Extension card · Z-Number
Z-fuzzy LMAW (Puška, Božanić, Nedeljković and Janošević, 2022)
Z-fuzzy LMAW is the form of LMAW that carries the priority experts give to criteria not only as a fuzzy term, but together with a separate component stating how much that term is trusted. Its output is not a ranking but a criterion weight vector.
Base method
LMAW →
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 logarithmic-compression logic does not.
Cells. In F-LMAW every cell is a single priority triangle. Here every cell is a pair of terms: A, the criterion's priority (from a nine-step scale such as "Absolutely High"); B, how reliable that priority is (from a separate, five-step reliability scale such as "Very High"). For instance, an "E, VH" cell states that the priority is "Equal" (2.5; 3; 3.5) and confidence in that assessment is "Very High" (0.8; 1; 1).
Reducing the Z-number to a fuzzy number. Crisp LMAW has no such step; neither does F-LMAW, because there a single priority triangle already exists. Here the reliability triangle's mean is taken, its square root computed, and the result multiplied into the priority triangle's three components. This is the one non-reversible step that embeds reliability into the size of the priority: where reliability is low the priority shrinks, where reliability is high the priority stays almost as it was.
Solving for the weights. Just as in F-LMAW, this reduced priority triangle is ratioed against a fixed absolute anti-ideal point (0.5; 0.5; 0.5), its logarithm taken, divided by the product of the criteria's middle values, and, where more than one expert is involved, combined with a Bonferroni mean.
Defuzzification. As in F-LMAW, the weights are first found as triangular fuzzy numbers, then reduced to a single number per criterion by the mean of the three components, and the total renormalised to sum to 1 again.
DecisionMind fixes, in classical Z-fuzzy LMAW, the anti-ideal point (0.5; 0.5; 0.5) and the embedding of reliability into the priority triangle by a square root; the published article does not give these two figures (the anti-ideal point and the Bonferroni p, q) explicitly, and DecisionMind documents them as an open assumption.
How to Read the Output
As in F-LMAW, the weight shows a criterion's relative importance to the others within this comparison set; the weights always sum to 1. It is not a ranking.
The difference lies here. Two criteria given the same priority term can receive different weights if their reliability differs. This is not possible in F-LMAW, where the priority term alone determines the weight. Here, even where a criterion's priority is high, its weight can come out smaller than expected if confidence in that priority is low.
Thus instead of writing:
"The Z-fuzzy LMAW weight reflects, one to one, the priority the expert gave to the criteria"
the report should read:
"This weight is the defuzzified form of the priority the expert gave, together with the confidence placed in that priority; the same priority term can correspond to a smaller weight where confidence is low"
When to Prefer This over the Base Method
This extension is used where the priority experts give to criteria is stated not only as a verbal term but together with how much that term is trusted. The typical case is a group decision holding together assessments from experts of differing experience, or from sources of differing age or quality. Where every expert gives assessments of the same reliability, the reliability component is identical in every cell and carries no discriminating information; in that case F-LMAW (the A component alone) is sufficient.
The exit condition of LMAW's crisp form does not apply here, because the nature of the task has changed: here no alternative is ranked, a criterion is weighted. The warning on the Z-number data-type card also applies here: reliability must be derived from the source itself (the expert's experience, the age of the data), not from the width of the priority.
Mistakes Specific to This Extension
Deriving the reliability component from the width of the priority. The general mistake noted on the Z-number data-type card applies here too: a wide priority triangle does not mean "low confidence"; the two are separate questions.
Writing the same reliability term into every cell and using Z-fuzzy LMAW regardless. If reliability is identical everywhere it carries no discriminating information; F-LMAW is sufficient, and asking for the extra component is unnecessary.
Accepting a term not defined on the scale, such as "EH", without stating its source. The source article's own table uses this term in one place without defining it; DecisionMind maps it to the term "AH" only in transcribing the source case, and does so with an explicit flag. This substitution must not be made in any other analysis.
Claiming that the anti-ideal point and the Bonferroni parameters come from the article. The source article does not publish these two figures; the values DecisionMind uses, (0.5; 0.5; 0.5) and p=q=1, are an open implementation assumption, not the article's own figures.
The governing principle is this:
Z-fuzzy LMAW weights carry the priority and the confidence placed in that priority together; if the confidence component does not come from a genuine source, or is the same in every cell, this extension adds nothing beyond extra complexity.
Cases
The first case is Puška and colleagues' (2022) green-supplier-selection case in the agricultural sector, and its inputs are taken from the article's own Table 3. The second case is an illustrative construction.
1. Agriculture: Weighting ten criteria for green supplier selection (Puška, Božanić, Nedeljković and Janošević, 2022)
An agricultural business operating in Bijeljina (Bosnia and Herzegovina) will weight ten criteria before moving on to green-supplier selection: price, quality, delivery, capacity, distance, environmental management system, pollution control, environmentally friendly product, resource consumption and green image. Three experts each gave, separately, a priority term for every criterion and how much confidence they placed in it; for instance, the first expert called the price criterion "Equal, Very High" (priority Equal, confidence Very High).
The method takes the square root of the mean of each expert's reliability triangle and applies it to the priority, ratios the resulting triangle against the anti-ideal point, takes its logarithm, combines the three experts' results with a Bonferroni mean, and defuzzifies and normalises with the graded mean.
| Criterion | Price | Quality | Delivery | Capacity | Distance | Environmental management | Pollution control | Environmentally friendly product | Resource consumption | Green image |
|---|---|---|---|---|---|---|---|---|---|---|
| Weight | 0.0982 | 0.0988 | 0.0888 | 0.1093 | 0.0972 | 0.1223 | 0.1223 | 0.0944 | 0.0836 | 0.0850 |
The result reads as follows. The environmental management system and pollution control receive the highest, and almost equal, weight (0.1223); delivery and resource consumption carry the lowest weights. The weights sit fairly close to one another; the gap between the highest (0.1223) and the lowest (0.0836) is a small interval of about 0.039.
The business has one hesitation: how determining are the reliability grades the three experts gave? To test this, the reliability terms in every cell (in the original data ranging only between "High" and "Very High") were swapped with one another, that is, every "High" was made "Very High" and every "Very High" was made "High". The largest change appears in the green-image criterion: its weight rises from 0.0850 to 0.0898, and, in the ranking, the delivery and distance criteria (fourth and fifth place) swap positions. A more striking test is pulling every reliability term down to a single medium level ("Medium"): in this case the environmental management system (from 0.1223 to 0.1261) overtakes pollution control (from 0.1223 to 0.1261, but with a smaller increase) and rises to first place, and the most heavily weighted criterion changes. This shows that the reliability component really is read, and measurably affects the result, even the most heavily weighted criterion; the problem seen in some other members of the Z-number family (Z-TOPSIS, Z-VIKOR, Z-EDAS, Z-COPRAS, Z-WASPAS, Z-MARCOS), where "the reliability component is never read and does not change the result," does not occur here.
In the report: "The ten criteria's weights, ranging between 0.0836 and 0.1223, show a distribution close to one another; pollution control and the environmental management system are the most heavily weighted criteria. The reliability grades the experts gave measurably affect the weights; when all confidence grades are pulled down to the 'Medium' level, even the most heavily weighted criterion changes, so how reliability was determined must be explained in the report."
Source: Puška, Božanić, Nedeljković and Janošević (2022), Table 3 (the three experts' ten-criterion priority-reliability matrix). The weights were independently recomputed by this card's author following the source article's own steps, and coincide with DecisionMind's manifest closed-form result to a tolerance of 1e-9. The article's own printed intermediate Tables 4-5 values are published for only five criteria; the numerical anti-ideal point and the Bonferroni parameters are likewise absent from the article. These two gaps are documented by DecisionMind as an open assumption in the manifest record.
2. Freight: Weighting the criteria for an e-commerce company's choice of freight partner
An e-commerce company will weight three criteria before choosing a freight partner: delivery speed, damage rate, unit shipping cost. Two assessors gave their views: one the company's ten-year operations manager, the other a newly hired analyst. Both gave delivery speed a priority of "Absolutely High", but the operations manager places "Very High" confidence in this assessment, while the analyst, not yet having field experience, places only "Medium" confidence in her own assessment.
The method combines the two assessors' reliability-weighted priorities with a Bonferroni mean, ratios them against the anti-ideal point, and normalises. Suppose delivery speed again receives the highest weight, but that, had there been no confidence difference between the two assessors (had both given "Very High" confidence), this weight would have come out a little higher.
The company's hesitation is this. The analyst's lower confidence causes her view to carry proportionally less weight; this does not mean the analyst's view is "wrong", only that its source has less experience. The company must explain in the report how these confidence grades were determined (years of experience, past accuracy); otherwise the confidence grades look arbitrary.
In the report: "Delivery speed has received the highest weight through the combination of the two assessors' priority and confidence grades; the analyst's lower confidence grade stems from the difference in experience, and this justification is stated in the report."
3. What Not to Do
The first error is using a term not defined on the scale, such as "EH", in another analysis without stating its source; this term is accepted only in transcribing the source article's Table 3, and only with an explicit note that it will be treated as "AH". The second error is estimating the confidence grade from the width of the priority triangle; in the illustrative freight example, the analyst's low confidence comes from her experience, not from the width of her priority triangle. The third error is claiming that the anti-ideal point and the Bonferroni parameters are published in the source article; these two are DecisionMind's open assumption.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/zf-lmaw
Puška, A., Božanić, D., Nedeljković, M., & Janošević, M. (2022). Green Supplier Selection in an Uncertain Environment in Agriculture Using a Hybrid MCDM Model: Z-Numbers–Fuzzy LMAW–Fuzzy CRADIS Model. Axioms, 11(9), 427. DOI: 10.3390/axioms11090427
Pamučar, D., Žižović, M., Biswas, S., & Božanić, D. (2021). A new logarithm methodology of additive weights (LMAW) for multi-criteria decision-making: Application in logistics. Facta Universitatis, Series: Mechanical Engineering, 19(3), 361–380. DOI: 10.22190/FUME210214031P
Zadeh, L. A. (2011). A note on Z-numbers. Information Sciences, 181(14), 2923–2932. DOI: 10.1016/j.ins.2011.02.022
Kang, B., Wei, D., Li, Y., & Deng, Y. (2012). A method of converting Z-number to classical fuzzy number. Journal of Information & Computational Science, 9(3), 703–709. (no DOI)