V-shape property for optimal schedules with monotone penalty functions

Consider a single-machine scheduling of n jobs with a common due date d. If a job is either completed before the due date or after the due date a penalty is imposed. The unit early penalty cost may be different from the unit late penalty cost. It is shown that any optimal schedule which minimizes the sum of the penalties (early or late as the case may be) for all the jobs is V-shaped when the penalty function for each job is ‘monotone’.