Robust Optimization with Multiple Ranges: Theory and Application to R & D Project Selection

We present a robust optimization approach when the uncertainty in objective coefficients is described using multiple ranges for each coefficient. This setting arises when the value of the uncertain coefficients, such as cash flows, depends on an underlying random variable, such as the effectiveness of a new drug. Traditional robust optimization with a single range per coefficient would require very large ranges in this case and lead to overly conservative results. In our approach, the decision-maker limits the number of coefficients that fall within each range; he can also limit the number of coefficients that deviate from their nominal value in a given range. Modeling multiple ranges requires the use of binary variables in the uncertainty set. We show how to address this issue to develop tractable reformulations and apply our approach to a R&D project selection problem when cash flows are uncertain. Furthermore, we develop a robust ranking heuristic, where the project manager ranks the projects according to densities (ratio of cash flows to development costs) or Net Present Values, while incorporating the budgets of uncertainty but without requiring any optimization procedure. While both density-based and NPV-based ranking heuristics perform very well in experiments, the NPVbased heuristic performs better; in particular, it finds the truly optimal solution more often.