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Unformatted text preview: Airline Scheduling Design and Analysis of Algorithms Andrei Bulatov Algorithms Airline Scheduling 132 The Problem An airline carrier wants to serve certain set of flights Example: Boston (6 am)  Washington DC (7 am), San Francisco (2:15pm)  Seattle (3:15pm) Philadelphia (7 am)  Pittsburg (8 am), Las Vegas (5 pm)  Seattle (6 pm) Washington DC (8 am) Los Angeles (11 am) Philadelphia (11 am)  San Francisco (2 pm) The same plane can be used for flight i and for flight j if the destination of i is the same as origin of j and there is enough time for maintenance (say, 1 hour) a flight can be added in between that gets the plane from the destination of i to the origin of j with adequate time in between Algorithms Airline Scheduling 133 Formalism Boston (6 am)  Washington DC (7 am), Philadelphia (11 am)  San Francisco (2 pm) Philadelphia (7 am)  Pittsburg (8 am), San Francisco (2:15pm)  Seattle (3:15pm) Washington DC (8 am) Los Angeles (11 am), Las Vegas (5 pm)  Seattle (6 pm) BOS 6 DC 7 DC 8 LAX 11 LAS 5 SEA 6 Flight j is reachable from flight i if it is possible to use the same plane for flight i, and then later for flight j as well. (or we can use a different set of rules, it does not matter) PHL 7 PIT 8 PHL 11 SFO 2 SFO 2:15 SEA 3:15 Algorithms Airline Scheduling 134 The Problem The Airline Scheduling Problem Instance : A set of flights to serve, and a set of pairs of reachable flights, the allowed number k of planes bjective Objective : Is it possible to serve the required flights with k planes Algorithms Airline Scheduling 135 The Idea Each airplane is represented by a unit of flow The required flights (arcs) have lower bound 1, and capacity 1 If and are arcs representing required flights i and j, then there is an arc connecting to ; we assign this arc capacity 1 ) , ( i i v u ) , ( j j v u i v j u Extend the network by adding an external source and sink BOS 6 DC 7 PHL 7 PIT 8 DC 8 LAX 11 PHL 11 SFO 2 SFO 2:15 SEA 3:15 LAS 5 SEA 6 Algorithms Airline Scheduling 136 Construction For each required flight i, the graph G has two nodes and G also has a distinct source s and a sink t For each i, there is an arc with a lower bound 1 and capacity 1 or each i and j such that flight j is reachable from flight i, there i v i u ) , ( i i v u For each i and j such that flight j is reachable from flight i, there is an arc with a lower bound 0 and a capacity 1 For each i there is an arc with a lower bound 0 and a capacity 1 For each i there is an arc with a lower bound 0 and a capacity 1 There is an arc (s,t) with lower bound 0 and capacity k The node s has demand k, and node t has demand k ) , ( j i u v ) , ( i u s ) , ( t v i Algorithms Airline Scheduling...
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This note was uploaded on 11/11/2009 for the course CS 405/705 taught by Professor Bulatov during the Fall '09 term at Simon Fraser.
 Fall '09
 Bulatov
 Algorithms

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