# Multi-attribute Utility/Value Methods

Multi-attribute utility methods (MAUT) (Keeney and Raiffa 1976) are based on the assumption that the decision maker's preferences are coherent with some increasing real function *U* called utility, which (s)he attempts to maximize. In other words, an option *a* is preferred over another option *b,* if and only if *U(a)> U(b).*

In the additive model, which is most commonly used, the utility of an option *a* is expressed as a sum of the partial utilities:

where *Ut* are single-attribute utility functions corresponding to the evaluation criteria *gt* and *n* is the number of criteria. An extensive literature has been dedicated to building the additive model, for example Fishburn (1967) and Jacquet-Lagreze and Siskos (1982). Fishburn (1967) has formulated sufficient and necessary conditions for the existence of an additive utility function. A necessary condition for the validity of the additive model is, for instance, that any subset of criteria is preferentially independent of the remaining criteria.1 Keeney (1992) provides other examples of relatively simple (for example, multi-linear) utility functions and the conditions for the validity of the corresponding models.

When uncertainties are not taken into account, the model becomes a multi-attribute *value* model. The additive value model can be formulated as the maximization of a value function *V* given by:

where the weights *wt* are scaling constants that indicate value trade-offs between criteria. These weights can be determined by various techniques, as illustrated for example in Hamalainen (2002).

The uncertainty and imprecision in MAUT models can be modelled by means of probability theory. It is interesting to note that the shape of the utility function has a direct relation with the attitude to risk of the decision makers. A concave utility function corresponds to *risk aversion,* a linear function to *risk neutrality* and a convex function to *risk proneness* (Keeney 1992).