Better approximation ratios for the single-vehicle scheduling problems on line-shaped networks

Authors

• Yoshiyuki Karuno,

Corresponding author
1. Department of Mechanical and System Engineering, Faculty of Engineering and Design, Kyoto Institute of Technology, Sakyo, Kyoto 606-8585, Japan
• Department of Mechanical and System Engineering, Faculty of Engineering and Design, Kyoto Institute of Technology, Sakyo, Kyoto 606-8585, Japan
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• Hiroshi Nagamochi,

1. Department of Information and Computer Sciences, Faculty of Engineering, Toyohashi University of Technology, Toyohashi, Aichi 441-8580, Japan
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• Toshihide Ibaraki

1. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Sakyo, Kyoto 606-8501, Japan
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Abstract

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 vV, has release time r(v) and handling time h(v). There is a single vehicle, which is initially situated at v1V, 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 [21], and there exists an approximate algorithm with the approximation ratio of 2 [12]. 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 vV and d(vn+1) for the vehicle. This problem is also NP-hard [13]. We improve the previous best-known approximation ratio of 2, which was obtained in [11], to 1.5. © 2002 Wiley Periodicals, Inc.