A Heuristic Decomposition Algorithm for Scheduling Problems on Mixed Graphs

What is it about?

We consider a scheduling problem where a set of jobs has to be processed on a set of machines and arbitrary precedence constraints between operations are given. Moreover, for any two operations i and j the minimal difference between the starting times of operations i and j is given when operation i is processed first. Often, the objective is to minimize the makespan but we consider also arbitrary regular criteria. Even the special cases of the classical job shop problem J//Cmax belong to the set of NP-hard problems. Therefore, approximation or heuristic algorithms are necessary to handle large-dimension problems. Based on the mixed graph model we give a heuristic decomposition algorithm for such a problem, i.e., the initial problem is partitioned into subproblems that can be solved exactly or approximately with a small error bound. These subproblems are obtained by a relaxation of a subset of the set of undirected edges of the mixed graph. The subproblems are successively solved and a proportion of the results obtained for one subproblem is kept for further subproblem definitions. Numerical results of the algorithm presented here are given.

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