MINIMIZING THE NUMBER OF VEHICLES

MINIMIZING THE NUMBER OF VEHICLES - AnAE- s ;~~~~RY N: S·E...

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Unformatted text preview: AnAE- s ;~~~~RY N: S·E DL :SM00n ~: :·-;"; " : 0 0; 0: 0 : ~: w or k i;s s p :~ : 0- -0- -:- E X ; a..;g. :: ~ ~ ~ 1: ,: :;: :Dd- :i .: : ::i f0 ; :-; :If ' 5:y.0:230003CtE~~~~~~~ ~ ~ ~~~~~~~: .· 0;0E0000:0000 ... _: :::.... .... ,, ,.... . .- ;.-.. -- ....... - . : ..-... -- :.:: ...- 0;.''0-';I f:~ ,00D000fff:0' Hf'0000 ::-.', : it00"ffE ; 0 t:05 St ' MASSecHUSETTS. INST iTUTE HNOF TECHOLOG M-M~ ~ i-i:~l::::i:: ;-::: ---- ~:::::- "--·!7 j:;- N T_·p:- ·:; -n-·I··· ·. ':· ,·--·:.· · :: r r ri- · ': · ! I 1 : :·':: :. :I: ': 'i 1··. i·· ? ·-· ri ··:! i ;r i .. . :· I- ·I.._i · . -- · _ --:· · · ·I .. .···; :; i r · .'·-- -·1 ·· : I . .. ;.-·\·:· : ··:- : Iii : ..- 1 MINIMIZING THE NUMBER OF VEHICLES TO MEET A FIXED PERIODIC SCHEDULE: AN APPLICATION OF PERIODIC POSETS By James B. Orlin OR 102-80 October 1980-1- 1. INTRODUCTION In 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 p.m. to Boston at 6 a.m.) so as to minimize the number of airplanes,.and deadheading between airports is allowed. This example may easily be extended to both bus scheduling and train scheduling. The finite horizon version of the above vehicle scheduling problem was solved by Dantzig and Fulkerson [DF]. The periodic version in which dead- heading is forbidden was solved by Bartlett [Ba] and by Bartlett and Charnes [BC]. (If deadheading is forbidden, the only question is how to start the schedule. Once in operation, a FIFO scheduling procedure is optimal). Finally, Orlin [02] has solved the more general problem of minimizing the average linear cost per day of flying a schedule subject to a fixed number of airplanes. This last paper involved a solution technique substantially different from the technique presented here, although both involve solutions induced from finite minimum-cost network flows. The periodic vehicle scheduling problem may be expressed in terms of task scheduling as follows: What is the minimum number of individuals to meet a fixed periodically repeating set of tasks? (For airplane scheduling, the tasks are flights and the individuals are airplanes.) This problem is shown in Section 4 to be a special case of the "minimum chain-cover problem for periodic partially ordered sets." This latter problem is formulated in Section 2 and a large subclass of this problem is solved as a minimum cost network flow problem in Section 3. The entire problem is solved in the _ ·_1_11_ -111111 ·._ _.__·__I__-2- appendix. These results generalize the work of Ford and Fulkerson [FF] who showed that the finite version of the above task scheduling problem may ,be solved as a special case of the minimum chain-cover problem for finite partially ordered sets....
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MINIMIZING THE NUMBER OF VEHICLES - AnAE- s ;~~~~RY N: S·E...

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