## Abstract

We study a scheduling problem with rejection on a set of two machines in a flow-shop scheduling system. We evaluate the quality of a solution by two criteria: the first is the makespan and the second is the total rejection cost. We show that the problem of minimizing the makespan plus total rejection cost is NP-hard and for its solution we provide two different approximation algorithms, a pseudo-polynomial time optimization algorithm and a fully polynomial time approximation scheme (FPTAS). We also study the problem of finding the entire set of Pareto-optimal points (this problem is NP-hard due to the NP-hardness of the same problem variation on a single machine [20]). We show that this problem can be solved in pseudo-polynomial time. Moreover, we show how we can provide an FPTAS that, given that there exists a Pareto optimal schedule with a total rejection cost of at most R and a makespan of at most K, finds a solution with a total rejection cost of at most (1ε)R and a makespan value of at most (1ε)K. This is done by defining a set of auxiliary problems and providing an FPTAS algorithm to each one of them.

Original language | English |
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Pages (from-to) | 1087-1096 |

Number of pages | 10 |

Journal | Computers and Operations Research |

Volume | 39 |

Issue number | 5 |

DOIs | |

State | Published - 1 May 2012 |

## Keywords

- Approximation algorithm
- Bicriteria optimization
- FPTAS
- Flow-shop scheduling
- NP - hard
- Pseudo-polynomial time algorithm
- Scheduling with rejection

## ASJC Scopus subject areas

- Computer Science (all)
- Modeling and Simulation
- Management Science and Operations Research