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 total link cost such that (a) each terminal node j has exactly one entering link; (b) for each terminal node j, a unique path from the central node to j exists; (c) for each terminal node j the number of links between the central node and j is limited to a predefined number hj, and (d) each terminal node has an associated outage cost, which is the economic cost incurred by the network user whenever that node is disabled due to failure of a link. We suggest a Lagrangian based heuristic to solve the integer programming formulation of this network problem.