Shortest-path problems with time windows on nodes and arcs

A dynamic programming formulation of a shortest path problem is presented when there are time windows on the nodes and arcs of a graph. A time window at a node i is given by [α(i), β(i)] and a time window at arc(i, j) is given by [α(ij), β(ij)]. The shortest path from source node 1 to sink N is formulated and satisfies all time window constraints, i.e., arriving at node i when time window is open and traveling along arc(i, j) when time window along arc is open.