Maximizing the weighted number of just-in-time jobs in several two-machine scheduling systems

The problem of maximizing the weighted number of just-in-time jobs in a two-machine ‡ow shop scheduling system is known to be NP-hard [4]. However, the question of whether this problem is strongly or ordinary NP-hard remains an open question. We provide a pseudo-polynomial time algorithm to solve this problem, proving that it is NP-hard in the ordinary sense. Moreover, we show how the pseudo-polynomial algorithm can be converted to a fully polynomial time approximation scheme (FPTAS). In addition, we prove that the same problem is strongly NP-hard for both a two-machine job shop scheduling system and a two-machine open shop scheduling system. This research was supported by THE ISRAEL SCIENCE FOUNDATION (grant No. 633/08). Partial support by the Paul Ivanier Center for Robotics and Production Management, Ben-Gurion University of the Negev is also gratefully acknowledged. ye-mails: [email protected] and [email protected]