Solving a stochastic single machine problem with initial idle time and quadratic objective

We study a static single machine scheduling problem in which processing times are stochastic, due-dates and penalties for not completing jobs on time are deterministic, and an initial fixed idle time is allowed to be inserted before the processing of the first job begins on the machine. The objective is to determine the optimal sequence and the optimal initial idle time that jointly minimize the expected value of the sum of a quadratic cost function of idle time and the weighted sum of a quadratic function of job lateness. The problem is NP-hard to solve; however, we develop an exact algorithm based on a precedence relation structure among adjacent jobs. Our extensive computational results show that the algorithm can solve large problem instances quickly. We also demonstrate that the proposed problem is general in the sense that its special cases reduce to new stochastic models while its limiting cases simplify to some deterministic models.