Abstract The design of distribution networks that simultaneously consider location and inventory decisions seeks to balance costs and product availability. The most commonly observed measure of product availability in practical settings is the fill‐rate service level. However, the optimal design of a distribution network that considers the fill rate to control shortages of fast‐moving consumer goods (FMCG) is considered intractable and has only been addressed by heuristic methods. This paper addresses the optimal design of a distribution network for FMCG able to provide high fill‐rate service level under a continuous review policy. Considering the exact formulation for the provided fill rate, we formulated a joint location–inventory model with fill‐rate service level constraints as a convex mixed integer nonlinear problem for which a novel decomposition‐based outer approximation algorithm is proposed. Numerical experiments have shown that our solution approach provides good‐quality solutions that are on average 0.15% and, at worst, 2.2% from the optimal solution.