Extension card · Z-Number
Z-number WASPAS (Jafarzadeh Ghoushchi et al., 2021)
This is the form of WASPAS for situations where the decision-matrix cells are not crisp numbers but Z-numbers, carrying a value together with how far that value is trusted. The logic of blending the sum with the product stays exactly the same; only the cells change.
Base method
WASPAS →
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?
Three things change; the logic of blending the sum with the product does not.
Cells. In crisp WASPAS every cell is a single number. Here every cell has two parts: A, the value itself (as a triangular fuzzy approximation); B, how far that value is trusted (again a triangular fuzzy number). A performance score an expert calls "roughly 0.7" is written together with a separate triangle showing how far that estimate is trusted.
Conversion to a crisp number. Before entering crisp WASPAS, every Z-number cell is converted into a single crisp number. This conversion is carried out by scaling the value triangle with the square root of the reliability triangle's centroid, and then taking the centroid of this scaled triangle. In this centroid-based conversion that DecisionMind uses, the scaling constant enlarges or shrinks only the membership height, not the position of the value. Because of this, once the centroid enters the calculation, this constant cancels out; the crisp result comes out equal only to the centroid of the value (A) triangle. This is an important finding, addressed further in the card's "Mistakes Specific to This Extension" section.
Scale equalisation, and the sum and product components. On the matrix once it has been converted to crisp numbers, exactly the same four steps run as in crisp WASPAS. Every column is scaled relative to its best value, the weighted sum (WSM) and weighted product (WPM) are calculated, and the two are then blended with λ.
DecisionMind holds this conversion rule, and the crisp WASPAS steps that follow it, fixed in classical Z-WASPAS; where λ is not specified, 0.5 is assumed.
How to Read the Output
The combined score says the same thing as in crisp WASPAS: the relative position within this alternative set, a summary of WSM and WPM blended with a given λ.
An important caveat is needed here. The promise of the Z-number is not to treat sources giving the same value at different reliability levels as equal. But in DecisionMind's current Z-WASPAS engine, the reliability (B) component never changes the crisp result at all, because the centroid-based conversion described above cancels B out. This has been independently tested and verified by this card's author, cell by cell. In the current engine's output, a reliability difference produces no effect worth reporting; the score depends only on the centroid of the value (A) triangle.
Thus instead of writing:
"This alternative's score has been found by also taking the reliability of the inputs into account"
the report should read:
"This score has been calculated from the centroid of the cells' value (A) component. Under the current conversion rule, the reliability (B) component does not change the crisp score, and no interpretation should therefore be built on a reliability difference"
When to Prefer This over the Base Method
If the assessments come from sources of differing reliability, and this difference is expected to show up in the report, the Z-number data type is logically appropriate. But because of the finding above, choosing Z-number WASPAS in DecisionMind for this purpose at present does not meet the expectation that a reliability difference will show up in the score. If a reliability difference genuinely needs to show up in the score, that difference should for now be carried into the criterion weight or into the value (A) itself. Where this is not possible, Fuzzy WASPAS (A alone, without any reliability claim) should be preferred.
The exit condition of crisp WASPAS applies here too. The data must contain no zero or negative value, the table must be fully filled, and this extension is not suitable if no compromise is acceptable on one criterion.
Mistakes Specific to This Extension
Assuming a difference in reliability (B) will change the score. As shown above, even if B is changed within the same cell from very low to very high, the crisp score does not change. A sentence such as "the reliability difference has affected the result" should not be written in the report; it would be an unverified claim.
Writing the same B value into every cell and treating this as the justification for using a Z-number. If B is the same everywhere, it already carries no discriminating information. Even entering different B values shows no effect in the current engine.
Confusing the width of the value (A) triangle with reliability. The width of A is the estimate's own uncertainty; B is the reliability of the source. This distinction is set out on the Z-number data-type card and applies here as well.
Leaving λ at its default of 0.5 without ever testing it. This is the same general mistake found in crisp WASPAS: unless the WSM and WPM components are tried separately, the choice of λ counts as untested.
The governing principle is this:
In Z-number WASPAS, the score comes from the centroid of the cells' value component. It has been verified that the reliability component does not change the score under DecisionMind's current conversion rule; the report must state this limit plainly.
Cases
The first case is DecisionMind's validation example. In the manifest, this three-alternative table does not come from the literature; it comes from a formula-faithful, hand-traceable synthetic construction. The second case is an illustrative construction.
1. Illustrative example: Three alternatives, three criteria (DecisionMind validation example)
Three alternatives are assessed on three criteria; the first two are "higher is better," the third is "lower is better." Every cell's value (A) is a triangle, and its reliability (B) is also a triangle; in this example, every cell's reliability is the same (0.7; 0.8; 0.9).
| Alternative | K1 (value) | K2 (value) | K3 (value, lower is better) |
|---|---|---|---|
| A1 | 0.65; 0.70; 0.75 | 0.45; 0.50; 0.55 | 0.55; 0.60; 0.65 |
| A2 | 0.75; 0.80; 0.85 | 0.55; 0.60; 0.65 | 0.35; 0.40; 0.45 |
| A3 | 0.55; 0.60; 0.65 | 0.65; 0.70; 0.75 | 0.45; 0.50; 0.55 |
| Weight | 0.40 | 0.35 | 0.25 |
The method first converts every cell to a crisp number; since reliability is the same across all cells in this example, the crisp value comes out equal to the midpoint of the value triangle. It then runs crisp WASPAS's four steps (scale equalisation, WSM, WPM, blending with λ=0.5).
| Alternative | Combined score | Rank |
|---|---|---|
| A2 | 0.9559 | 1 |
| A3 | 0.8528 | 2 |
| A1 | 0.7698 | 3 |
The result reads as follows. A2 holds the highest value on K1, the heaviest criterion, and carries the lowest value on K3 (cost); its advantage on these two criteria more than offsets its relative weakness on K2 and carries A2 into first place.
The board's first question is this: does the ranking change if the cells' reliability (B) is lowered? This has been tested independently: even if the reliability (0.7; 0.8; 0.9) of A2's K1 cell is pulled down to a very low level (0.05; 0.1; 0.15), the combined scores (0.7698; 0.9559; 0.8528) come out exactly the same. This is because, under DecisionMind's current conversion rule, reliability never affects the crisp value at all. When λ is varied too (tried from 0 to 1), the ranking never changes; the WSM and WPM components separately give the same order too.
In the report: "A2 has the highest combined score (0.9559), and this ranking is robust against the choice of λ. The cells' reliability (B) component does not affect the crisp score in DecisionMind's current Z-WASPAS engine; the result should therefore be interpreted based only on the value (A) component."
Source: DecisionMind's validation example for the Z-WASPAS engine; the matrix, weights and λ value were produced as a hand-traceable synthetic example, not taken from a paper. The scores and the reliability sensitivity test have been independently recomputed by this card's author using the DecisionMind engine.
2. Logistics: A courier company's choice of delivery-vehicle fleet
A courier company will add one of three vehicle models to its fleet for urban delivery. Three criteria apply: fuel-efficiency score, load capacity, and annual maintenance cost (this last one "lower is better"). The values are the field team's estimates, and how far each estimate is trusted has been noted separately: some estimates rest on past records for a long-used model, others on the vendor's own statement for a new model.
The method converts every cell to a crisp number and runs crisp WASPAS's four steps. Suppose the vehicle with the highest fuel efficiency also has the lowest maintenance cost, and comes out first.
The company's hesitation: the reliability of estimates resting on the vendor's own statement for the new model appears low, yet this low reliability does not affect the score in DecisionMind's current engine. The company should add, separately from the score, a note in the report on how far the vendor-statement estimates can be trusted.
In the report: "The vehicle with the highest fuel efficiency has the highest combined score. Some of this vehicle's inputs rest on the vendor's own statement and can be considered of low reliability; but since this reliability difference is not reflected in the score, the decision-maker is advised to assess this risk separately."
3. What Not to Do
In the illustrative example, showing very low reliability for A2's K1 cell and reporting "this value is unreliable, so its score should also come out low" is wrong; as shown above, reliability never changes the score at all. The second error is writing different-looking but identical reliability terms (say, "high" for all of them) into every cell and thinking this justifies the use of a Z-number; where there is no discriminating information, Fuzzy WASPAS is sufficient. The third error is reporting A2's score of 0.9559 as "96 per cent suitable"; the score only ranks these three alternatives against one another.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/z-waspas
Jafarzadeh Ghoushchi, S., Dorosti, S., Ab Rahman, M. N., Khakifirooz, M., & Fathi, M. (2021). Theory-Based Failure Modes and Effect Analysis for Medication Errors. Journal of Healthcare Engineering, 2021, 5533208, 1–14. DOI: 10.1155/2021/5533208
Zavadskas, E. K., Turskis, Z., Antuchevičienė, J., & Zakarevičius, A. (2012). Optimization of weighted aggregated sum product assessment. Elektronika ir Elektrotechnika, 122(6), 3–6. DOI: 10.5755/j01.eee.122.6.1810
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)