Extension card · Linguistic
Probabilistic Linguistic EDAS (Wei, Wei and Guo, 2021)
This is the form of EDAS for situations where expert scores are given not as a single word but as the probabilities of several linguistic terms. Every cell is first reduced to an expected linguistic value, and the ranking against the average runs on these values.
Base method
EDAS →
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 EDAS every cell is a single number. Here every cell is a probabilistic linguistic term set: one or several terms, each given together with its own probability. The term set has five terms and is ordered, running from lowest to highest as s0 to s4. If an expert says "80 per cent s3, 20 per cent s4," the opinion is not rounded to a single term; both terms are carried together with their probabilities. Weights remain crisp numbers; EDAS does not generate weights, it takes them from outside.
Scale equalisation. Classical EDAS builds every criterion's average directly from the numbers. Here every cell is first reduced to a single number, the expected linguistic value. DecisionMind finds this value by multiplying each term's index by its probability and summing the products. An opinion spread across two terms and one concentrated on a single term both yield a single number at this step. Once the expected values are found, EDAS's average reference is built from these numbers, exactly as in crisp EDAS.
Deviation from the average and score. Once the expected values have been obtained, the calculation follows exactly the same path as crisp EDAS. The method measures how far above and how far below this average each alternative falls, sums these two deviations with the weights, and normalises. The result is again a single evaluation score between 0 and 1.
DecisionMind fixes, for probabilistic linguistic EDAS, the expected-value formula and the crisp EDAS steps that follow it. Weights are taken from outside as crisp numbers; the method does not generate weights.
How to Read the Output
The output is an evaluation score and a ranking, as in crisp EDAS; it is read the same way. It is not a percentage, it is not compared with an EDAS score from a different analysis, and the average and the scores change when the alternative set changes.
The difference is here. Beneath the score lies a linguistic opinion spread across probabilities, but this opinion has already dissolved into a single number as early as the expected-value step. Two experts can reach the same expected value through different distributions. One might say "definitely s3" while another says "half s2 and half s4"; both give the same expected value, but the second expert's opinion is more scattered. The score itself does not show this difference.
Thus instead of writing:
"Because the experts gave probabilistic terms, the EDAS score fully reflects the uncertainty"
the report should read:
"Every cell has first been reduced to its expected linguistic value, and EDAS has been computed on these values; how scattered the probability distribution was disappears once it is folded into the expected value"
When to Prefer This over the Base Method
Use this extension when experts give several terms with their probabilities rather than a single word, and rounding this distribution to one term would lose an opinion. The typical situation is a committee decision in which several assessors give different terms for the same criterion and the proportions among these terms need to be preserved.
The situation calling for a return to base EDAS is a measured criterion. If an expert gives a single term with no probability distribution, classical linguistic EDAS is sufficient; the probabilistic form adds an unnecessary layer. DecisionMind's single-data-type condition also applies here: if a measured criterion exists, it too is written in the same term set, as a single-term cell with probability one.
The exit condition is the same as for crisp EDAS: if no compromise is acceptable on one criterion, the EDAS family, including this form, is compensatory and will not eliminate anything below a threshold.
Mistakes Specific to This Extension
Leaving the probabilities in a cell summing to more than one. The probabilities of the terms in a cell must sum to one. If the sum exceeds this, the expected-value calculation comes out wrong and the error slips silently into the ranking.
Changing the defuzzification method without stating it. The expected value, the sum of the products of term index and probability, is the canonical choice but not the only option. A different defuzzification can give a different ranking; which method was used must be stated in the report.
Changing the term set during the analysis. The five-term set must stay fixed for all experts and all alternatives. If one expert's set has six terms, the terms' positions do not correspond one to one with the others.
The governing principle is this:
Probabilistic linguistic EDAS reduces an expert's opinion, spread across terms, to its expected value, and runs the rest of EDAS on these values; leaving probabilities that exceed one, or changing the term set afterwards, breaks this average-reference logic.
Cases
The first case is DecisionMind's validation example; the numbers are taken from the manifest, and the engine has produced the same result. The second case is an illustrative construction.
1. Illustrative example: Three alternatives, three criteria (DecisionMind validation example)
This example is not a literature case; it is a small table built to let the engine's steps be followed by hand. The term set has five terms, ordered s0 to s4. Three alternatives are evaluated on three criteria with probabilistic linguistic terms; the first two criteria are higher is better, the third is lower is better.
| Alternative | K1 | K2 | K3 (cost) |
|---|---|---|---|
| A1 | s2(0.2) + s3(0.8) | s2(1.0) | s2(0.6) + s3(0.4) |
| A2 | s3(0.8) + s4(0.2) | s2(0.6) + s3(0.4) | s1(0.4) + s2(0.6) |
| A3 | s2(0.6) + s3(0.4) | s2(0.2) + s3(0.8) | s2(1.0) |
| Weight | 0.40 | 0.35 | 0.25 |
The method first finds every cell's expected value: it multiplies each term's index by its probability and sums the products. For A1 this comes to 2.8 on K1, 2.0 on K2 and 2.4 on K3; for A2, 3.2, 2.4 and 1.6 respectively; for A3, 2.4, 2.8 and 2.0. Every criterion's average is computed from these expected values: 2.8 for K1, 2.4 for K2, 2.0 for K3. Classical EDAS then proceeds: the parts of each alternative that lie above and below the average are summed with the weights, normalised, and combined into a single score.
| Alternative | Evaluation score | Rank |
|---|---|---|
| A2 | 1.0000 | 1 |
| A3 | 0.5085 | 2 |
| A1 | 0.0000 | 3 |
The result reads as follows. A2's expected value sits on the favourable side of the average on K1 (3.2) and on K3 (1.6, low cost); only on K2 (2.4) does it sit exactly on the average (2.4), neither favourable nor unfavourable. Being strong on the two most heavily weighted criteria, K1 (0.40) and K3 (0.25), carries A2 into first place. A3 carries expected values close to the average on all three criteria, and its score also sits right in the middle. A1 finishes last, since it falls on the unfavourable side of the average on both K1 and K3.
The committee's hesitation is this: does the ranking change if K2's weight is raised from 0.35 to 0.60, with K1 and K3 each reduced by 0.20? When the same calculation is independently re-run in Python, A3 reaches 0.8980 and moves ahead of A2 at 0.8429. A3's expected value on K2 (2.8) is higher than A2's (2.4); once K2's weight grows large enough, this advantage more than offsets A3's shortfall on the other two criteria.
In the report: "With the weights given (K1=0.40, K2=0.35, K3=0.25), A2 is in the most advantageous position relative to the set's average (1.0000). If K2's weight is raised to 0.60 with K1 and K3 each lowered by 0.20, A3 moves ahead (0.8980); the ranking is sensitive to K2's relative weight."
Source: DecisionMind's PL-EDAS validation example; the steps follow Keshavarz Ghorabaee et al.'s (2015) EDAS definition combined with Pang, Wang and Xu's (2016) probabilistic linguistic term set. The expected values and the weight-change scenario were obtained by this card's author by independently running the DecisionMind engine.
2. Museology: Choosing a temporary exhibition theme for a museum
A city museum will choose one of three candidate temporary exhibition themes for the coming year. Three criteria apply: potential to attract visitors, suitability for cooperation with educational institutions, and installation cost (lower is better). The curatorial board scored the themes not with a single number but with the spread of opinion that emerged during its internal review meeting. For one theme, a two-term opinion formed: "most likely high interest, small probability of moderate interest." The board gave the highest weight to visitor-attraction potential.
The method reduces each theme's probabilistic opinion on each criterion to its expected value, builds the criterion averages from these values, and measures each theme's position relative to the average. Suppose the theme seen as strongest on educational cooperation also turns out to be the most expensive to install, and still ranks first, because its advantage on visitor-attraction potential was won on the most heavily weighted criterion.
The board's hesitation is this. One curator's "definitely high interest" for a theme was seen by the other three curators as "half high, half moderate." The expected value can reduce both opinions to the same number, but the second group's opinion is more scattered, and this dispersion should be noted separately in the report; otherwise the board gives outside stakeholders the impression of a firmer consensus than actually exists.
In the report: "With the highest weight given to visitor-attraction potential, the theme seen as strong on this criterion comes out ahead; its weakness on installation cost has had limited effect on the ranking because of this criterion's low weight. The breadth of the curators' disagreement should also be noted."
3. What Not to Do
Had K3's probabilities in the illustrative example mistakenly been entered so that they summed to 1.2 (for example, s1(0.4) and s2(0.8) for A2), the expected value would come out higher than it should, and A2's low-cost advantage would be concealed. The second error is reading A2's two-term opinion on K1 as "the highest term is s4, so A2 is best on K1" before defuzzifying it; looking at a single term without factoring in probability ignores the distribution. The third error is reporting A2's score of 1.0000 as "definitely the best supplier"; the score only compares these three alternatives against the set's own average.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pl-edas
Wei, G., Wei, C., & Guo, Y. (2021). EDAS method for probabilistic linguistic multiple attribute group decision making and their application to green supplier selection. Soft Computing, 25(14), 9045–9053. DOI: 10.1007/s00500-021-05842-x
Keshavarz Ghorabaee, M., Zavadskas, E. K., Olfat, L., & Turskis, Z. (2015). Multi-criteria inventory classification using a new method of evaluation based on distance from average solution (EDAS). Informatica, 26(3), 435–451. DOI: 10.15388/Informatica.2015.57
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