Multicommodity Supply and Transportation Networks with Resource Constraints

The Generalized Multicommodity Flow Problem

by Richard D. Wollmer


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A method for treating multicommodity network flows in which limited resources are shared among several arcs instead of only one. Applications include solving practical problems of vehicle scheduling, military interdiction, and two-way traffic flow. This study extends the previous solution methods for networks with individual arc capacity constraints to cover joint constraints. This formulation can handle one or more joint capacity constraints in a multicommodity network; with some adjustment in the objective function, it can maximize a linear combination of commodity flows and find a feasible routing to meet flow requirements. An arc subset-chain incidence matrix is formulated and solved by a simplex multiplier method. Alternatively, a node-arc incidence matrix can be solved by the Dantzig-Wolfe decomposition algorithm. The procedures turn out to be identical. This approach can solve several times as many problems as the individual arc constraint method, yet surprisingly little change is required in some of the solution algorithms.

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