Flow shop and open shop scheduling with a critical machine and two operations per job

This paper develops polynomial time algorithms to solve the m machine flow shop and open shop makespan problems with a critical machine (that processes all jobs) and at most two operations per job under certain assumptions. In the flow shop case, an O(m(nlogn + logm)) algorithm is developed (where n is the number of jobs) assuming that all jobs that require processing on the same subset of machines are contiguously processed. In the open shop case, O(n + mlogm) algorithm is developed assuming that the load on the critical machine exceeds the load on any other machine.