Compact scheduling of zero–one time operations in multi-stage systems Original Research Article

A problem of no-wait scheduling of zero–one time operations without allowing inserted idle times is considered in the case of open, flow and mixed shop. We show that in the case of open shop this problem is equivalent to the problem of consecutive coloring the edges of a bipartite graph G. In the cases of flow shop and mixed shop this problem is equivalent to the problem of consecutive coloring the edges of G with some additional restrictions. Moreover, in all shops under consideration the problem is shown to be strongly NP-hard. Since such colorings are not always possible when the number of processors m>3 for open shop (m>2 for flow shop), we concentrate on special families of scheduling graphs, e.g. paths and cycles, trees, complete bipartite graphs, which can be optimally colored in polynomial time.