In emergency situations, disaster relief organizations are faced with the difficult decision of how to allocate scarce resources in an efficient manner in order to provide the best possible relief action. This paper aims to provide an analytical model that will help relief organizations in reducing human suffering following a disaster while maintaining an acceptable level of cost efficiency. A mathematical model is introduced to optimize the relief distribution problem which considers the social cost -the total sum of logistics and deprivation costs. The fuzzy nature of the deprivation cost function is addressed with possibilistic mixed integer programming with fuzzy objectives to reflect variation in deprivation costs perceptions. The model is solved using the Rolling Horizon method in a sequence of iterations. In each iteration, part of the planning horizon is modeled in detail and the rest of the time horizon is represented in an aggregated manner. The model is tested both empirically and on a case study of internal displacement in northwest Syria. Computational results showed that considering the demographic structure in affected areas and reflecting it to the deprivation cost function helped to reach better prioritization in distribution of commodities. The rolling horizon methodology is also found to be efficient in solving large scale instances and in capturing the dynamic changes in demand and supply parameters.