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Dominance conditions for particular single machine scheduling problems with nested execution intervals

This paper takes interest in some particular one machine scheduling problems. The input is a set of n jobs with fixed processing time and temporal execution interval associated with each job. Preemption is not allowed. We assume all along this paper that the temporal execution intervals of the jobs are nested (none of them overlaps the other). Two kinds of objective are investigated: the minimization of the lateness and the minimization of the number of late jobs. In this paper, some dominance conditions are established and optimal sequencing rules are given for some particular cases.