Minimum-Peak-Cost Flows Over Time

When planning transportation whose operation requires non-consumable resources, the peak demand for allocated resources is often of higher interest than the duration of resource usage. For instance, it is more cost-effective to deliver parcels with a single truck over eight hours than to use two trucks for four hours, as long as the time suffices. To model such scenarios, we introduce the novel minimum peak cost flow over time problem, whose objective is to minimise the maximum cost at all points in time rather than minimising the integral of costs. We focus on minimising peak costs of temporally repeated flows. These are desirable for practical applications due to their simple structure. This yields the minimum-peak-cost temporally repeated flow problem (MPC-TRF). We show that the simple structure of temporally repeated flows comes with the drawback of arbitrarily bad approximation ratios compared to general flows over time. Furthermore, our complexity analysis shows the integral version of MPC-TRF is strongly NP-hard, even under strong restrictions. On the positive side, we identify two benign special cases: unit-cost series-parallel networks and networks with time horizon at least twice as long as the longest path in the network (with respect to the transit time). In both cases, we show that integral optimal flows if the desired flow value equals the maximum flow value and fractional optimal flows for arbitrary flow values can be found in polynomial time. For each of these cases, we provide an explicit algorithm that constructs an optimal solution.