Robust Optimization with Multiple Ranges: Theory and Application to Pharmaceutical 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. This is particularly useful for the manager of a pharmaceutical company who aims at constructing a portfolio of R&D projects, or drugs to be developed and tested for commercialization. Modeling multiple ranges requires the use of binary variables in the uncertainty set. We show how to develop tractable reformulations using a concept called total unimodularity 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 manager ranks the projects according to densities, while incorporating the budgets of uncertainty but without requiring any optimization procedure, to help the manager gain insights into the solution.