We study a predisaster relief network design problem with uncertain demands. The aim is to determine the prepositioning and reallocation of relief supplies. Motivated by the call of the International Federation of Red Cross and Red Crescent Societies (IFRC) to leave no one behind, we consider three important practical aspects of humanitarian operations: shortages, equity, and uncertainty. We first employ a measure that we call the "Shortage Severity Measure" to evaluate the severity of the shortage caused by uncertain demand. Because shortages often raise concerns about equity, we then formulate a mixed-integer lexicographic optimization problem with non-convex objectives and propose a new branch-and-bound algorithm to identify the exact solution. We also propose two approaches for identifying optimal postdisaster adaptable resource reallocation: an exact approach and a conservative approximation that is more computationally efficient. Numerical studies on the 2010 Yushu earthquake positively demonstrate the value of our methodology in alleviating geographical inequities and reducing shortages. Our case study also provides several other interesting insights that may be useful for humanitarian organizations for disaster preparedness.