Stochastics and Statistics Dynamic shortest path in stochastic dynamic networks: Ship routing problem Amir Azaron a, * , Farhad Kianfar b a Department of Industrial Engineering, University of Bu-Ali-Sina, P.O. Box 65176, Ghobare Hamadani Boulevard, Hamadan, Iran b Department of Industrial Engineering, Sharif University of Technology, Azadi Street, Tehran, Iran Received 28 March 2000; accepted 4 October 2001 Abstract In this paper, we apply the stochastic dynamic programming to find the dynamic shortest path from the source node to the sink node in stochastic dynamic networks, in which the arc lengths are independent random variables with exponential distributions. In each node there is an environmental variable, which evolves in accordance with a con- tinuous time Markov process. The parameter of the exponential distribution of the transition time of each arc is also a function of the state of the environmental variable of its initiative node. Upon arriving at each node, we can move toward the sink node through the best outgoing arc or wait. At the beginning, it is assumed that upon arriving at each node, we know the state of its environmental variable and also the states of the environmental variables of its adjacent nodes. Then we extend this assumption such that upon arriving at each node, we know the states of the environmental variables of all nodes. In the ship routing problem, which we focus in this paper, the environmental variables of all nodes are known, but it is shown that the complexity of the algorithm becomes exponential in this case. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Dynamic programming; Stochastic processes; Markov processes 1. Introduction This paper develops an algorithm to find the dynamic shortest path from the source node to the sink node in acyclic networks with the following specifications. It is assumed that the arc lengths are inde- pendent random variables with exponential distributions. In each node (except the sink node) there is an environmental variable, which evolves in accordance with a continuous time Markov process. The transitions of each environmental variable influence the parameters of the exponential distributions of the lengths of the related outgoing arcs and consequently the expected lengths of the outgoing arcs. It is also assumed that all continuous time Markov processes corresponding to the environmental variables are independent. At the beginning, it is assumed that upon arriving at each node, we know the state of its en- vironmental variable and also the states of the environmental variables of its adjacent nodes that all together European Journal of Operational Research 144 (2003) 138–156 www.elsevier.com/locate/dsw * Corresponding author. E-mail addresses: aazaron@is.dal.ca (A. Azaron), fkianfar@sina.sharif.ac.ir (F. Kianfar). 0377-2217/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII:S0377-2217(01)00385-X