A Survey of Linear Cost Multicommodity Network Flows

Abstract

This exposition presents a state-of-the-art survey of results and algorithms for linear multicommodity network flow problems.

Dynamic-Programming Approximations for Stochastic Time-Staged Integer Multicommodity-Flow Problems

“…Early survey papers by Assad (1978) and Kennington (1978) describe various solution approaches to the linear version. The integer version has seen increased attention due to its wide applicability to fields such as airline fleet assignment (Hane et al 1995), car and container distribution in the railroad and maritime industries (Jordan and Turnquist 1983, Shan 1985, Chih 1986, Crainic et al 1993, Holmberg et al 1998, and dynamic fleet management (Powell et al 1995).…”

Solution of Large-Scale Multicommodity Network Flow Problems Via a Logarithmic Barrier Function Decomposition

“…The stipulation that binds the flow of several commodities to joint capacities is given by (2) and constitutes the set of "complicating constraints". Without those constraints, the problem would reduce to independent, bounded, single-commodity flow problems.…”

“…Without those constraints, the problem would reduce to independent, bounded, single-commodity flow problems. The duals u-, corresponding to (2) are nonpositive. The constraints (4) are redundant, but they ensure that x is bounded when the constraints in (2) are relaxed.…”

“…The duals u-, corresponding to (2) are nonpositive. The constraints (4) are redundant, but they ensure that x is bounded when the constraints in (2) are relaxed. This approach can further be divided into resource-directive and price-directive algorithms.…”

“…The overlapping joint capacity constraints in (2) are placed into the barrier term of the objective function in order to enable the successive solutions of independent single commodity problems in a sequence of subproblems . The derivation shows that the use of a logarithmic barrier function is not arbitrary since it has a very natural origin.…”