Extension card · Z-Number
Z-number COPRAS
This is the form of COPRAS for situations where every criterion value is given together with how far that value can be trusted. The output is a relative-significance value that comes from combining the benefit and cost totals.
Base method
COPRAS →
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 decision logic does not.
Cells. In crisp COPRAS every cell is a single number. Here every cell is a Z-number: Z = (A, B). A is a triangular fuzzy number stating the criterion value. B is a separate triangular fuzzy number stating how far that value can be trusted.
Scale equalisation. DecisionMind reduces the Z-number to a single crisp number before the linear sum normalisation. This reduction follows the method of Kang and colleagues (2012). A's membership curve is scaled by a coefficient drawn from B's centroid, and the centroid of this scaled curve is then taken. Classical COPRAS's own division-by-column-total step and weighting run unchanged on the crisp values obtained this way.
This reduction has a verified consequence. In the Kang transformation, B's contribution only scales the height of A's membership curve; it does not shift the position of the centroid. The crisp value obtained is therefore equal only to A's centroid. This has been verified by running DecisionMind's Z-COPRAS engine: in the illustrative example below, changing B drastically leaves the relative-significance values exactly the same.
The result and defuzzification. There is a second difference from base COPRAS here. Once base COPRAS combines the benefit and cost totals, it converts the result into a percentage relative to the best alternative; the best alternative always comes out at 100. DecisionMind's current Z-COPRAS engine does not yet apply this final step. The output is the raw relative-significance value (Q); it is not a percentage between 0 and 100. This has been verified by reading the engine's source code; the code notes this point explicitly.
DecisionMind holds the Kang transformation, classical COPRAS's column-total normalisation and weighting fixed in this extension. The percentage-conversion step is not applied in the current engine.
How to Read the Output
The output is a summary of the relative balance between the alternatives' benefit and cost totals; the highest value shows the best alternative. But unlike base COPRAS, this value is not a percentage relative to the best. It is a raw relative-significance figure and cannot be compared with a Z-COPRAS value from a different analysis.
The difference also sits on the reliability side. The purpose of the Z-number data structure is to carry trust in the source into the decision. But in the current engine, this trust is eliminated before scale equalisation.
Thus instead of writing:
"According to Z-COPRAS, A2 is the best alternative at 100 per cent"
the report should read:
"With the weights given, A2 has the highest relative-significance value (0.370); this value is not a percentage based on 100 as in base COPRAS, and the degree of reliability is not reflected in the result by the current engine"
When to Prefer This over the Base Method
If your data is genuinely Z-number in structure, this extension is the formally correct choice. But the result should not be expected to read as "a percentage relative to the best," the way base COPRAS does; the current engine does not yet apply this final step. This limit, and the fact that reliability is not reflected in the result, should be stated before the analysis begins.
The exit point is the same as for crisp COPRAS: if no compromise is acceptable on one criterion, this extension is compensatory too; a negative value in your data must be transformed first.
Mistakes Specific to This Extension
Mistaking the relative-significance value for a percentage. In the current engine, the output has not been converted into a percentage based on 100 relative to the best; it is a raw relative-significance value. Reading it as "A2 is 100 per cent suitable" is wrong.
Assuming reliability (B) will change the result. In the current engine, B is eliminated before scale equalisation. This is not a software fault; it is a mathematical consequence of the Kang transformation, and it should be stated in the report.
Feeding the table without transforming it despite a negative value. Division by the column total gives a meaningless result with negative values; this is a constraint that also holds for base COPRAS.
The governing principle is this:
Z-number COPRAS exists to record the criterion value and the trust placed in that value separately. But in the current engine reliability makes no contribution to the result, and the output has not yet been converted into base COPRAS's percentage form; both points must be stated plainly in the report.
Cases
The first case is DecisionMind's validation example, built from synthetic 3x3 fixed data. The second case is an illustrative construction.
1. Illustrative example: Scoring three alternatives on three criteria as Z-numbers (DecisionMind validation example)
Three alternatives are scored as Z-numbers on three criteria. Every cell gives the A (value) triangle; reliability B is the same across all three criteria and all three alternatives: (0.70; 0.80; 0.90).
| Alternative | C1 (A) | C2 (A) | C3 (cost, A) |
|---|---|---|---|
| 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 |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method reduces every cell's Z-number to a crisp number by the Kang transformation, divides by the column total to obtain a share, and multiplies by the weights. It sums the weighted values on C1 and C2 (benefit) into a benefit total for each alternative, keeps the weighted value on C3 (cost) as a separate cost total, and combines the two totals to build the relative-significance value.
| Alternative | Relative-significance value (Q) | Rank |
|---|---|---|
| A2 | 0.370 | 1 |
| A3 | 0.331 | 2 |
| A1 | 0.298 | 3 |
The result reads as follows. A2 holds the highest value on C1, the most heavily weighted criterion, and the lowest value on C3 (cost); it trails A3 only on C2. These two advantages make A2 first. A1 is last, because it is the weakest alternative on two of the three criteria (C1 and C3).
The decision's hesitation is this. When the weights are changed to C1=0.20, C2=0.55, C3=0.25, the ranking does not change, but the gap between A2 (0.361) and A3 (0.352) narrows to 0.009. This shows how sensitive the ranking between A2 and A3 is to the relative weight of C1 against C2.
Reliability B has no effect at all on these results. Even if B is lowered, say, to (0.01; 0.02; 0.03), the relative-significance values come out exactly the same under both weighting scenarios. This has been verified by running the kernel.
In the report: "With weights C1=0.40, C2=0.35, C3=0.25, A2 has the highest relative-significance value (0.370); this value is not a percentage based on 100. As the weight of C2 is brought closer to that of C1, the gap between A2 and A3 narrows to 0.009. The degree of reliability is not reflected in the result by the current engine."
Source: DecisionMind's Z-COPRAS validation example; built as synthetic 3x3 fixed data, following the Z-number framework of Zadeh (2011) and Kang et al. (2012). The relative-significance values and the sensitivity scenario have been independently recomputed by this card's author using the same algorithm, and match the manifest's expected values exactly (tolerance 1e-9).
2. Fisheries: Choosing among three proposals for a cooperative's cold-storage facility investment
A fishing cooperative will choose one of three engineering proposals for a new cold-storage facility. The criteria are storage capacity, installation cost and energy efficiency. Capacity and efficiency are "higher is better," cost is "lower is better." Energy efficiency is entered as a Z-number because it rests on an estimate from each proposal's own technical file. A is the estimated efficiency value; B is whether that estimate rests on an independent test report or only on the manufacturer's own declaration.
The method reduces the three proposals' Z-numbers to crisp numbers by the Kang transformation and applies classical COPRAS. Suppose the result places first the proposal with the highest efficiency estimate, where that estimate rests only on the manufacturer's own declaration.
The cooperative's hesitation is this. If this proposal's efficiency declaration has not passed independent testing, it should be known that the current engine does not reflect this gap in the result at all. The cooperative should require an independent performance test before signing the contract.
In the report: "With the weights given, the proposal with the highest efficiency estimate has the highest relative-significance value. This proposal's efficiency declaration has not passed an independent test, and the current calculation does not reflect this gap; an independent performance test is recommended before the contract is signed."
3. What Not to Do
In the illustrative example, sharply lowering A2's reliability on the C1 criterion and expecting "A2's value should now fall" is wrong: the value does not change at all, because the current engine does not read B. The second error is reporting A2's value of 0.370 as "37 per cent suitable" or "100 per cent of the best"; the current engine does not perform this percentage conversion. The third error is feeding a table into COPRAS without transforming it while it contains a negative value; division by the column total produces a meaningless result.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/z-copras
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)
Zavadskas, E. K., & Kaklauskas, A. (1996). Determination of an efficient contractor by using the new method of multicriteria assessment. In International Symposium for the Organization and Management of Construction: Shaping Theory and Practice, Vol. 2: Managing the Construction Project and Managing Risk (CIB W65), 94–104. (no DOI)
Zavadskas, E. K., & Turskis, Z. (2011). Multiple criteria decision making (MCDM) methods in economics: An overview. Technological and Economic Development of Economy, 17(2), 397–427. DOI: 10.3846/20294913.2011.593291