X - UWSpace - University of Waterloo

Benders komposition Method
In the Benders decomposition method, the master pmblem allocates shared resources to
the subproblems. The subproblems, then. react by utilizing the resources and report to the master
problem the prices which reflect how well the subproblems use the shared resources. Then. the
master problern adjusts and reallocates the resources according to the price information.
Benders decomposition is efficient for the cornplicated problem which cm be parritioned
into two parts: one major part containing linear variables and another part containing the
complicating variables such as integer or nonlinear vanables. Benden decomposition cm be
cdled the primai decomposition method.
The Nested Primai Decomposition Method for MulüStage Models
Dantzig [L980i applied a nested decomposition algorithm to the dual of stYrcase Linear
Pro_mrnming modeis. which cm be considered as a nested decomposition of Benders method.
In nested primal decomposition. the resource mechanism is used for the coordination between
periods. The master problem. which is the preceding subproblems, receives cuts from the
following subproblems and sen& a resource vector to the following subproblems.
Cross Decomposition Method by Van Roy
Cross decornpsi tion was first introduced by Van Roy [1983] for mixed integer pro*pms.
The cross decomposition method cm be described as simultaneously using Benden
decomposition (primai decomposition) and Dantzig-Wolfe decornposition (dud decomposition)