Extension card · Intuitionistic
Intuitionistic fuzzy TOPSIS (Boran, Genç, Kurt and Akay, 2009)
This is the form of TOPSIS that expresses criterion assessment as a degree of support for, and a degree of rejection of, a judgement; this pair is called an intuitionistic fuzzy number. The method combines the views of several decision-makers and builds the ideal and anti-ideal points from these pairs. It measures distance through the degree of membership, the degree of non-membership and hesitancy, and ranks the result with a single closeness score.
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 decision logic does not.
Cells. In crisp TOPSIS every cell is a single number. Here every cell consists of a pair of numbers: a degree of support (μ) for a judgement and a degree of rejection (ν). The sum of these two degrees does not exceed 1. Criterion weights take the same paired form.
Unlike most of the fuzzy extensions in DecisionMind, IF-TOPSIS builds the group decision into the method's own steps. Several decision-makers' scores and weights are combined into a single pair using the intuitionistic fuzzy weighted average operator (IFWA), together with each decision-maker's own importance weight. With a single decision-maker, this step does not come into play, and the calculation proceeds directly with the single pair.
Scale equalisation. Crisp TOPSIS carries a separate scale-equalisation step; a column is divided by the root of the sum of its squares. Because intuitionistic fuzzy pairs are already defined within the 0-1 range and under the μ+ν≤1 constraint, IF-TOPSIS carries no separate normalisation step. Weighting is instead carried out with Atanassov's intuitionistic fuzzy multiplication. This operation differs from crisp multiplication and keeps the result a valid (μ, ν) pair. If crisp multiplication is used instead, that is, if μ is multiplied directly by the crisp weight, the pair falls outside the intuitionistic fuzzy axioms.
Distance. Crisp TOPSIS calculates a straight-line, Euclidean, distance between two numbers. Here the Szmidt-Kacprzyk normalised Euclidean distance is used. On every criterion, three components enter the calculation together: the μ difference, the ν difference, and the difference in the hesitancy margin (π = 1−μ−ν); the total is normalised by dividing by twice the number of criteria. This is a shift from a single-component crisp difference to a three-component one. The hesitancy margin is not asked for separately; it enters the calculation as what remains from the two degrees.
Result and defuzzification. The definition here is the same as elsewhere: the result is calculated as the ratio of the distance to the anti-ideal over the sum of the two distances. The result is again a single number between 0 and 1. DecisionMind fixes IFWA group aggregation, weighting by Atanassov multiplication, and the Szmidt-Kacprzyk normalised distance in this extension. Depending on criterion direction, the ideal and anti-ideal pair take the maximum μ value and the minimum ν value; this applies to a benefit criterion, and the reverse applies to a cost criterion.
How to Read the Output
The output is a closeness score and a rank, as in crisp TOPSIS, and it is read the same way: it is not a percentage, and it is not compared with a different analysis.
The difference is here: beneath the score lies support, rejection and hesitancy information all at once, yet the score compresses this into a single number. Because the input comes not from a single crisp score but from a support and rejection vote on a judgement, the robustness of a score gap depends on how decisive these votes were. Where support and rejection sit close to one another, that is, where the hesitancy margin is large, a small score gap can easily close once the votes become more decisive in a later round.
Thus instead of writing:
"IF-TOPSIS accounts for uncertainty, so the result is more reliable"
the report should read:
"Because the assessment comes from a support and rejection vote on a judgement, uncertainty has been carried through the calculation; A1 is ahead at 0.566, and this gap closes entirely if A1's support on the first criterion falls by one unit"
When to Prefer This over the Base Method
Use this method when the assessment comes from a judgement, and the opposing view can also be measured from a separate information source: situations where the support given to a judgement such as "this supplier is reliable" and the reservation held about the same judgement need to be recorded separately; board decisions where several decision-makers' scores must be combined by weighting rather than reduced to an average. If the expert only states "how suitable" and there is no separate source for the opposing view, that is, if rejection is to be calculated by subtracting support from 1, the principle underlying the Intuitionistic fuzzy data-type card is violated, and the intuitionistic fuzzy structure's one contribution disappears. Crisp or fuzzy TOPSIS is then sufficient.
Stay with the base method in the same situation as always: where criteria are measured. Turning a measured value into a judgement and inventing a support/rejection pair adds no information, as the Intuitionistic fuzzy data-type card warns. DecisionMind requires a single data type if the table is mixed; a measured criterion can also be written as (support, 1-support), but this zeroes the hesitancy margin and becomes indistinguishable from crisp data. If no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold.
Mistakes Specific to This Extension
Confusing the intuitionistic fuzzy pair (μ, ν) with a triangular fuzzy number (a, b, c). IF-TOPSIS inputs must be a pair of two numbers, under the constraint μ+ν≤1; three-number triangular fuzzy values do not fit this structure and belong to a different extension (Fuzzy TOPSIS).
Using crisp multiplication. Applying crisp multiplication such as v_ij = w_j · r_ij in weighting breaks the intuitionistic fuzzy axioms and produces invalid (μ, ν) pairs; Atanassov's intuitionistic fuzzy multiplication must be used.
Forgetting the normalisation factor. The normalised form of the Szmidt-Kacprzyk distance (dividing by twice the total number of criteria) must be used; the unnormalised form produces figures on a different scale and does not match Boran et al.'s (2009) results.
Deriving rejection from support. Writing ν as 1-μ is the fundamental error the data-type card warns against: the hesitancy margin is zeroed and the intuitionistic fuzzy structure is reduced to crisp data.
The governing principle is this:
Support and rejection degrees must come from separate sources, Atanassov multiplication must replace crisp multiplication, and the normalised distance must be used; if any one of these is skipped, the result stops being the IF-TOPSIS that Boran et al. (2009) defined.
Cases
The first case is DecisionMind's validation example: a small, hand-traceable intuitionistic fuzzy fixture with a single decision-maker (the full example in Boran et al.'s 2009 paper itself, with five alternatives and four decision-makers, is held in a separate DecisionMind validation record, not here). The second case is an illustrative construction.
1. Illustrative example: Intuitionistic fuzzy scoring of three candidates on two criteria
A single assessor scores three candidates on one benefit and one cost criterion; each score is a support-rejection pair for a judgement.
| Candidate | Criterion 1 (support, rejection) | Criterion 2 (support, rejection) |
|---|---|---|
| A1 | (0.80; 0.10) | (0.70; 0.20) |
| A2 | (0.60; 0.30) | (0.50; 0.40) |
| A3 | (0.40; 0.50) | (0.30; 0.60) |
| Weight | (0.80; 0.10) | (0.60; 0.30) |
| Direction | higher is better | lower is better |
Because there is a single assessor, the method skips the group-aggregation step, weights with Atanassov multiplication, chooses the ideal and anti-ideal pair on each criterion according to criterion direction, calculates the Szmidt-Kacprzyk normalised distance, and finds the closeness score.
| Candidate | Closeness score | Rank |
|---|---|---|
| A1 | 0.566 | 1 |
| A2 | 0.500 | 2 |
| A3 | 0.434 | 3 |
The result reads as follows. A1 has received the highest support and the lowest rejection vote on both criteria; A3 is exactly the opposite. A2 sits exactly in the middle, and its score is exactly 0.500.
A hesitation: if A1's support on the first criterion fell by one unit, that is, if (0.80; 0.10) were entered as (0.70; 0.20) instead, the order reverses completely. A2 moves ahead at 0.579, A3 comes second at 0.506, and A1 falls to third at 0.494. If the weights swapped, with the first criterion taking (0.60; 0.30) and the second (0.80; 0.10), the order reverses again and A3 takes A1's place at 0.566, because the table has deliberately been built to be symmetric. In this example the order is fragile to both a single-score change and a weight swap, and the report should say so plainly.
In the report: "The assessment was given as a support and rejection vote on a judgement. A1, at 0.566, is closest to the ideal; however, if its support on the first criterion falls by one unit, or if the criterion weights swap, the order reverses completely."
Source: DecisionMind's IF-TOPSIS validation example (a small, hand-calculated, closed-form fixture with a single decision-maker); the algorithm is taken from Boran, Genç, Kurt and Akay's (2009) eight-step IF-TOPSIS method. The closeness scores and sensitivity values were independently recomputed.
3. What Not to Do
In the illustrative example, reducing A1's (0.80; 0.10) pair to a single crisp score, such as 0.80, and running crisp TOPSIS on it: the order may not change, but the support-rejection distinction and the hesitancy-margin information are lost entirely, and the fragility of the gap becomes invisible. The second error is using crisp multiplication instead of Atanassov multiplication in weighting (multiplying μ directly by the crisp weight); this produces invalid (μ, ν) pairs, and the result no longer matches the method Boran et al. (2009) defined. The third error is presenting A3's rejection in the (0.40; 0.50) pair as if it had been "calculated" directly as 1-0.40=0.60, rather than genuinely coming from an independent source (counter-votes); its coming out equal here is coincidental, it does not mean rejection was derived from support, and this distinction must not be blurred in the report.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/if-topsis
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
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
Boran, F. E., Boran, K., & Menlik, T. (2012). The evaluation of renewable energy technologies for electricity generation in Turkey using intuitionistic fuzzy TOPSIS. Energy Sources, Part B: Economics, Planning, and Policy, 7(1), 81–90. DOI: 10.1080/15567240903047483