This paper investigates a shortest-path network problem using an annealed ant system algorithm, in which an annealing strategy is embedded to calculate the probabilities to decide which path the ants will select next. The shortest-path problem is to determine the shortest route between a source and a destination in a transportation-network topology. In this approach, according the concrete problems of shortest routing, we construct two globally optimizing annealed ant algorithms that are Concentrated Model and Distributed Model. The Concentrated Model (CM) means all ants are initially concentrated in the source node while all ants randomly select a node except the destination as their starting point initially and at least one must appear in the source node for the Distributed Model (DM). The experimental results show that the proposed annealed ant algorithm with the Roulette wheel selection can obtain better performance than that generated by the traditional ant strategy with/without the Roulette wheel selection.