Time-varying shortest path problems with constraints

The shortest path problem is considered with yet another extension. The transit times b(x,y,u) and the arc costs c(x,y,u) on arc (x,y) vary as a function of the departure time u at the beginning vertex of the arc. The problem considered is to find the shortest path between a pair of nodes such that the total traverse time is lesser than some specified value T. Waiting times at nodes are considered as decision variables in this paper as well. Three variations of the problem are analyzed in which the waiting times are allowed to be arbitrary, the waiting times are zero and the waiting time has an upper bound at each vertex. Algorithms are also discussed to solve the problem. Some of the applications of the problem are encountered in communication networks and freight delivery problems. Suppose that some freight is to be sent from one city to another in a network before a deadline T. Different types of freight services with different costs and transit times are available between the two cities. An optimal solution would have to specify the route as well as the waiting times of the freight at each city so that the overall cost is minimum. A dynamic programming approach is used to compute the shortest path from the origin node to the destination node. Three cases are considered for the waiting times and an algorithm is suggested for each of those cases. This problem is obviously an NP-Complete problem as it is more difficult to solve than a problem with constant transit times and arc costs such as the previous problem which is NP-Complete. Some definitions: Waiting time w(x i) at a vertex x i is a nonnegative integer with an upper bound U x >= 0. Let P be a path from vertex x 1 to x r. The departure time T(x i) at a vertex x i can be expressed as below: T(x 1) = w(x 1) The departure time for all other vertices on the path can be expressed recursively as : The time of path P is defined as T(x r) where x r is the final node on the path..