Linear and Stochastic versions of the Min Cost Multi-Commodity Flow (MMCF) Problem

In the Multi-Commodity Network Flow Problem problem, a directed graph G(N,A), where N a set of nodes is and A is a set of Arcs is given. A set of K commodities has to be routed on G at minimal total cost while satisfying the usual flow-conservation constraints at the nodes.
See Symmetric and Asymmetric Parallelization of a Cost-Decomposition Algorithm for Multicommodity Flow Problems for details
Parameters

• : cost of assigning commodity k to arc ij
• : capacity of arc ij
• : demand/supply of commodity k at node i

Decision variable

• flow of commodity k on arc ij

Stochastic Multi-commodity network flow problem (Stochastic-MMCF)
The popular Sample Average Approximation method is used to solve the stochastic MMCF. Also instances from the OR-library of min cost multi-commodity problems is used for comparison purposes.