Motivated by a real-world application at a large international pharmaceutical company, we tackle an integrated location-inventory-transportation problem under demand uncertainty. The supply chain network of this problem comprises multiple plants, distribution centers (DCs), and customers. The decision-making process involves simultaneously determining the facility locations, inventory planning, and transportation volumes. Apart from the computational complexity resulting from this integration, other practical challenges arise from the fact that the planner must determine inventory policies that account for safety stock consolidation, whereas transportation is charged based on volume-based piecewise linear costs. To this end, we propose an exact and an approximate solution framework to solve this problem. The exact approach is based on a logic-based Benders decomposition (LBBD) framework enhanced by a piecewise-linear lower-bound function and efficient logic cuts. We then improve the scalability by leveraging an approximate model with a piecewise linear approximation for safety stock computation. Finally, using the instances derived from real-world data, we empirically demonstrate the benefit of the integrated model, which yields up to 9% of potential cost savings.