We consider two variants of the single-vehicle scheduling problem on line-shaped networks. Let L = (V, E) be a line, where V = {v1, v2,. . ., vn} is a set of n vertices and E = {{vi, vi+1}|i = 1, 2, . . ., n − 1} is a set of edges. The travel times w(u, v) and w(v, u) are associated with each edge {u, v} ∈ E, and each job, which is also denoted as v and is located at vertex v ∈ V, has release time r(v) and handling time h(v). There is a single vehicle, which is initially situated at v1 ∈ V, and visits all vertices to process the jobs before it returns back to v1. The first problem asks to find an optimal routing schedule of the vehicle that minimizes the completion time. This is NP-hard, and there exists an approximate algorithm with the approximation ratio of 2. In this paper, we improve this ratio to 1.5. On the other hand, the second problem minimizes the maximum lateness, under the assumption that all release times r(v) are zero, but there are due times d(v) for v ∈ V and d(vn+1) for the vehicle. The problem is also NP-hard. We improve the previous best-known approximation ratio of 2, to 1.5.
