Extension card · Fuzzy
Complex Fuzzy TOPSIS
The form of TOPSIS for situations where the degree of support for, and rejection of, a judgement carry two components, amplitude and phase. It measures distance to the ideal and anti-ideal point through these two components and ranks the result with a 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)
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?
Three things change; the five-step skeleton and the decision logic do not.
Cells. In crisp TOPSIS every cell is a single number. Here every cell consists of four numbers: a support amplitude r_μ and support phase ω_μ, and a rejection amplitude r_ν and rejection phase ω_ν. Amplitudes lie between 0 and 1; phases are angles between 0 and 2π. Amplitude measures the same thing as the support and rejection degree in intuitionistic fuzzy. Phase is a separate dimension; it carries where the judgement sits on a second axis. On a cost criterion the support/rejection pair swaps places, together with its amplitude and phase. Criterion weights are taken from outside as crisp numbers.
The ideal and anti-ideal point. Crisp TOPSIS builds the ideal point from each column's best value. Here the ideal point (CIFPIS) is built, on every criterion, from the separate best of amplitude and of phase: the highest support amplitude and phase, and the lowest rejection amplitude and phase. The anti-ideal (CIFNIS) is exactly the reverse. This step works on the same logic as crisp TOPSIS's third step, only across four components.
Distance and closeness score. The distance between two values is a Euclidean distance into which both the amplitude differences and the normalised phase differences (phase divided by 2π) enter together. Each alternative's distance to the ideal and to the anti-ideal is the weighted sum of these distances across criteria. The closeness score is, as in crisp TOPSIS, the ratio of the distance to the anti-ideal against the sum of the two distances.
Phase's contribution is real only when it is given genuinely independently of amplitude. This card's author independently recomputed the manifest's own validation example with the kernel: in that example the phases are given proportional to amplitude (ω=2π·r). In that case each distance term's phase component produces exactly the same difference as its amplitude component, and the total distance grows only by a fixed factor (√2). Because this factor is reflected equally in both the distance to the ideal and the distance to the anti-ideal, it cancels out in the closeness score's ratio: the closeness scores computed with and without phase come out identical to the decimal place in this example (A1=0.2000, A2=0.8018, A3=0.5380). This changes once phase comes from a source genuinely independent of amplitude: this card's author, changing only one cell's phase in the same table independently of amplitude (raising A1's phase on the first criterion above its amplitude-proportional value), found the closeness scores shift to A1=0.2873, A2=0.6864, A3=0.4850; the order is preserved in this small example, but the scores change markedly.
DecisionMind fixes, for this extension, the amplitude-phase Euclidean distance and the equally-weighted skeleton. Weights are taken from outside; the method does not generate weights.
How to Read the Output
The closeness score is read as in crisp TOPSIS: a position relative to the set's own ideal and anti-ideal points, not a percentage or a probability. The difference here is that both amplitude and phase enter this score together, but if phase is given proportional to amplitude it adds no extra information to the score at all. Where phase genuinely comes from an independent source, part of the score comes from this second dimension, and the score remains half unreadable unless what phase represents is explained.
Thus instead of writing:
"Complex fuzzy TOPSIS found this alternative through a richer uncertainty model"
the report should read:
"This score carries the amplitude and phase components together; where phase does not come from a source independent of amplitude (for instance where it is proportional to amplitude), the result is indistinguishable from intuitionistic fuzzy TOPSIS"
When to Prefer This over the Base Method
This extension is worth considering when the support or rejection given to a judgement genuinely comes from two independent dimensions: amplitude answering "how strong," phase answering "on which second axis, at which phase." If these two dimensions are not independent, that is, if phase is derived from amplitude, the complex structure adds nothing to intuitionistic fuzzy TOPSIS and only makes the model harder to explain. Turning a measured criterion into an amplitude-phase pair does not model uncertainty, it manufactures it. Crisp TOPSIS's exit condition applies here too: where no compromise is acceptable on a criterion, this extension is compensatory as well and does not screen out anything below a threshold.
Mistakes Specific to This Extension
Giving phase proportionally to amplitude, or arbitrarily. If phase is derived from amplitude (a rule such as ω=2π·r), the closeness score comes out identical to the one intuitionistic fuzzy TOPSIS gives; this has been verified by this card's author with the kernel. Phase must come from its own independent source.
Mistaking the phase angle for degrees and entering it as such rather than radians. The engine expects ω in radians between 0 and 2π; entering a value on a 0–360 degree scale directly violates the value range.
Value-range violation. Every amplitude r_μ, r_ν must lie in [0,1]; entering the calculation without this check invalidates the method.
Not stating in the report what phase represents. Since part of the score comes from phase, presenting a score without explaining which second dimension phase measures leaves it incomplete.
The governing principle is this:
Complex fuzzy TOPSIS's phase adds something only when it genuinely comes from a source independent of amplitude; a phase that is proportional to, or arbitrarily set against, amplitude produces a result indistinguishable from intuitionistic fuzzy TOPSIS, and the report must not conceal this.
Cases
The first case is DecisionMind's validation example; the synthetic 3×3 table in the manifest is built faithfully to the formulas, carries no literature page, and its phases are given proportional to amplitude (ω=2π·r). The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): A publishing decision among three magazine designs
A publisher compares three cover-and-interior design proposals on three criteria: reader appeal (more is better), print cost (less is better), and brand fit (more is better). Each proposal's support and rejection on each criterion is given as an amplitude-phase pair; in this example the phases are proportional to the amplitudes (ω=2π·r), meaning phase adds no information beyond what amplitude already carries.
| Proposal | Reader appeal (r_μ; r_ν) | Print cost (r_μ; r_ν, less is better) | Brand fit (r_μ; r_ν) |
|---|---|---|---|
| T1 | (0.85; 0.15) | (0.75; 0.15) | (0.65; 0.15) |
| T2 | (0.85; 0.05) | (0.55; 0.15) | (0.75; 0.15) |
| T3 | (0.75; 0.15) | (0.65; 0.15) | (0.85; 0.15) |
| Weight | 0.40 | 0.25 | 0.35 |
The method swaps the support-rejection pair, together with its amplitude and phase, on the cost criterion, builds the ideal and anti-ideal point on each criterion from the separate best of amplitude and phase, sums the weighted amplitude-phase distances, and computes the closeness score.
| Proposal | Closeness score | Rank |
|---|---|---|
| T2 | 0.802 | 1 |
| T3 | 0.538 | 2 |
| T1 | 0.200 | 3 |
The result reads as follows. T2 holds the highest support and lowest rejection degree on reader appeal, the most heavily weighted criterion, and also holds the lowest degree on print cost. T1 stands out on no criterion and finishes last.
The publisher's hesitation: because phase is given proportional to amplitude in this table, the closeness scores come out identical to the decimal place whether or not phase is factored in at all (T1=0.200, T2=0.802, T3=0.538). Only when T1's phase on reader appeal is changed independently of amplitude (same amplitude, different phase) do the scores shift to T1=0.287, T2=0.686, T3=0.485; the order holds in this small example, but the scores change markedly. This shows that phase carries no genuine information in this example; it is only a reflection of amplitude.
In the report: "With the given amplitudes and phases, T2 is the most advantageous proposal (0.802). In this example phase is given proportional to amplitude, so the closeness scores come out the same using amplitude alone; the complex structure's phase carries no extra information here."
Source: DecisionMind's CF-TOPSIS validation example. The TOPSIS skeleton rests on the formulas in the manifest; the complex intuitionistic fuzzy structure rests on Alkouri and Salleh's (2012) definition, and the underlying idea of the complex fuzzy set on Ramot and colleagues' (2002) work. The closeness scores and the shift between independent and proportional phase were computed independently by this card's author by running the kernel directly.
2. Publishing: A publisher's evaluation of foreign-rights translation offers
A publisher is evaluating three foreign novels to acquire translation rights for. There are three criteria: expected sales potential, translation difficulty (less is better), and the likelihood of collaboration with the author on future works. The publisher looks at each novel on two separate axes: how strongly this assessment is supported (amplitude), and which market cycle, an emerging genre, a stagnant one, or a rising one, this support is valid for (phase). These two axes are genuinely independent, because support of the same strength can point to a different market phase.
The method computes the weighted amplitude-phase distance for each criterion, and builds the closeness score against the ideal and anti-ideal point. Suppose two novels share the same sales-potential amplitude, but one receives a different score because it points to the phase of a rising genre; this difference would not be visible from the sales-potential amplitude alone.
The publisher's hesitation is this: unless phase's meaning here, which market phase, is clearly defined, the source of the score difference cannot be explained to the editor. The publisher must state in the report what the phase angle represents and how it was determined.
In the report: "Part of the closeness score comes from the strength of support, part from the market phase it points to; the second component represents the emerging/stagnant/rising genre cycle, and without this definition the score cannot be fully read."
3. What Not to Do
In the illustrative example, making up phases independently of amplitude, for instance writing an arbitrary phase value for T1, makes the score's origin unclear and produces a result that cannot be reproduced. The second error is entering a phase angle in degrees (say, 180) directly and mistaking it for radians; the engine expects a value between 0 and 2π. The third error is reading T2's score of 0.802 as "eighty-two per cent suitable"; the score is only a comparison against this set's own ideal and anti-ideal points, among these three proposals.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/cf-topsis
Ramot, D., Milo, R., Friedman, M., & Kandel, A. (2002). Complex fuzzy sets. IEEE Transactions on Fuzzy Systems, 10(2), 171–186. DOI: 10.1109/91.995119
Alkouri, A. M. J. S., & Salleh, A. R. (2012). Complex intuitionistic fuzzy sets. AIP Conference Proceedings, 1482(1), 464–470. DOI: 10.1063/1.4757515
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