Influence diagrams represent decision-making problems with interdependencies between random events, decisions, and consequences. Traditionally, they have been solved using algorithms that determine the expected utility-maximizing decision strategy. In contrast, state-of-the-art solution approaches convert influence diagrams into a mixed-integer linear programming (MILP) model, which can be solved with powerful off-the-shelf MILP solvers. From a computational standpoint, the existing MILP formulations can be efficiently solved when applied to influence diagrams that represent periodic (or sequential) decision processes, which can be cast as partially observable Markov Decision Processes. However, they are inefficient in problems that lack a periodic structure or if the nodes in the influence diagram have large state spaces, thus limiting their practical use. In this paper, we present an efficient MILP formulation that is specifically designed for influence diagrams that are challenging for the earlier MILP formulation-based methods. Additionally, we present how the proposed formulation can be adapted to maximize conditional value-at-risk and how chance and logical constraints can be incorporated into the formulation, thus retaining the modeling flexibility of the MILP-based methods. Finally, we perform computational experiments addressing problems from the literature and compare the computational efficiency of the proposed formulation against the available MILP formulations for the reported influence diagrams. We find that the MILP models based on the proposed formulations can be solved significantly more efficiently compared to the state-of-the-art when solving influence diagrams that cannot be cast as partially observable Markov decision processes.