Accurate outbound volume prediction of refined oil depots is vital for maintaining the supply-demand balance and optimizing transportation efficiency. However, the large number of refined oil depots, each typically storing multiple types of oil products with interrelated outbound volumes, poses a challenge for existing methods, which struggle to deliver accurate results due to overlooking these latent correlations and often suffer from low efficiency. This paper introduces an innovative multi-task learning framework for jointly predicting the outbound volume of different refined oil products in the same depot. To capture comprehensive features, trends in multiple outbound volume data are extracted using Ensemble Empirical Mode Decomposition, and the overall features are carefully selected and reconstructed. Auxiliary variables that may influence outbound volume are also included. After in-depth analysis of the impact of different features on multiple outbound volume, an improved Multi-gate Mixture-of-Experts (MMOE) framework is developed to integrate the coupled multidimensional temporal features of the outbound volume of refined oil depots. Eventually, several real-world cases from different types of oil depots are used for model verification and performance comparisons. Combined with Ensemble Empirical Mode Decomposition, the effects of different MMOE structures on prediction accuracy are tested. The results demonstrate that the proposed framework achieves a more accurate prediction than other time series prediction models, with a significant decrease of 58% in Root Mean Square Error and 64% in Mean Absolute Percentage Error, and an increase in R-squared to over 0.95.