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This paper considers the

Steel-making is a multistage process in which melted iron is converted into steel products sequentially by the processes of converting furnace, heating furnace, and rolling mill. Distinctly, this is a very typical model of the flow shop. Differently, in the steel production, the hot work-in-processes can not wait between two successive operations. For example, a slab has to reach a rolling temperature through the heat furnace before it can be processed by the rolling mill. If a heated slab waits for a long time in front of the machine, its temperature will drop significantly. Once the temperature of a slab falls below the rolling temperature, the reheating must be executed, which will consume a lot of energy. Furthermore, the size of the work-in-process in steel-making is especially large, which limits the storage capacity of the buffer between two successive machines. As minimizing the criterion of total completion time (TCT; the detail about TCT objective can be found in [

With the standard scheduling notation suggested by Graham et al. in 1979 [

In this paper, the asymptotic optimality of the Shortest Processing Time (SPT) first SPT rule is proved for problem

The remainder of the paper is organized as follows. The problem is formulated in Section

In no-wait flow shop problem, a set of

The SPT rule is actually a heuristic in which the jobs are scheduled in nondecreasing order according to value

Let the processing times

The jobs to be scheduled are index in the SPT sequence. For arbitrary job number

As the stronger NP-hardness of the problem, the lower bound is usually a substitution of the optimal solution. In 2013, Bai and Ren [

This lower bound can deal with problem

Gantt chart of Example

There is three-machine flow shop problem with three jobs. The processing times of these jobs are

For any instance of problem

Consider a given optimal schedule for

In this section, we designed a series of computational experiments to reveal the convergence of the SPT rule and the effectiveness of the new lower bound in different size problems. Firstly, we compare the SPT rule with LB

The ratios SPT/LB

Results of SPT/LB(3).

Distribution | Uniform | Normal | ||||
---|---|---|---|---|---|---|

5 | 10 | 15 | 5 | 10 | 15 | |

Machine | ||||||

100 jobs | 1.11284 | 1.07673 | 1.04071 | 1.07082 | 1.05216 | 1.05210 |

200 jobs | 1.10885 | 1.06516 | 1.02421 | 1.06953 | 1.04967 | 1.04013 |

500 jobs | 1.10405 | 1.06016 | 1.01848 | 1.06874 | 1.04820 | 1.03421 |

1000 jobs | 1.10193 | 1.05846 | 1.01507 | 1.06835 | 1.04715 | 1.02225 |

1500 jobs | 1.09988 | 1.05631 | 1.01291 | 1.06585 | 1.04639 | 1.01098 |

The ratios LB

Results of LB(3)/LB(2).

Distribution | Uniform | Normal | ||||
---|---|---|---|---|---|---|

5 | 10 | 15 | 5 | 10 | 15 | |

Machine | ||||||

100 jobs | 1.0632 | 1.0369 | 1.0362 | 1.0392 | 1.0310 | 1.0493 |

200 jobs | 1.0313 | 1.0305 | 1.0271 | 1.0438 | 1.0321 | 1.0380 |

500 jobs | 1.0453 | 1.0709 | 1.0254 | 1.0363 | 1.0248 | 1.0346 |

1000 jobs | 1.0298 | 1.0456 | 1.0245 | 1.0340 | 1.0269 | 1.0259 |

1500 jobs | 1.0199 | 1.0304 | 1.0220 | 1.0273 | 1.0309 | 1.0238 |

In this paper, we investigate a very useful scheduling problem in steel production, the no-wait flow shop minimizing total completion time. The asymptotic optimality of the classical SPT rule is proven with the tool of asymptotic analysis when the problem scale is large enough. For the promotion of numerical simulation, a new lower bound with performance guarantee is given. Computational results show that the SPT rule and the new lower bound work well for large scale problems.

This research is partly supported by the National Natural Science Foundation of China (61104074), Fundamental Research Funds for the Central Universities (N110417005), China Postdoctoral Science Foundation (2013T60294, 20100471462), Scientific Research Foundation for Doctor of Liaoning Province of China (20101032), and the 2011 Plan Project of Shenyang City (F11-264-1-63).