Applying Dijkstra's algorithm for general shortest path problem with normal probability distribution arc length

We present a general treatment of shortest path problems by dynamic program in networks having normal probability distributions as arc lengths. We consider a network that may contains cycles. Two operators of sum and comparison need to be adapted for the proposed Dijkstra's algorithm. Because of inefficiency of the methods such as maximum likelihood estimation and moment generating function due to long computational efforts and inaccurate solution results, convolution approach is used to sum two normal probability distributions being employed in the Dijkstra's algorithm. Generally, stochastic shortest path problems are treated using expected values of the arc probabilities, but in the proposed method using distributed observed past data as arc lengths, an integrated value is obtained as the shortest path length. The objective of this paper is to extend the general shortest path problem in dynamic and stochastic networks where link travel times are defined as normal probability distributions. We solve a numerical example to show efficiency of our proposed approach and after finding the shortest path in the network we obtain the shortest path's density function, too.