^{1}

^{2}

In this paper, we consider scheduling problems with general truncated job-dependent learning effect on unrelated parallel-machine. The objective functions are to minimize total machine load, total completion (waiting) time, total absolute differences in completion (waiting) times respectively. If the number of machines is fixed, these problems can be solved in time respectively, where m is the number of machines and n is the number of jobs.

In modern planning and scheduling problems, there are many real situations where the processing time of jobs may be subject to change due to learning effect. An extensive survey of different scheduling models and problems with learning effects could be found in Biskup [

Recently, Wang et al. [

duled in the rth position of a sequence, where

There are n independent jobs

, (1)

where

Let

dule so that the following objective functions is to be minimized: the total machine load

Let

Lemma 1. For a given permutation

If the vector

where

Now, the question is how many vectors

Theorem 1. For a given constant

Proof. As discussed above, to solve the problem

Note that if the number of machines

Algorithm 1

Step 1. For each possible vector

Step 2. The optimal solution for the problem is the one with the minimum value of the objective function

The following example illustrates the working of Algorithm 1 to find the optimal solution for the problem

Example 1. There are

Solution. When

When

60 | 38.45135 | 25.32933 | 16.80000 | 8.40000 | |

50 | 34.58149 | 23.81912 | 14.94849 | 7.13208 | |

45 | 29.03910 | 19.20685 | 12.60000 | 6.30000 | |

90 | 54.19170 | 36.87501 | 22.94328 | 11.20000 | |

40 | 27.09585 | 18.43750 | 11.20000 | 5.60000 |

15 | 48 | 28.83852 | 16.88622 | 8.40000 | |

11 | 40 | 25.93612 | 15.87942 | 7.13208 | |

14 | 36 | 21.77933 | 12.80457 | 6.30000 | |

3 | 64 | 40.64377 | 24.58334 | 11.20000 | |

9 | 32 | 19.62965 | 11.63469 | 5.60000 |

(2)-(5) to obtain that the optimal schedule on machine

When

When

When

When

30 | 12.78952 | 36 | 19.22568 | 8.44311 | |

22 | 8.81177 | 30 | 17.29074 | 7.93971 | |

28 | 11.77255 | 27 | 14.51955 | 6.40228 | |

6 | 2.35375 | 48 | 27.09585 | 12.29167 | |

18 | 7.51579 | 24 | 13.08643 | 5.81735 |

45 | 25.57905 | 11.65074 | 24 | 9.61284 | |

33 | 17.62354 | 7.739517 | 20 | 8.64537 | |

42 | 23.54510 | 10.63770 | 18 | 7.25978 | |

9 | 4.70751 | 2.10000 | 32 | 13.54792 | |

27 | 15.03158 | 6.76380 | 16 | 6.54322 |

60 | 38.36857 | 23.30147 | 10.90479 | 12 | |

44 | 26.43531 | 15.47903 | 7.70000 | 10 | |

56 | 35.31765 | 21.2754 | 9.89950 | 9 | |

12 | 7.06126 | 4.20000 | 2.10000 | 16 | |

36 | 22.54737 | 13.52761 | 6.30000 | 8 |

75 | 51.15809 | 34.95221 | 21.80959 | 10.50000 | |

55 | 35.24707 | 23.21855 | 15.40000 | 7.70000 | |

70 | 47.09020 | 31.91310 | 19.79899 | 9.80000 | |

15 | 9.41501 | 6.30000 | 4.20000 | 2.10000 | |

45 | 30.06317 | 20.29141 | 12.60000 | 6.30000 |

Hence, the optimal schedule on machine

Hsu was supported by the Ministry Science and Technology of Taiwan under Grant MOST 104-2221-E-252- 002-MY2.

Jibo Wang,Chou-Jung Hsu, (2016) Unrelated Parallel-Machine Scheduling Problems with General Truncated Job-Dependent Learning Effect. Journal of Applied Mathematics and Physics,04,21-27. doi: 10.4236/jamp.2016.41004