Delay in inpatient discharge processes reduces patient satisfaction and increases hospital congestion and length of stay. Further, flow congestion manifests as patient boarding, where new patients awaiting admission are blocked by bed unavailability. Finally, length of stay is extended if the discharge delay incurs an extra overnight stay. These factors are often in conflict, thus, good hospital performance can only be achieved through careful balancing. We formulate the discharge planning problem as a two-stage stochastic program with uncertain discharge processing and bed request times. The model minimizes a combination of discharge lateness, patient boarding, and deviation from preferred discharge times. Patient boarding is integrated by aligning bed requests with bed releases. The model is solved for different instances generated using data from a large hospital in Texas. Stochastic decomposition is compared with the extensive form and the L-shaped algorithm. A shortest expected processing time heuristic is also investigated. Computational experiments indicate that stochastic decomposition outperforms the L-shaped algorithm and the heuristic, with a significantly shorter computational time and small deviation from optimal. The L-shaped method solves only small problems within the allotted time budget. Simulation experiments demonstrate that our model improves discharge lateness and patient boarding compared to current practice.