Extension card · Intuitionistic
Cubic intuitionistic fuzzy TOPSIS (Garg & Kaur, 2018)
This is the form of TOPSIS for situations where a cell carries, together, both an interval-valued intuitionistic fuzzy pair AND a single-point intuitionistic fuzzy pair layered on top of it. It processes both layers and ranks alternatives with a single closeness coefficient.
Base method
TOPSIS →
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 idea of closeness to the ideal does not.
Cells. In crisp TOPSIS every cell is a single number; in intuitionistic fuzzy TOPSIS it is a support-rejection pair. In cubic intuitionistic fuzzy TOPSIS every cell consists of six numbers: an interval-valued intuitionistic fuzzy pair (support lower bound, support upper bound, rejection lower bound, rejection upper bound), plus a single-point intuitionistic fuzzy pair on top (point support, point rejection). The interval part answers "how much do I support, how much do I reject, but stated as a band"; the point part answers the same judgement with a single representative value. The method supports more than one decision-maker; each decision-maker's own table is processed separately and combined at the end with priority weights (λ_k, summing to 1). With a single decision-maker this weight is 1 and the combination step does not come into play.
The ideal and anti-ideal point. Crisp TOPSIS's third step works here in two layers. For a benefit criterion, the ideal point is built from the largest of the interval part's support bounds and the smallest of its rejection bounds, together with the largest of the point part's support and the smallest of its rejection; the anti-ideal is the exact opposite. For a cost criterion, these roles swap.
Distance. The distance between two cubic intuitionistic fuzzy numbers is computed as the q-th root of the weighted average of the q-th powers of the differences across all six components (four interval bounds, two point values). DecisionMind uses q=2 (Euclidean) for this family; Garg and Kaur's (2018) own article also treats this value as canonical. Choosing q=1 turns it into a Hamming-like distance and the results can change.
Combining more than one decision-maker. Every decision-maker's distance to the ideal and anti-ideal is computed separately; these distances are then summed with the decision-makers' priority weights (λ_k), and the closeness coefficient is built from a single combined distance pair. With a single decision-maker this step is skipped and λ_1=1 is taken.
DecisionMind fixes, for this extension, the q=2 distance and the direct summation of decision-maker weights. Criterion weights are taken from outside; the method does not generate weights.
How to Read the Output
The closeness coefficient is read as in crisp TOPSIS: it states how close an alternative sits to the ideal for this alternative set and these weights, and is not compared with a different analysis. The difference lies here: beneath this coefficient sit both a band (interval) and a point assessment. When the gap between two alternatives' coefficients is small, that gap can be sensitive to both the interval's width and the point assessment; the report should show which component drives the difference.
Thus instead of writing:
"Cubic intuitionistic fuzzy TOPSIS uses richer data, so the result is more reliable"
the report should read:
"This coefficient carries, together, the judgement the decision-maker gave both as a band and as a point; where the gap between two alternatives is small, which component drives that gap should also be shown"
When to Prefer This over the Base Method
This extension suits situations where experts give a judgement both as a band and as a representative point, that is, a two-layered assessment such as "support is between 0.30 and 0.45, but the most likely value is 0.55." It is also used where more than one expert's table must be summed separately and then combined with a priority weight. Where only a band or only a point is given, plain intuitionistic fuzzy or interval-valued intuitionistic fuzzy TOPSIS is sufficient; the cubic structure adds no extra information there. Crisp TOPSIS's exit condition applies here too: where one criterion cannot be traded away, this extension remains compensatory as well and does not screen out anything below a threshold.
Mistakes Specific to This Extension
Deriving the interval and the point from one another. Automatically computing the point value as the interval's midpoint effectively erases the cubic structure's second layer; the point should come from its own judgement.
Writing an invalid CIFN. Every cell must satisfy ζL≤ζU, ϑL≤ϑU, ζU+ϑU≤1 (interval part) and ζ+ϑ≤1 (point part); feeding this into the calculation without checking it invalidates the method.
Reporting without stating q. q=2 (Euclidean) is the canonical choice; q=1 (Hamming) gives different distances. The q used must be stated in the report.
Not giving decision-maker weights (λ_k) that sum to 1. With a single decision-maker, λ_1=1 should be taken; with more than one, the weights must sum to 1.
The governing principle is this:
Cubic intuitionistic fuzzy TOPSIS exists to carry interval and point assessments together, from separate sources; if one is derived from the other, the second layer adds no information, it only complicates the calculation.
Cases
The first case is the 4x4 decision table from Garg and Kaur's (2018) article (Tables 1-4). DecisionMind's engine runs this table in the single-decision-maker setting (λ=1, q=2); the second case is an illustrative construction.
1. Illustrative example: Choosing among four suppliers (Garg & Kaur, 2018)
A business will choose among four suppliers. Four criteria apply (C1-C4, all higher is better); the weights are 0.17, 0.30, 0.13 and 0.40. Every cell consists of an interval-valued intuitionistic fuzzy pair and a point intuitionistic fuzzy pair.
| Supplier | C1 (interval; point) | C2 (interval; point) | C4 (interval; point) |
|---|---|---|---|
| A1 | [0.15;0.30],[0.35;0.40]; (0.20;0.65) | [0.13;0.25],[0.40;0.45]; (0.30;0.60) | [0.10;0.30],[0.25;0.35]; (0.11;0.20) |
| A2 | [0.10;0.15],[0.35;0.40]; (0.40;0.17) | [0.15;0.22],[0.27;0.30]; (0.15;0.29) | [0.50;0.60],[0.15;0.20]; (0.35;0.19) |
| A3 | [0.14;0.25],[0.35;0.65]; (0.10;0.40) | [0.35;0.45],[0.15;0.20]; (0.30;0.50) | [0.30;0.50],[0.10;0.30]; (0.20;0.35) |
| A4 | [0.30;0.35],[0.25;0.45]; (0.20;0.30) | [0.20;0.55],[0.40;0.45]; (0.20;0.45) | [0.10;0.29],[0.40;0.50]; (0.30;0.40) |
(Column C3 is not shown in the table; its weight of 0.13 is included in the calculation.) The method builds the ideal point from the interval and point components' separate best values in each column, the anti-ideal from their worst values, sums the q=2 distance with the weights, and computes the closeness coefficient.
| Supplier | Closeness coefficient | Rank |
|---|---|---|
| A2 | 0.645 | 1 |
| A3 | 0.524 | 2 |
| A4 | 0.439 | 3 |
| A1 | 0.378 | 4 |
The result reads as follows. A2 holds a strong position, on both the interval and the point, on C4, the heaviest criterion; A1 does not stand out on any criterion and finishes last.
The business's hesitation: when the weight is concentrated on the second criterion (0.10; 0.60; 0.10; 0.20) and recomputed, A3 (0.5685) overtakes A2 (0.5641); this shows that A2's first place depends on the weight given to C4 and C1 (independently recomputed with the kernel).
A caveat is also needed: DecisionMind's engine runs this table in the single-decision-maker (λ=1) setting. Garg and Kaur's (2018) article processes the same data as a priority-weighted combination of three decision-makers (0.20; 0.30; 0.40; 0.10) and reports a different order (A3 first) in its own Table 5. This card reports DecisionMind's single-decision-maker engine output, not the article's multi-expert result; the distinction is explained further in the verification notes.
In the report: "With the single-decision-maker calculation, A2 holds the highest closeness coefficient (0.645); when the weight is concentrated on the second criterion (0.10; 0.60; 0.10; 0.20), A3 moves ahead. This calculation differs from the article's own multi-expert Table 5 result, because that uses a priority-weighted combination of three decision-makers."
Source: Garg and Kaur (2018), Scientia Iranica E, Tables 1-4 (decision table). DecisionMind's cubic intuitionistic fuzzy TOPSIS engine independently recomputes this table in the single-decision-maker (λ=1, q=2) setting; the result has been verified by this card's author running the kernel directly.
2. Livestock: A cooperative's choice of dairy-cattle breed
A dairy cooperative will switch to one of three cattle breeds. Three criteria apply: milk yield, disease resistance and feed-conversion ratio (lower is better). The veterinary panel scores every breed both with a band ("yield falls somewhere between low and high") and with a single representative point judgement ("my most likely assessment is this"); the band and the point come from separate observations.
The method builds the ideal breed from the interval and point components' separate best values on every criterion, the anti-ideal from their worst, and sums the q=2 distance with the weights. Suppose the breed with the highest milk yield also carries the widest uncertainty band; it still comes out first on the closeness coefficient, because the weight is concentrated on yield.
The panel's hesitation is this: this breed's wide band means real-world yield could vary more than expected. The panel should look not only at the closeness coefficient but also at which breed's band is narrower, and hence more predictable.
In the report: "With the weight on milk yield, the first breed stands out; this breed's assessment band is wider than the others', so band width should be separately evaluated before the field trial."
3. What Not to Do
In the illustrative example, automatically computing the point value as the interval's midpoint, for instance writing A1's point support on C1 as (0.15+0.30)/2=0.225: the cubic structure's second layer stops adding information, because the point is now derived from the interval. The second mistake is reporting "cubic intuitionistic fuzzy TOPSIS found A2 first" without stating q; the calculation with q=1 can give different distances. The third mistake is presenting the article's multi-expert Table 5 result (A3 first) as though it were DecisionMind's single-expert engine output; the two calculations rest on different input assumptions.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/cubic-topsis
Garg, H., & Kaur, G. (2018). Extended TOPSIS method for multi-criteria group decision-making problems under cubic intuitionistic fuzzy environment. Scientia Iranica E. DOI: 10.24200/sci.2018.5307.1194
Jun, Y. B., Kim, C. S., & Yang, K. O. (2012). Cubic sets. Annals of Fuzzy Mathematics and Informatics, 4(1), 83–98. (no DOI)
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI: 10.1016/S0165-0114(86)80034-3
Hwang, C. L., & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications, A State-of-the-Art Survey. Lecture Notes in Economics and Mathematical Systems, Vol. 186. Springer-Verlag. DOI: 10.1007/978-3-642-48318-9
Kaur, G., & Garg, H. (2018). Multi-attribute decision-making based on Bonferroni mean operators under cubic intuitionistic fuzzy set environment. Entropy, 20(1), 65. DOI: 10.3390/e20010065