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lec3_mcf_keypaths_col_and_row_generation_2003

# lec3_mcf_keypaths_col_and_row_generation_2003 -...

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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 = Σ ij ∈ p c ij k δ 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 Σ k ∈ K Σ p ∈ P k d k c p f p subject to: Σ p ∈ P k Σ k ∈ K d k f p δ ij p ≤ u ij 2200 ij ∈ A Σ 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 = Σ k ∈ K d k δ ij p(k) c r p(k) : = c r – c p(k) = Σ...
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lec3_mcf_keypaths_col_and_row_generation_2003 -...

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