We are interested in evaluating the strength of a network by determining the maximum number of failures that it can face. This can be done by solving a multi-commodity flow blocker problem. Given a directed graph and a set of commodities (or demands), the multi-commodity flow problem (MCFP) consists in finding a set of arcs, called multicommodity flow, routing flow to satisfy all demands, with respect to flow conservation constraints and capacity constraints. The MCFP aims to minimize the total flow cost. We study the blocker problem applied to the multi-commodity flow problem, which is called the multi-commodity flow blocker problem (MCFBP). The MCFBP consists in finding a minimum set of arcs to remove from the graph in such a way that the minimum cost of the multi-commodity flow in the remaining graph is greater than or equal to a given threshold, called target cost value. We introduce a new IP model with an exponential number of constraints, called cover constraints, to solve the multi-commodity flow blocker problem. Our goal is to provide a thin and well-understood formulation. We examine the polyhedral structure of the model proposed and give new valid inequalities to strengthen the formulation. Using this we develop a Branch-and-cut algorithm to solve the model.