This paper investigates a shortest path problem on a network in which a stochastic number is assigned to each arc length. The stochastic optimization models of the shortest path are constructed according to expected value and chance-constrained programming. An efficient method is proposed to conversion of the stochastic optimization problems to their respective deterministic equivalents, and Dijkstra's algorithm is used to deal with the problems. Finally, a numerical experiment is given to show the effectiveness of the proposed method.