Extension card · Stochastic
Probabilistic ARAS
Probabilistic ARAS is the form of ARAS for situations where a criterion value comes from a probability distribution and this distribution is reduced to a single representative number (for instance, the expected value). The calculation itself is identical to crisp ARAS; only the source of the number that enters the cell changes.
Base method
ARAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Stochastic →
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?
Honesty is required here: DecisionMind's current Probabilistic ARAS engine runs crisp ARAS's steps (building the optimal alternative, scaling to the column sum, weighting, summation, utility degree) unchanged, exactly as they are. At no step is sampling from a distribution, a confidence interval or an acceptability ratio computed.
Cells. In crisp ARAS every cell is a single number, either measured directly or judged. Here every cell is likewise a single number, but this number is a representative value derived from a distribution: the most common choice is the expected value (the distribution's mean), and sometimes it is the value of a single prominent scenario. The cell itself appears in the calculation not as a distribution but as a crisp number.
Scale equalisation, weighting, summation. None of these three steps differs from crisp ARAS: the same scaling to the column sum, the same weighted summation, the same definition of the utility degree (K = S/S₀) are used. There is no distance, score or aggregation rule that differs.
Result and interpretation. The utility degree K carries the same definition as in crisp ARAS: percentage utility relative to the optimal alternative. But the number beneath K is now the summary of a distribution; the cell's own uncertainty (standard deviation, confidence interval) enters the calculation at no point and does not appear in the result. If two alternatives' expected values are close but one comes from a much wider distribution, this difference is not reflected in K at all.
DecisionMind's current version does not apply the full method described on the probabilistic data-type card, that is, an acceptability analysis (SMAA-type methods) in which a large number of scenarios is scanned and the proportion of times each alternative comes out on top is computed. What is applied is crisp ARAS, operating on a single representative scenario or expected value. This is a significant difference between the method the data-type card promises and the work the engine actually does today, and this card does not hide that difference.
How to Read the Output
The utility degree K is read here as in crisp ARAS: the best alternative is taken as 100, and the others receive a percentage relative to it; this percentage holds only for this alternative set and these weights. The difference is this: K has been computed on a number derived from a distribution, but the width of that distribution does not appear in the report. If K = 0.95 and K = 0.85 both come from means of wide distributions, the gap between them is far less reliable than a gap of the same size coming from narrow distributions; but the engine itself does not show this difference in reliability, the person writing the report must state it separately.
Thus instead of writing:
"According to Probabilistic ARAS, A2 is the best alternative with 95 per cent probability"
the report should read:
"With the given expected values and weights (0.40; 0.35; 0.25), A2 has the highest utility degree (K = 0.950); this number is not a probability, it is a proportional utility measure computed over expected values, and when the weight is shifted to the second criterion (0.10; 0.80; 0.10), A3 moves into first place"
When to Prefer This over the Base Method
This method is suitable when a criterion value is the summary of a distribution arising from a measurement error, a sampling error, or a historical series. It should be accepted that using this summary, usually the expected value, as a single representative number is sufficient. Examples: an average return computed from a historical return series, an average output computed from a historical production series.
If the decision maker's need is a genuine probability answer to the question "which alternative comes out best, and with what probability," today's Probabilistic ARAS engine does not meet that need; the engine is crisp ARAS operating on a single representative number, and it does not account for the width of the distribution. In that case a full acceptability analysis (SMAA-type methods) or a method supported by Monte Carlo simulation is required; DecisionMind's current PROB-ARAS is not that. Crisp ARAS's exit condition applies here too: if no compromise is acceptable on a criterion, this extension is also fully compensatory and will not eliminate anything below a threshold.
Mistakes Specific to This Extension
Reporting the expected value as if it were a "probability." However K is computed, it is not a probability; here, in addition, the number entering the cell is itself only the summary of a distribution, not the distribution itself. The sentence "K = 0.95, best with 95 per cent probability" is wrong twice over.
Inventing an expected value with no source. The data-type card's rule applies here too: the expected value must be computed from a series, a simulation or a document; writing a number as "approximately this much" and presenting it as an expected value represents a judgement, not a probabilistic structure, and belongs in the fuzzy structure instead.
Assuming the engine has performed an acceptability analysis. A user or report reader may, from the name "Probabilistic ARAS," expect a statement such as "first place with such-and-such per cent probability" in the result. Today's engine does not produce this; all it produces is crisp ARAS's utility degree, computed on representative numbers. This limit must be stated explicitly in the report.
Ignoring the width of the distribution. If two alternatives' expected values are close and one comes from a far more variable source, looking only at K conceals this risk. The standard deviation should be reported separately for the alternative with higher variability.
The governing principle is this:
Probabilistic ARAS today is crisp ARAS operating on a distribution's representative number; any use that presents this as a full acceptability analysis, or that ignores the width of the distribution, misrepresents what the engine actually does.
Cases
The first case is DecisionMind's validation example: a small table over three alternatives and three criteria, built with representative numbers; it involves no sampling from a distribution, because the engine involves none either. The second case is an illustrative fiction.
1. Illustrative example: Comparing three investment alternatives with expected values (DecisionMind validation example)
An investor is comparing three alternatives on three criteria; the first and second criteria are benefit criteria, the third (a risk indicator) is a cost criterion. Every cell is an expected value computed from a five-year historical series. The investor has given the three criteria weights of 0.40, 0.35 and 0.25 respectively.
| Alternative | Expected return | Expected growth | Risk indicator (cost) |
|---|---|---|---|
| A1 | 0.7 | 0.5 | 0.6 |
| A2 | 0.8 | 0.6 | 0.4 |
| A3 | 0.6 | 0.7 | 0.5 |
| Weight | 0.40 | 0.35 | 0.25 |
The method reverses the third criterion, scales every column to its own sum (including the optimal alternative), multiplies by the weights and sums, then divides to obtain the utility degrees; none of the steps differs from crisp ARAS.
| Alternative | Utility degree (K) | Rank |
|---|---|---|
| A2 | 0.950 | 1 |
| A3 | 0.850 | 2 |
| A1 | 0.765 | 3 |
The result reads as follows. A2 has both the highest value on expected return, the heaviest criterion, and simultaneously the lowest risk indicator; being strong on two heavy criteria at once, it ranks first. A1 is best on no criterion and comes last.
The investor has one hesitation. When the weight is concentrated on the second criterion (0.10; 0.80; 0.10) and the calculation is redone, the utility degrees come out at 0.955 for A3, 0.886 for A2 and 0.725 for A1, and the ranking is COMPLETELY reversed: A3 rises to first place (computed by independently running the same algorithm in Python). This shows that A2's first place is entirely dependent on the weight given to the expected-return and risk criteria; moreover, this ranking has been made only over expected values, and how much the series fluctuates from year to year has not been taken into account.
In the report: "With the given expected values and weights (0.40; 0.35; 0.25), A2 has the highest utility degree (K = 0.950); this number has been computed over expected values, and the width of the distribution has not been separately assessed. When the weight is concentrated on expected growth (0.10; 0.80; 0.10), the ranking is completely reversed and A3 rises to first place."
Source: DecisionMind's validation example for the Probabilistic ARAS engine; the algorithm is identical to Zavadskas and Turskis's (2010) crisp ARAS definition, because the engine today processes not the distribution but only the representative number. The matrix and the weight-swap figures were independently computed by this card's author using the same algorithm.
2. Insurance: An insurance company's choice of policy provider
An insurance company is to choose among three providers for a reinsurance agreement. There are two criteria: the provider's payment performance over the past five years (a benefit criterion) and the provider's price quote (a cost criterion). The company's actuaries have given the providers' past performance not as a single crisp score but as an expected performance score computed from the distribution of past years; behind these scores lie distributions of different widths, because one provider's performance varies greatly from year to year while another's is more stable.
The method compares the three providers: it reverses the price criterion, scales to the column sum, multiplies by the weights and sums. Suppose the provider with the highest expected performance score also has the most variable history; its performance has been very good in some years and weak in others. It nonetheless comes first on the utility degree, because the calculation looks only at the expected value.
The company has one hesitation. The first provider's high variability means its performance next year could deviate significantly from the expected value. The company should look not only at the utility degree but also at which provider's past performance has been more stable; the actuaries should present this difference in stability in a separate table.
In the report: "The first provider stands out according to the expected performance score; however, this provider's past performance is more variable than the others', so the difference in stability should be separately assessed before the agreement."
3. What Not to Do
The first error is reporting the illustrative example's A2 utility degree of 0.950 as "A2 is the best alternative with 95 per cent probability"; K is not a probability, and the cells themselves are not a distribution but the summary of one. The second error is inventing an expected value with no source: writing a number not based on any series as an expected value by saying "it comes to approximately 0.7." The third error is assuming, from the name "Probabilistic ARAS," that the engine produces an acceptability percentage; today's engine is crisp ARAS operating only on representative numbers and does not account for the width of the distribution.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/prob-aras
Zavadskas, E. K., & Turskis, Z. (2010). A new additive ratio assessment (ARAS) method in multicriteria decision-making. Technological and Economic Development of Economy, 16(2), 159–172. DOI: 10.3846/tede.2010.10
Lahdelma, R., Hokkanen, J., & Salminen, P. (1998). SMAA – Stochastic multiobjective acceptability analysis. European Journal of Operational Research, 106(1), 137–143. DOI: 10.1016/S0377-2217(97)00163-X
Lahdelma, R., & Salminen, P. (2001). SMAA-2: Stochastic multicriteria acceptability analysis for group decision making. Operations Research, 49(3), 444–454. DOI: 10.1287/opre.49.3.444.11220