The hop constrained min-sum arborescence with outage costs problem consists of selecting links in a network so as to connect a set of terminal nodes N={2,3,…,n} to a central node with minimal expected annual cost. The maximum number of links between the central node and each terminal node j is limited to a predefined number h_j (the hop constraint). Each terminal node in the network has an associated outage cost, which is the economic cost borne by the network user whenever that node is disconnected from the central node due to failure of a link. We formulate this problem as an integer programming problem and suggest a Lagrangian relaxation-based heuristic to get a good solution to this network problem. Lower bounds found as a byproduct of the solution procedure are used to assess the quality of the heuristic solutions. Computational results over a wide range of problem structures involving up to 100 nodes are given showing the effectiveness of the proposed approach.