Let $G=(V,A)$ be a directed graph and $F$ be a set of items. The Location-Dispatching Problem consists of determining subsets $L_i \subseteq F$ located at nodes $i \in V$, minimizing the sum of two costs: a piecewise linear installation cost associated with $L_i$ and an access cost for each node of $V$ to reach a copy of each item of $F$. We formulate this problem as a linear program with binary variables $x$ and integer variables $z$. We propose a facial study of the associated polytope and we introduce the so-called integrity hop cost inequalities that force $z$ to be an integer as soon as $x$ is binary. Using this, we devise a branch-and-cut algorithm and report some experimental results. This algorithm has been used to solve Content Delivery Network instances in order to optimize a Video On Demand (VoD) system.