The purpose of this paper is to analyze a special case of the non-pre-emptive single machine scheduling problem where the distinct due dates for each job are related to processing times according to the Equal-Slack rule. The scheduling objective is to minimize the sum of earliness and tardiness penalties. After determining some properties of the problem, the unrestricted case is shown to be equivalent to a polynomial time solvable problem, whereas the restricted case is shown to be NP-hard, and suggestions are made for further research.