Scheduling unit time arc shutdowns to maximize network flow over time: Complexity results

Abstract

We study the problem of scheduling maintenance on arcs of a capacitated network so as to maximize the total flow from a source node to a sink node over a set of time periods. Maintenance on an arc shuts down the arc for the duration of the period in which its maintenance is scheduled, making its capacity zero for that period. A set of arcs is designated to have maintenance during the planning period, which will require each to be shut down for exactly one time period. In general this problem is known to be NP-hard. Here we identify a number of characteristics that are relevant for the complexity of instance classes. In particular, we discuss instances with restrictions on the set of arcs that have maintenance to be scheduled; series parallel networks; capacities that are balanced, in the sense that the total capacity of arcs entering a (non-terminal) node equals the total capacity of arcs leaving the node; and identical capacities on all arcs.

Scheduling arc shut downs in a network to maximize flow over time with a bounded number of jobs per time period

“…For instance, C |J| sp is the class of instances with a series-parallel network and no limit on the number of jobs per time period, and C |J|/3 contains the instances in which at most one third of all jobs can be scheduled per time period. As proved in [2], the classes C |J| aa and C |J| sp ∩ C |J| bal are trivial: it is always optimal to schedule all jobs at the same time. In contrast, the restriction of the problem to C |J| bal is still strongly NP-hard, and the restriction to C |J| sp is NP-hard, but for fixed T it can be solved in pseudopolynomial time using dynamic programming.…”

“…Boland et al [4,5] introduced a general network optimization problem in which arc maintenance jobs need to be scheduled so as to maximize the total flow in the network over time. A simplified version of the problem in which all jobs have unit processing time was studied in [2], and the complexity was determined taking into account certain instance characteristics, such as special network structures and restrictions on the set of jobs.…”

“…Let C be the class of all MFASS instances. With an upper index K we denote the class of all instances with an upper bound of K on the number of jobs scheduled per time period, and a lower index indicates additional restrictions as introduced in [2].…”

Scheduling network maintenance jobs with release dates and deadlines to maximize total flow over time: Bounds and solution strategies

“…This yields the property that if the data is integer, an optimal solution with integer start times is assured. 3. We demonstrate that when storage is allowed, even with integer data, non-integer job start times may be needed in an optimal solution.…”

“…In order to tackle a real problem in which maintenance jobs must be timed to within 15-minute intervals over a planning horizon of a year, the integer programming model presented in [4] is formulated in terms of a sparse set of possible start times selected heuristically for each job. The problem variants tackled in [3,5] (all without storage) simply state the problem as one in which jobs must start at integer times. Whilst this property of start times is intuitively reasonable in the case without storage, it has not yet been formally proved; the first such proof is one of the key contributions of this paper.…”