making methods in picture fuzzy environment. International Journal on Interactive Design and Manufacturing (IJIDeM), 19, 7669–7694. DOI: 10.1007/s12008-025-02315-5
Zavadskas, E. K., Turskis, Z., Antuchevičienė, J., & Zakarevičius, A. (2012). Optimization of weighted aggregated sum product assessment. Elektronika ir Elektrotechnika, 122(6), 3–6. DOI: 10.5755/j01.eee.122.6.1810
Cuong, B. C., & Kreinovich, V. (2013). Picture fuzzy sets – A new concept for computational intelligence problems. 2013 Third World Congress on Information and Communication Technologies (WICT 2013), 1–6. DOI: 10.1109/WICT.2013.7113099
Cuong, B. C. (2014). Picture fuzzy sets. Journal of Computer Science and Cybernetics, 30(4), 409–420. DOI: 10.15625/1813-9663/30/4/5032