Two bipatite matching problems arising in Vehicle Scheduling are considered: the capacitated matching and the multicommodity matching. For the former, given a reasonable cost structure, we can exhibit a polynomial time algorithm, while the general case is conjectured to be NP-hard. The latter problem is shown to be NP-hard. A heuristic algorithm based on Lagrangean relaxation for the capacitated version of the multicommodity matching is also presented together with experimental results.
VEHICLE SCHEDULING AND MATCHINGIn this paper two particular Matching Problems are considered, which arise in Vehicle Scheduling. With the term Vehicle Scheduling we refer to that broad class of optimization problems [l], [2], [5] in which vehicles must be assigned to time-tabled trips in such a way that: (i) each trip is run by one vehicle; (ii) a given set of constraints is satisfied; (iii) a cost function is minimized.Let V = {uI,ut, . . . ,u,} be the set of trips. The ordered pair of trips (ui,uj) is said to be a compatible pair of trips if the same vehicle can run trips ui and ui in the sequence. The compatibility depends on the starting and ending times and terminals of the trips. It is quite reasonable to assume that the set V with the compatibility relation be a partially ordered set (poser). We assume throughout that there is only one vehicle type, and that each vehicle can run each trip.The simplest case of vehicle scheduling problem is the single depot problem; in this problem all the vehicles are housed at the same depot, and no "a priori"
AbstractÐHard-real-time systems require predictable performance despite the occurrence of failures. In this paper, fault tolerance is implemented by using a novel duplication technique where each task scheduled on a processor has either an active backup copy or a passive backup copy scheduled on a different processor. An active copy is always executed, while a passive copy is executed only in the case of a failure. First, the paper considers the ability of the widely-used Rate-Monotonic scheduling algorithm to meet the deadlines of periodic tasks in the presence of a processor failure. In particular, the Completion Time Test is extended so as to check the schedulability on a single processor of a task set including backup copies. Then, the paper extends the well-known Rate-Monotonic First-Fit assignment algorithm, where all the task copies, included the backup copies, are considered by Rate-Monotonic priority order and assigned to the first processor in which they fit. The proposed algorithm determines which tasks must use the active duplication and which can use the passive duplication. Passive duplication is preferred whenever possible, so as to overbook each processor with many passive copies whose primary copies are assigned to different processors. Moreover, the space allocated to active copies is reclaimed as soon as a failure is detected. Passive copy overbooking and active copy deallocation allow many passive copies to be scheduled sharing the same time intervals on the same processor, thus reducing the total number of processors needed. Simulation studies reveal a remarkable saving of processors with respect to those needed by the usual active duplication approach in which the schedule of the non-fault-tolerant case is duplicated on two sets of processors.
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