This paper is the first attempt to successfully design efficient approximation algorithms for the single-machine maximum lateness minimization problem when jobs have different release dates and tails (or delivery times) under the no idle time assumption (i.e., the schedule cannot contain any idle time between two consecutive jobs on the machine). Our work is motivated by interesting industrial applications to the production area (Chrétienne (2008) [3]). Our analysis shows that modifications of the classical algorithms of Potts and Schrage can lead to the same worst-case performance ratios obtained for the relaxed problem without the no idle time constraint. Then, we extend the result developed by Mastrolilli (2003) [13] for such a relaxed problem and we propose a polynomial time approximation scheme with efficient time complexity.