Maximization Problems in Single Machine Scheduling

Rapport interne.Problems of scheduling $n$ jobs on a single machine to maximize regular objective functions are studied. Precedence constraints may be given on the set of jobs and the jobs may have different release times. Only semi-active schedules are of interest, i.e., those for which the jobs cannot be shifted to start earlier without changing the job sequence or violating the feasibility. Such maximization problems arise in predictive-reactive scheduling. Their solutions are used to evaluate the worst-case performance of a flexible schedule, which is characterized by a succession of groups containing partially permutable operations. The most general problem of maximizing maximum cost is solved in $O(n^3)$ time. When all release times are equal to zero, it is shown that the problem of maximizing the total weighted completion time or the weighted number of late jobs is equivalent to its minimization counterpart with precedence constraints reversed with respect to the original ones. If there are no precedence constraints, the problem of maximizing arbitrary regular function reduces to $n$ similar problems of scheduling $n-1$ jobs available at the same time. Algorithmic and computational complexity results for several maximization problems follow from these statements