Extension card · Fuzzy
Scenario fuzzy GRA
This is the form of GRA for situations where criterion scores are given as triangular fuzzy numbers: it applies crisp GRA separately to the triangle's lower, middle and upper ends, then combines the results. Its output is again a score between 0 and 1 and a ranking based on that score.
Base method
GRA →
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?
Cells. In crisp GRA every cell is a single number. Here every cell is three numbers: lowest, most likely, highest. The expert either gives this triple directly or converts it from a word using a pre-declared scale.
Calculation path. Most fuzzy methods, such as Fuzzy TOPSIS, compute a single "vertex distance" that holds all three components together. This profile does not work that way. All of crisp GRA's steps (direction-aware normalisation, absolute difference, the discrimination-coefficient grey relational coefficient, weighted sum) are run three times, independently of one another: once on the matrix at the triangle's lower end, once at its middle value, once at its upper end. The same weights and the same discrimination coefficient (ρ) are used in all three.
Separate rescaling. The three separate sets of grey relational degrees produced by the three components are each rescaled within themselves before being combined: within each component, the lowest degree is pulled to 0 and the highest to 1. The three rescaled sets are then combined by simple averaging. This separate rescaling step equalises each component's own distribution; averaging the unrescaled degrees directly gives a different result, and the manifest's own warning specifically says not to confuse the two.
The discrimination coefficient is open to user adjustment here. In the family's other extensions (Plithogenic, Pythagorean, q-Rung, Spherical fuzzy GRA), ρ is fixed inside the engine at 0.5. In this profile, ρ can be entered as an external parameter; the default is still 0.5, but if it is changed this must be stated in the report.
DecisionMind fixes, in this profile, the independent component-by-component GRA computation and the rescaling of each component within itself; ρ can be changed by the user.
How to Read the Output
The final score lies between 0 and 1 and is the average of the three components' separately rescaled degrees; like crisp GRA's grey relational degree, it is only a ranking within this set of alternatives.
The difference is here. This score does not directly carry the triangle's width, that is, the size of the uncertainty. It carries only the average of each of the three points' (lower, middle, upper) relative position within its own column. Two alternatives with very different triangle widths can still end up with a similar average score.
Thus instead of writing:
"This score is a result of the standard fuzzy GRA in the literature, comparable with classical vertex-distance methods"
the report should read:
"This score is a DecisionMind-specific profile, produced by computing the three components independently, rescaling each separately and averaging them; it cannot be compared directly with fuzzy GRA calculations based on vertex distance"
When to Prefer This over the Base Method
Today this profile cannot be recommended in DecisionMind's live product, because the engine does not yet run this method. Mechanically, it is a profile worth considering when expert scores are given as triangular fuzzy numbers and a reading is wanted that processes the three scenarios (pessimistic, middling, optimistic) separately before combining them. The general logic of fuzzy data is on the Fuzzy data-type card.
Opening up a measured criterion into a triangle is producing uncertainty, not modelling it; this principle holds here too. Crisp GRA's exit condition also applies here: if no compromise is acceptable on one criterion, this profile is compensatory as well.
Mistakes Specific to This Extension
Presenting this as a standard fuzzy GRA. The manifest explicitly states that this profile does not rest on a canonical source; it must not be introduced in the report as "the fuzzy GRA in the literature."
Averaging the components without rescaling them separately. Rescaling each component to 0-1 within itself is a mandatory step; skipping it produces a different profile and a different result.
Changing the discrimination coefficient without reporting it. Because ρ can be changed by the user in this profile, any departure from the default value must be clearly stated.
Applying the direction label to only one component. A "lower is better" criterion's direction must be applied identically across all three of the triangle's lower, middle and upper components.
Presenting this method to the user as an option that can be run through the product today. DecisionMind's current version rejects this method for lack of a canonical engine; until that situation changes, it must not be written into a report as a "usable method."
The governing principle is this:
This profile has no founding paper; it is DecisionMind's own derivation and cannot currently be run within the product. The card documents only what is computed at code level, through an independently run kernel.
Cases
The first case is taken from DecisionMind's test record. This record is not an example carried over from a paper; it is the engine's own verification fixture, built with four alternatives on three criteria so that it can be traced in closed form. Because the product API rejects this method today, the figures were produced independently by calling the kernel file directly in Python rather than through the engine's API. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's test record): Four alternatives, triangular fuzzy scoring on three criteria
Four alternatives have been assessed with triangular fuzzy numbers on three criteria. C1 and C3 are "higher is better", C2 is "lower is better". Weights: C1 = 0.40, C2 = 0.35, C3 = 0.25; ρ = 0.5.
| Alternative | C1 (higher is better) | C2 (lower is better) | C3 (higher is better) |
|---|---|---|---|
| A1 | (2; 3; 4) | (7; 8; 9) | (4; 5; 7) |
| A2 | (4; 5; 7) | (5; 6; 7) | (6; 7; 8) |
| A3 | (6; 8; 9) | (2; 3; 4) | (3; 4; 6) |
| A4 | (3; 4; 6) | (6; 7; 8) | (8; 9; 10) |
The method reverses C2's direction, runs crisp GRA separately on the lower, middle and upper components, rescales each component's grey relational degrees within itself to 0-1, and averages the three.
| Alternative | Score | Rank |
|---|---|---|
| A3 | 1.0000 | 1 |
| A4 | 0.4166 | 2 |
| A2 | 0.3031 | 3 |
| A1 | 0.0000 | 4 |
The result reads as follows. A3 has the highest triple on C1, the most heavily weighted criterion, and the lowest (hence best) triple on the reversed C2. These two together give it a clear lead.
The board's hesitation lies in the weights. If C3's weight is raised from 0.25 to 0.60, with C1 pulled down from 0.40 to 0.25 and C2 from 0.35 to 0.15, A4 (1.0000) overtakes A3 (0.5914) when verified with an independent Python calculation, because A4's C3 triple (8; 9; 10) is the highest value in the table. In other words, A3's lead depends on the weight given to C1 and C2.
In the report: "With the given weights (0.40; 0.35; 0.25), A3 is clearly first on score (1.0000). When C3's weight is markedly increased (0.60), first place shifts to A4 (1.0000); the report must therefore justify which criterion is treated as the priority. This score is a DecisionMind-specific, component-based profile, and should not be compared with standard fuzzy GRA results."
Source: DecisionMind's test record (manifest J, "disproportionately ordered triangular fuzzy number fixture"), not an example carried over from a paper. The scores and weight sensitivity were independently computed by this card's author by calling the kernel file directly in Python, and matched the manifest's recorded result (A3 > A4 > A2 > A1, identical decimal values, 1e-12 tolerance) exactly.
2. Disaster management: A provincial disaster-coordination centre's choice of temporary shelter site
A provincial disaster-coordination centre will choose among three candidate sites for a temporary shelter to be set up after an earthquake. Three criteria are used: accessibility of access roads and ground stability (higher is better), setup cost (lower is better). Because damage assessment is not yet complete, every criterion is given as a "worst case, most likely case, best case" three-scenario estimate, that is, a triangular fuzzy number.
The method evaluates the three candidates separately by scenario (lower, middle, upper), averaging each scenario's own rescaled degrees. Suppose the candidate with the highest ground stability in the most-likely scenario comes out first despite its relatively high cost, because ground stability carries more weight than cost.
The centre's hesitation is this. This score does not separately show which candidate leads in the worst-case scenario; the three scenarios have been averaged. The centre should examine not only the average score but also the ranking under the worst-case (lower component) scenario separately, because in disaster planning the worst case can be the deciding factor.
In the report: "With the high weight given to ground stability, the candidate strong in the most-likely scenario has come out first. This score is the average of three scenarios; the ranking under the worst-case scenario has been examined separately, and the decision has been made taking both findings together."
3. What Not to Do
The first error in the illustrative example is averaging the three components' grey relational degrees directly, without rescaling each within itself first; the manifest specifically warns that this is a different profile and gives a different result. A second error is presenting this profile as "DecisionMind's standard fuzzy GRA"; it has no founding paper, and the manifest explicitly forbids this. A third error is offering this method to the user as an option that can be run through the product today; DecisionMind's live version currently rejects this method.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/scenario-fuzzy-gra
This extension has no founding paper in the literature; it has been derived from DecisionMind's internal reference implementation (see the explanation above).
Deng, J. L. (1989). Introduction to grey system theory. The Journal of Grey System, 1(1), 1–24. (no DOI)
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI: 10.1016/S0019-9958(65)90241-X
Kuo, Y., Yang, T., & Huang, G. W. (2008). The use of grey relational analysis in solving multiple attribute decision-making problems. Computers & Industrial Engineering, 55(1), 80–93. DOI: 10.1016/j.cie.2007.12.002