Extension card · Intuitionistic
Intuitionistic fuzzy GRA
IF-GRA is the form of GRA that works with intuitionistic fuzzy numbers when criteria are assessed through degrees of support for and rejection of a judgement. It computes distance to the reference from these two degrees and ranks the result again by a grey relational grade.
Base method
GRA →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Intuitionistic →
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 a degree of support (μ) and a degree of rejection (ν); μ + ν ≤ 1 must hold, and the remaining share is indecision, not written into the cell separately. Weights stay crisp numbers; GRA does not generate weights, it takes them from outside.
Reversal on a cost criterion. Crisp GRA reverses a cost criterion within the normalisation formula. Here there is no normalisation; instead, in every cell of a "less is better" criterion, the degrees of support and rejection swap places (μ ↔ ν). This is the complement rule used by Boran et al. (2009) in intuitionistic fuzzy MCDM: what is "less is better" on one criterion becomes a "more is better" judgement when viewed from the opposite direction.
Reference and distance. In crisp GRA the reference sequence is a fixed (1; 1; …; 1) normalised value, and distance is a plain absolute difference. Here the reference is an intuitionistic fuzzy ideal r* = (max μ, min ν), built from each column's own highest degree of support and lowest degree of rejection. This reference is not fixed; it is built from the data of that particular analysis. Distance is likewise not a single difference; it is the Szmidt-Kacprzyk (2000) intuitionistic fuzzy distance: the Euclidean-like distance of every cell to this ideal, in terms of both μ and ν, is the square root of half the sum of the squares of the two components.
Grey relational coefficient and grade. These distances are converted, using the same formula as crisp GRA (a discrimination coefficient of ζ = 0.5, fixed in DecisionMind), into a grey relational coefficient, then into a grey relational grade through a weighted sum. These final two steps work with exactly the same logic as crisp GRA; only the input is an intuitionistic fuzzy distance.
DecisionMind fixes the discrimination coefficient (ζ = 0.5) and the distance definition (Szmidt-Kacprzyk) in Intuitionistic Fuzzy GRA; weights are taken from outside as crisp numbers, not given in intuitionistic fuzzy form.
How to Read the Output
The grey relational grade, as in crisp GRA, shows an alternative's relative closeness to this analysis's own intuitionistic fuzzy reference; it cannot be compared with another analysis.
The difference lies here. The reference itself is a composite point seeking both "the strongest support" and "the lowest rejection." Even if an alternative has high support (μ) on a criterion, if it does not also give a low rejection (ν) on that same criterion, that is, if its share of indecision is high, its distance to the reference grows. For this reason, the report should state not only the grey relational grade, but also which criterion, through support or through rejection, pulled the result away from the reference.
Thus instead of writing:
"According to IF-GRA, A3 came out as the most reliable alternative"
the report should read:
"A3's grey relational grade is highest at 0.733; this superiority comes from A3's degrees of support and rejection on the second criterion matching the reference exactly, despite falling behind A2 on the first criterion"
When to Prefer This over the Base Method
Use this extension when criteria come from a judgement, and the expert's opposing view also carries separate information. This is appropriate, for example, when the support and rejection given to propositions such as "this supplier is reliable" or "this expert witness is convincing" can be gathered from mutually independent sources, such as evidence for and evidence against.
If criteria are measured, the base method should be used. Turning a measured score directly into μ and computing ν as 1 − μ does not produce intuitionistic fuzzy data; the share of indecision is set to zero and the structure becomes indistinguishable from crisp data. If the table is mixed, that is, if some criteria are crisp and others intuitionistic fuzzy, DecisionMind requires a single data type. A measured criterion is then written as (μ, 0), that is, full support and zero indecision. The exit condition of the base GRA applies here in exactly the same way: if a criterion carries a threshold that can never be compromised, GRA's additive structure, even in this form, does not protect it.
Mistakes Specific to This Extension
Value-space violation. In every cell, μ, ν ∈ [0,1] and μ + ν ≤ 1 must hold; if the share of indecision π = 1 − μ − ν comes out negative, the cell is invalid and cannot be calculated.
Writing ν as 1 − μ. This reduces intuitionistic fuzzy data to crisp data; the share of indecision is set to zero in every cell, and the structure's one contribution to GRA, keeping the gap between support and rejection separate, disappears.
Forgetting to reverse a cost criterion. If the degrees of support and rejection are not swapped (μ ↔ ν) on a "less is better" criterion, the reference sequence is built from the wrong end on that criterion, and the worst alternative on that criterion appears the best.
Never questioning the discrimination coefficient. ζ = 0.5 is DecisionMind's fixed value; in tables where the deviations (distances) are close to one another, this choice affects the result to a greater or lesser degree.
Changing the distance definition. DecisionMind uses the Szmidt-Kacprzyk distance as fixed; other intuitionistic fuzzy distance definitions also exist in the literature (such as Hamming, normalised Euclidean), and these can produce a different ranking. Which definition was used should be stated in the report.
The governing principle is this:
In intuitionistic fuzzy GRA, the reference is built from that analysis's own strongest support and lowest rejection values; deriving ν from 1 − μ, or skipping the reversal on a cost criterion, erases the structure's one contribution to GRA.
Cases
The first case is DecisionMind's validation example. The manifest's source citation gives no page or table number, and its authorship cannot be verified (see the approval notes). This case is therefore not a literature case, but an illustrative example that is faithful to the formulas and traceable by hand. The second case is an illustrative fiction.
1. Illustrative example: A mining firm's equipment-rental reliability assessment
A mining company will choose among three firms for heavy-equipment rental. Three judgements are assessed. "This firm keeps to the contract schedule" (C1) and "this firm meets maintenance standards" (C2) are "more is better" criteria. "This firm has a history of workplace accidents" (C3) is "less is better," that is, high support for this judgement is a bad sign. The degree of support (μ) and rejection (ν) for each judgement have been derived separately from past contract audits.
| Firm | C1: keeps to schedule | C2: meets standard | C3: has accident record |
|---|---|---|---|
| F1 | (0.70; 0.20) | (0.60; 0.30) | (0.40; 0.50) |
| F2 | (0.80; 0.10) | (0.50; 0.40) | (0.30; 0.60) |
| F3 | (0.50; 0.40) | (0.70; 0.20) | (0.20; 0.70) |
| Direction | more is better | more is better | less is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method swaps support and rejection on C3 (high support for accident record now sits in the unfavourable pole), builds the reference from each column's highest-support/lowest-rejection pair, measures the deviations with Szmidt-Kacprzyk distance, and converts them into a grey relational grade with ζ = 0.5.
| Firm | Grey relational grade | Rank |
|---|---|---|
| F3 | 0.733 | 1 |
| F2 | 0.700 | 2 |
| F1 | 0.557 | 3 |
The result reads as follows. F3 matches the reference exactly on C2 (0.70; 0.20) and on the reversed C3 (0.70; 0.20). Although it is furthest from the reference (0.80; 0.10) on C1, at (0.50; 0.40), this criterion's weight (0.40) is not enough to drop F3 to last place. F2 is the reference itself on C1, but stays in the middle on C2 and C3.
The firm's hesitation lies in the weights. If C1's weight is raised from 0.40 to 0.45 and C2's is lowered from 0.35 to 0.30, with C3 fixed at 0.25, F2 (0.729) overtakes F3 (0.700) by a very small margin. In other words, F3's lead is sensitive to the relative weight given to contract-schedule adherence.
In the report: "With the given weights (0.40; 0.35; 0.25), F3 is the firm closest to the reference (grey relational grade 0.733). When C1's weight is raised to 0.45, F2 overtakes F3 by a very small margin (0.729 / 0.700). The ranking is sensitive to the balance between the schedule and standards criteria."
Source: A hand-traceable illustrative example of 3 firms × 3 judgements; DecisionMind's validation example for the Intuitionistic Fuzzy GRA engine. Because the manifest's source citation is ambiguous, this case has not been presented as a literature case (see the approval notes). The grey relational grades and the weight sensitivity were independently computed by this card's author in Python; the results match the DecisionMind manifest's validation record exactly (F3 > F2 > F1, identical decimal values).
2. Law: Choosing an expert witness for an arbitration case
A law firm will choose among three outside expert witnesses for a major arbitration case. Three separate judgements are assessed: "this expert's technical opinion will be accepted in court" (C1, more is better), "this expert stays consistent under cross-examination" (C2, more is better), and "this expert has a history of contradictory testimony" (C3, less is better). Degrees of support and rejection have been derived separately from the firm's past case files.
| Expert | C1: opinion accepted | C2: consistent under cross | C3: has contradictory record |
|---|---|---|---|
| U1 | (0.65; 0.25) | (0.60; 0.30) | (0.30; 0.60) |
| U2 | (0.75; 0.15) | (0.55; 0.35) | (0.45; 0.45) |
| U3 | (0.55; 0.35) | (0.70; 0.20) | (0.20; 0.70) |
| Direction | more is better | more is better | less is better |
| Weight | 0.45 | 0.30 | 0.25 |
The method reverses C3, builds the reference, measures the deviations with Szmidt-Kacprzyk distance, and computes the grey relational grades.
| Expert | Grey relational grade | Rank |
|---|---|---|
| U3 | 0.723 | 1 |
| U2 | 0.670 | 2 |
| U1 | 0.556 | 3 |
The result reads as follows. U3 matches the reference on cross-examination consistency (C2) and on a clean record (the reversed C3). U2 holds the highest support figure on technical-opinion acceptance (C1), but this alone is not enough to overtake U3.
The firm has a hesitation. If the weight of the technical grounds (C1) is raised from 0.45 to 0.55, and cross-examination consistency (C2) is lowered from 0.30 to 0.25 and the contradictory record (C3) from 0.25 to 0.20, U2 (0.730) overtakes U3 (0.662). In other words, which expert is chosen depends on the relative priority given to technical grounds.
In the report: "With the current weights (0.45; 0.30; 0.25), U3 is the expert closest to the reference (grey relational grade 0.723). When the technical-grounds weight is raised to 0.55, U2 moves ahead (0.730 / 0.662). Whether the firm prioritises technical grounds or cross-examination consistency determines the choice."
3. What Not to Do
The first mistake is leaving C3's (accident record, less is better) degrees of support and rejection unreversed in the illustrative example. In that case the reference sequence is built from the alternative with the most support for an accident record, and the ranking becomes meaningless.
The second mistake, in the expert-witness example, is "correcting" U2's (0.45; 0.45) pair on C3 to (0.45; 0.55) using the ν = 1 − μ logic. This pair already comes from two separate sources, records for and against; deriving one from the other distorts the true information in the record.
The third mistake is converting the grey relational grade into a percentage, reporting it as "U3 is 72 per cent reliable." The grade is neither a probability nor a percentage; it shows only relative closeness within this particular analysis.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/if-gra
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI: 10.1016/S0165-0114(86)80034-3
Deng, J. L. (1989). Introduction to grey system theory. The Journal of Grey System, 1(1), 1–24. (no DOI)
Wei, G. W. (2010). GRA method for multiple attribute decision making with incomplete weight information in intuitionistic fuzzy setting. Knowledge-Based Systems, 23(3), 243–247. DOI: 10.1016/j.knosys.2010.01.003
Boran, F. E., Genç, S., Kurt, M., & Akay, D. (2009). A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method. Expert Systems with Applications, 36(8), 11363–11368. DOI: 10.1016/j.eswa.2009.03.039