Tackling poverty has been one of the greatest global challenges and a prerequisite to sustainable development of countries. Countries implement nationally appropriate social protection systems and measures to address poverty. This paper addresses an aid system adopted by the government in Turkey where significant amounts of coal is distributed to poor families each year. The objective of the coal aid system is to complete the delivery of coal to poor families by the start of winter. However, an analysis of the data from previous years indicates that the distribution to many families cannot be completed on time. This results from the fact that planning is done manually and by trial-and-error as there is no system that can be used for distribution planning. This paper describes the planning problem encountered and develops a mathematical model to solve it. The proposed model is a multimodal, multicommodity, and multiperiod linear programming (LP) model. The model can be used to develop and update a distribution plan as well as to answer several what-if questions with regard to capacities, time constraints, and so forth. The model is solved using CPLEX for several problem instances obtained under different scenarios using data for the year 2012. The results show that at least 9% cost savings and about 40% decrease in distribution completion time can be achieved when the model is used. We analyze scenario results qualitatively and quantitatively and provide several insights to the decision makers. As a part of quantitative analysis, we develop regression models to predict optimal costs based on several factors. Our main contribution is to provide an efficient and effective tool to handle a large-scale real-world problem. The model has also helped to prove that the organization responsible for distribution planning may move from the current planning practice to an all-encompassing top-down approach.