Most of previous works related to survivable networks have been focused on unicast communications. In this paper, we address the problem of network survivability in the context of anycast communications, which has not attracted much attention recently. Anycast transmission is defined as one-to-one-of-many. To assure the network survivability, we use the single backup path approach, i.e., each demand has two node-disjoint paths to provide protection against a single link failure. We formulate new ILP models to find the optimal paths for anycast and unicast connections, as well as to find the optimal location of replica servers. To test the proposed approach, we run extensive numerical experiments using CPLEX solver on four example network topologies with different scenarios of replica server count and the proportion between unicast and anycast traffic. Results prove the efficiency of our solution in terms of various parameters including cost, path length, resource utilization, and time of computation.