Soft due window assignment and scheduling of unit-time jobs on parallel machines

We study problems of scheduling n unit-time jobs on m identical parallel machines, in which a common due window has to be assigned to all jobs. If a job is completed within the due window, then no scheduling cost incurs. Otherwise, a job-dependent earliness or tardiness cost incurs. The job completion times, the due window location and the size are integer valued decision variables. The objective is to find a job schedule as well as the location and the size of the due window such that a weighted sum or maximum of costs associated with job earliness, job tardiness and due window location and size is minimized. We establish properties of optimal solutions of these min-sum and min-max problems and reduce them to min-sum (traditional) or min-max (bottleneck) assignment problems solvable in O(n5/m2) and O(n4.5 log0.5 n/m2) time, respectively. More efficient solution procedures are given A. Janiak (B) Institute of Computer Engineering, Control and Robotics, Wrocław University of Technology, Janiszewskiego 11/17, 50-372 Wrocław, Poland e-mail: [email protected] W. Janiak Faculty of Computer Science and Management, Wrocław University of Technology, Łukasiewicza 5, 50-371 Wrocław, Poland e-mail: [email protected] M. Y. Kovalyov United Institute of Informatics Problems, National Academy of Sciences of Belarus, Surganova 6, 220012 Minsk, Belarus e-mail: [email protected] F. Werner Faculty of Mathematics, Otto-von-Guericke-University Magdeburg, PSF 4120, 39016 Magdeburg, Germany e-mail: [email protected]