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This paper considers a single-machine due-window assignment scheduling problem with position-dependent weights, where the weights only depend on their position in a sequence. The objective is to minimise the total weighted penalty of earliness, tardiness, due-window starting time, and due-window size of all jobs. Optimal properties of the problem are given, and then, a polynomial-time algorithm is provided to solve the problem. An extension to the problem is offered by assuming general position-dependent processing time.

Conventionally, in scheduling theory, due-windows are job-dependent either if they are dictated by the customer (i.e., given constants) or they are decision variables (i.e., due-window assignment). A due-window for job

Common due-window (CON-DW) assignment method (Liman et al. [

Slack due-window (SLK-DW) assignment method (Mosheiov and Oron [

Different due-windows’ (DIF-DW) assignment method: it is assumed that the job

In a recent paper, Wang et al. [

The remainder of the paper is organized as follows. In Section

A set of

Using the three-field notation (Graham et al. [

Obviously, there exists an optimal sequence

There exists an optimal sequence such that

We consider two cases that contradict this optimal property:

Case i: if

We shift

Hence, Case i is not an optimal due-window assignment.

Case ii: if

We shift

Hence, Case ii is not an optimal due-window assignment.

To summarise, we have

For a given sequence

When

When

When

When

From Lemma

Case i: if

Case ii: if

To summarise, if

Similarly, cases (2) and (3) can be proved.

For a given sequence

When

When

When

When

The proof is similar to the proof of Lemma

The optimal sequence of the problem

From Lemmas

Let

And

Obviously,

From Lemmas

Step 1: obtain the optimal sequence by the SPT rule (see Lemma

Step 2: calculate the completion time of each job under the optimal sequence, and determine the optimal due-window locations

Step 3: obtain the optimal due-window size by setting

Algorithm

Optimality can be guaranteed by Lemmas

In order to illustrate Algorithm

Results of the optimal due-window location.

Job | Job | ||
---|---|---|---|

The data are as follows:

Now, we can solve the problem

Step 1: according to Lemma

Step 2: for the optimal sequence

Step 3: the optimal due-window sizes are

In this section, the problem

From (

From (

Based on the above analysis, the solution procedure of the problem

Step 1: solve assignment problem (

Step 2: calculate the completion time of each job under the optimal sequence, and determine the optimal due-window locations

Step 3: obtain the optimal due-window size by setting

Algorithm

Optimality is guaranteed by Lemmas

The data are as follows:

This study addressed the due-window (DIF-DW) assignment scheduling problem under the consideration of position-dependent weights. The goal is to determine the optimal sequence, the optimal due-window location, and size such that the total penalty (including the earliness, tardiness, due-window starting time, and due-window size of all jobs) is minimized. It was proved that the problem can be solved in polynomial time. The proposed model was also extended to the general position-dependent processing time, and the polynomial-time solution was provided. Further extensions are considering the above problems in the setting of

Date of Example

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
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6 | 7 | 11 | 5 | 6 | 22 | 13 | 21 | |

8 | 14 | 7 | 8 | 11 | 17 | 12 | 6 | |

9 | 11 | 13 | 32 | 7 | 10 | 12 | 8 | |

17 | 22 | 19 | 10 | 5 | 9 | 13 | 7 | |

16 | 8 | 15 | 14 | 11 | 17 | 13 | 14 | |

18 | 17 | 31 | 14 | 8 | 23 | 15 | 20 | |

15 | 12 | 18 | 19 | 8 | 16 | 21 | 13 | |

13 | 17 | 24 | 16 | 18 | 16 | 13 | 15 |

Step 1: by (

Step 2: for the optimal sequence

Step 3: the optimal due-window sizes are

Results of the optimal due-window location.

Job | Job | ||
---|---|---|---|

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This work was supported by the Natural Science Foundation of Liaoning Province (2020-MS-233).