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In this paper we deal with a single-vehicle scheduling problem on a tree-shaped road network. Let T = (V, E) be a tree, where V is a set of n vertices and E is a set of edges. A task is located at each vertex v E V, which is also denoted as v. Each task v has its due date d(v) and processing time p(v). The travel times w(u, v) and w(v, U,) are associated with each edge (U, v) E E. The vehicle starts from an initial vertex VQ E V, visits all tasks v E V for their processing (no preemption is allowed) and returns to VQ. The problem asks to find a routing schedule of the vehicle that minimizes the maximum lateness from the due dates of tasks. This problem is called TREE-VSP(Lmax). We prove that TREE-VSP(Lmax) is strongly NP-hard in general, but show that it can be solved in polynomial time if only depth-first routing is allowed for the vehicle.

VEHICLE SCHEDULING ON A TREE TO MINIMIZE MAXIMUM LATENESS