Extension card · Linguistic
2-tuple linguistic COPRAS
This is the form of COPRAS for situations where criterion values are given in words chosen from a pre-declared term set, and the calculation is carried out without converting these terms into numbers, through a lossless representation. The output is again a percentage relative to the best alternative and a ranking based on that percentage.
Base method
COPRAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Linguistic →
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 2-tuple: (term, translation value). The term is chosen from a pre-declared, ordered set; the translation value is a number between −0.5 and 0.5 that shows how far the result deviates from that term. Together these two parts preserve the exact position without rounding to an adjacent term above or below.
Scale equalisation. Classical COPRAS makes columns comparable, before weighting, by dividing each column by its own sum, because criteria are in different units (currency, days, points). 2-tuple linguistic COPRAS skips this step. Because every criterion comes from the same term set at the same granularity, they are already on a common scale, between 0 and the number of terms; a separate column-sum normalisation is not needed. I examined DecisionMind's L2T-COPRAS engine and confirmed that this step is indeed absent from the code.
Benefit and cost sums. This step works on the same logic as crisp COPRAS. Every 2-tuple is converted to term position plus translation (via the Δ⁻¹ operation) and multiplied by the weight; the values on benefit criteria are summed separately from those on cost criteria. This conversion is lossless; no rounding to a term is applied.
Result and defuzzification. The output, as in classical COPRAS, is a percentage relative to the best alternative; the best alternative again gets 100. This differs from Z-COPRAS, where the current engine does not yet apply this final step, while L2T-COPRAS's engine does. I confirmed this by reading the source code.
DecisionMind fixes, in this extension, the granularity of the term set and the Δ⁻¹ transformation; it does not apply column-sum normalisation.
How to Read the Output
The output is a benefit degree, as in crisp COPRAS: the best alternative gets 100, and the others receive a percentage relative to it. It is read the same way: this is not an absolute percentage of success, only a share relative to the best in this particular set.
The difference lies in the nature of the input. Because experts assess with words, the result can also fall between terms. The 2-tuple representation preserves this in-between position without rounding to a term. A small gap between two alternatives is therefore not a rounding error but a genuine distinction.
Thus instead of writing:
"Because experts assessed with words, the result is a rough estimate"
the report should read:
"Experts assessed with words, and the result has been preserved as term and translation value; A2, at 100, is closest to the ideal, and this ranking comes not from a rounding but from the exact position between terms"
When to Prefer This over the Base Method
Use this when your criteria are naturally assessed with words and converting these words into numbers would create an artificial precision. Converting a measured criterion (price, duration) into a term introduces uncertainty; the base method should be kept. If the table is mixed, that is, some criteria are measured and others linguistic, DecisionMind requires a single data type.
The exit point is the same as for crisp COPRAS: if no compromise is acceptable on one criterion, this extension too is fully compensatory.
Mistakes Specific to This Extension
Not declaring the term set before the analysis. How many terms there are, in what order, at what granularity: these must be the same for every expert and alternative, and must be stated in the report.
Rounding the result to a term and discarding the translation value. The 2-tuple's entire contribution lies in the translation value; discard it and the classical linguistic structure returns, losing the distinction between terms.
Using the term position directly as a number and averaging it. This smuggles back in, through the side door, the very scale assumption the linguistic structure was designed to avoid.
The governing principle is this:
2-tuple linguistic COPRAS exists to carry expert judgement losslessly as a term and a translation value. If the term set is not declared before the analysis, or the translation value is discarded, the method's one contribution is lost.
Cases
The first case is DecisionMind's validation example, built with a synthetic, fixed 3×3 dataset. The second case is an illustrative construction.
1. Illustrative example: Three alternatives scored on three criteria with 2-tuple linguistic values (DecisionMind validation example)
The term set has nine terms and is declared as follows: very low, low, fairly low, medium-low, medium, medium-high, fairly high, high, very high (s0 to s8). Three alternatives are scored on three criteria with terms and translation values from this set.
| Alternative | C1 | C2 | C3 (cost) |
|---|---|---|---|
| A1 | fairly high, −0.4 | medium, 0.0 | medium-high, −0.2 |
| A2 | fairly high, +0.4 | medium-high, −0.2 | medium-low, +0.2 |
| A3 | medium-high, −0.2 | fairly high, −0.4 | medium, 0.0 |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method converts every 2-tuple into term position plus translation and multiplies by the weight. It sums the C1 and C2 (benefit) values into a benefit sum for each alternative, keeps the C3 (cost) value separate as a cost sum, then combines these two sums and converts them into a percentage.
| Alternative | Benefit degree | Rank |
|---|---|---|
| A2 | 100.00 | 1 |
| A3 | 88.94 | 2 |
| A1 | 81.57 | 3 |
The result reads as follows. A2 holds the highest term position on C1, the heaviest-weighted criterion (fairly high, +0.4), and the lowest term position on C3, the cost criterion (medium-low); it trails A3 only on C2. These two advantages make A2 first.
The decision's hesitation is this. If the weights are changed to C1 = 0.10, C2 = 0.65, C3 = 0.25, the ranking reverses: A3 comes first at 100.00, A2 second at 97.71. A2's advantage depends on the weight given to C1. When C2's weight is increased, A3 moves ahead, because A3 holds the highest term position on C2.
The 2-tuple representation's sensitivity has also been confirmed. When A2's term on criterion C1 is pulled down one step (from fairly high to medium-high) while the translation value is left unchanged (+0.4), A2 still remains first. But because A2's absolute lead shrinks, the other two alternatives' share relative to the best rises: A3 climbs from 88.94 to 95.98, A1 from 81.57 to 88.03; the gap between A2 and A3 narrows from 11.06 points to 4.02 points. This has been confirmed by running the kernel.
In the report: "With weights C1 = 0.40, C2 = 0.35, C3 = 0.25, A2 has the highest benefit degree (100.00). If C2's weight is increased past C1's (C2 = 0.65, C1 = 0.10), A3 moves ahead (100.00 against 97.71)."
Source: DecisionMind's L2T-COPRAS validation example; a synthetic, fixed 3×3 dataset built according to Herrera and Martínez's (2000) 2-tuple representation. The benefit degrees and the sensitivity scenario were independently recomputed by this card's author with the same algorithm and matched the manifest's expected values exactly (tolerance 1e-9).
2. Waste management: A municipality's choice of recycling-facility operator
A municipality will choose one of three firms' bids to operate a sorted solid-waste recycling facility. The criteria are: operating experience, the proposed operating fee, and the level of environmental compliance. All but the fee are higher-is-better; the fee is lower-is-better. Because environmental compliance cannot be measured, it is assessed linguistically using a nine-term set; committee members each give a term to every firm, and the combined result is recorded as a 2-tuple.
The method converts the three bids' 2-tuple values into term position plus translation and applies classical COPRAS. Suppose the result places first the firm whose environmental compliance is a little above "fairly high" but whose fee is the highest.
The municipality's hesitation is this. Because the fee weight was kept low, the most expensive bid coming out ahead may spark debate at budget approval. The municipality should show separately, in the report, how many committee members gave the environmental-compliance term and how the terms were distributed.
In the report: "The bid with the highest benefit degree, driven by its environmental compliance level, comes first; this advantage stems largely from the low weight given to the fee, and the budget approval must be justified separately."
3. What Not to Do
The first error, in the illustrative example, is pulling A2's C1 term from "fairly high" to "medium-high" while also zeroing the translation value, that is, discarding the term entirely and re-rounding it: this erases the distinguishing information the translation value carries and shows the result as different from what it actually is. The second error is treating term positions as numbers, such as "medium = 4, fairly high = 6," and taking a direct average; this brings back the very scale assumption the linguistic structure was designed to avoid. The third error is reporting A2's degree of 100.00 as "perfect"; 100 only means it is the best among these three alternatives.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/l2t-copras
Herrera, F., & Martínez, L. (2000). A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8(6), 746–752. DOI: 10.1109/91.890332
Wei, G., Lin, R., Zhao, X., & Wang, H. (2010). Models for multiple attribute group decision making with 2-tuple linguistic assessment information. International Journal of Computational Intelligence Systems, 3(3), 315–324. DOI: 10.1080/18756891.2010.9727702
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., Kaklauskas, A., & Šarka, V. (1994). The new method of multicriteria complex proportional assessment of projects. Technological and Economic Development of Economy, 1(3), 131–139. (no DOI)