In this paper, a single-machine scheduling problem with simple linear deterioration was explored. By simple linear deterioration, we mean that the processing time of a job is a simple linear function of its execution starting time and its deterioration rate. The objective of this study was to find a schedule that minimizes the absolute value of maximum lateness, where the due date of jobs are a common due date. Given that the cases of common due date is a given constant and variable, we showed that the problem can be solved in polynomial time, respectively. Some extensions of the problem were also delineated.