UTA (Additive Utility Method)

UTA answers the question: "instead of asking the decision-maker directly for weights or a utility formula, how can I extract that information from the way they rank a handful of reference alternatives?" Its output is a marginal utility curve for every criterion, a total utility score for every alternative, and the rank that score produces. It was proposed by Jacquet-Lagrèze and Siskos in 1982 and is the founding method of the preference disaggregation approach. It is used in fields such as consumer preference modelling, supplier selection and product evaluation, where the decision-maker is more comfortable ranking concrete examples than assigning abstract weights. It has many extensions, including UTASTAR, UTADIS and STOCHASTIC-UTA.

Most multi-criteria decision methods, such as AHP and SWARA, ask the decision-maker directly for weights or pairwise comparisons. UTA's philosophy reverses this. Many people find it difficult to assign abstract weights, but are more comfortable ranking concrete examples; saying "I prefer this car to that one" is easier than saying "I give speed a weight of thirty per cent." UTA asks the decision-maker to rank a handful of reference alternatives, then searches mathematically for a utility function consistent with that ranking; this is called preference disaggregation.

An analogy makes this clear. A wine taster may not be able to explain in a formula why they like a wine, but can confidently rank several wines. UTA works backwards from the taster's ranking to infer how much value they place on which characteristics (acidity, tannin, alcohol content). The philosophical consequence is this: UTA is additive and compensatory, and assumes independence between criteria. It also measures the decision-maker's own consistency directly, through an error term; if this error term is zero, the decision-maker is fully consistent, and if it is greater than zero, there is an internal contradiction, which the method does not hide but expresses as a number.

The method proceeds through four steps.

First, defining the breakpoints. For every criterion, a range is set from worst to best and divided into a fixed number of equal breakpoints. These breakpoints are the points at which the criterion's utility curve will be defined piecewise-linearly.

Second, evaluating the reference alternatives. Each ranked reference alternative's value on every criterion is expressed linearly between the two nearest breakpoints. The alternative's total utility is written as the sum of the criterion utilities plus a small error term.

Third, building and solving the linear programming model. A model is built to minimise the total error term; the model carries three types of constraint. Preference constraints require the utility gap between every consecutive pair in the reference ranking to exceed a set minimum threshold; for pairs ranked as equally preferred, this gap must be zero. Monotonicity constraints ensure that, for a criterion, the next breakpoint never gives lower utility than the previous one. Normalisation constraints fix the worst breakpoint's utility at zero and the sum of all criteria's best-breakpoint utilities at one. The model is solved and the smallest total error term (F*) is found.

Fourth, robustness analysis. The linear programming solution may not be unique; more than one utility function can produce the same minimum error term. A broad solution region is therefore defined, allowing an error term close to F*. Within this region, the smallest and largest utility values that each criterion's best breakpoint can take are separately computed; averaging these solutions gives a representative utility function that does not depend on a single arbitrary solution.

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

The total utility score shows how good an alternative is according to the decision-maker's inferred preference logic; the best alternative generally receives a value close to 1. But more important than this score is the error term (F*). If F* is zero, the decision-maker's 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 a calculation error, but UTA revealing the decision-maker's own inconsistency. Because the solution may also not be unique, presenting a single LP solution as the definite truth without a robustness analysis is misleading.

Thus instead of writing:

"UTA found the decision-maker's true preference with certainty"

the report should read:

"This utility function is the most consistent model possible with the reference ranking the decision-maker gave; if the error term is zero the ranking is fully consistent, and if it is greater than zero this inconsistency must be shown in the report"

UTA works with crisp data: one number per cell. DecisionMind holds 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; the worst and best extreme value for every criterion. You also need a reference ranking in which the decision-maker has ranked all, or a representative subset, of the reference alternatives. UTA does not require weights; it produces a utility function itself (and, implicitly, a weight distribution). The recommended number of reference alternatives is between five and thirty, and a comfortable range for the number of criteria is three to twelve.

UTA is a sound choice if the decision-maker can confidently rank concrete examples but struggles to assign abstract weights. If the decision-maker has no reference alternatives to rank, or has no clear preference among them, the method lacks the data it needs to run; in that case, methods that ask directly for weights (AHP, BWM, SWARA) should be used instead. If the reference ranking is seriously inconsistent (the error term is very large), UTA's single-error-variable structure may prove inadequate.

The decision-maker can rank examples but cannot assign abstract weights → UTA

No reference ranking exists, or it is unreliable → AHP, BWM, SWARA (methods that ask directly for weights)

The reference ranking is seriously inconsistent, the error term is large → UTASTAR (a separate error variable for every pair)

Not a ranking but only weights are needed → Entropy, CRITIC (objective) or AHP, BWM, SWARA (subjective)

UTA's core strength is that it asks the decision-maker for an intuitive task, ranking examples, rather than abstract weights; in practice this yields more reliable information. The error term (F*) measures the decision-maker's own consistency directly and transparently. The robustness analysis avoids dependence on a single arbitrary solution, and how well the resulting ranking matches the reference ranking can be shown concretely with Kendall's tau coefficient; in the founding paper's own example, this coefficient comes out at 1.0.

Its limitations stem from the structure of the model. First, the linear programming solution may not be unique; without a robustness analysis, an arbitrary solution can be mistaken for the true one (Siskos, Grigoroudis and Matsatsinis, 2005). Second, the piecewise-linear utility assumption can remain a crude approximation if the true preference curve is strongly non-linear (threshold-driven, S-shaped); increasing the number of breakpoints partly addresses this but increases the computational load. Third, the result depends directly on the size and quality of the reference set; a small or carelessly chosen reference set can produce an unreliable utility function. Fourth, it has been shown that preference disaggregation methods need additional assumptions when working with uncertain or incomplete preference information (Wu and Liao, 2023).

The most common mistake is entering the reference ranking as a dictionary in the form {alternative: rank} and then processing it by the order in which items were entered into the dictionary, rather than by the rank value; this unknowingly builds the wrong model. A second mistake is reversing the direction of a cost criterion by rearranging the criterion range itself (swapping the worst and best ends) and then reversing it a second time separately; this double reversal breaks the direction again. A third mistake is presenting the result as definite fact when the error term (F*) is greater than zero; in that case, the inconsistency should be stated in the report, or a more flexible model such as UTASTAR should be used instead. A fourth mistake is presenting a single LP solution as an immutable utility function without a robustness analysis.

The governing principle is this:

A UTA result is the most consistent additive model possible with the reference ranking the decision-maker gave; ignoring the error term and the uniqueness of the solution creates a false impression of certainty.