Minimizing maximum cost for a single machine under uncertainty of processing times

Abstract

One of the most known results in the machine scheduling is Lawler’s algorithm to minimize the maximum cost of jobs processed by a single machine subject to precedence constraints. We consider an uncertain version of the same min-max cost scheduling problem. The cost function of each job depends on the job completion time and on an additional generalized numerical parameter, which may be a tuple of parameters. For each job, both, its processing time and the additional parameter are uncertain, only intervals of possible values of these parameters are known.We analyse certain classes of cost functions and develop polynomial algorithms which construct min-max regret solutions. The considered problems cover the most general range of studied cases of interval uncertainty.In the only two papers that present algorithms for minimizing the maximum regret for the problem with uncertain job processing times, the algorithms are based on extremal scenarios, where some uncertain parameters take their maximum values, while all others take their minimum possible values.We show that it is impossible to always limit the search to extremal scenarios. Our approach is based on new ideas different from those underlying previous work.Finally, we show that our approach outperforms all known results for constructing min-max regret solutions for the min-max cost scheduling problem under uncertainty of job processing times.