Extension card · Linguistic
Probabilistic linguistic VIKOR (Li et al., 2021)
This is the form of VIKOR for situations where an expert gives several terms together with their probabilities. It reduces every term distribution to a single expected value, then ranks the result with a compromise proposal as usual.
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 is a list of terms together with their probabilities. An expert, or a panel of experts, judges a criterion with probabilities spread across several terms. Criterion weights and the compromise coefficient v remain crisp numbers; the method does not generate weights.
Scale equalisation. The method first reduces every term list to an expected value: each term's rank within the set is multiplied by its probability, and the products are summed. Once these expected values are obtained, crisp VIKOR's linear normalisation is applied: the best and worst expected value on every criterion are found, and alternatives are scaled by their position between these two extremes. This is the same two-stage structure as 2-tuple linguistic VIKOR; the difference is that the first stage summarises a probability distribution, not a single term plus a shift.
Group utility and individual regret. The computation of S and R uses the same formula as crisp VIKOR; as input it takes the normalised distances derived from the expected values.
Compromise index and result. The computation of Q and the testing of the two conditions are the same as in crisp VIKOR. Defuzzification differs here from 2-tuple linguistic VIKOR: reducing to an expected value discards the shape of the distribution, that is, the probability spread across the terms. Two different distributions can give the same expected value and lead to the same S, R and Q; whereas in the 2-tuple case, the passage from a term to a number is one-to-one and loses no information.
DecisionMind fixes, for classical probabilistic linguistic VIKOR, the reduction to expected value followed by crisp VIKOR's linear normalisation; the compromise coefficient v defaults to 0.5.
How to Read the Output
The output is three columns, as in crisp VIKOR, and is read the same way: the real result is not the Q ranking but the decision produced by the two conditions. The difference is this: the distances that build S and R come from the expected value of a probability distribution spread across terms. The report should therefore state not only Q, but also which distribution the expected value came from and how concentrated or spread out that distribution is.
Thus instead of writing:
"According to probabilistic linguistic VIKOR, this is the best alternative"
the report should read:
"With these weights and v = 0.5, this alternative is the sole compromise solution; this result rests on the expected value of the experts' term distributions, and the distribution itself should also be shown"
When to Prefer This over the Base Method
Use this extension when an expert, or a panel of experts, judges a criterion not with a single term but with a probability spread across several terms, and the decision sits in a compromise setting where the interests of more than one party conflict.
It must not be confused with the other member of the same family. If the expert selects a single term and the fusion result is to be kept as a shift rather than rounded to a term, 2-tuple linguistic VIKOR is used. If the expert gives a range such as "at least medium" and no probability, the hesitant linguistic structure is used.
The condition for staying with the base method is the same: if a criterion is measured, it remains measured. Crisp VIKOR's exit conditions apply here as well.
Mistakes Specific to This Extension
Skipping the two conditions and declaring the alternative with the smallest Q the winner. VIKOR's defining feature is the compromise set, not a single winner.
Leaving the term list's probabilities unreconciled. The probabilistic linguistic values on the same criterion must have their probabilities sum to one and be brought to equal-length term lists.
Interpreting the expected value on its own. Two alternatives reaching the same expected value through different distributions must remain visible to the decision.
Choosing the coefficient v without justification, or pulling it to an extreme. At 0 or at 1, the method stops being a compromise method.
The governing principle is this:
The result of probabilistic linguistic VIKOR is not a ranking but a conditional compromise proposal; this proposal rests on the expected value of the experts' term distributions, and a report that conceals the shape of the distribution is incomplete.
Cases
The first case is DecisionMind's validation example. In the manifest, this 3x3 table is recorded as a synthetic fixture without a page reference; the figures have been independently recomputed with the expected-value and VIKOR formulas. The second case is an illustrative fiction.
1. Illustrative example (DecisionMind's validation example): Choosing among three machine-maintenance contract proposals
A factory is to choose among three maintenance-contract proposals. The criteria are the contracting firm's response speed, the technical team's competence, and the contract fee; the last is a cost criterion. Every criterion is expressed as a term distribution drawn from the opinions of the factory's different shift supervisors; the term set has five steps: very low, low, medium, high, very high.
| Proposal | Response speed | Team competence | Contract fee |
|---|---|---|---|
| A1 | medium (0.2), high (0.8) | medium (1.0) | medium (0.6), high (0.4) |
| A2 | high (0.8), very high (0.2) | medium (0.6), high (0.4) | low (0.4), medium (0.6) |
| A3 | medium (0.6), high (0.4) | medium (0.2), high (0.8) | medium (1.0) |
| Direction | benefit | benefit | cost |
| Weight | 0.40 | 0.35 | 0.25 |
The method reduces every distribution to an expected value, finds the best and worst value on every criterion, computes the weighted group utility (S) and individual regret (R), and produces Q with v = 0.5.
| Proposal | 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 the criterion where it fares worst, less bad than the others. Both conditions are satisfied: the Q gap between A2 and A3 is 0.780, and with three alternatives the acceptance threshold is 0.5; A2 also ranks first on both S and R individually. The sole compromise solution is A2.
The factory's hesitation is this: if A3's contract-fee distribution had, at best, matched A2's, that is, if A3's fee also carried the distribution "low (0.4), medium (0.6)," by how much would A3's Q fall? Once computed, A3's Q falls from 0.780 to 0.680, but A2 still ranks first on both S and R, and the gap stays above the acceptance threshold. The report can state that this result holds even under the supplier's most optimistic price scenario.
In the report: "Proposal scores rest on the expected value of the shift supervisors' term distributions. A2 is the sole compromise solution, and this result is robust even against optimistic scenarios for the contract fee."
Source: DecisionMind's PL-VIKOR validation example. The term set and the maintenance-contract scenario were constructed for this card; the figures were taken from the manifest's synthetic fixture and independently recomputed with the expected-value and VIKOR formulas.
2. Fisheries: A compromise choice among three fishing-fleet investment proposals
A cooperative is to choose among three fishing-fleet investment proposals. The criteria are the fleet's catch capacity, fuel efficiency and investment cost; the last is a cost criterion. Cooperative members assess every fleet with a term distribution, because a single member's opinion is not considered sufficient.
The method computes every fleet's total distance (S) and its distance on the worst criterion (R). Suppose the fleet with the highest catch capacity also turns out to be the most expensive; another fleet, with high fuel efficiency, is balanced on cost and efficiency but lags on capacity. The Q index places the two close together, and the acceptable-advantage condition is not satisfied; the compromise set consists of two fleets.
The cooperative's hesitation is this: if the members' term distributions on catch capacity differ widely from one another, that is, some members say "very high" while others say "medium," the expected value conceals this disagreement. How spread out this distribution is should be shown to the members separately before a decision is made.
In the report: "Fleet scores rest on the expected value of the members' term distributions. The acceptable-advantage condition is not satisfied, and the compromise set consists of two fleets; the spread of opinion on catch capacity has been noted separately."
3. What Not to Do
In the illustrative example, if the contract-fee criterion had been marked as a benefit criterion, the most expensive proposal would be treated as ideal and the ranking would become meaningless. The second error is finding A2's Q to be 0 and writing only "A2 won," without showing the S and R columns or the distribution behind the expected value. The third error is allowing different shift supervisors to use term sets with a different number of steps; "high" in a five-step set does not correspond to the same expected value as "high" in a nine-step set.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pl-vikor
Li, L., Chen, Q., Li, X., & Gou, X. (2021). An improved PL-VIKOR model for risk evaluation of technological innovation projects with probabilistic linguistic term sets. International Journal of Fuzzy Systems, 23(2), 419–433. DOI: 10.1007/s40815-020-00971-1
Pang, Q., Wang, H., & Xu, Z. (2016). Probabilistic linguistic term sets in multi-attribute group decision making. Information Sciences, 369, 128–143. DOI: 10.1016/j.ins.2016.06.021
Opricovic, S. (1998). Multicriteria Optimization of Civil Engineering Systems (Višekriterijumska optimizacija sistema u građevinarstvu). PhD thesis, Faculty of Civil Engineering, University of Belgrade. (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