Extension card · Fuzzy
Fuzzy AROMAN
Fuzzy AROMAN is the form of AROMAN used when the values in the decision table are not crisp numbers but an approximation drawn from expert judgement or estimation. It carries the calculation through triangular fuzzy numbers and still ranks the result with a single score.
Base method
AROMAN →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Fuzzy →
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?
Four things change. The decision logic stays the same.
Cells. In crisp AROMAN every cell is a single number. Here every cell carries three numbers: the lowest, the most likely and the highest plausible value. Criterion weights are also entered as triangles. If a crisp weight is entered, the method treats all three of its components as the same number. DecisionMind automatically brings the sum of the weights' centre values to 1; the centre is the average of the three components. The user does nothing extra for this.
Scale equalisation. Crisp AROMAN has two normalisations: linear and vector. Here both are applied separately to the triangle's three components. Linear normalisation looks at the range between the column's lowest lower value and highest upper value. Vector normalisation divides by the magnitude computed from the column's centre values. As in the crisp method, the two are blended by averaging.
Score and aggregation. The blended triangles are multiplied by the criterion weight, again component by component. After this step every cell descends to its centre value; the centre is the average of the three components. The centre values of the benefit criteria are summed, and those of the cost criteria are summed separately. As in crisp AROMAN, these two sums are combined with a balance parameter (λ) into a single score. However, the form of this final combination differs between the two engines: DecisionMind's crisp AROMAN engine sums two powers, whereas this fuzzy engine takes an exponential difference. Both claim to rest on the final step of the same paper, but the closing formulas are not strictly identical. This is a consistency note between the two engines, open for scientific review.
Result. Defuzzification happens not at the very end but immediately before the benefit-cost sums are built. The triangles descend to their centre value at that point. The output is a single crisp number. As in crisp AROMAN, a higher score is better.
DecisionMind fixes, in this fuzzy extension, the two parameters that the user can adjust in crisp AROMAN: the normalisation blending coefficient and the balance parameter λ, both set at 0.5. In this extension neither is a user input.
How to Read the Output
The output is a single score, as in crisp AROMAN, and is read the same way: it is not a percentage or a probability. It shows only a ranking relative to the other alternatives computed with the same triangular inputs and the same fixed parameters.
The difference is this. Beneath the score there is still a three-cornered uncertainty. This uncertainty is lost the moment the triangles descend to their centre value while the benefit-cost sums are built. The score gap between two alternatives may be robust or fragile depending on the width of the input triangles. The report must show this separately.
Thus instead of writing:
"According to the Fuzzy AROMAN score, A2 is the best alternative"
the report should read:
"With these triangular inputs, A2 has taken the highest score (1.342). The gap to A3 (1.204) is robust; this ranking does not change unless A2's cost triangle worsens markedly"
When to Prefer This over the Base Method
This extension is suitable when criteria come from expert judgement or estimation, and equally when reducing that approximation to a single number would create an artificial precision. Which situations can be fuzzified, and the principle of not converting measured data, are explained on the Fuzzy data-type card. DecisionMind requires a single data type. If the table is mixed, a measured criterion is also written as a triangle, with all three components the same number.
Crisp AROMAN's exit condition applies here too. This extension is also compensatory. Where no compromise is acceptable on one criterion, that is, where veto logic is required, it is not suitable.
Mistakes Specific to This Extension
Violating the value-range constraint. In every triangle the lowest value must be less than or equal to the most likely value, and the most likely less than or equal to the highest; none may be negative. If this is violated, normalisation and the sums produce invalid numbers.
Changing the defuzzification method without stating it. The canonical choice is the centre value, that is, the average of the three components. A different defuzzification, such as taking only the most likely value, can change the ranking. If the choice is not stated in the report, the comparison becomes misleading.
Defuzzifying first and then running crisp AROMAN. Reducing the triangles to their centre value from the outset and running the crisp method is not Fuzzy AROMAN. The uncertainty is erased in the very first step.
Assuming the fixed parameters are adjustable. In crisp AROMAN, the blending coefficient and λ are in the analyst's hands. In this fuzzy extension both are fixed at 0.5 and cannot be changed. The report should say "with this extension's fixed parameters", not "with the default parameters".
The governing principle is this:
Fuzzy AROMAN exists to carry the approximation in expert judgement through to the benefit-cost sum. Any application that crystallises the input from the outset, or fails to state the defuzzification method, erases the method's sole contribution.
Cases
The first case is DecisionMind's validation fixture. No published example of this fuzzy extension exists in the literature that can be traced to a page number. Instead, a synthetic triangular matrix with three alternatives and three criteria has been built, faithful to the manifest's steps and traceable by hand. The engine's steps have been independently recomputed in Python. The second case is an illustrative construction.
1. Illustrative example: Triangular fuzzy scoring of three alternatives on three criteria
Three alternatives are assessed with triangular fuzzy numbers on two benefit criteria and one cost criterion. The criterion weights are also triangular.
| Alternative | Criterion 1 | Criterion 2 | Criterion 3 |
|---|---|---|---|
| A1 | (0.65; 0.70; 0.75) | (0.45; 0.50; 0.55) | (0.55; 0.60; 0.65) |
| A2 | (0.75; 0.80; 0.85) | (0.55; 0.60; 0.65) | (0.35; 0.40; 0.45) |
| A3 | (0.55; 0.60; 0.65) | (0.65; 0.70; 0.75) | (0.45; 0.50; 0.55) |
| Weight | (0.35; 0.40; 0.45) | (0.30; 0.35; 0.40) | (0.20; 0.25; 0.30) |
| Direction | higher is better | higher is better | lower is better |
The method equalises every column with linear and vector normalisation, averages the two, and multiplies by the triangular weight. It then reduces every cell to its centre value. It builds the sum of the benefit columns (Criterion 1, Criterion 2) and the sum of the cost column (Criterion 3) separately. It combines the two with the fixed balance parameter; λ is 0.5 here.
| Alternative | Fuzzy AROMAN score | Rank |
|---|---|---|
| A2 | 1.342 | 1 |
| A3 | 1.204 | 2 |
| A1 | 1.103 | 3 |
The result reads as follows. A2 holds the highest triangle on Criterion 1, the heaviest criterion. It also holds the lowest, that is the best, triangle on the cost criterion. The combination of these two keeps A2 ahead. A3 is best on Criterion 2 but trails A2 on Criterion 1 and sits second. A1 is best on no criterion and finishes last.
The gap between A2 and A3 is not small; it is 0.138. But this gap rests entirely on A2's cost advantage. A2's Criterion 3 triangle currently stands at (0.35; 0.40; 0.45). If this triangle worsens by roughly 0.17 units, to (0.52; 0.57; 0.62), A3 moves ahead: A3 reaches 1.250 while A2 stays at 1.245. Below this threshold A2 continues to hold first place. Changing the criterion weights across a wide range, for instance raising Criterion 1's centre weight from 0.40 to 0.65, does not on its own change the ranking. The ranking is robust to the weights but fragile to A2's cost estimate.
In the report: "With triangular fuzzy inputs, A2 has taken the highest score (1.342). This ranking is robust to the criterion weights. However, A3 moves ahead if A2's triangle on the cost criterion worsens by more than roughly 0.17 units."
Source: DecisionMind's Fuzzy AROMAN validation fixture. The computational logic rests on Bošković, Švadlenka, Jovčić, Dobrodolac, Simić and Bačanin's (2023) crisp AROMAN algorithm, applied here to triangular fuzzy numbers component by component. This table and its numbers are not taken from a paper's page. The engine's steps have been independently recomputed in Python for this card and verified to match DecisionMind's recorded ranking (A2, A3, A1) exactly.
3. What Not to Do
The first error is reducing the illustrative example's triangles to their centre value from the outset and running crisp AROMAN, using single numbers such as 0.70/0.50/0.60 for A1. The ranking may not change in this example, but the uncertainty information carried by the triangles' width is lost, and the report claims a precision it does not have. The second error is reducing A2's Criterion 3 triangle to a single number, "around 0.40", and entering it that way; the method expects every cell to be a triangle. If a crisp value is to be entered, all three components must be written as the same number. The third error is reporting the blending coefficient or λ as "I chose 0.6", as in crisp AROMAN; in this fuzzy extension both are fixed, not user input.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/fuzzy-aroman
Bošković, S., Švadlenka, L., Jovčić, S., Dobrodolac, M., Simić, V., & Bačanin, N. (2023). An alternative ranking order method accounting for two-step normalization (AROMAN) — a case study of the electric vehicle selection problem. IEEE Access, 11, 39496–39507. DOI: 10.1109/ACCESS.2023.3265818
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI: 10.1016/S0019-9958(65)90241-X
Chen, C.-T. (2000). Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets and Systems, 114(1), 1–9. DOI: 10.1016/S0165-0114(97)00377-1
Xiang, H., Farid, H. M. A., & Riaz, M. (2024). Linear programming-based fuzzy alternative ranking order method accounting for two-step normalization for comprehensive evaluation of digital economy development in provincial regions. Axioms, 13(2), 109. DOI: 10.3390/axioms13020109