Extension card · Linguistic
Probabilistic linguistic TODIM (Liu & Teng, 2017)
Probabilistic linguistic TODIM is the form of TODIM for situations where an expert judges a criterion not with a single word but with an opinion spread across several terms. It reduces the term distribution to an expected value and ranks alternatives with the same loss-aversion logic.
Base method
TODIM →
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?
Four things change; the losing side is still magnified by the loss-aversion coefficient, but in this extension both that coefficient and the reference criterion are held fixed.
Cells. In crisp TODIM every cell is a single number. Here every cell is a set of terms together with their probabilities. An expert may judge a criterion with an opinion spread across more than one term, such as "20 per cent low, 80 per cent high." The probabilities of the terms in a cell never exceed 1, and the term set (how many terms, in what order) stays the same across every criterion and alternative. Weights remain crisp numbers. DecisionMind does not support group decisions in this extension; every cell belongs to a single assessment.
Scale equalisation. Crisp TODIM derives a share by dividing every column by its own sum, because the raw data are in different units. There is no such division here. The method first reduces every term distribution to an expected value: each term's rank within the set is multiplied by that term's probability, and the products are summed. Because this expected value already sits on a scale common to every criterion, namely the term set's own ranking, crisp TODIM's step of dividing by the column sum is not needed here.
Comparison. In crisp TODIM the difference between two values is a direct subtraction. The same logic continues here, but the subtraction now runs between the expected values of two term distributions. Which alternative "wins" on a criterion is decided from the magnitude of these expected values and the criterion's direction; for a cost criterion the direction of the comparison is reversed.
Result. The global value is again a single number normalised between 0 and 1. The information carried by a term distribution disappears once it is reduced to an expected value; two different distributions can reach the same expected value and will not be distinguished in the global value.
DecisionMind fixes the expected-value reduction (Pang, Wang & Xu, 2016) in this extension. In base TODIM the user may choose the loss-aversion coefficient θ and the reference criterion. In this extension neither can be changed from the interface: θ is fixed at 1, and the reference criterion is always the criterion with the highest weight.
How to Read the Output
The global value is read as in crisp TODIM: the lowest overall dominance takes 0, the highest takes 1, and this holds only for the given alternative set. The difference is this: beneath this value now lies a probability distribution, and this distribution disappears once it is reduced to an expected value. Two alternatives reaching the same expected value through different distributions will not be distinguished in the global value.
Thus instead of writing:
"According to the probabilistic linguistic TODIM result, A2 comes first"
the report should read:
"The term distributions have been reduced to expected values, the reference criterion has been fixed to the criterion with the highest weight, and ranking was carried out with θ = 1; A2 has the highest global value"
When to Prefer This over the Base Method
Use this extension when a criterion's natural measure is a word and the expert, or the panel of experts, judges the criterion with an opinion spread across several terms rather than a single one. The typical situation is one where multiple assessors score the same criterion with different terms, and this spread is to be preserved rather than collapsed into a single average.
If the expert selects a single term and gives no probability, the classical linguistic structure is sufficient. If the expert gives a range such as "at least medium," the hesitant linguistic structure is used. If the criteria are measured, crisp TODIM remains the right choice; in DecisionMind the table must be of a single type, and measured and linguistic criteria cannot be written together. TODIM's exit condition applies unchanged: if no compromise is acceptable on a criterion, elimination is applied first.
Mistakes Specific to This Extension
Keeping the term set different from expert to expert. The number and order of terms must be identical across every expert and alternative; "high" in a five-term set does not correspond to the same expected value as "high" in a seven-term set.
Not checking that the probabilities sum correctly. The probabilities of the terms on the same criterion must sum to 1; otherwise the expected value stops being comparable.
Trying to change θ or the reference criterion. Neither can be changed from the interface in this extension. A report that compares a different θ or a different reference criterion is in fact presenting a scenario the engine did not compute.
The "more advanced" fallacy. Probabilistic linguistic TODIM does not produce a "more correct" ranking than classical linguistic TODIM; it only carries an expert opinion spread across several terms without losing that spread.
The governing principle is this:
Probabilistic linguistic TODIM reduces the expert's opinion, spread across terms, to an expected value; this reduction discards the shape of the distribution, and θ and the reference criterion are fixed in this extension, not changeable by the user.
Cases
The first case is DecisionMind's validation example: a hand-traceable table over three alternatives, three criteria and a five-term set. It is not the table from Liu and Teng's (2017) paper; it has been constructed synthetically, faithfully to the formulas. The second case is an illustrative fiction.
1. Illustrative example (DecisionMind's validation example): Scoring three candidates on three criteria with term distributions
Three alternatives are evaluated on three criteria, using a five-term set: very low, low, medium, high, very high. The first and second criteria are benefit criteria, the third is a cost criterion. Every cell carries an opinion the expert has spread across one or more terms.
| Alternative | Criterion 1 | Criterion 2 | Criterion 3 |
|---|---|---|---|
| A1 | low (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 (reference) | 0.35 | 0.25 |
The method reduces every term distribution to an expected value (A1's distribution on the first criterion comes to 2.8, A2's to 3.2, A3's to 2.4), takes the first criterion as reference to build the relative weights, compares every pair of alternatives, and scales the global value to the 0–1 range.
| Alternative | Global value | Rank |
|---|---|---|
| A2 | 1.000 | 1 |
| A3 | 0.519 | 2 |
| A1 | 0.000 | 3 |
The result reads as follows. A2 has the highest expected value on the first criterion, which is both the heaviest and the reference criterion. Even its low expected value on the third, cost-direction, criterion does not close this gap. A3 is best on no single criterion but comes second with a balanced profile. A1 comes third because it has the lowest expected value on the first criterion; its global value of 0 means the lowest relative dominance among these three alternatives, not an absolute judgement.
The panel's hesitation is this: if the weights of C1 and C3 were swapped (C1 from 0.40 to 0.25, C3 from 0.25 to 0.40, C2 unchanged at 0.35), the ranking does not change, A2 stays first and A1 stays third. The reference criterion, however, is no longer C1 but C2, because C2 (0.35) now carries the highest weight; A3's global value falls from 0.519 to 0.458. This shows how tightly the reference criterion is bound to the weight ranking.
In the report: "With the highest weight given to the first criterion, A2 is clearly ahead; when the weights of C1 and C3 are swapped, the A2-A3-A1 ranking is preserved, but the reference criterion shifts to C2, and A3's global value falls accordingly."
Source: This case is DecisionMind's validation example for the PL-TODIM engine (fixture status SYNTHETIC_VERIFIED); it has been constructed faithfully to Pang, Wang and Xu's (2016) rules for the five-term probabilistic linguistic term set, and is not a table from Liu and Teng's (2017) paper. The global values and the weight-swap scenario were verified by this card's author by independently running DecisionMind's PL-TODIM engine (method_runner.py PL-TODIM).
2. Public transport: A municipality's choice of bus operator
A municipality is to choose among three bus operators for a new route contract. There are three criteria: schedule punctuality and vehicle comfort (both benefit criteria), and complaint volume (a cost criterion). Passenger surveys judge every operator on these criteria not with a single term but with an opinion spread across several terms; one operator, for instance, might receive "30 per cent medium, 70 per cent high" on schedule punctuality. The municipality gives the highest weight (the reference criterion) to schedule punctuality.
The method reduces every term distribution to an expected value, builds the relative weights, compares the three operators pairwise and computes the global value. Suppose the operator with the highest expected value on schedule punctuality also has the lowest expected value on complaint volume, and comes first in the global value.
The municipality's hesitation is this: this operator's term distribution on the vehicle-comfort criterion is spread across two terms, meaning some passengers reported low comfort. The global value does not show this spread. The municipality should separately clarify a vehicle-renewal commitment before the contract is signed.
In the report: "With the highest weight given to schedule punctuality, this operator is clearly ahead; there is a spread among passenger assessments on the vehicle-comfort criterion, and this should be addressed separately before the contract is signed."
3. What Not to Do
In the illustrative example, writing A1's distribution on the first criterion, "low (0.2), high (0.8)," as simply "high" without stating the probability, is wrong: the expected value rises from 2.8 to 3, and A1's position is calculated differently from its true value. The second error is changing θ or the reference criterion and reporting that "the same result holds under a different scenario"; neither can be changed from the interface in this extension, both are fixed. The third error is marking a cost criterion such as complaint volume as a benefit criterion; in that case the operator with the most complaints would be counted as advantaged on this criterion, and the result becomes meaningless.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pl-todim
Liu, P., & Teng, F. (2017). Probabilistic linguistic TODIM approach for multiple attribute decision-making. Granular Computing, 2, 333–342. DOI: 10.1007/s41066-017-0047-4
Gomes, L. F. A. M., & Lima, M. M. P. P. (1992). TODIM: Basics and application to multicriteria ranking of projects with environmental impacts. Foundations of Computing and Decision Sciences, 16, 113–127. (no DOI)
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
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