We consider a class of problems of scheduling n jobs on m identical, uniform, or unrelated parallel machines with an objective of minimizing an additive criterion. We propose a decomposition approach for solving these problems exactly. The decomposition approach first formulates these problems as an integer program, and then reformulates the integer program, using Dantzig-Wolfe decomposition, as a set partitioning problem. Based on this set partitioning formulation, branch-and-bound exact solution algorithms can be designed for these problems. In such a branch-and-bound tree, each node is the linear relaxation problem of a set partitioning problem. This linear relaxation problem is solved by a column generation approach where each column represents a schedule on one machine and is generated by solving a single machine subproblem. Branching is conducted on variables in the original integer programming formulation instead of variables in the set partitioning formulation such that single machine subproblems ...
['Zhi-Long Chen', 'Warren B. Powell']
Solving Parallel Machine Scheduling Problems by Column Generation