Extension card · Fuzzy
Complex Fuzzy MARCOS
The form of MARCOS for situations where the degree of support for, and rejection of, a judgement each carry two components, amplitude AND phase. The output is still a final utility degree and a rank.
Base method
MARCOS →
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 extended table, the ideal/anti-ideal ratio and the final utility degree logic do not.
Cells. In crisp MARCOS 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, carrying where the judgement sits on a second axis. Weights are taken from outside as crisp numbers; the method does not generate weights.
The extended table and the ideal/anti-ideal. In crisp MARCOS, the ideal and anti-ideal rows are built from the best and worst observed number. Here these two rows are built from the best and worst of each column's amplitude AND phase components separately; in a benefit column both amplitude and phase go to their highest, in a cost column both go to their lowest.
Aggregation and defuzzification. Crisp MARCOS multiplies and sums every row by the weights. Here every row is first combined into a single complex intuitionistic fuzzy value with the complex intuitionistic fuzzy weighted average (CIFWA). This value then reduces to a single number through a score function: s = ((r_μ−r_ν) + (ω_μ−ω_ν)/(2π)) / 2. Half the score comes from the amplitude difference, half from the phase difference. Everything after this is crisp MARCOS itself: the ratio to the ideal and anti-ideal (K+, K−), the two utility functions, and their combination into the final utility degree are built through the same steps.
Phase's contribution is real only when it is given independently. This card's author verified it by running the kernel directly: with the same amplitude pair (r_μ=0.85, r_ν=0.15) but different phases, A1's final utility degree varies between 0.4869 and 0.5999. But if phase is given proportional to amplitude (a rule such as ω=2π·r), the CIFWA aggregation carries this proportionality through and the score reduces algebraically to the intuitionistic fuzzy MARCOS score (r_μ−r_ν); the complex structure produces no difference at all. DecisionMind's own validation example (Case 1 below) uses this proportional phase; 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, the CIFWA aggregation operator, and the equal (half-and-half) weighting between amplitude and phase.
How to Read the Output
The final utility degree is read as in crisp MARCOS: a position relative to this set's own ideal and anti-ideal references, not a percentage or a probability (see the MARCOS card). The difference here is that half of this degree comes from amplitude, half from phase. Two alternatives having the same amplitude difference does not mean they will have the same final utility degree; 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:
"Complex fuzzy MARCOS put this alternative first because it found it most balanced"
the report should read:
"This final utility degree 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 degree 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: how strong it is (amplitude) must be known separately from which second axis it holds under (phase). If these two dimensions are not independent, that is, if phase is derived from amplitude, the complex structure adds nothing to intuitionistic fuzzy MARCOS 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 MARCOS's exit condition applies here too: where no compromise is acceptable on a criterion, this extension is compensatory as well.
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 CIFWA aggregation preserves this proportionality and the final utility degree comes out indistinguishable from intuitionistic fuzzy MARCOS; 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 half the final utility degree comes from phase, presenting a degree without explaining which second dimension phase measures leaves half of it unexplained.
The governing principle is this:
Complex fuzzy MARCOS'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 MARCOS, 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): Comparing three distribution-route plans
An e-commerce firm compares three distribution-route plans on three criteria: delivery-speed score (more is better), customer-satisfaction score (more is better), and shipping-cost score (less is better). Each plan'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.
| Route | Delivery speed (r_μ; r_ν) | Satisfaction (r_μ; r_ν) | Cost (r_μ; r_ν, less is better) |
|---|---|---|---|
| R1 | (0.85; 0.15) | (0.65; 0.15) | (0.75; 0.15) |
| R2 | (0.85; 0.05) | (0.75; 0.15) | (0.55; 0.15) |
| R3 | (0.75; 0.15) | (0.85; 0.15) | (0.65; 0.15) |
| Weight | 0.40 | 0.35 | 0.25 |
The method carries each cell's amplitude-phase pair into the extended table (with its ideal and anti-ideal rows), combines rows with CIFWA, ratios each row's score to the ideal and anti-ideal, and merges the two ratios into a single final utility degree.
| Route | Final utility degree | Rank |
|---|---|---|
| R2 | 0.7504 | 1 |
| R3 | 0.6361 | 2 |
| R1 | 0.5911 | 3 |
The result reads as follows. R2 reaches the ideal (0.85) exactly on delivery speed, the most heavily weighted criterion, and holds the lowest value on the cost criterion (less is better); only on satisfaction does it sit mid-table. R1 shows exactly the opposite profile and finishes last.
The decision's hesitation: raising satisfaction's weight from 0.35 to 0.65 while lowering delivery speed from 0.40 to 0.20 and cost from 0.25 to 0.15 changes the order. The same calculation puts R3 first at 0.7181 and R2 second at 0.6999. This shows that the superiority between R2 and R3 is sensitive to the weight on the satisfaction criterion.
In the report: "With the given weights (delivery speed 0.40, satisfaction 0.35, cost 0.25), R2 has the highest final utility degree (0.7504). Once satisfaction's weight rises to 0.65, R3 moves ahead (0.7181); the order should therefore be treated as sensitive to the weight on the satisfaction criterion. In this example phase was given proportional to amplitude, so the entire final degree comes from the amplitude difference."
Source: DecisionMind's CF-MARCOS validation example. The score function, the CIFWA operator and the MARCOS skeleton rest on the formulas in the manifest; the complex intuitionistic fuzzy structure rests on Alkouri and Salleh's (2012) definition, the underlying idea of the complex fuzzy set on Ramot and colleagues' (2002) work, and MARCOS itself on Stević and colleagues' (2020) work. The weight-trade-off scenario and phase-sensitivity figures were computed independently by this card's author by running the kernel directly.
2. Museology: A museum's choice of artefact-conservation workshop
A state museum will hand a fragile collection's conservation work to one of three specialist workshops. There are three criteria: conservation-quality score, turnaround time (less is better), and insurance/coverage score. The museum evaluates each workshop on two axes. The first is how strongly the workshop's success on past work is evidenced (amplitude). The second is which period of the collection this evidence relates to (phase), for instance early-period paper works or late-period textiles. These two axes are independent, because evidence of the same strength may have been gathered for a different period.
The method carries each workshop's amplitude-phase pair into the extended table, combines it with CIFWA, ratios it to the ideal and anti-ideal, and computes the final utility degree. Suppose two workshops share the same quality evidence, but one receives a different degree because its evidence is closer to the museum's currently prioritised period; this difference would not be visible from a quality comparison alone.
The museum's hesitation is this: unless phase's meaning here, which period the evidence relates to, is clearly defined, half the degree cannot be explained to the board. The museum must state separately in the report what the phase angle represents and how it was determined.
In the report: "Half of the final utility degree comes from the strength of past-work evidence, half from the collection period that evidence relates to; the second component represents the early/late-period distinction, and without this definition the degree 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 R1, makes the final degree's origin unclear and produces a result that cannot be reproduced. The second error is entering a phase angle in degrees directly and mistaking it for radians; the engine expects a value between 0 and 2π. The third error is reading R2's degree of 0.7504 as "75 per cent suitable"; this value is only a comparison against this set's own ideal/anti-ideal axis, among these three routes.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/cf-marcos
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
Stević, Ž., Pamučar, D., Puška, A., & Chatterjee, P. (2020). Sustainable supplier selection in healthcare industries using a new MCDM method: Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS). Computers & Industrial Engineering, 140, 106231. DOI: 10.1016/j.cie.2019.106231