Minimizing the Number of Vehicles to Meet a Fixed Periodic Schedule: An Application of Periodic Posets

In this paper we consider countably infinite partially ordered sets (posets) in which the order relations occur periodically. We show that the problem of determining the minimum number of chains (completely ordered subsets) needed to cover all of the elements may be solved efficiently as a finite network flow problem. A special case of the chain-cover problem for periodic posets is the problem of minimizing the number of individuals to meet a fixed periodically repeating set of tasks with setup times between tasks. For example, if we interpret tasks as flights and individuals as airplanes, the resulting problem is to minimize the number of airplanes needed to fly a fixed daily repeating schedule of flights, where deadheading between airports is allowed. TN THIS PAPER we consider and solve in polynomial time the problem of minimizing the number of vehicles to meet a fixed periodic schedule. For example, consider an airline that wishes to assign airplanes to a set of fixed daily repeating flights (e.g., San Francisco at 10:00 p.m. to Boston at 6:00 a.m.) in order to minimize the number of airplanes where dead-heading between airports is allowed. The finite horizon version of the above vehicle scheduling problem was solved by Dantzig and Fulkerson [1954]. The periodic version in which deadheading is forbidden was solved by Bartlett [1957] and by Bartlett and Charnes [1957]. (If deadheading is forbidden, the only question is how to start the schedule. Once in operation, a FIFO scheduling procedure is optimaL) Dantzig [1962], in collaborative consulting work for United Airlines, solved the problem of minimizing the number of vehicles to meet a fixed periodic schedule under the restriction that the schedule of deadheading flights also repeats daily. Simpson [1969] describes Dantzig's solution technique (alongwith a host of solution techniques forrelated problems), and Orlin [1981a] proves that Dantzig's solution is also optimal if the stationarity constraint on deadheading is relaxmed Recently Wollmer Subject classification: 432 anapplication of periodic posets, 484 periodic vehicle scheduling via network flows, 832 the minimum number of vehicles to meet a periodic schedule.