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

Abstract

The problem of maximizing the weighted number of just-in-time jobs in a two-machine flow shop scheduling system is known to be $\mathcal {NP}$ -hard (Choi and Yoon in J. Shed. 10:237---243, 2007). However, the question of whether this problem is strongly or ordinarily $\mathcal{NP}$ -hard remains an open question. We provide a pseudo-polynomial time algorithm to solve this problem, proving that it is $\mathcal{NP}$ -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 $\mathcal{NP}$ -hard for both a two-machine job shop scheduling system and a two-machine open shop scheduling system.