We formulate a robust optimization model for the multi-mode resource-constrained project scheduling problem (MRCPSP) with uncertain activity durations. The objective is to define a project plan that includes selected modes, resource allocations and a project schedule that minimize the worst-case project duration, under polyhedral uncertainty sets. A Benders decomposition approach is proposed to solve the robust counterpart of the suggested model. The solution approach involves iterative solutions of two optimization problems: A master problem that selects activity modes and allocates resources to activities to minimize the project duration using nominal activity durations, and a subproblem that uses the master solutions to find the longest network path for a given uncertainty set. We develop valid and optimality cuts, which are added in each iteration to the master problem and guarantee convergence to the optimal solution. Within a budgeted uncertainty set, characterized by a parameter that controls the level of conservativeness, the subproblem is polynomially solvable.
We conduct computational experiments for analyzing the price of robustness under varying levels of uncertainty. The results provide managerial insights regarding to the performance of robust policies under various conditions, compared to their respective utopian and deterministic policies. These demonstrate the conditions under which the robust approach may be favorable with respect to deterministic policies and the general conditions under which the price of robustness is relatively low.