In this paper, we study two versions of the multicast routing problem in multirate loss networks: complete and partial. In the complete version of the multicast routing problem, the identities of all destination nodes are available to the multicast routing algorithm at once. Conversely, in the partial version of the multicast problem, the identities of the destination nodes are revealed to the routing algorithm one by one. Although the complete version of the multicast routing problem, also known as the Steiner tree problem, has been well studied in the literature, less attention has been paid for the definition of link costs and evaluating the performance of multicast routing algorithm from the network revenue point of view. Therefore, in this paper, we first propose two approaches, namely, the Markov Decision Process-based (MDP-based) and Least Loaded Routing-based (LLR-based) approaches, for defining link costs. Several heuristic multicast routing algorithms are then proposed for both fully connected networks and sparsely connected networks. We have also proposed a new performance metric, referred to as fractional reward loss, for evaluating the performance of multicast routing algorithms. Our simulation results indicate that algorithms based on partial destination information yield worse performance than those based on complete information. We also found that, for fully connected networks, algorithms that use LLR-based link costs yield very competitive performance as compared to those that use MDP approach. However, for sparsely connected networks, LLR-based algorithms yield significantly worse performance as compared to the MDP-based algorithms.