Minimum time paths in a network with mixed time constraints

Abstract

The time-constrained shortest path problems are generalized from and more realistic than the classical shortest path problem. These problems are most important subproblems in many large-scale real-world problems in the field of transportation. The problem considered in this paper is to find the shortest path in a new kind of time-constrained network, called a mixed time-schedule network, in which departures from some nodes are only allowed at some discrete points. This problem arises in many practical situations. For example, a shipping or transportation from one place to another may be through different kinds of vehicles such as (1) scheduled flights, ocean liners, trains, or buses and (2) rental cars, own cars or own trucks, where the vehicles in (1) have to follow predetermined departure-time schedules and the vehicles in (2) do not have time constraints. The time-constrained shortest path problem is an important generalization of the shortest path problem and has attracted much research interest in recent years. In a recent paper, a new time constraint, namely time-schedule constraint, is introduced. This time constraint assumes that every node in the network has a list of pre-specified departure times and requires that departure from a node take place only at one of these departure times. Therefore, when a time-schedule constraint is considered, the total time in a network includes traveling time and waiting time. In this paper, we consider a network consisting of two types of nodes in terms of their time constraints. The first type of nodes are subject to time-schedule constraints, but the second type is not. For such a network, a set of minimum time (shortest) path problems is studied, including minimization of total time, minimization of total time subject to a total traveling time constraint, minimization of total traveling time subject to a total time constraint and minimization of a weighted sum of total time and total traveling time.