Critical path in an activity network with time constraints

An acyclic graph with nonnegative weights and with a unique source and destination is called an activity network. A project comprised of a set of activities and precedence relationships can be represented by an activity network and the mathematical analysis of the network can provide useful information for managing the project. In a traditional activity network, it is assumed that an activity can begin any time after all of its preceding activities have been completed. This assumption does not adequately describe many practical situations, in which some kinds of time constraint are usually associated with an activity. In this paper, we investigate two types of time constraint commonly encountered in project management. The first is the time-window constraint, which assumes that an activity can begin its execution only in a specified time interval. The second is the time-schedule constraint, which requires that an activity begin only at one of pre-specified beginning times. An efficient, linear time algorithm for finding the longest path (critical path) and for analyzing the flow time of each arc is developed for activity networks with these time constraints.