### Abstract

In this note, we investigate the time complexity of non-preemptive shop scheduling problems with two jobs. First we study mixed shop scheduling where one job has a fixed order of operations and the operations of the other job may be executed in arbitrary order. This problem is shown to be binary NP-complete with respect to all traditional optimality criteria even if distinct operations of the same job require different machines. Moreover, we devise a pseudo-polynomial time algorithm which solves the problem with respect to all non-decreasing objective functions. Finally, when the job with fixed order of operations may visit a machine more than once, the problem becomes unary NP-complete. Then we discuss shop scheduling with two jobs having chain-like routings. It is shown that the problem is unary NP-complete with respect to all traditional optimality criteria even if one of the jobs has fixed order of operations and the jobs cannot visit a machine twice.

Original language | English |
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Pages (from-to) | 37-49 |

Number of pages | 13 |

Journal | Computing (Vienna/New York) |

Volume | 69 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 19 2002 |

### Keywords

- Computational complexity
- Mixed shop scheduling
- Pseudo-polynomial time algorithm

### ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Numerical Analysis
- Computer Science Applications
- Computational Theory and Mathematics
- Computational Mathematics