For the widely use of multistage decision-making problem in our normal life such as in the new research area of dynamic selection of composite web services, this paper exerts all its effort on proposing a new approach to solve such problem. Motivated by transiently chaotic neural networks' high parallel performance and powerful computation, a novel transiently chaotic neural network is proposed in this paper for this task. In order to make this algorithm more adaptable for multistage decision-making problem, a new neural network structure for implementing the algorithm is proposed which is a modification to the one used by Thomopoulos or Rauch and Winarske.