Extension card · Picture
Picture fuzzy MOORA
This is the form of MOORA for situations where criterion scores come from a board's or a survey's yes-abstain-no vote distribution. It merges the beneficial and harmful criteria separately, then reduces their difference to a single net score.
Base method
MOORA →
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 MOORA every cell holds a single number. Here every cell holds three degrees: support (yes), abstention, and rejection (no); 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 and aggregation. Crisp MOORA first divides every column by its own magnitude, then weights it and sums the beneficial and harmful criteria separately. There is no column-based division in this family; picture fuzzy triples are already a structure between 0 and 1. Instead, all the "higher is better" criteria are reduced, in a single operation, through a picture fuzzy weighted aggregation, directly to one aggregate value (B_i); all the "lower is better" criteria are likewise reduced to a separate aggregate value (C_i). The weights are used exponentially inside this aggregation; there is no separate column normalisation as in crisp MOORA.
Score and ratio system. The aggregate values B_i and C_i are each reduced to a single number with a score based on the average of the three degrees. This score is one-third of the support share plus the not-abstaining share plus the not-rejecting share. The ratio-system score is the difference between these two numbers (y_i = s_B − s_C). This is the same logic as crisp MOORA's "beneficial total minus harmful total"; the difference is that the totals are built here through picture fuzzy aggregation.
Result and defuzzification. Defuzzification is done in a separate step, after the aggregate values B_i and C_i have been built. DecisionMind does not implement the reference-point approach in this family; only the ratio system exists. The reason is that the founding source's own decision-table example carries a large number of degrees of freedom, whereas DecisionMind has chosen to implement a simplified, closed-form ratio-system version.
DecisionMind fixes, in classic picture fuzzy MOORA, the benefit-cost grouping order, the aggregation form, and the average-based score function. Weights are taken from outside as crisp numbers; the method does not generate weights.
How to Read the Output
The output is a net score and a ranking, as in crisp MOORA, and reads the same way. The difference is this. B_i and C_i reduce the vote distribution of more than one criterion to a single combined vote; in this reduction, whichever criterion carries the higher weight determines the aggregate value more strongly. If the net-score gap between two alternatives is small, it should be asked which criterion's weight, changed slightly, could close this gap.
Thus instead of writing:
"Picture fuzzy MOORA reflects the board's opinion in full, so the result is more reliable"
the report should read:
"The vote distribution of the beneficial criteria has been merged into a single aggregate value, that of the harmful criteria merged separately, and the difference between the two taken; the report should show which criterion's weight is determining the order"
When to Prefer This over the Base Method
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; DecisionMind requires a single data type. The exit condition is the same as for crisp MOORA: if no concession is acceptable on a criterion, this extension is compensatory too, and will not eliminate anything below a threshold.
Mistakes Specific to This Extension
Value-space violation. In every cell, the sum of support, abstention and rejection must not exceed 1. This check must be made before aggregation; aggregate values built while a cell exceeds this are invalid too.
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.
Mixing beneficial and harmful criteria into the same aggregate value. B_i is built only from "higher is better" criteria, C_i only from "lower is better" criteria. Putting a criterion in the wrong group reverses its direction.
Defuzzifying first and then running crisp MOORA. 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 MOORA exists to carry a board's or a survey's abstention share through to the construction of B_i and C_i. Any application that invents a degree or crispens the input from the outset erases this contribution.
Cases
The first case is DecisionMind's validation example. Tian and colleagues' (2022) picture fuzzy MULTIMOORA paper uses a real investment case. This case, however, is too complex to follow by hand, using a seven-term linguistic scale and three sub-methods together. DecisionMind has therefore built a small, hand-traceable table with the same score function. The second case is an illustrative fiction.
1. Illustrative example: Three alternatives, three criteria (DecisionMind validation example)
Three alternatives are assessed on three criteria. The first two criteria are "higher is better," the third is "lower is better." Every cell carries an assessment board's support, abstention, rejection vote distribution.
| Alternative | C1 (higher is better) | C2 (higher is better) | C3 (lower is better) |
|---|---|---|---|
| A1 | (0.40; 0.30; 0.20) | (0.50; 0.20; 0.20) | (0.20; 0.30; 0.40) |
| A2 | (0.60; 0.20; 0.10) | (0.30; 0.30; 0.30) | (0.40; 0.20; 0.30) |
| A3 | (0.50; 0.20; 0.20) | (0.40; 0.20; 0.30) | (0.30; 0.20; 0.40) |
| Weight | 0.50 | 0.30 | 0.20 |
The method merges C1 and C2 into a single aggregate value (B_i), keeps C3 in a separate aggregate value (C_i), scores both, and takes their difference.
| Alternative | MOORA score | Rank |
|---|---|---|
| A1 | 0.4439 | 1 |
| A2 | 0.4378 | 2 |
| A3 | 0.4314 | 3 |
The result reads as follows. A1 does not have the highest support on C1, the heaviest criterion (0.40; A2 has 0.60), but has the highest support on C2 (0.50) and the lowest support on C3, the cost criterion (0.20). Although C1's weight is high, this advantage on C2 and C3 carries A1 into the lead. The gaps between the three alternatives are small.
The board's hesitation: if C1's weight is pulled from 0.50 to 0.60 and C2's from 0.30 to 0.20 (C3 held at 0.20), A2 moves ahead of A1 with 0.4599 against A1's 0.4354. Once C1's weight crosses a certain threshold, A2's strong support on this criterion (0.60) becomes decisive.
In the report: "With the given weights, A1 has the highest net score (0.4439). When C1's weight is raised to 0.60, A2 takes first place; the gap between the three alternatives is small, and the ranking is sensitive to C1's weight."
Source: DecisionMind PIF-MOORA manifest, validation example. The score function rests on Definition 3 in Tian and colleagues' (2022) picture fuzzy MULTIMOORA paper; the paper's own decision table has not been used in this card. The reason is that the table requires a seven-term linguistic scale and the joint application of three sub-methods (the ratio system, the reference point, the full multiplicative form). DecisionMind implements only the ratio system, with a simplified picture fuzzy weighted aggregation. The MOORA scores and the weight sensitivity have been independently recomputed by this card's author with the same algorithm, matching the kernel code exactly.
2. Public transport: A local government's choice of new bus-route corridor
A local government's transport department is to choose a new bus route among three corridor options. Three criteria are used: meeting passenger demand, impact on existing traffic (this last is "lower is better"), and support from neighbourhood headmen. Every criterion is scored with the relevant stakeholder group's (passenger survey, traffic committee, headmen's council) yes-abstain-no vote distribution on the same judgement.
The method merges passenger demand and headmen's support into a single aggregate value, keeps traffic impact separate, scores both and takes their difference. Suppose the corridor with the highest passenger demand is also the one with the highest traffic impact; it still comes out first, because the combined weight of passenger demand and headmen's support exceeds that of traffic impact.
The local government's hesitation: if the weight of traffic impact is increased, that is, if the department comes to see this criterion as more critical, the ranking can change; this should be reported not with a single weighting scenario, but by showing at what point the order turns as the weight is increased.
In the report: "With the given weights, the corridor with the highest passenger demand is first on the net score; at what point this ranking changes as the weight of traffic impact is increased has been shown separately."
3. What Not to Do
In the illustrative example, reducing C1, C2 and C3's vote distribution to the support ratio alone from the outset and running crisp MOORA erases the abstention and opposing-vote information, and inflates or shrinks the score gaps unrealistically. The second error is wrongly including C3 (the cost criterion) in B_i (the beneficial aggregate value); this reverses the direction of the cost and makes the ranking meaningless. The third error is reporting A1's score of 0.4439 as "44 per cent of the board supports it"; the score only ranks these three alternatives relative to each other.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pif-moora
Brauers, W. K. M., & Zavadskas, E. K. (2006). The MOORA method and its application to privatization in a transition economy. Control and Cybernetics, 35(2), 445–469. (no DOI)
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
Tian, C., Peng, J.-J., Zhang, Z.-Q., Wang, J.-Q., & Goh, M. (2022). An extended picture fuzzy MULTIMOORA method based on Schweizer-Sklar aggregation operators. Soft Computing, 26(7), 3435–3454. DOI: 10.1007/s00500-021-06690-5