Extension card · Z-Number
Complex Fuzzy Z-Number MARCOS (Shahid et al., 2026)
The form of MARCOS for situations where criterion values are given both as an amplitude-phase pair and as a reliability degree attached to that pair. 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)
Z-Number →
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 MARCOS logic of carrying a closeness coefficient, derived from distance to the ideal and anti-ideal, into the extended table does not.
Cells. In crisp MARCOS every cell is a single number. Here every cell consists of four numbers: an amplitude (σ) and a phase ratio (τ), accompanied by a reliability amplitude (ϖ) and a reliability phase ratio (R). σ and τ carry the judgement itself; ϖ and R carry how reliable that judgement is, that is, the B component of the Z-number. All four lie between 0 and 1. Weights are taken from outside as crisp numbers; the method does not generate weights.
Ideal and anti-ideal. In crisp MARCOS the ideal and anti-ideal are built from the best and worst observed number. Here an ideal point is built, for every column, from the separate best of the four components (σ, τ, ϖ, R), and an anti-ideal point from their separate worst; in a benefit column all four go to their highest, in a cost column all four go to their lowest.
Distance and closeness coefficient. This is a step MARCOS generally omits. Every cell's distance to the ideal and to the anti-ideal is computed by multiplying value and reliability together (σ·ϖ and τ·R); this product ensures a low-reliability judgement enters the distance calculation with less weight. A closeness coefficient (the ratio of the distance to the anti-ideal against the sum) is built from the two distances, and this coefficient reduces every cell to a single crisp number. Crisp MARCOS's extended table is then built on these closeness coefficients rather than on the cells themselves.
Outcome and defuzzification. Defuzzification is an early step here: the four-number cell reduces to a closeness coefficient before it ever enters the extended table. What follows is crisp MARCOS itself: normalise, multiply by weight, ratio against the ideal and anti-ideal (Ψ+, Ψ−), combine the two utility functions, and compute the final utility degree (Γ).
Reliability's contribution is real only when it differs across alternatives. This card's author verified it by running the kernel directly: if a criterion's reliability (ϖ, R) is fixed at the SAME value across every alternative, whether 0.95 or 0.05, the closeness coefficient, and with it the final utility degree, comes out exactly identical; because the ideal and anti-ideal points are then also built from this same fixed reliability, and this constant factor cancels out in the distance ratio. By contrast, when only a single cell's reliability is changed, for instance when one alternative's reliability on one criterion alone is lowered, the final utility degree and the rank genuinely change; as the illustrative example below shows, this difference can be large enough to reverse the ranking.
DecisionMind fixes, for this extension, the distance formula, the closeness coefficient, and MARCOS's canonical final utility function.
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 (see the MARCOS card). The difference here is that this degree passes through a distance calculation carrying, separately for every criterion, both the judgement itself and its reliability. Two alternatives sharing the same judgement value (σ, τ) does not mean they will share the same final utility degree; a reliability difference between them comes into play as well.
Thus instead of writing:
"In complex fuzzy Z-number MARCOS, reliability is purely indicative and does not change the result"
the report should read:
"Reliability has no effect only when it is identical across alternatives; where different alternatives rest on sources of different reliability, this difference genuinely carries through to the final utility degree and the rank"
When to Prefer This over the Base Method
This extension is worth considering when a judgement's amplitude-phase pair and its reliability are each known separately, and this reliability genuinely differs across alternatives. The typical case is where judgements from different sources, some a verified measurement, some an expert estimate, are combined in the same table, and the source's reliability is meant to carry through into the decision. Where every cell's reliability is the same (all coming from the same source, at the same confidence), this extra dimension contributes nothing, and Complex Fuzzy MARCOS suffices. 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
Writing the same reliability into every cell and calling it "using a Z-number." If reliability never differs across alternatives, it cancels out algebraically in the distance ratio and has no effect on the result whatsoever; this has been verified by this card's author with the kernel.
Putting value and reliability through a squared difference separately. The distance formula uses the products (σ·ϖ) and (τ·R); treating σ and ϖ separately gives the wrong distance.
Forgetting that the closeness coefficient is an early defuzzification. The four-number cell reduces to a single crisp number before it ever enters the extended table; after this point, which component, the value or the reliability, is driving the result is no longer separately visible.
Accepting the weights from the paper as given. This extension's founding paper did not publish criterion weights explicitly; DecisionMind uses equal weights, and this is an assumption, not data.
The governing principle is this:
Reliability changes the final utility degree only when it genuinely differs across alternatives; writing the same reliability into every cell preserves the appearance of a Z-number structure but contributes nothing to the calculation.
Cases
The first case is anchored to Shahid, Ashraf and Chohan's (2026) augmented-reality decision-making case study. As the paper did not publish criterion weights, DecisionMind uses equal weights; when this card's author ran the engine independently, the ranking matched the paper's, but the magnitude of the final utility degrees did not match the paper's published figures, and this discrepancy is reported openly below. The second case is an illustrative construction.
1. Illustrative example (anchored to source): Comparing four augmented-reality platforms (Shahid et al., 2026)
The paper compares four augmented-reality (AR) platforms (U1–U4) on four criteria (Y1–Y4, all more is better); the paper does not name in detail what these four criteria are, only giving their values. Every cell is a complex fuzzy Z-number: an amplitude (σ), a phase ratio (τ), a reliability amplitude (ϖ) and a reliability phase ratio (R). Because weights are not specified in the paper, equal weights (0.25) are used.
| Platform | Y1 (σ; τ; ϖ; R) | Y2 | Y3 | Y4 |
|---|---|---|---|---|
| U1 | 0.3; 0.6; 0.3; 0.6 | 0.1; 0.5; 0.7; 0.2 | 0.1; 0.5; 0.2; 0.3 | 0.2; 0.5; 0.4; 0.2 |
| U2 | 0.2; 0.3; 0.2; 0.4 | 0.8; 0.7; 0.2; 0.9 | 0.5; 0.6; 0.6; 0.4 | 0.3; 0.6; 0.4; 0.2 |
| U3 | 0.6; 0.8; 0.1; 0.1 | 0.7; 0.1; 0.1; 0.7 | 0.3; 0.4; 0.8; 0.6 | 0.1; 0.2; 0.1; 0.7 |
| U4 | 0.2; 0.7; 0.5; 0.2 | 0.9; 0.6; 0.1; 0.4 | 0.3; 0.6; 0.4; 0.3 | 0.1; 0.2; 0.4; 0.9 |
The method computes each cell's distance to the ideal and anti-ideal through the value-times-reliability product, reduces it to a closeness coefficient, builds the extended table, normalises and weights it, and computes the final utility degree (Γ).
| Platform | Final utility degree (Γ) | Rank |
|---|---|---|
| U2 | 0.7630 | 1 |
| U1 | 0.4006 | 2 |
| U4 | 0.3077 | 3 |
| U3 | 0.2466 | 4 |
The result reads as follows. U2, though not the highest-weighted, holds a strong value-times-reliability product on criteria Y2 and Y3 and departs least from the ideal; it ranks first. U3 finishes last, weighed down by criteria where it holds either low reliability or a low value.
An important disclosure is needed here: the final utility degrees above (0.7630; 0.4006; 0.3077; 0.2466) were obtained by DecisionMind's engine independently recomputing this table today. The engine's ranking matches the order published in the paper (U2>U1>U4>U3), but the magnitudes published in the paper's Table 16 (U1=0.498; U2=0.638; U3=0.283; U4=0.455) do not match the figures the engine produces; the engine's own internal quality gate flags this discrepancy as "failed" (the 0.02 absolute tolerance was exceeded). This has been logged separately as a finding (see the approval notes).
The board's hesitation: U3's reliability on criterion Y1 (ϖ=0.1) was kept low; what happens if this reliability is raised from 0.1 to 0.8? This scenario was recomputed in Python using the engine's own algorithm. When only U3's reliability on Y1 (ϖ and R together) is raised from 0.1 to 0.8, its final utility degree rises from 0.2466 to 0.4627 and U3 moves from last place to second (new order U2>U3>U1>U4); by contrast, changing the reliability on all four criteria together, by the same factor, for instance setting all four to the same constant, does not change the result at all, because a constant reliability cancels out in the distance ratio.
In the report: "By the engine's current computation, U2 has the highest final utility degree (0.7630); this ranking matches the order published in the paper. However, the magnitude of the degrees does not match the paper's Table 16 figures, and this discrepancy must be reported separately. U3's last place does not come from a low value on criterion Y1 alone, but also from low reliability on that criterion; raising the reliability moves U3 up to second place."
Source: Shahid, A., Ashraf, S., & Chohan, M. S. (2026), Spectrum of Operational Research, 3(1), Table 1 (input) and Table 16 (output). The ranking matches the paper; the magnitude of the final utility degrees differs between DecisionMind's current engine output and the paper's published figures (detail in the approval notes). The reliability-sensitivity scenario was computed independently by this card's author.
2. Livestock farming: A dairy operation's choice of forage supplier
A large dairy operation will choose one of three companies as its roughage supplier. The criteria, all more is better, are nutritional-value score, delivery-reliability score, and price-stability score. The operation's veterinarian assesses each supplier's nutritional value both through laboratory analysis (high reliability) and through the supplier's own declaration (low reliability); these two sources are reflected separately into the complex fuzzy Z-number's value and reliability components.
The method computes each supplier's distance to the ideal and anti-ideal through the value-times-reliability product, builds the closeness coefficient, and gives the final utility degree. Suppose the supplier that appears to have the highest nutritional value rests this value solely on its own declaration (low reliability).
The operation's hesitation is this: if this supplier's nutritional value is confirmed by laboratory analysis, its reliability will rise and its lead in the ranking could firm up, but if it is not confirmed, the current lead rests on a single low-reliability declaration. The operation should verify this value with an independent laboratory analysis before signing the contract.
In the report: "The supplier with the highest nutritional-value score appears to lead in the current evaluation; however, this value rests on an unconfirmed declaration without laboratory backing, and independent analysis should be requested before the contract decision."
3. What Not to Do
The first error, in the illustrative example, is setting all four criteria's reliability to a single constant and reporting "reliability rose, so the result changed"; a uniform, constant reliability shift does not change the result at all, whereas the result does change when only a single alternative's reliability on a single criterion changes; the two are not the same thing. The second error is reporting the paper's Table 16 figures (such as U1=0.498) as though they were the engine's current output; the engine's ranking matches the paper but the magnitude of the degrees does not, and this discrepancy must not be concealed. The third error is assuming that value and reliability remain separately visible after reduction to the closeness coefficient; from that step onward the two have merged into a single crisp number.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/cfzn-marcos
Shahid, A., Ashraf, S., & Chohan, M. S. (2026). Complex Fuzzy MARCOS and WASPAS Approaches with Z-Numbers for Augmented Reality Decision Making. Spectrum of Operational Research, 3(1), 40–62. DOI: 10.31181/sor31202637
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
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
Zadeh, L. A. (2011). A note on Z-numbers. Information Sciences, 181(14), 2923–2932. DOI: 10.1016/j.ins.2011.02.022