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We consider a problem that marries network flows and scheduling, motivated bythe need to schedule maintenance activities in infrastructure networks, such asrail or general logistics networks. Network elements must undergo regularpreventive maintenance, shutting down the arc for the duration of the activity.Careful coordination of these arc maintenance jobs can dramatically reduce theimpact of such shutdown jobs on the flow carried by the network. Schedulingsuch jobs between given release dates and deadlines so as to maximize the totalflow over time presents an intriguing case to study the role of timediscretization. Here we prove that if the problem data is integer, and no flowcan be stored at nodes, we can restrict attention to integer job start times.However if flow can be stored, fractional start times may be needed. This makestraditional strong integer programming scheduling models difficult to apply.Here we formulate an exact integer programming model for the continuous timeproblem, as well as integer programming models based on time discretizationthat can provide dual bounds, and that can - with minor modifications - alsoyield primal bounds. The resulting bounds are demonstrated to have small gapson test instances, and offer a good trade-off for bound quality againstcomputing time.

Scheduling Network Maintenance Jobs with Release Dates and Deadlines to Maximize Total Flow Over Time: Bounds and Solution Strategies