lec3_mcf_keypaths_col_and_row_generation_2003

lec3_mcf_keypaths_col_and_row_generation_2003 - 1...

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Unformatted text preview: 1 .206J/16.77J/ESD.215J Airline Schedule Planning Cynthia Barnhart Spring 2003 1.206J/16.77J/ESD.215J Multi-commodity Network Flows: A Keypath Formulation Outline Path formulation for multi-commodity flow problems revisited Keypath formulation Example Keypath solution algorithm Column generation Row generation Path Notation Sets A:set of all network arcs K:set of all commodities N: set of all network nodes Parameters u ij : total capacity on arc ij d k : total quantity of commodity k P k : set of all paths for commodity k, for all k Parameters (cont.) c p : per unit cost of commodity k on path p = ij p c ij k ij p : = 1 if path p contains arc ij; and = 0 otherwise Decision Variables f p : fraction of total quantity of commodity k assigned to path p The Path Formulation Revisited MINIMIZE k K p P k d k c p f p subject to: p P k k K d k f p ij p u ij 2200 ij A p P k f p = 1 2200 k K f p 0 2200 p P k , 2200 k K The Keypath Concept The path formulation for MCF problems can be recast equivalently as follows: Assign all flow of commodity k to a selected path p, called the keypath, for each commodity k K Often the keypath is the minimum cost path for k The resulting flow assignment is often infeasible One or more arc capacity constraints are violated If the resulting flows are feasible and the keypaths are minimum cost, the flow assignment is optimal Solve a linear programming formulation to minimize the cost of adjusting flows to achieve feasibility Flow adjustments involve removing flow of k from its keypath p and placing it on alternative path p P k , for Additional Keypath Notation Parameters p(k) : keypath for commodity k Q ij : total initial (flow assigned to keypaths) on arc ij = k K d k ij p(k) c r p(k) : = c r c p(k) =...
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This note was uploaded on 11/08/2011 for the course AERO 16.72 taught by Professor Hansman during the Fall '06 term at MIT.

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lec3_mcf_keypaths_col_and_row_generation_2003 - 1...

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