A due-window determination in minmax scheduling problems

In several recent scheduling studies the notion of due-date is generalized: it is assumed that jobs completed within a certain time interval (rather than at a time point) are considered as being on time whereas jobs completed outside this interval are penalized. We study a scheduling problem on parallel identical machines, in which the schedule as well as the common due-window are to be determined. The relevant cost components are: maximum earliness, maximum tardiness, due-window starting time and due-window length. The objective is of a minmax type, i.e. we look for the schedule and due-window with minimum cost of the worst scheduled job, with respect to all cost components. We solve the problem to optimality on a single machine, and we introduce an efficient (and asymptotically optimal) heuristic algorithm and a simple lower bound for the general multi-machine case. An extension to non-linear cost functions as well as the special case of unit processing times are also studied. We conclude with an extensive numerical study which indicates that both the heuristic and the lower bound produce very close-to-optimal results under various job and machine environments.