Extension card · Picture
Picture fuzzy CIMAS (Kara et al., 2024)
Picture fuzzy CIMAS is the form of CIMAS used when experts' importance judgements and criterion scores are given as degrees of yes, abstention and no. The output remains a weight vector: every judgement is reduced to a single crisp score in the first step of the calculation, and the rest of the computation runs exactly as in crisp CIMAS.
Base method
CIMAS →
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?
Two things change; the six-step skeleton and the decision logic do not.
Cells. In crisp CIMAS, both an expert's own importance judgement and every expert's score for every criterion are single numbers. Here both are picture fuzzy triples: degrees of support (μ, yes), abstention (η) and rejection (ν, no), whose sum does not exceed 1. An expert's judgement of "how important is this criterion" and the judgement of "how experienced or reliable is this expert" are both expressed with this triple.
The point of defuzzification comes at the very start. Crisp CIMAS already begins with single numbers. Here, every (μ, η, ν) triple is reduced to a single crisp score in the first step of the calculation, through the formula Sc = (μ + (1−η) + (1−ν)) / 3. This is done the same way both when the expert importance weight (w_h) is computed and when every expert-criterion cell is computed. Once the score has been calculated, the remaining five steps (dividing by the column sum, multiplying by the expert weight, finding the column's highest and lowest values, taking the difference, converting it into a weight) run exactly as in crisp CIMAS. The picture fuzzy structure changes only the input layer; the rest of the algorithm is crisp CIMAS.
DecisionMind fixes the Sc formula for this extension and applies the second-round reliability-index threshold (RI < 0.1) in the same way as in crisp CIMAS; when no second-round data is supplied, RI is not calculated.
How to Read the Output
The weight, as in crisp CIMAS, is an experience-weighted summary of how much the experts diverge. The lowest weight does not mean "unimportant"; it means the experts agree on that criterion.
The difference is here: both an expert's own importance judgement and the criterion score are picture fuzzy judgements, and their abstention and rejection shares stay in the calculation until they enter the Sc score. Two experts can reach the same Sc score through very different (μ, η, ν) triples; one, for instance, through strong support and low abstention, the other through moderate support and high abstention. From the moment the score is calculated, this distinction is lost; the two experts can no longer be told apart.
Thus instead of writing:
"Picture fuzzy CIMAS is more reliable because it also folds the expert's indecision into the weight"
the report should read:
"The abstention and rejection shares are carried only as far as the Sc score. Once the score has been calculated, the weight is processed exactly as in crisp CIMAS. The robustness of the result must still be tested separately: how far does the weight distribution shift when a single expert's own importance statement changes?"
When to Prefer This over the Base Method
Use this extension when the experts' own importance statements or criterion scores come from a spread of opinion or a judgement rather than from a measurement. For that, though, the support, abstention and rejection shares of the judgement must be collectable separately, as in a vote or a survey. Computing rejection as 1 − support, or leaving the abstention share at zero, carries none of this extension's contribution; crisp CIMAS is then already sufficient.
Crisp CIMAS's exit conditions hold here too: if you only have the view of a single decision-maker, or if the experts diverge from bias rather than from knowledge, this extension is not suitable either.
Mistakes Specific to This Extension
Inventing a triple for an expert's own importance statement from a single proportion. If an expert says "70 per cent important," writing this as (0.70; 0; 0.30) sets the abstention share to zero and erases the structure's contribution.
Feeding an invalid triple with μ + η + ν > 1 into the calculation unnoticed. This breaches the definition of the picture fuzzy set and makes the Sc score meaningless.
Treating the result as final without ever calculating the reliability index (RI) when no second-round data exists. RI shows how much the first round's weights shift under a second elicitation; when it is not calculated, this check is missing.
Treating the Sc score as a "separate step" and looking for μ/η/ν again in the remaining calculations. Sc is a defuzzification applied at the very start of the calculation; everything after it is crisp CIMAS itself.
The governing principle is this:
Picture fuzzy CIMAS reduces picture fuzzy judgements to the Sc score at the very start, then processes the rest with the same six steps as crisp CIMAS. A small change in a single component of an expert's own importance statement, particularly in their self-confidence, can radically change the weight distribution here, exactly as it can in crisp CIMAS.
Cases
The first case is DecisionMind's validation example. The numerical application in Kara and colleagues' (2024) paper, a website performance assessment across eight sites × twelve criteria × eight experts, could not be accessed during the build. DecisionMind has therefore used a synthetic table, faithful to these formulas, of three experts × three criteria, giving fractional results. The second case is an illustrative fiction.
1. Illustrative example (DecisionMind's validation example): Three experts weighting three criteria
A team is to weight three criteria (C1, C2, C3). Three experts (E1, E2, E3) have expressed both their own importance and their score for each criterion as a picture fuzzy triple (μ, η, ν).
| Expert (own importance) | C1 | C2 | C3 |
|---|---|---|---|
| E1 (0.70; 0.10; 0.10) | (0.70; 0.10; 0.10) | (0.60; 0.20; 0.10) | (0.50; 0.20; 0.20) |
| E2 (0.60; 0.20; 0.10) | (0.50; 0.20; 0.20) | (0.70; 0.10; 0.10) | (0.60; 0.20; 0.10) |
| E3 (0.50; 0.20; 0.20) | (0.60; 0.20; 0.10) | (0.50; 0.20; 0.20) | (0.70; 0.10; 0.10) |
The method first reduces each expert's own importance triple to the Sc score and converts it into an expert weight (w_h). It then reduces every cell to the Sc score, divides it by the column sum, multiplies it by the expert weight, finds the column's highest-lowest difference, and converts these differences into weights that sum to 1.
| Criterion | Weight |
|---|---|
| C1 | 0.5071 |
| C2 | 0.4786 |
| C3 | 0.0143 |
The result reads as follows: C1 receives the highest weight, C2 sits close behind it in second place, and C3 receives a weight close to zero; this means the experts gave each other very similar scores on C3, that is, they are almost unanimous on this criterion.
The team's hesitation: this distribution depends heavily on E1's own importance statement (0.70; 0.10; 0.10), that is, on the member carrying the panel's highest expert weight. If E1 had stated their confidence more moderately, choosing, say, the same triple as E3, (0.50; 0.20; 0.20), the weights would come out at C1≈0.159, C2≈0.508, C3≈0.333, as confirmed by an independent Python calculation. That is, C1 falls from the highest to the lowest weight, while C3 rises from near zero to about a third. A single expert's own statement of self-confidence can overturn the weight distribution completely.
In the report: "Based on the three experts' panel scores, C1 receives the highest weight (50.7 per cent) and C3 the lowest (1.4 per cent), because the experts are almost unanimous on this criterion. This distribution, however, depends heavily on E1's own importance statement; had E1 stated their confidence more moderately, C1 would have received the lowest weight and C3 the highest."
Source: DecisionMind's picture fuzzy CIMAS validation example. It is faithful to the Sc score and CIMAS step order published by Kara, Yalçın, Kaygısız, Simić, Örnek and Pamučar (2024), but this table is synthetic because the paper's own numerical application could not be accessed during the build (see the approval notes). The weights and the sensitivity scenario have been verified with an independent Python calculation.
2. Publishing: Weighting criteria for a new book-series decision
A publishing house's editorial board is to weight three criteria before launching a new book series: originality of content, sales potential, and translation/editorial complexity. The board has collected, from five editors, both a self-assessment of their own editorial experience and their views on these three criteria in picture fuzzy form, as in a vote: support, abstain, oppose.
The method reduces each editor's self-assessment to the Sc score and converts it into an editor weight, processes the editors' views on each criterion the same way, and converts the difference between the column's extreme values into a weight. Suppose translation/editorial complexity receives the highest weight; this means the editors diverge most on this criterion, while largely agreeing on originality of content.
The board's hesitation: one of the five editors has stated their own experience markedly higher than the others. Without removing this editor from the panel and recalculating the weights, there is no way to know whether translation/editorial complexity really is the most divisive criterion, or whether this single editor's weight is driving the result.
In the report: "Translation/editorial complexity receives the highest weight according to the current five-editor panel; how far this result depends on the panel's single most heavily weighted member should be tested separately."
3. What Not to Do
The first error, in the illustrative example, is writing one expert's importance statement of "I support this 70 per cent" as (0.70; 0; 0.30); if the abstention share was never measured, it is not written as zero — a judgement that was not measured should not be collected at all. The second error is interpreting C3's weight of nearly zero as "an unimportant criterion" and dropping the criterion from the table, when this only shows that the experts agree on it. The third error is claiming, without ever testing E1's weight over the panel, that "C1 is the most important criterion"; as shown, this ranking is sensitive to E1's own importance statement.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pif-cimas
Kara, K., Yalçın, G. C., Kaygısız, E. G., Simić, V., Örnek, A. Ş., & Pamučar, D. (2024). A picture fuzzy CIMAS-ARTASI model for website performance analysis in human resource management. Applied Soft Computing, 162, 111826. DOI: 10.1016/j.asoc.2024.111826
Bošković, S., Jovčić, S., Simić, V., Švadlenka, L., Dobrodolac, M., & Bacanin, N. (2025). A new criteria importance assessment (CIMAS) method in multi-criteria group decision-making: Criteria evaluation for supplier selection. Facta Universitatis, Series: Mechanical Engineering, 23(2), 335–349. DOI: 10.22190/FUME230730050B
Cường, B. C. (2014). Picture fuzzy sets. Journal of Computer Science and Cybernetics, 30(4), 409–420. DOI: 10.15625/1813-9663/30/4/5032
Yalçın, G. C., & Kara, K. (2026). CIMAS: Criteria importance assessment for multi-attribute decision-making. In Encyclopedia of Multi-Attribute Decision Making (MADM) (pp. 461–468). DOI: 10.1016/B978-0-443-33275-3.00009-9