Extension card · Spherical
Spherical fuzzy GRA
This is the form of GRA used when criterion scores are given as three separate numbers: a judgement's degree of support, rejection and hesitancy. Its output remains a grey relational grade and the rank that grade produces.
Base method
GRA →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Spherical →
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 weighted-sum ranking step does not.
Cells. In crisp GRA every cell is a single number. Here every cell consists of three degrees: support (μ), rejection (ν) and hesitancy (π). The constraint is that the sum of the three degrees' squares must not exceed 1.
Complementation on a cost criterion. On a "lower is better" criterion, every cell's degrees of support and rejection swap places; hesitancy (π) stays unchanged.
The reference is built component by component, not taken from a real alternative. This is where SF-GRA departs from the family's SF-TOPSIS extension. In SF-TOPSIS the ideal point is the triple of a real alternative that comes out best under a score function. In SF-GRA the engine uses no score function at all; the reference is built from each column's highest support, lowest rejection and lowest hesitancy values, with the three components taken independently of one another. These three values can come from different rows, and the reference is a point that does not actually exist, carrying the same logic as crisp GRA's hypothetical reference sequence.
Distance is calculated over the squares of the components. The distance the engine uses is a Euclidean-like distance calculated not over the raw support-rejection-hesitancy differences but over the differences of these three components' squares. This is a definition separate from SF-TOPSIS's distance, which rests on the raw component difference; the two methods' distance figures cannot be compared with one another.
DecisionMind fixes, for this extension, the component-based single-positive-ideal reference, the squared Euclidean distance and the distinguishing coefficient (ρ = 0.5).
How to Read the Output
As in crisp GRA, the grey relational grade shows an alternative's relative closeness to the reference in this analysis.
An independent Python test carried out for this card produced an important finding. The validation figures recorded in DecisionMind's manifest (A1 = 0.6200; A2 = 0.7667; A3 = 0.4155) differ from the figures the engine actually produces today (A1 = 0.6664; A2 = 0.7948; A3 = 0.5200). The ranking (A2, then A1, then A3) is the same in both calculations, but the scores are not. The manifest's own note states that this recorded figure comes not from a published paper but from DecisionMind's internal audit tool, from a simplified single-score calculation. This card uses the figures the engine actually produces today.
Thus instead of writing:
"Figures such as 0.6200 recorded in the manifest are the engine's actual output"
the report should read:
"The engine's actual output today is 0.7948 for A2, 0.6664 for A1 and 0.5200 for A3; the figures recorded in the manifest come from an internal audit tool's simplified calculation and do not exactly match the engine's real output"
When to Prefer This over the Base Method
This extension is used when experts give a judgement's degree of support, rejection and hesitancy separately, and these three pieces of information are too high to fit the picture fuzzy constraint. The detail is on the Spherical fuzzy data-type card.
If hesitancy is not separately measured, a Pythagorean or intuitionistic fuzzy structure is sufficient. Crisp GRA's exit condition applies here too: the matrix must be of a single type, and if no compromise is acceptable on one criterion, this extension is also compensatory.
Mistakes Specific to This Extension
Assuming the reference is "a real alternative's triple," as in SF-TOPSIS. In SF-GRA the reference is built component by component; it need not belong to an alternative that actually exists.
Mistaking the figure recorded in the manifest for the engine's output. As shown above, in this extension there is a numerical gap between the recorded validation figure and the engine's actual output; the engine's actual output must be written into the report.
Assuming a score function (support squared minus rejection squared) is used. No score function at all is used in choosing the reference for this method; only the component-based highest and lowest values are taken.
Also changing hesitancy in cost complementation. Only support and rejection swap places; hesitancy (π) stays fixed.
Never questioning the distinguishing coefficient. ρ = 0.5 is DecisionMind's fixed value.
The governing principle is this:
In spherical fuzzy GRA the reference is a hypothetical point built component by component, not a real alternative's triple as in SF-TOPSIS. This card's figures are the engine's present-day actual output; the validation figure recorded in the manifest comes from an internal audit tool and does not exactly match the engine.
Cases
The first case is DecisionMind's validation example; it is the same 3x3 spherical fuzzy table used in common across the SF family's sixteen extensions, identical to the table on the SF-TOPSIS card. The figures here are the engine's actual present-day output, not the old manifest record; the difference is explained above. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's engine output): Three alternatives assessed on three criteria in spherical fuzzy form
Three alternatives are assessed on three criteria with spherical fuzzy triples. All three are higher-is-better criteria. Weights are C1 = 0.40, C2 = 0.35, C3 = 0.25.
| Alternative | K1 | K2 | K3 |
|---|---|---|---|
| A1 | (0.70; 0.20; 0.50) | (0.80; 0.10; 0.40) | (0.60; 0.30; 0.50) |
| A2 | (0.90; 0.10; 0.30) | (0.60; 0.30; 0.60) | (0.80; 0.20; 0.40) |
| A3 | (0.50; 0.40; 0.60) | (0.70; 0.20; 0.50) | (0.70; 0.10; 0.50) |
The method builds a component-by-component reference from each column's highest support, lowest rejection and lowest hesitancy values, measures distance over the squares, and converts this to the grey relational grade with ρ = 0.5.
| Alternative | Grey relational grade | Rank |
|---|---|---|
| A2 | 0.7948 | 1 |
| A1 | 0.6664 | 2 |
| A3 | 0.5200 | 3 |
The result reads as follows. A2, with (0.90; 0.10; 0.30) on K1, the most heavily weighted criterion, has the support-rejection pair closest to the reference. A1 is strong on K2 (0.80; 0.10; 0.40) but ranks second because it trails A2 on the most heavily weighted criterion.
The decision's hesitation lies in the weights. When verified with an independent Python calculation, if K2's weight is raised from 0.35 to 0.60 and K1's weight is lowered from 0.40 to 0.20 (K3 is pulled to 0.20), A1 (0.7966) overtakes A2 (0.6679). A2's lead therefore depends on the high weight given to K1.
In the report: "With the weights given (K1=0.40, K2=0.35, K3=0.25), A2 is first on the grey relational grade (0.7948). If K2's weight is raised enough to overtake K1 (K2=0.60, K1=0.20), A1 moves ahead (0.7966); how the weights were set must therefore be justified in the report. These figures are the engine's present-day actual output."
Source: DecisionMind's SF-GRA validation example; the 3x3 spherical fuzzy fixture used in common across the SF family's sixteen extensions. The grey relational grades and the weight sensitivity were independently calculated by this card's author running the kernel directly. The validation figure recorded in the manifest (A1=0.6200; A2=0.7667; A3=0.4155) does not match this calculation; details are in the approval notes.
2. Food Production: A dairy plant's choice of raw-milk supplier
A dairy plant will choose a raw-milk supplier among three farms. Three criteria are used: cold-chain reliability and milking-hygiene score (higher is better), and delivery price (lower is better). The audit team has separately given a degree of support, rejection and hesitancy for these criteria for every farm; hesitancy has been recorded separately for periods that could not be observed during the audit.
The method compares the three farms: it swaps support and rejection on the price criterion, builds the component-based reference, measures distance over the squares, and computes the grey relational grades. Suppose the farm with the strongest support-rejection pair on cold-chain reliability comes out first despite its relatively high price, because cold-chain reliability carried more weight than price.
The plant's hesitation is here. If the weight on milking hygiene is raised, a farm with a lower price but weaker cold-chain performance could move ahead. The plant should factor this sensitivity separately into the supply decision.
In the report: "With the high weight given to cold-chain reliability, the farm strong on this criterion ranks first; its relative disadvantage on price does not change this advantage. If the weight on milking hygiene is raised, the ranking may change."
3. What Not to Do
The first mistake in the illustrative example is writing the old figure recorded in the manifest (such as A1=0.6200) into the report as if it were the engine's present-day output; the engine actually produces different figures. The second mistake is assuming the reference is "the real alternative with the best score," as in SF-TOPSIS, and looking for a score function; in SF-GRA the reference is built component by component. The third mistake is also reversing hesitancy on the price criterion (lower is better); only support and rejection swap places.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/sf-gra
Ashraf, S., Abdullah, S., & Mahmood, T. (2018). GRA method based on spherical linguistic fuzzy Choquet integral environment and its application in multi-attribute decision-making problems. Mathematical Sciences, 12(4), 263–275. DOI: 10.1007/s40096-018-0266-0
Kutlu Gündoğdu, F., & Kahraman, C. (2019). Spherical fuzzy sets and spherical fuzzy TOPSIS method. Journal of Intelligent & Fuzzy Systems, 36(1), 337–352. DOI: 10.3233/JIFS-181401
Deng, J. L. (1989). Introduction to grey system theory. The Journal of Grey System, 1(1), 1–24. (no DOI)
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