Scheduling Capacitated One-Way Vehicles on Paths with Deadlines
Jun Uchida, Yoshiyuki Karuno, Hiroshi Nagamochi · SICE Journal of Control Measurement and System Integration · 2009
In this paper, we deal with a scheduling problem of minimizing the number of employed vehicles on paths. Let G=(V,E) be a path with a set V={vi|i=1,2,...,n} of vertices and a set E={{vi,vi+1}|i=1,2,...,n-1} of edges. Vehicles with capacity b are initially situated at v1. There is a job i at each vertex vi∈V, which has its own handling time hi and deadline di. With each edge {vi,vi+1}∈E, a travel time wi,i+1 is associated. Each job is processed by exactly one vehicle, and the number of jobs processed by a vehicle does not exceed the capacity b. A routing of a vehicle is called one-way if the vehicle visits every edge {vi,vi+1} exactly once (i.e., it simply moves from v1 to vn on G). Any vehicle is assumed to follow the one-way routing constraint. The problem asks to find a schedule that minimizes the number of one-way vehicles, meeting the deadline and capacity constraints. A greedy heuristic is proposed, which repeats a dynamic programming procedure for a single one-way vehicle problem of maximizing the number of non-tardy jobs. We show that the greedy heuristic runs in O(n3) time, and the approximation ratio is at most ln b+1.