Extension card · Z-Number
Spherical Fuzzy Z-Number CRITIC (Niu, 2024)
This is the form of CRITIC for situations where criterion values are given as spherical fuzzy triples (support, rejection, hesitancy), and each of these three degrees is further accompanied by its own reliability. The output is again a weight vector.
Base method
CRITIC →
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; CRITIC's "spread × distinctiveness" logic does not.
Cells. In crisp CRITIC every cell is a single number. Here every cell is six numbers: a support (ε), rejection (ν) and hesitancy (∂) degree for a judgement, under the spherical fuzzy constraint (the sum of the squares does not exceed 1); each of these three degrees is further accompanied by its own reliability (τ_ε, τ_ν, τ_∂). The method supports group decisions: several experts' matrices can be merged, with expert weights, into a single collective matrix. The output is still a weight, not a ranking; that is CRITIC's own identity, and it does not change here either.
Score function. Crisp CRITIC works directly with the numerical cell value. Here every cell is first reduced to a single score: the score is two times (support × reliability), plus a constant, minus (rejection × reliability), minus (hesitancy × reliability), the whole divided by three. This step converts the six-number spherical fuzzy Z-number cell into the single number that the rest of crisp CRITIC can process.
Scale equalisation. Crisp CRITIC uses min–max scaling, and so does this extension, but it now runs on the score computed in the previous step rather than on the raw cell value; for a cost criterion, the scaling direction is reversed.
Spread, correlation, information content, weight. These four steps are identical, term for term, to their definition in crisp CRITIC: standard deviation, correlation, "spread × conflict with the rest," and normalised weight. The difference is only that the input is now a scored and scaled number; the formulas themselves do not change.
DecisionMind fixes the score function and the min–max scaling for this extension.
How to Read the Output
The weight measures a criterion's share of distinctive information added to the table, exactly as in base CRITIC; it is read the same way (see the CRITIC card).
The difference is here. Reliability enters CRITIC's spread and correlation calculation through the score function, but whether this entry shows up in the weight is conditional. This has been verified by running DecisionMind's SFZN-CRITIC engine directly: if every cell of a criterion (across every alternative) carries the SAME reliability, min–max scaling cancels this shared factor out and the weight barely moves. But if reliability differs from alternative to alternative, that is, if one source is known to be more reliable for some alternatives than for others, the weight genuinely changes. In a small example, lowering a criterion's reliability for only one alternative moved that criterion's weight from 0.2809 to 0.2821, and the other two criteria's weights shifted by correspondingly small amounts.
Thus instead of writing:
"Reliability enters the calculation in spherical fuzzy Z-number CRITIC, so any difference in reliability changes the weight"
the report should read:
"Reliability only shows up in the weight when it differs from alternative to alternative; writing the same reliability into every alternative adds no discriminating information, because min–max scaling cancels this shared factor out"
When to Prefer This over the Base Method
This extension is suitable when the criterion list is long and contains repetitive indicators, the weights are wanted to derive objectively from the data, and the criterion values themselves come both as a support-rejection-hesitancy triple and with a separate reliability attached to each degree of that triple. If reliability is identical across all alternatives (for instance, if a single expert's every assessment shares the same source reliability), this extension's reliability component adds no discriminating information; the Spherical Fuzzy data-type card points to a simpler extension.
Crisp CRITIC's exit conditions apply unchanged: if the number of alternatives is small (fewer than five), correlation becomes unreliable; if criteria were chosen to be independent beforehand, the correlation measure adds nothing.
Mistakes Specific to This Extension
Dropping the §_j (standard deviation) factor from the Γ_j calculation. The formula as typeset in Niu's (2024) paper gives only Σ(1−Υ_jl); but Diakoulaki et al.'s (1995) standard CRITIC definition requires Γ_j = §_j · Σ(1−Υ_jl). Without the §_j factor, the paper's Table V weights cannot be reproduced.
Skipping direction-sensitive normalisation for cost criteria. If a cost criterion is normalised as though it were a benefit, the sign of the correlations flips and the repetition penalty is applied to the wrong criterion.
Dividing the standard deviation by the sample size (m−1) rather than the population size (m). Niu's formula divides by m; dividing by m−1 produces a different spread, and so a different weight.
Writing the same reliability into every cell and expecting it to change the weight. As shown above, reliability only shows up in the weight when it genuinely differs from alternative to alternative; a constant reliability is cancelled out under min–max scaling.
The governing principle is this:
Reliability enters the calculation here through CRITIC's score function, but whether it shows up in the weight depends on whether reliability genuinely differs across alternatives; writing a constant reliability wastes the effort spent collecting it.
Cases
The first case is anchored to Niu's (2024) case study on evaluating English-teacher performance. The second case is an illustrative construction.
1. Illustrative example (anchored to a source): Weighting four teachers on four criteria (Niu, 2024)
A school's English department wants to derive, from the data, the weight of four criteria (C1–C4: lesson planning, classroom management, student feedback, professional development) on which four teachers (Y1–Y4) are assessed. Every cell is a spherical fuzzy Z-number; in this example the assessments of three experts have already been merged into a single collective matrix, and the rejection and reliability degrees have been fixed (rejection at zero, reliability at one); only the support and hesitancy degrees vary. All four are "higher is better."
| Teacher | C1 (support; hesitancy) | C2 | C3 | C4 |
|---|---|---|---|---|
| Y1 | 0.517; 0.461 | 0.577; 0.451 | 0.511; 0.709 | 0.378; 0.467 |
| Y2 | 0.378; 0.439 | 0.476; 0.539 | 0.378; 0.423 | 0.366; 0.439 |
| Y3 | 0.697; 0.366 | 0.584; 0.468 | 0.533; 0.524 | 0.503; 0.342 |
| Y4 | 0.552; 0.370 | 0.570; 0.470 | 0.539; 0.341 | 0.322; 0.311 |
| Direction | higher is better | higher is better | higher is better | higher is better |
The method reduces every cell to a single score, applies direction-sensitive min–max scaling, computes spread and correlations, and derives information content and weight.
| Criterion | Weight |
|---|---|
| C2 | 0.323 |
| C3 | 0.321 |
| C4 | 0.208 |
| C1 | 0.148 |
The result reads as follows. C2 and C3 receive the highest weights, being the criteria that most distinguish the alternatives and least repeat one another; C1 ends up with the lowest weight because it overlaps more with the other criteria. The order (C2 > C3 > C4 > C1) matches the order published in Niu's (2024) paper.
An important disclosure is needed here: the weights above were obtained by independently recomputing this table with today's DecisionMind engine. The engine's own internal verification gate reports that these weights do not exactly match the numbers published in the paper (C1=0.18; C2=0.26; C3=0.36; C4=0.20); the order matches the paper, the magnitude of the weights does not (detail in the verification notes).
The department's hesitation: C1's low weight does not mean this criterion is unimportant; C1 shares its informational contribution because it correlates more highly with the other criteria. Also, because reliability is held constant (at one) across every cell here, how reliable the three experts' assessments were is not reflected in these weights at all; this is not an error but the natural consequence of the data carrying no reliability difference.
In the report: "The weights (C2=0.323; C3=0.321; C4=0.208; C1=0.148) match, in order, the order published by Niu (2024) (C2 > C3 > C4 > C1); the magnitude of the weights differs between DecisionMind's current engine output and the numbers published in the paper. Reliability was held constant across all cells in this dataset and so is not separately reflected in the weights."
Source: Niu (2024), IJACSA 15(3), Tables I–IV (input) and Table V (output). The order matches the paper; the magnitude of the weights does not. The sensitivity observation (that reliability shows up in the weight only when it differs from alternative to alternative) was independently computed by this card's author with a small synthetic example in Python.
2. Quality management: Weighting a manufacturing plant's supplier-audit criteria
A manufacturing plant wants to derive, from the data, the weight of five criteria (raw-material quality consistency, delivery timeliness, production-capacity flexibility, environmental-compliance certification, price stability) it uses to audit its suppliers. For each criterion, three auditors (a quality engineer, a procurement specialist, an independent audit firm) give support, rejection and hesitancy degrees separately; reliability is derived from how many times each auditor has inspected that supplier on site. Some suppliers have been inspected only once (low reliability), others are inspected regularly for years (high reliability).
The method computes the five criteria's scores, derives the spreads and correlations together with reliability, and produces the weights. Suppose delivery timeliness and raw-material quality consistency come out highly correlated and both receive a moderate weight; environmental-compliance certification, moving independently of the others, receives the highest weight.
The plant's hesitation is this: because reliability here genuinely differs from supplier to supplier (some inspected once, some many times), the weights are affected not only by the support-rejection-hesitancy degrees but also by this reliability difference. If the data for suppliers inspected only once is weak in reliability, the plant should not treat the weights as final without confirming these suppliers through an additional inspection round.
In the report: "The weights have been calculated from the three auditors' support-rejection-hesitancy assessments together with reliability degrees derived from inspection frequency; the low reliability of suppliers inspected only once has been reflected in the weights, and results for these suppliers should not be treated as final without an additional inspection."
3. What Not to Do
In the illustrative example, writing the same reliability (one) into every cell and then reporting "reliability was taken into account, so the result is more robust" is wrong: reliability carries no discriminating information here, because min–max scaling cancels out the shared factor. The second error is reporting the weights from Niu's paper (such as C1=0.18) as though they were today's engine output; the order matches but the magnitude does not. The third error is claiming to use "a more standard statistic" by dividing by m−1 instead of m in the standard deviation; Niu's formula divides by m, and this is a precondition for reproducing the paper's weights.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/sfzn-critic
Niu, J. (2024). Spherical Fuzzy Z-Numbers-based CRITIC CRADIAS and MARCOS Approaches for Evaluating English Teacher Performance. International Journal of Advanced Computer Science and Applications, 15(3). DOI: 10.14569/IJACSA.2024.01503115
Diakoulaki, D., Mavrotas, G., & Papayannakis, L. (1995). Determining objective weights in multiple criteria problems: The CRITIC method. Computers & Operations Research, 22(7), 763–770. DOI: 10.1016/0305-0548(94)00059-H
Kutlu Gündoğdu, F., & Kahraman, C. (2019). Spherical fuzzy sets and spherical fuzzy TOPSIS method. Journal of Intelligent & Fuzzy Systems, 36(1), 337–352. DOI: 10.3233/JIFS-181401
Zadeh, L. A. (2011). A note on Z-numbers. Information Sciences, 181(14), 2923–2932. DOI: 10.1016/j.ins.2011.02.022