Stochastic bicriteria single machine scheduling with quadratic cost functions

In real world scheduling systems, job attributes are stochastic and schedulers are required to consider multiple criteria before arriving at some decisions. Moreover, schedulers incorporate their risk taking behaviour, characterised by their cost (or disutility) functions, into scheduling decisions. This paper deals with a single machine scheduling problem in which job attributes are random variables and a scheduler's cost function for sequence evaluation is quadratic in two criteria. The objective is to determine the optimal sequence that minimises the scheduler's expected cost of the criteria. The problem is NP-hard; however, it can be solved exactly when at least one of the criteria is a regular measure. We show that some cases have exact polynomial time solutions, while the rest can be formulated as quadratic assignment problems that are solvable either exactly or approximately. The results demonstrate that the schedulers' risk taking behaviour, the stochasticity of job attributes, and the two criteria affect scheduling decisions. Our computational experiments on the cases with quadratic assignment formulations indicate that the proposed heuristic performs well in producing good sequences within reasonable amounts of time.