A variety of statistical graphical models have been defined to represent the conditional independences underlying a random vector of interest. Similarly, many different graphs embedding various types of preferential independences, such as, for example, conditional utility independence and generalized additive independence, have more recently started to appear. In this paper, we define a new graphical model, called a directed expected utility network, whose edges depict both probabilistic and utility conditional independences. These embed a very flexible class of utility models, much larger than those usually conceived in standard influence diagrams. Our graphical representation and various transformations of the original graph into a tree structure are then used to guide fast routines for the computation of a decision problem’s expected utilities. We show that our routines generalize those usually utilized in standard influence diagrams’ evaluations under much more restrictive conditions. We then proceed with the construction of a directed expected utility network to support decision makers in the domain of household food security.