Integer programming formulations in sequencing with total earliness and tardiness penalties, arbitrary due dates, and no idle time: A concise review and extension

•A concise review of mathematical programming formulations for earliness-tardiness scheduling with and without controllable processing times and no idle time between jobs provided.•Improvement and/or alternative formulations are provided.•Further research directions are provided. Scheduling with arbitrary due dates and no idle time permitted between jobs with the objective of minimizing the total earliness and tardiness has a variety of industry applications. We review mathematical programming formulations for single and parallel machine problems with fixed and controllable processing and setup times. Weaknesses of the formulations are identified, corrections and/or improvements are provided, and further research directions are suggested.