Alternative Formulations for o-D Commodity case Node-Arc Formulation Decision variables: flow of commodity k on each arc ii Path formulation Decision variables: flow of commodity k on each path for k Tree?' or Sub-network?", Formulation Define: super commodity: set of all(O-D)commodities with the same origin o(or destination d Decision variables: quantity of the super commodity k as signed to each“ tree or“sub- network>, for k Formulations are equivalent 2/212021 Barnhart 1.206J/16.77J/ESD. 15J2/21/2021 Barnhart 1.206J/16.77J/ESD.215J 11 Alternative Formulations for O-D Commodity Case • Node-Arc Formulation – Decision variables: flow of commodity k on each arc ij • Path Formulation – Decision variables: flow of commodity k on each path for k • “Tree” or “Sub-network” Formulation – Define: super commodity: set of all (O-D) commodities with the same origin o (or destination d) – Decision variables: quantity of the super commodity k’ assigned to each “tree” or “sub-network” for k’ • Formulations are equivalent