Extension card · Neutrosophic
Neutrosophic GRA (Biswas, Pramanik & Giri, 2014)
N-GRA is the form of GRA that works with neutrosophic triples for situations where criteria are assessed independently by degrees of truth, indeterminacy and falsity. It computes the distance to a reference from these three components, and ranks the result once again with a grey relational degree.
Base method
GRA →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Neutrosophic →
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 reference-deviation-coefficient skeleton does not.
Cells. In crisp GRA every cell is a single number. Here every cell is one degree each of truth (T), indeterminacy (I) and falsity (F); the three come from independent sources, and their sum is not forced to 1. Weights remain crisp numbers; GRA does not generate weights, it takes them from outside. In Biswas, Pramanik and Giri's (2014) founding paper, these weights are derived from the data by the entropy method.
Complementing for a cost criterion. Crisp GRA reverses a cost criterion within the normalisation formula. Here there is no normalisation; instead, the neutrosophic complement is taken for every cell of a "lower is better" criterion: truth and falsity swap, indeterminacy is subtracted from 1, (T, I, F) → (F, 1−I, T). A low value on a "lower is better" criterion thus becomes high truth once complemented.
Reference and distance. In crisp GRA the reference sequence is a fixed (1; 1; …; 1) normalised value, and distance is the absolute difference. Here, the reference is a neutrosophic ideal α₀ = (max T, min I, min F), built from each column's own highest truth, lowest indeterminacy and lowest falsity. This reference is not fixed; it is built from that analysis's own data. The distance itself is not a single difference either, but the square root of one-third of the sum of the squared differences of the three components; a Euclidean-like neutrosophic distance.
Grey relational coefficient and degree. These distances are converted, using the same formula as crisp GRA (discrimination coefficient ρ = 0.5, fixed in DecisionMind), into a grey relational coefficient, and then, by weighted summation, into a grey relational degree. These last two steps work by exactly the same logic as crisp GRA; only the neutrosophic distance differs as an input.
DecisionMind's Neutrosophic GRA follows the manifest's F.steps to the letter. The reference is built against a single positive ideal only (α₀); unlike Biswas-Pramanik-Giri's (2014) paper, a second grey relational degree against a negative ideal is not computed and combined with the first. This produces an important difference in outcome in Case 1 below, explained there in more detail.
How to Read the Output
The grey relational degree, as in crisp GRA, shows an alternative's relative closeness to the neutrosophic reference in this analysis; it cannot be compared with another analysis.
The difference is here. The reference is built from three independent components, truth, indeterminacy and falsity, and if an alternative departs from the reference on any one of these three, its total distance grows. Whether it is missing evidence or opposing evidence that has enlarged the distance is not visible in the result, only in the intermediate table. For this reason, the report should state not only the grey relational degree but also which criterion and which component the gap between two alternatives comes from.
Thus instead of writing:
"By N-GRA, A1 came out as the best investment alternative"
the report should read:
"A1's grey relational degree is highest at 0.8897, but the gap to A3 (0.8863) is very small; the ranking between these two alternatives is sensitive to a small change in the criterion weights"
When to Prefer This over the Base Method
This extension is used when information about a criterion is incomplete, inconsistent or contradictory, and this matters for the decision itself. Examples: a new investment alternative with no track record, expert opinions from conflicting sources, situations where "there is both strong favourable and strong unfavourable evidence" applies.
If criteria are measured, or experts do not express a separate indeterminacy share, the base method should be kept. Converting a measured value into a triple such as (T, 0, 1−T) to "make it look more comprehensive" means declaring no indeterminacy, and adds nothing that the structure brings to GRA. If the table is mixed, that is, some criteria crisp and others neutrosophic, DecisionMind requires a single data type; a measured criterion is embedded as (t, 0, 1−t), and this is stated in the report as an honest embedding that adds no new information. Base GRA's exit condition applies here in exactly the same way: if a criterion carries a threshold on which no compromise is ever possible, GRA's additive structure does not protect it.
Mistakes Specific to This Extension
Value-space violation. In every cell T, I, F ∈ [0,1] and 0 ≤ T+I+F ≤ 3 must hold; a triple outside this range cannot be computed.
Deriving indeterminacy from T and F. Computing I as 1 − T − F reduces neutrosophic data to intuitionistic fuzzy data; indeterminacy must come from its own source (a lack or contradiction of evidence).
Forgetting to complement for a cost criterion. If the neutrosophic complement (T↔F swap, I → 1−I) is not taken for a "lower is better" criterion, the reference sequence is built from the wrong end on that criterion.
Never questioning the discrimination coefficient. ρ = 0.5 is DecisionMind's fixed value; in tables where the distances are close to one another, this choice affects the result by a small or large amount.
Confusing DecisionMind's single-ideal implementation with the literature's dual-ideal (positive+negative) combined result. As Case 1 below shows, the founding paper's own algorithm computes two separate grey relational degrees against a positive and a negative ideal and combines them; DecisionMind's manifest computes against the positive ideal only. The two approaches can give different rankings, and this difference should not be read as "the method is running incorrectly" but stated clearly in the report as which algorithm was run.
The governing principle is this:
In neutrosophic GRA, truth, indeterminacy and falsity must remain independent of one another, and the reference must be built from that analysis's own best triple; unless the report states against which ideal(s) it was computed, a single positive one or positive plus negative, it becomes unclear which algorithm the result belongs to.
Cases
The first case comes from the literature. The investment-selection problem in Biswas, Pramanik and Giri's (2014) paper; the table is identical to the data in their Section 5. The second case is an illustrative construction.
1. Business: Choosing among four investment alternatives (Biswas, Pramanik & Giri, 2014)
An investor will choose among four alternatives: an automobile company (A1), a food company (A2), a computer company (A3), an arms-industry company (A4). Three criteria, all "higher is better": risk analysis (C1), growth analysis (C2), environmental impact (C3). The weights are computed from the data by the entropy method in the paper.
| Alternative | C1: risk analysis | C2: growth analysis | C3: environmental impact |
|---|---|---|---|
| A1 | (0.40; 0.30; 0.20) | (0.60; 0.10; 0.20) | (0.70; 0.00; 0.10) |
| A2 | (0.50; 0.30; 0.20) | (0.50; 0.20; 0.30) | (0.30; 0.20; 0.30) |
| A3 | (0.50; 0.20; 0.20) | (0.60; 0.10; 0.20) | (0.60; 0.10; 0.20) |
| A4 | (0.20; 0.20; 0.50) | (0.40; 0.20; 0.30) | (0.40; 0.20; 0.30) |
| Direction | higher is better | higher is better | higher is better |
| Weight | 0.2958 | 0.4325 | 0.2697 |
Because all three criteria are "higher is better," the method applies no complementing; it builds the reference α₀ = (max T, min I, min F) from each column (C1: 0.50/0.20/0.20 · C2: 0.60/0.10/0.20 · C3: 0.70/0.00/0.10), measures each alternative's neutrosophic distance from this reference, and converts it to a grey relational degree with ρ = 0.5.
| Alternative | Grey relational degree | Rank |
|---|---|---|
| A1 | 0.8897 | 1 |
| A3 | 0.8863 | 2 |
| A2 | 0.5533 | 3 |
| A4 | 0.4250 | 4 |
The result can be read as follows. A1 is, on C3, (0.70; 0; 0.10), the reference itself, and also coincides with the reference on C2; its lagging behind A3's (0.50; 0.20; 0.20) on C1 lowers its total degree only slightly. A3 coincides with the reference on C1 but lags behind A1 on C3, (0.60; 0.10; 0.20). The gap between the two alternatives (0.8897 − 0.8863 = 0.0034) is one of the smallest decimal differences that DecisionMind applies.
The investor's hesitation lies exactly in this closeness. If the weight on risk analysis (C1) rises from 0.2958 to 0.32, and environmental impact (C3) falls from 0.2697 to 0.25, with growth analysis held roughly constant at ≈0.43, A3 (0.896) overtakes A1 (0.883). That is, A1's first place depends on a small tilt in the risk-environment weight balance.
An important caveat applies here. Biswas, Pramanik and Giri's (2014) own paper uses an algorithm that computes two separate grey relational degrees against both a positive and a negative ideal and combines them, and reports the final ranking there as A4 > A2 > A3 > A1. DecisionMind's manifest instead works against the positive ideal only (the calculation above) and produces the reverse ranking (A1 > A3 > A2 > A4). This is not a calculation error; it is a different algorithm. The report must always state which one DecisionMind runs, the single positive ideal (see the approval notes).
In the report: "By DecisionMind's single-positive-ideal formula, A1 is closest to the reference (grey relational degree 0.8897); its gap to A3 (0.0034) can close with a small shift in the risk-environment weight balance. This result differs from the source paper's own dual-ideal (positive+negative) combination algorithm, which places A4 first in the original table."
Source: Biswas, P., Pramanik, S., & Giri, B. C. (2014), Section 5, the table in Equation (21); the data is taken directly from the paper. This card's author independently computed the grey relational degrees and weight sensitivity in Python; the results match the recorded kernel-recomputation in the DecisionMind manifest exactly (A1 > A3 > A2 > A4, the same decimal values).
2. Media: Choosing a production company for a series project
A broadcaster will choose among three production companies for a new series project. Three criteria, all "higher is better": creative team strength (C1), track record of budget discipline (C2), track record of meeting delivery deadlines (C3). For each criterion, truth (favourable evidence), indeterminacy (missing or outdated records) and falsity (unfavourable evidence) are gathered from separate sources: reference projects, financial audits and delivery records.
| Production company | C1: creative team | C2: budget discipline | C3: delivery record |
|---|---|---|---|
| P1 | (0.60; 0.20; 0.20) | (0.50; 0.30; 0.30) | (0.70; 0.10; 0.10) |
| P2 | (0.70; 0.10; 0.20) | (0.40; 0.30; 0.40) | (0.50; 0.20; 0.30) |
| P3 | (0.50; 0.20; 0.30) | (0.60; 0.10; 0.20) | (0.60; 0.20; 0.20) |
| Direction | higher is better | higher is better | higher is better |
| Weight | 0.40 | 0.30 | 0.30 |
The method builds the reference α₀ = (0.70; 0.10; 0.20 for C1 · 0.60; 0.10; 0.20 for C2 · 0.70; 0.10; 0.10 for C3) and computes the grey relational degrees.
| Production company | Grey relational degree | Rank |
|---|---|---|
| P1 | 0.6445 | 1 |
| P3 | 0.6157 | 2 |
| P2 | 0.6098 | 3 |
The result can be read as follows. P1 lags behind the reference on creative team (0.60 against 0.70), but is the reference itself on delivery record, (0.70; 0.10; 0.10); this offsets its shortfall on creative team, the heaviest criterion (0.40). P2, despite having the best evidence on creative team, comes last because it has the weakest evidence on budget discipline (0.40; 0.30; 0.40).
The broadcaster has a hesitation here. If new evidence on P2's budget discipline arrives, for example an audit report from its most recent project, and the favourable evidence (0.40; 0.30; 0.40) shifts to (0.55; 0.20; 0.25), P2 (0.665) moves to first place and P1 (0.620) drops to second. That is, P1's lead rests on outdated and incomplete records about P2's budget history.
In the report: "On current evidence, P1 is the production company closest to the reference (grey relational degree 0.6445); if up-to-date audit evidence on P2's budget discipline is added, P2 could move to first place (0.665). The gap between these three alternatives is small, and the decision should not be finalised before the incomplete record on P2 is completed."
3. What Not to Do
The first mistake is to ignore the 0.0034 gap between A1 and A3 in the investment example and report "A1 is a clear first." This gap closes with a small shift in the risk-environment weight, and this sensitivity must not be hidden from the report.
The second mistake is to present DecisionMind's single-positive-ideal result (A1 > A3 > A2 > A4) as though it were the source paper's dual-ideal result (A4 > A2 > A3 > A1). Unless the report states which algorithm was run, the two cannot be used interchangeably.
The third mistake is to read P2's high indeterminacy (I=0.30) on budget discipline (0.40; 0.30; 0.40) in the media example as "P2 is a poor alternative." High indeterminacy means "information is missing," not "the evidence is unfavourable"; the ranking can change once new evidence about P2 arrives.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/n-gra
Biswas, P., Pramanik, S., & Giri, B. C. (2014). Entropy based grey relational analysis method for multi-attribute decision-making under single valued neutrosophic assessments. Neutrosophic Sets and Systems, 2, 102–110. DOI: 10.5281/zenodo.22459
Smarandache, F. (1998). Neutrosophy: Neutrosophic Probability, Set, and Logic. American Research Press, Rehoboth. (no DOI)
Deng, J. L. (1989). Introduction to grey system theory. The Journal of Grey System, 1(1), 1–24. (no DOI)
Ye, J. (2013). Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. International Journal of General Systems, 42(4), 386–394. DOI: 10.1080/03081079.2012.761609