Finding optimal tour schedules on transportation paths under extended time window constraints

This paper addresses time-critical routing on a given path under release dates and deadline restrictions, while the minimization of the total weighted completion time is pursued. Since there may be destinations with flexible picking area resources that enable a delivery of goods before the defined release date at no additional costs, the well-known Line-Traveling Repairman problem (Line-TRPTW) is extended into the Line-TRPeTW. The Line-TRPeTW has various practical applications, such as, for example, the tour planning of an inland container ship along a river or that of a vehicle along a coastline. Although the feasibility variant of the Line-TRPTW/Line-TRPeTW is known to be strongly NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {NP}}$$\end{document}-hard, a first practically applicable Branch&Bound procedure is generated. By making use of different dominance rules and lower bounds, a comprehensive computational study proves that instances comprising up to 200 requests and locations are solved to optimality in reasonable time. Moreover, the paper analyzes the complexity of the simpler variant with relaxed release dates at all customer locations. This relaxed variant provides tight lower bounds. Furthermore, the complexity analysis shows that the relaxed problem variant is binary NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {NP}}$$\end{document}-hard even if deadlines are ignored, but can be efficiently solved by a specific Dynamic Programming approach that runs in pseudo-polynomial time.