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13 - Airline Scheduling Design and Analysis of Algorithms...

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Airline Scheduling Design and Analysis of Algorithms Andrei Bulatov

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Algorithms – Airline Scheduling 13-2 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 13-3 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

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Algorithms – Airline Scheduling 13-4 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 Objective : Is it possible to serve the required flights with k planes
Algorithms – Airline Scheduling 13-5 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

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Algorithms – Airline Scheduling 13-6 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 - For each i and j such that flight j is reachable from flight i, there i v i u ) , ( i i v u 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 13-7 The Result Theorem There is a way to perform all flights using at most k planes if and only if there is a feasible circulation in the network G.

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