A Polynomial Time Approximation Scheme for the Multi-vehicle Scheduling Problem on a Path with Release and Handling Times

Abstract

In this paper, we consider a scheduling problem of vehicles on a path. Let G = (V,E) be a path, where V= {v ₁, v ₂, . . . , vn} is its set of n vertices and E{{v _j , v _j }+1} j = 1, 2, . . . , n . 1 is its set of edges. There are m vehicles (1 ≤ m ≤ n). The travel times w v _j , v _j +1) (=w v _j +1, v _j are associated with edges v j, v _j +1∈ E. Each job j which is located at each vertex vj ∈ GBV has release time rj and handling time hj . Any job must be processed by exactly one vehicle. The problem asks to find an optimal schedule of m vehicles that minimizes the maximum completion time of all the jobs. The problem is known to be NP-hard for any fixed m ≥ 2. In this paper, we present a polynomial time approximation scheme A _ε to the problem with a fixed m. Our algorithm can be extended to the case where G is a tree so that a polynomial time approximation scheme is obtained if m and the number of leaves in G are fixed