Single-machine scheduling to stochastically minimize maximum lateness

We study the problem of scheduling a set of jobs on a single machine, to minimize the maximum lateness ML or the maximum weighted lateness MWL under stochastic order. The processing time Pi, the due date Di, and the weight Wi of each job i may all be random variables. We obtain the optimal sequences in the following situations: (i) For ML, the {Pi} can be likelihood-ratio ordered, the {Di} can be hazard-rate ordered, and the orders are agreeable; (ii) For MWL, {Di} are exponentially distributed, {Pi} and {Wi} can be likelihood-ratio ordered and the orders are agreeable with the rates of {Di}; and (iii) For ML, Pi and Di are exponentially distributed with rates μi and νi, respectively, and the sequence {νi(νi+μi)} has the same order as {νi(νi+μi+A0)} for some sufficiently large A0. Some related results are also discussed.