In this paper, we consider an assignment problemin a health care facility where doctors with different working volumes have to be assigned to continuous shifts. A doctor is not allowed to work two consecutive shifts, and sometimes both the first and the last shifts in the month cannot be served by the same doctor. We show that this problem is equivalent to minimising the makespan Cmax of scheduling n unit-time conflicting jobs on m parallel uniform machines with speeds s1 ≥ s2 ≥ . . . ≥ sm. The conflict is modelled by a graph G, in which two adjacent jobs cannot be processed on the same machine. Herein, for the sake of the assignment problem, G is restricted to a path Pn and a cycle Cn. We present polynomial time algorithms to solve both problems to optimality.



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