Extension card · Fuzzy
Complex Fuzzy EDAS
The form of EDAS for situations where the degree of support for, and rejection of, a judgement each carry two components, amplitude AND phase. These two components merge into a single score first, and the rest of the calculation runs exactly as crisp EDAS does.
Base method
EDAS →
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 six-step skeleton and the decision logic do not.
Cells. In crisp EDAS 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: how much I agree with a judgement, how much I do not. Phase is a separate dimension, carrying where the judgement sits on a cyclical or second axis, for instance which seasonal phase an assessment is strongest in. Criterion weights are taken from outside as crisp numbers.
The point of defuzzification. These four numbers collapse into a single score at the very first step of the calculation: s = ((r_μ − r_ν) + (ω_μ − ω_ν)/(2π)) / 2. Half of this score comes from the amplitude difference, half from the phase difference; the two contribute with equal weight. Once the score is calculated, everything that follows is crisp EDAS itself: the column average is taken, each cell's positive and negative deviation from that average is measured, weighted and summed, normalised, and merged into a single appraisal score.
Phase's contribution is real only when it is given independently. Phase affects the score separately from amplitude when it genuinely carries information independent of amplitude; this card's author verified it by running the kernel directly: with the same amplitude pair (r_μ=0.70, r_ν=0.20) but different phases (ω_μ=3.0 and ω_μ=5.5), the score changes from 0.409 to 0.608. But if phase is given proportionally to amplitude (a rule such as ω=2π·r), the formula simplifies algebraically and the score reduces exactly to r_μ−r_ν; the complex structure then produces no difference at all from intuitionistic fuzzy EDAS. DecisionMind's own validation example (Case 1 below) uses precisely this second situation: phases are proportional to amplitudes, and phase's contribution in that example is zero. This is not a general limitation of the engine, only a consequence of that example's choice of data.
DecisionMind fixes, for this extension, the score function and the equal (half-and-half) weighting between amplitude and phase. Weights are taken from outside; the method does not generate weights.
How to Read the Output
The appraisal score is read as in crisp EDAS: a position relative to the set's own average, not a percentage or a probability. The difference here is that half of this score comes from amplitude, half from phase. Two alternatives having the same amplitude difference does not mean they will have the same score; the phase difference also comes into play. Where phase's meaning has not been clearly defined, this second contribution looks to the reader like a repeated score of unclear origin.
Thus instead of writing:
"CF-EDAS chose this alternative because it found it most advantageous"
the report should read:
"This score carries the difference in both the support/rejection amplitude and the phase angle with equal weight; phase here represents (X), and without that definition half the score remains unreadable"
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. For example, how strong an assessment is (amplitude) and which cyclical phase or second criterion it holds under (phase) might be known separately. If these two dimensions are not actually independent, that is, if phase is derived from amplitude, the complex structure adds nothing to intuitionistic fuzzy EDAS 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 EDAS'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 score reduces algebraically to r_μ−r_ν, and the complex structure adds no extra information; 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 and the calculation will fail.
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 half the score comes from phase, presenting a score without explaining which second dimension phase measures leaves half of it unexplained.
The governing principle is this:
Complex fuzzy EDAS'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 EDAS, 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): Choosing a campaign timing strategy
An e-commerce company compares three campaign timing strategies on three criteria: expected conversion rate (more is better), brand perception score (more is better), and cancellation/refund rate (less is better). Each strategy'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.
| Strategy | Conversion rate (r_μ; r_ν) | Brand perception (r_μ; r_ν) | Cancellation rate (r_μ; r_ν, less is better) |
|---|---|---|---|
| S1 | (0.85; 0.15) | (0.65; 0.15) | (0.75; 0.15) |
| S2 | (0.85; 0.05) | (0.75; 0.15) | (0.55; 0.15) |
| S3 | (0.75; 0.15) | (0.85; 0.15) | (0.65; 0.15) |
| Weight | 0.40 | 0.35 | 0.25 |
The method computes each cell's score; in this example s = r_μ − r_ν, because phase is proportional to amplitude. It then finds the column average, measures each strategy's positive and negative deviation from that average, sums them by weight, and merges them into a single appraisal score.
| Strategy | Appraisal score | Rank |
|---|---|---|
| S2 | 1.000 | 1 |
| S3 | 0.508 | 2 |
| S1 | 0.000 | 3 |
The result reads as follows. S2 sits above average on conversion rate, the most heavily weighted criterion, and below average on cancellation rate (less is better), meaning it is favourably positioned there too; only on brand perception does it fall below average. S1 shows exactly the opposite profile and finishes last.
The company's hesitation is sensitive to the weight on brand perception. If the conversion and cancellation weights are shrunk proportionally and brand perception's weight is raised from 0.35 to about 0.60, S3 moves ahead (S3=0.8655, S2=0.8297); as long as brand perception's weight stays below 0.55, S2 stays ahead (at 0.55, S2=0.9046, S3=0.8434). This shows that the S2–S3 order is fairly sensitive to the weight on brand perception.
In the report: "With the given weights (conversion 0.40, brand perception 0.35, cancellation 0.25), S2 is the most advantageous position relative to the set's average (1.000); once brand perception's weight rises to about 0.60, S3 moves ahead, so the order is sensitive to this weight. In this example phase was given proportional to amplitude, so the entire score comes from the amplitude difference."
Source: DecisionMind's CF-EDAS validation example. The score function and EDAS skeleton rest 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 appraisal scores and the weight-trade-off scenario were computed independently by this card's author by running the kernel directly.
2. Cybersecurity: A company's choice of threat-intelligence provider
A company's security operations centre will contract with one of three threat-intelligence providers. There are three criteria: detection accuracy, false-alarm rate (less is better), and ease of integration. The centre evaluates each provider on two separate axes: how strong the evidence is behind an alert it raises (amplitude), and which phase of the attack cycle, reconnaissance, intrusion or exfiltration, that alert points to (phase). These two axes are genuinely independent, because evidence of the same strength can point to an earlier or a later phase of an attack.
The method computes, for each criterion, a score combining the amplitude and phase difference with equal weight. It measures each provider's positive and negative deviation from the column average and merges them into a single appraisal score. Suppose two providers share the same strength of evidence, but one receives a different score because it points to a later phase of an attack; this difference would not be visible from a comparison of evidence strength alone.
The centre's hesitation is this: unless phase's meaning here, which attack phase, is clearly defined, half the score difference cannot be explained to the team. The centre must state separately in the report what the phase angle represents and how it is measured; otherwise analysts cannot answer why a provider received a higher score.
In the report: "Half of the appraisal score comes from an alert's evidence strength, half from the attack phase it points to; the second component represents the reconnaissance/intrusion/exfiltration 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 a random ω=1.0 for S1, 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 S2's score of 1.000 as "one hundred per cent suitable"; the score is only a comparison against this set's own average, among these three strategies.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/cf-edas
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
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
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI: 10.1016/S0165-0114(86)80034-3
Keshavarz Ghorabaee, M., Zavadskas, E. K., Olfat, L., & Turskis, Z. (2015). Multi-criteria inventory classification using a new method of evaluation based on distance from average solution (EDAS). Informatica, 26(3), 435–451. DOI: 10.15388/Informatica.2015.57