Extension card · Linguistic
Probabilistic Linguistic MARCOS
This is the form of MARCOS 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 utility ratio against the ideal and anti-ideal is built on these values.
Base method
MARCOS →
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 MARCOS 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. Weights remain crisp numbers; MARCOS does not generate weights, it takes them from outside.
The extended table and scale equalisation. Classical MARCOS builds its ideal and anti-ideal rows from the observed best and worst numbers. Here every cell is first reduced to an expected linguistic value; DecisionMind finds this value by multiplying each term's index by its probability and summing the products. The ideal and anti-ideal rows are then built from these expected values, using each criterion's observed best and worst expected value. Crisp MARCOS's normalisation and weighting steps then run on these numbers exactly as before.
Utility ratios and the final degree. Once the expected values and the extended table are obtained, the calculation follows exactly the same path as crisp MARCOS. Each alternative's total score is separately compared with the ideal's and the anti-ideal's total score, these two ratios are passed through utility functions, and combined into a single final utility degree.
DecisionMind fixes, for probabilistic linguistic MARCOS, the expected-value formula and the crisp MARCOS 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 a final utility degree and a ranking, as in crisp MARCOS; it is read the same way. It is not compared with a MARCOS degree from a different analysis, because the ideal and anti-ideal are built, in every analysis, from that analysis's own expected values.
The difference is here. Beneath the degree 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; the degree itself does not show this difference, it carries only the difference between the expected values.
Thus instead of writing:
"Because the experts gave probabilistic terms, the MARCOS degree fully reflects the uncertainty"
the report should read:
"Every cell has first been reduced to its expected linguistic value, and the ideal and anti-ideal have been built from 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 MARCOS is a measured criterion. If an expert gives a single term with no probability distribution, classical linguistic MARCOS is sufficient. 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 MARCOS: if no compromise is acceptable on one criterion, the MARCOS 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 ideal and anti-ideal, and hence a different ranking.
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 MARCOS reduces an expert's opinion, spread across terms, to its expected value, and builds the ideal and anti-ideal from these values; leaving probabilities that exceed one, or changing the term set afterwards, breaks this proportional-position 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. 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. The ideal row is built from these expected values: the highest on K1 (3.2), the highest on K2 (2.8), and, on K3, the lowest, meaning the best cost (1.6). The anti-ideal row takes the lowest on K1 (2.4), the lowest on K2 (2.0), and, on K3, the highest, meaning the worst cost (2.4). Crisp MARCOS then proceeds: each alternative is compared with the ideal's and the anti-ideal's total score, these two ratios are passed through utility functions, and combined into a single degree.
| Alternative | Final utility degree | Rank |
|---|---|---|
| A2 | 0.7376 | 1 |
| A3 | 0.6518 | 2 |
| A1 | 0.5832 | 3 |
The result reads as follows. A2's expected value equals the ideal exactly on K1 (3.2) and equally on K3 (1.6); only on K2 (2.4) does it sit closer to the anti-ideal (2.0). Reaching the ideal on the most heavily weighted criterion, K1 (0.40), and on the cost criterion K3 (0.25), more than offsets its weakness on K2. A3 draws a balanced profile with middling values on all three criteria and stays in second place. A1 finishes last, since, although it sits close to the ideal on K2 (2.8), it stays close to the anti-ideal 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 0.68, with K1 and K3 each reduced by 0.16? When the same calculation is independently re-run in Python, A3 reaches 0.7098 and moves ahead of A2 at 0.7032, but the gap is only 0.0066. This shift occurs at a larger weight swing than the one seen in PL-EDAS; MARCOS's ideal-anti-ideal ratio has proved somewhat more resistant here than EDAS's average-deviation approach, though this resistance is not guaranteed for every data set.
In the report: "With the weights given (K1=0.40, K2=0.35, K3=0.25), A2 has the highest final utility degree (0.7376). If K2's weight is raised to 0.68 with K1 and K3 each lowered by 0.16, A3 moves ahead (0.7098), but the gap between A2 and A3 in this scenario is only 0.0066."
Source: DecisionMind's PL-MARCOS validation example; the steps follow Stević et al.'s (2020) MARCOS definition combined with Pang, Wang and Xu's (2016) probabilistic linguistic term set. A systematic literature search found no independent source specific to PL-MARCOS that first proposes this combination; DecisionMind has built this extension by following the same expected-value reduction rule as PL-EDAS. The expected values and the weight-change scenario were obtained by this card's author by independently running the DecisionMind engine.
2. Archival science: Choosing a digitisation partner for an institutional archive
An institutional archive will choose one of three candidate service providers for digitising its paper records. Three criteria apply: scanning quality, delivery time (lower is better), and experience working with old or damaged documents. The archive committee scored the providers not with a single number but with the spread of opinion that emerged from a sample-document test. For one provider, a two-term opinion formed: "most likely high quality, small probability of moderate quality." The committee gave the highest weight to scanning quality.
The method reduces each provider's probabilistic opinion on each criterion to its expected value, builds the ideal and anti-ideal rows from these values, and computes each provider's ratio to these two references. Suppose the provider seen as strongest on experience with damaged documents also has the longest delivery time, and still ranks first, because its advantage on scanning quality was won on the most heavily weighted criterion.
The committee's hesitation is this. One assessor's "definitely high quality" for a provider was seen by the other two assessors 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. The committee should consider reducing this scatter with a second sample test before signing the contract.
In the report: "The provider seen as strong on scanning quality, to which the highest weight was given, reaches the highest final utility degree relative to the ideal and anti-ideal references; its weakness on delivery time has remained limited because of this criterion's low weight."
3. What Not to Do
Had K3's probabilities in the illustrative example mistakenly been entered so that they summed to 1.2, 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 at the ideal on K1" before defuzzifying it; looking at a single term without factoring in probability ignores the distribution. The third error is reporting A2's degree of 0.7376 as "74 per cent suitable"; the degree only compares these three alternatives along the set's own ideal-anti-ideal axis.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pl-marcos
Stević, Ž., Pamučar, D., Puška, A., & Chatterjee, P. (2020). Sustainable supplier selection in healthcare industries using a new MCDM method: Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS). Computers & Industrial Engineering, 140, 106231. DOI: 10.1016/j.cie.2019.106231
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
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