Heuristics for a scheduling problem of minimizing the number of one-way vehicles on paths with deadlines

In this paper, we deal with a vehicle scheduling problem on paths with handling times and deadlines. Let G = (V, E) be a path with a set V = {v(i) vertical bar i = 1, 2,..., n} of vertices and a set E = {{v(i), v(i+1)} vertical bar i = 1,2,..., n - 1} of edges. Vehicles are initially situated at v(1). There is a job i at each vertex v(i) is an element of V, which has its own handling time h(i) and deadline d(i). With each edge {v(i), v(i+1)} exactly once (i.e., it simply moves from v(1) to v(n) on G). Any vehicle is assumed to move on G under the one-way routing constraint. The problem asks to find a schedule of one-way vehicles that minimizes the number of vehicles, meeting the deadline constraints for all jobs. In this paper, we propose an iterative DP heuristic algorithm to the problem, and show that it runs in O(n(3)) time and the approximation ratio is O(log n).