Inventory planning has traditionally been modeled with the main objective of minimizing supply chain operational costs. However, in the context of humanitarian aid for disaster relief, it becomes vital to simultaneously manage the supply of food products, medical equipment as well as clothes and personal supplies, whilst minimizing the expected delivery time. The quantity of resources available is as important as the delivery time, especially in cases of extreme disasters. Additionally, since warehouse space is limited, there exists a tradeoff when determining the inventory quantities of each product class. This paper presents a two-stage multi-objective stochastic programming model to account for the different disaster scenarios that could occur for demand at different locations as well as possible warehouse disturbances. The presented model considers different penalties of having a shortage of each product class and the perishability of some of the products, aiming to minimize the penalized shortage of each product class as well as the expected delivery time and storage costs. The importance of considering a single integrated model for all product classes is analyzed with respect to available warehouse capacities. Finally, different techniques are also investigated for solving the presented model and case studies are used to demonstrate the efficacy.