Extension card · Linguistic
2-tuple linguistic VIKOR (Ju and Wang, 2013)
This is the form of VIKOR for situations where expert scores are chosen from a pre-declared term set, and where the aggregation result is preserved with its shift rather than rounded to a term.
Base method
VIKOR →
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?
Cells. In crisp VIKOR every cell is a single number. Here every cell consists of a term and a shift. An expert score is chosen from the term set; if the aggregation result falls between two terms, it is not rounded to the nearest one but kept with its shift. Criterion weights and the compromise coefficient v are crisp numbers; the method does not generate weights, it takes them from outside.
Scale equalisation. Crisp VIKOR finds every criterion's best and worst value, then scales each alternative's position between these two extremes. 2-tuple linguistic VIKOR does the same operation, but first converts every term, with its shift, into a single number. The best and worst values are chosen from these numbers according to the criterion's direction; the only difference from crisp VIKOR's linear normalisation is that the input has moved from a term to a number, while the form of the scaling stays the same.
Group utility and individual regret. The calculation of S and R uses exactly the same formula as crisp VIKOR: the sum of the weighted distances on every criterion gives S, the largest of them gives R. The only difference is that this distance is now computed on a value that comes from the term-to-number conversion.
Compromise index and result. The calculation of Q and the testing of the two conditions (acceptable advantage, acceptable stability) are the same as in crisp VIKOR. Defuzzification here loses no information: the transformation that converts a term into a number is one-to-one and reversible. This differs from extensions that work with triangular fuzzy numbers, where an interval is compressed to a single number during defuzzification; here a fine shift around a single term becomes a number.
DecisionMind fixes, for classical 2-tuple linguistic VIKOR, a nine-level term set and the term-to-number conversion; the compromise coefficient v defaults to 0.5.
How to Read the Output
The output, as in crisp VIKOR, is three columns: S, R and Q; it is read the same way. The real result is not the Q ranking but the decision given by the two conditions. The difference is here: the distances that build S and R now come from levels within the term set, so the question "how many levels of term change does the ranking rest on" is the sensitivity test here.
Thus instead of writing:
"According to 2-tuple linguistic VIKOR, this is the best alternative"
the report should read:
"With these weights and v = 0.5, this alternative is the single compromise solution; it is first in both S and R, and the gap to the second alternative exceeds the acceptance threshold"
is correct; if one of the conditions is not met, the compromise set is reported as it stands.
When to Prefer This over the Base Method
Use this method when experts assess a criterion with a verbal term, and when the aggregation result needs to be preserved without rounding to a term, particularly in a compromise setting where several parties' interests conflict and the alternative that "attracts the least objection" is being sought.
It should not be confused with the other member of the same family. If an expert gives not a single term but several terms with probabilities, probabilistic linguistic VIKOR is used; there the expected value summarises a distribution, whereas here there is a single term and its shift. If the expert has an "approximately this number" in mind, fuzzy VIKOR is appropriate.
The case for staying with the base method is unchanged: if a criterion is measured, it stays measured. Crisp VIKOR's exit conditions (no compromise acceptable at all on one criterion, criteria strongly linked to one another) apply here too.
Mistakes Specific to This Extension
Skipping the two conditions and reporting only the alternative with the smallest Q as the winner. VIKOR's defining feature is the compromise set, not a single winner; this does not change for linguistic data either.
Dropping the shift and rounding to the nearest term. Dropping the shift before computing S and R defeats the method's property of not losing information and hides how robust the ranking is.
Reversing the best and worst term on cost criteria. On a "lower is better" criterion, the term at the lowest level is the best one; reversing this makes an expensive or risky alternative look ideal.
Choosing the coefficient v without justification or pushing it to an extreme. At 0 or 1 the method stops being a compromise method; which value was chosen and why must be stated in the report.
The governing principle is this:
2-tuple linguistic VIKOR's result is not a ranking but a conditional compromise proposal; any application that erases the shift when converting a term to a number, or that skips the two conditions, is misusing the method.
Cases
The first case is DecisionMind's validation example. In the manifest, this 3x3 table is recorded as a synthetic fixture with no page reference; the figures have been independently recomputed with the VIKOR formulas. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Choosing among three consultancy bids
A company will choose among three consultancy firms' bids. Criteria: the firm's sector experience, the competence of the proposed team, and the project fee; the last is a "lower is better" criterion. The assessment board scores each firm with a word chosen from the term set; the term set has nine levels: very low, low, quite low, slightly low, medium, slightly high, quite high, high, very high.
| Firm | Sector experience | Team competence | Project fee |
|---|---|---|---|
| A1 | quite high, slight downward shift | medium | slightly high, slight downward shift |
| A2 | quite high, slight upward shift | slightly high, slight downward shift | slightly low, slight upward shift |
| A3 | slightly high, slight downward shift | quite high, slight downward shift | medium |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method converts every term, with its shift, into a number, finds each criterion's best and worst value, computes the weighted group utility (S) and individual regret (R), and combines the two with v = 0.5 to produce Q.
| Firm | S | R | Q |
|---|---|---|---|
| A2 | 0.175 | 0.175 | 0.000 |
| A3 | 0.525 | 0.400 | 0.780 |
| A1 | 0.800 | 0.350 | 0.889 |
The result reads as follows. A2 is both closest to the ideal overall and, on its single weakest criterion, less weak than the others. Neither the side that cares about the fee nor the side that cares about experience can raise a serious objection to it. Both conditions are met: the Q gap between A2 and A3 is 0.780, and the acceptance threshold for three alternatives is 0.5; A2 is also first in both S and R. The single compromise solution is A2.
The company's hesitation is this: if A3's team competence rose by one level, from quite high to high, would the gap between A2 and A3 narrow? When computed, A3's Q falls from 0.780 to 0.712, but A2 still stays first in both S and R, and the gap remains above the acceptance threshold. Even if the compromise coefficient v were pulled to 0.3 or to 0.7, the result would not change, because A2's lead in S and R is wide. The report can state that this result is robust, that a single one-level change does not upset the compromise.
In the report: "Board scores were chosen from the term set, and the aggregation result was preserved with its shift. A2 is the single compromise solution; it is first in both S and R, and this result is robust against single one-level changes in scores."
Source: DecisionMind's L2T-VIKOR validation example. The term set and the consultancy scenario were constructed for this card; the figures were taken from the manifest's synthetic fixture and independently recomputed with the VIKOR formulas.
2. Livestock farming: A compromise choice among three feed suppliers
A cattle-farming operation will choose among three feed suppliers. Criteria: the feed's nutritional value, the supplier's delivery regularity and price; the last is a "lower is better" criterion. The farm owner and the veterinarian each assess every supplier with a word chosen from the term set, leaving the compromise coefficient at 0.5.
The method computes every supplier's total distance (S) and its distance on the single worst criterion (R). Say the supplier with the highest nutritional value also turned out the most expensive; another supplier, with medium delivery regularity, was balanced on price and regularity but lagged on nutritional value. The Q index gave the two suppliers close scores, and the acceptable-advantage condition was not met; the compromise set consisted of two suppliers.
The operation's hesitation is this: if the veterinarian says "there can be no compromise on nutritional value," VIKOR does not fully accommodate this; the regret measure makes weakness visible but does not eliminate it. In this case, a nutritional-value threshold should be applied first, and the remaining suppliers ranked with this method.
In the report: "Supplier scores were chosen from the term set and aggregated. The acceptable-advantage condition was not met, and the compromise set consists of two suppliers; applying a lower bound for nutritional value could narrow the set to a single supplier."
3. What Not to Do
Had the project-fee criterion in the illustrative example been marked "higher is better," the most expensive bid would have been treated as ideal and the ranking would have become meaningless. The second error is finding A2's Q at 0 and simply writing "A2 won" without showing the S and R columns; the board needs to see why A2 came out ahead. The third error is choosing the compromise coefficient at 0 or 1 and presenting this as a compromise solution; at these extremes the method looks only at regret or only at the sum.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/l2t-vikor
Ju, Y., & Wang, A. (2013). Extension of VIKOR method for multi-criteria group decision making problem with linguistic information. Applied Mathematical Modelling, 37(5), 3112–3125. DOI: 10.1016/j.apm.2012.07.035
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
Opricovic, S. (1998). Multicriteria Optimization of Civil Engineering Systems (Višekriterijumska optimizacija sistema u građevinarstvu). PhD thesis, University of Belgrade, Faculty of Civil Engineering. (no DOI)
Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning—I. Information Sciences, 8(3), 199–249. DOI: 10.1016/0020-0255(75)90036-5