STOCHASTIC-UTA (Additive Utility Method under Uncertainty)

STOCHASTIC-UTA answers, together, the questions "how is a decision-maker's implicit evaluation logic inferred from the way they ranked a few alternatives" and "how does that logic stay reliable when the criterion values are not known precisely." Its output is a utility, or acceptability, score for every alternative and the rank that score produces. Stavrou, Ventikos and Tsoukalas proposed the method in 2018 for risk assessment in maritime operations such as ship-to-ship fuel transfer. It can also be used in other fields involving uncertainty, such as energy technology selection, supplier selection and healthcare system evaluation. It combines UTA's (Jacquet-Lagrèze and Siskos, 1982) classical approach of inferring utility from preference (preference disaggregation) with the logic of SMAA (Stochastic Multicriteria Acceptability Analysis), which measures acceptability under uncertainty.

UTA's philosophy is to observe how the decision-maker ranks a few alternatives, rather than asking them directly for a formula or a set of weights, and to reverse-engineer a utility function consistent with that ranking. STOCHASTIC-UTA adds a layer of uncertainty to this. The criterion values themselves may not be known precisely; a measurement error margin or an expert estimate may be involved. In that case, the method repeatedly samples these values from a probability distribution instead of treating them as single numbers, and redoes the utility calculation for every sample.

An analogy makes this clear. A doctor, rather than trusting a single test result, accounts for the margin of measurement error. They assess the same test result under many possible error scenarios and ask "in what percentage of the plausible error range does this diagnosis hold." STOCHASTIC-UTA works the same way: the result is not a single definite ranking but an acceptability rate of the kind "in what percentage of the samples did this alternative come out best." The philosophical consequence is that the method is both additive and compensatory, and that it turns uncertainty explicitly into the language of probability rather than hiding it.

The method proceeds through nine steps; the first six hold even without uncertainty, and the last three add the uncertainty sampling.

First step, direction correction. Cost criteria are reversed so that every criterion can be treated as "higher is better"; each criterion's best and worst value is recorded.

Second step, dividing the criterion range into stages. Each criterion's range from worst to best is divided into a set number of equal stages. These stages are the points at which the utility function will be defined piecewise, linearly.

Third step, the linear relationship between stages. Each alternative's value on each criterion is expressed linearly between the two nearest stages; this allows the utility function to be written as a combination of the unknown stage values.

Fourth step, building the linear programming model. A model is built that minimises the total error while remaining consistent with the decision-maker's reference ranking. The model carries three kinds of constraint. Preference constraints require the utility difference between every consecutive pair in the reference ranking to exceed a set threshold. Monotonicity constraints ensure that, for a given criterion, no later stage yields a lower utility than the one before it. Normalisation constraints fix the worst stage's utility at zero and the sum of every criterion's best-stage utilities at one.

Fifth step, solving the model. The linear programming model is solved to find the utility value at every stage of every criterion; the solution gives the utility function most consistent with the reference ranking.

Sixth step, computing the total utility. Each alternative's total utility is found by summing the utility values corresponding to its criterion values.

Seventh step, uncertainty sampling. If the criterion values are not precise, each value is drawn at random from its own probability distribution; steps three through six are then repeated many times over these samples. If the values are precise, this step reduces to a single computation.

Eighth step, computing the acceptability rates. For every alternative, the number of samples in which it comes out in each rank is counted; from this a rank-acceptability rate and an overall acceptability score are derived for every alternative.

Ninth step, ranking. Alternatives are ranked from largest to smallest acceptability score; when no uncertainty sampling has been carried out, this ranking is identical to the ranking by total utility.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

When no uncertainty sampling has been carried out (the deterministic mode), the score is the total utility computed from the utility function consistent with the decision-maker's reference ranking; the best alternative takes a value close to 1, the worst close to 0. When uncertainty sampling has been carried out, the score is an acceptability rate: a summary of how often, and in which rank, the alternative appeared across the samples. In both cases the score is meaningful only for this particular alternative set and this reference ranking.

The error margin (F*) that emerges from solving the model should also not be overlooked. If F* is zero, the reference ranking is fully consistent with an additive utility function. If F* is greater than zero, the decision-maker's preferences do not fit an additive model exactly; this is not an error, it is the method's own way of quantifying that inconsistency.

For this reason:

"STOCHASTIC-UTA found the most accurate ranking"

should be written instead as:

"This ranking is the product of the utility function consistent with the reference ranking the decision-maker supplied; where uncertainty sampling was used, the score is an acceptability rate, not a guarantee of a definite ranking"

STOCHASTIC-UTA works with crisp data: one number per cell, and optionally a probability distribution. DecisionMind holds STOCHASTIC-UTA only in this base form; there is no separate data-type extension.

You need alternatives in rows, criteria in columns, one number per cell; direction information for every criterion. You also need a reference ranking from the decision-maker covering at least a few alternatives, whether the full set or a representative subset. STOCHASTIC-UTA produces weights rather than requiring them from outside; the utility function takes their place. If the uncertainty mode is to be used, a probability distribution is also needed for every criterion cell. The recommended number of alternatives is between three and thirty, with a comfortable range of four to fifteen; the comfortable range for the number of criteria is three to ten.

STOCHASTIC-UTA is a suitable choice when the decision-maker can reliably rank a few alternatives and a general evaluation logic (a utility function) needs to be inferred from that ranking. When the criterion values are uncertain and the effect of that uncertainty on the result needs to be shown, the method's Monte Carlo mode answers exactly that need. The method is unsuitable when the decision-maker cannot give a consistent reference ranking, or when strong non-additive interactions exist between criteria, that is, when one criterion's effect changes entirely with the value of another.

A reference ranking exists, a utility function is to be inferred → STOCHASTIC-UTA

Criterion values are uncertain, that uncertainty must be carried through into the result → STOCHASTIC-UTA (Monte Carlo mode)

Criterion values are precise, no uncertainty → classical UTA suffices

The reference ranking comes out inconsistent, the error margin is too large → UTASTAR (a more flexible form using a separate error variable for each pair)

STOCHASTIC-UTA's core advantage is that it works from concrete examples rather than asking the decision-maker for an abstract weight, a task many decision-makers find easier than assigning weights directly. Turning uncertainty explicitly into an acceptability rate, rather than hiding it, strengthens the honesty of the report. The error margin (F*) offers a clear indicator of how reliable the result is.

Its limitations stem from the complexity of the model. First, the linear programming model's solution is not unique; different optimal solutions can give different utility functions at the same error margin, which is why post-optimality analysis, averaging several solutions, is recommended (Jacquet-Lagrèze and Siskos, 1982). Second, uncertainty sampling substantially increases the computational burden; run time grows with the number of samples. Third, the quality of the probability distributions lies outside the method itself; sampling from a poorly calibrated distribution produces a misleading acceptability rate (Durbach, Lahdelma and Salminen, 2016).

The most common mistake is forgetting to include some of the alternatives in the reference ranking; this leaves the linear programming model under-constrained and the solution meaningless. A second mistake is setting the utility-difference threshold between consecutive alternatives (δ*) far larger than the natural utility gaps allow; this can make the model unsolvable and forces a large error margin. A third mistake is feeding a cost criterion into the model without direction correction, that is, without negating it; this reverses the criterion's true effect and penalises a low-cost alternative. A fourth mistake is reporting the result as "definite" when the error margin (F*) is greater than zero; F*>0 is a sign of inconsistency and must be shown in the report.

The governing principle is this:

A STOCHASTIC-UTA result reflects the reference ranking the decision-maker supplied and, where present, the criterion uncertainty; ignoring the error margin and the acceptability rate gives the result a false impression of certainty.