The principal focus of this research effort is to investigate a maritime stochastic transportation problem. In this case, crude oil shipments are to be transported from multiple sources to multiple destinations. The demand at the destinations is normally distributed and the violation of certain specified lower and upper daily storage limits can lead to various types of penalties. We aim to utilize (stochastic) mathematical programming approaches to minimize the overall cost of such fleet operation by optimizing the vessel schedules and maintaining acceptable daily storage levels considering the stochastic demand structures. This research effort signifies that the nature of daily demand distributions in the multiple sources-destinations scheduling-inventory scenario significantly impacts the overall fleet schedules and the total expected cost. Therefore, it is crucial to grasp essential stochastic aspects of the daily demands to avoid potential misrepresentation of the operational costs. The robustness of the adopted approach is illustrated by presenting computational results that are based on a wide range of test problems. Moreover, our computational study also examines the impact of variations in demand and the probability of meeting demands on the cost structures.