In this paper the authors consider an n job one machine sequencing problem in which all jobs have a common due date and unequal penalties occur when a job is completed before or after its due date. The objective is to determine an optimal sequence and the corresponding common due date that yield the global minimum value of a penalty function. All per unit costs involved are assumed to be linear and depend on the position in the schedule in which the job appears. The purpose of the present paper is to show that: (i) such a problem is equivalent to the problem of minimization of weighted mean absolute deviation of the completion times of those n different jobs from the common due date, and (ii) a sequence that minimizes the penalty function globally is a V-shaped sequence.
