Extension card · Fuzzy
Triangular Intuitionistic Fuzzy CODAS (Daami Remadi and Frikha, 2023)
This is the form of CODAS for situations where criterion scores are given both as a triangular support and as a separate, wider triangular rejection region; it still ranks the result with an assessment score.
Base method
CODAS →
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?
Four things change. The decision logic stays the same.
Cells. In crisp CODAS every cell is a single number. An ordinary triangular fuzzy number has a single triangle: lowest, most likely, highest. A triangular intuitionistic fuzzy number (TIFN) adds a second triangle on top of this. It defines a second rejection-region triangle that shares the same most-likely point (a2) but has wider ends: {(a1, a2, a3); (a'1, a2, a'3)}. The first triangle answers "how much do I support this", the second "how much do I reject this"; the rejection triangle wraps the support triangle at its ends (a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3). This carries the intuitionistic fuzzy structure (membership and non-membership degree) over into a triangular fuzzy number. In DecisionMind, criterion weights are crisp numbers; in the method's own literature the weight, too, can be a TIFN, but DecisionMind accepts only crisp weights in this family.
Scale equalisation. Crisp CODAS divides every column by its largest or smallest value. In TIFN-CODAS, normalisation divides every TIFN's six components by the column's largest upper value (x_j+) for a benefit criterion; for a cost criterion, it ratios the column's smallest lower value (x_j-) to each component. Every TIFN is then scaled by the criterion's crisp weight.
Defuzzification and distance. This is the largest difference from base CODAS. The weighted TIFN matrix and the TIFN negative-ideal are first reduced to a single number by the Gani-Abbas (2014) weighted-average rule. The negative-ideal is chosen, on each criterion, from the TIFN with the lowest lower bound of the rejection triangle (a'1). After defuzzification the calculation returns entirely to crisp CODAS: the Euclidean distance (E) and the Taxicab distance (T) are calculated on the defuzzified numbers.
Group aggregation. In the method's own literature, scores calculated separately by several decision-makers are combined into a weighted average using decision-maker weights (λ). DecisionMind currently runs this step in single-decision-maker mode: the user enters a single, already-merged TIFN matrix, and λ = 1 is assumed. Entering several decision-makers' own matrices separately and taking their weighted average is not supported in this version; the detail is in the verification notes.
DecisionMind fixes the Gani-Abbas (2014) defuzzification rule and the threshold value (psi_threshold, default 0.02) for this family; the threshold can be changed by the user.
How to Read the Output
The assessment score is read as in crisp CODAS: it is a relative score, a negative value is not a failure, and the negative-ideal shifts when the alternative set changes. The difference is here: underneath the score there is both a support uncertainty and a separate, wider rejection uncertainty; the two collapse into a single number at the defuzzification step and become invisible in the report.
Thus instead of writing:
"Triangular intuitionistic fuzzy CODAS carries a double uncertainty, so the result is more reliable"
the report should read:
"The support and rejection triangles have been reduced to a single number at the defuzzification step; the wider the gap between the two triangles, the coarser a summary this single number is"
When to Prefer This over the Base Method
Use this extension when an expert gives a judgement both as how much they support it and, separately, as how much they reject it, through two distinct triangles. If a judgement is given with only a single triangle (lowest, most likely, highest), the ordinary fuzzy extension is enough; TIFN is not needed if there is no second, rejection triangle. A measured criterion should not be expanded directly into two triangles. Base CODAS's exit condition applies here too: if no compromise at all is acceptable on one criterion, this compensatory family is not the right choice.
Mistakes Specific to This Extension
Confusing the ordering constraint. Every TIFN must satisfy a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3; the support triangle must stay inside the rejection triangle. If this order is broken, the TIFN is invalid.
Defuzzifying before weighting. Defuzzification is only carried out after the weighted matrix has been built. Reducing the raw matrix to a single number first and only then weighting it loses the scale information the TIFN carries.
Reversing the triangles for a cost criterion. The method normalises a cost criterion by a min/max ratio; it does not swap the support and rejection triangles for one another (there is no complementing operation such as μ↔ν). Confusing these two operations produces a wrong negative-ideal.
Assuming group aggregation works here. DecisionMind runs this version with a single decision-maker. Entering several decision-makers' scores and reporting that "the method combines them with a weighted average" is wrong; the user must merge the scores themselves.
The governing principle is this:
Triangular intuitionistic fuzzy CODAS exists to carry support and rejection in two separate triangles. Any application that confuses the two triangles, or defuzzifies in the wrong order, erases this contribution.
Cases
The J example in the manifest does not come from the literature page; it comes from DecisionMind's own validation example, built with three alternatives on three criteria, using TIFNs chosen from Daami Remadi and Frikha's (2023) seven-term linguistic scale. The first case is this example. The second case is an illustrative construction.
1. Illustrative example: A TIFN assessment of three alternatives on three criteria (DecisionMind's validation example)
All three criteria are benefit-type, with weights of 0.40, 0.35 and 0.25 respectively. The scores are chosen from Daami Remadi and Frikha's (2023) seven-term linguistic scale; each term corresponds to a predefined TIFN.
| Alternative | K1 | K2 | K3 |
|---|---|---|---|
| A1 | high {(7,9,10);(6,9,10)} | medium {(3,5,7);(2,5,8)} | very low {(0,1,3);(0,1,4)} |
| A2 | medium {(3,5,7);(2,5,8)} | high {(7,9,10);(6,9,10)} | medium {(3,5,7);(2,5,8)} |
| A3 | very low {(0,1,3);(0,1,4)} | medium {(3,5,7);(2,5,8)} | high {(7,9,10);(6,9,10)} |
| Direction | higher is better | higher is better | higher is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method divides every column by its largest upper bound, scales it by the crisp weights, chooses the negative-ideal from the lowest lower bound of the rejection triangle, defuzzifies with the Gani-Abbas (2014) rule, and calculates crisp CODAS's two distances. The threshold value is τ = 0.02.
| Alternative | Assessment score | Rank |
|---|---|---|
| A1 | 0.2221 | 1 |
| A2 | 0.2083 | 2 |
| A3 | -0.4304 | 3 |
The result reads as follows. A1 obtains the best score on the most heavily weighted criterion (K1, high) and beats A2 by a small margin (0.0138). A3 has failed to make up for its weak score on the first criterion (very low) with the other criteria and finishes last.
The decision's hesitation: had A1's score on K1 been one term lower, "medium-high", the 0.0138 gap between A1 and A2 would most likely close or reverse. This shows how sensitive the two alternatives are to a single-term change on the most heavily weighted criterion.
In the report: "A1, which scores high on the most heavily weighted criterion (K1), is first; the gap with A2 is small (0.0138) and is sensitive to a single-term change in this criterion's score."
Source: DecisionMind's validation example for the TIFN-CODAS engine; it follows Daami Remadi and Frikha's (2023) 9-step algorithm (S1–S9), but is a reduced 3x3 example rather than the 4-alternative, 8-criterion application table they present. The numbers were calculated by independently rewriting the engine in Python and match the engine's own output.
2. Energy: A municipal company's choice of renewable-energy technology
A municipal energy company will choose among three renewable technologies for a new plant: wind (R), solar (G) and biomass (B). Three criteria have been set: investment cost (lower is better), production capacity and ease of maintenance. The evaluation board has chosen its pre-merged judgement from Daami Remadi and Frikha's (2023) linguistic scale.
| Technology | Investment cost (lower is better) | Production capacity | Ease of maintenance |
|---|---|---|---|
| R | medium {(3,5,7);(2,5,8)} | high {(7,9,10);(6,9,10)} | medium-high {(5,7,9);(4.5,7,9.5)} |
| G | medium-low {(1,3,5);(0.5,3,5.5)} | medium-high {(5,7,9);(4.5,7,9.5)} | high {(7,9,10);(6,9,10)} |
| B | high {(7,9,10);(6,9,10)} | medium {(3,5,7);(2,5,8)} | medium {(3,5,7);(2,5,8)} |
| Weight | 0.40 | 0.35 | 0.25 |
The method normalises the cost column by a min/max ratio, weights it, builds the negative-ideal, defuzzifies, and calculates the two distances.
| Technology | Assessment score | Rank |
|---|---|---|
| G | 0.4126 | 1 |
| R | 0.3292 | 2 |
| B | -0.7418 | 3 |
The result reads as follows. G is strong both on low cost and on high ease of maintenance, and beats R by a small margin (0.0834).
The board's hesitation: if weight is shifted towards production capacity (capacity 0.65, cost 0.15, maintenance 0.20), R moves to first place, its score rising to 0.6566; G drops to second (0.2578). R's advantage on production capacity (high) becomes decisive once weight shifts there.
In the report: "With the weight given to cost and ease of maintenance, G comes out ahead; the gap with R is small (0.0834). When weight is shifted to production capacity, R moves to first place; which criterion the board prioritises determines the result."
3. What Not to Do
In the illustrative example, if A3's "high" {(7,9,10);(6,9,10)} score on K3 were changed to {(7,9,10);(8,9,10)}, so that the rejection triangle no longer wraps the support triangle, a'1=8 would exceed a1=7 and the ordering constraint would be broken; this is an invalid TIFN. The second error is defuzzifying the raw linguistic scores first and only then weighting them; this loses the scale difference the TIFN carries (for instance, the width difference between "high" and "medium-high"). The third error is, as in the energy example, summing several decision-makers' separate scores and presenting it as "the method took a group average"; DecisionMind expects a single, already-merged matrix in this version.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/tifn-codas
Daami Remadi, F., & Frikha Moalla, H. (2023). The triangular intuitionistic fuzzy numbers CODAS method for solving green material selection problem. International Journal of Operational Research. DOI: 10.1504/IJOR.2023.129409
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI: 10.1016/S0165-0114(86)80034-3
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI: 10.1016/S0019-9958(65)90241-X
Keshavarz Ghorabaee, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2016). A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Economic Computation and Economic Cybernetics Studies and Research, 50(3), 25–44. (no DOI)