Stochastic single machine scheduling problem with V-shaped or Λ-shaped optimal sequences

In this paper, we study a problem of scheduling n jobs with stochastic processing times on a single machine which is subject to stochastic breakdowns to minimize the expectation of the linear combination of three functions of completion times: (i) the sum of squares, (ii) the square of mean and (iii) the mean. The deterministic equivalent of the stochastic scheduling problem is established when the point process N(t) related to the machine uptimes is a generalized Poisson process, and sufficient conditions under which the optimal sequences of the scheduling problem is V-shaped or Λ-shaped in terms of means of processing times are derived