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We consider a common due-window assignment scheduling problem jobs with variable job processing times on a single machine, where the processing time of a job is a function of its position in a sequence (i.e., learning effect) or its starting time (i.e., deteriorating effect). The problem is to determine the optimal due-windows, and the processing sequence simultaneously to minimize a cost function includes earliness, tardiness, the window location, window size, and weighted number of tardy jobs. We prove that the problem can be solved in polynomial time.

In most scheduling studies, job processing times are treated as constant numbers; however, in many practical situations, job processing times are affected by the learning effects and/or deteriorating (aging) effects. Learning effects and deteriorating (aging) effects are important for production and scheduling problems. For details on this line of the scheduling problems with learning effects (deteriorating effects), the reader is referred to a comprehensive survey by Biskup [

J.-B. Wang and M.-Z. Wang [

The recent paper Li et al. [

The following notations will be used throughout the paper:

Consider a nonpreemptive single machine setting. There are

Our task of this paper is to determine the optimal earliest due date

For a given schedule

If

Suppose

An optimal schedule exists in which the due-window starting time (i.e.,

Suppose that there exists a schedule starting at time zero and containing jobs at the

When we shift

When we shift

When we shift

Again, a shift of

Therefore, an optimal schedule exists such that both

An optimal schedule exists in which the index of the job completed at the due-window starting time is

Using the classical small perturbation technique (see J.-B. Wang and C. Wang [

We shift

From

For the problem

By Lemmas

If

Equation (

If the number of

We define

Based on the above analysis, we have the following result.

The scheduling problem

For a given

Consider the instance with

Now we apply Algorithm

The optimal objective value is

The optimal objective value is

By the same way as in the previous subsection, we consider the following scheduling problem:

For a given schedule

For the problem

By Lemmas

If

Equation (

If we fix the number of

It is similar to the proof of Theorem

Similar to Section

The scheduling problem

We have considered the single machine due-window assignment scheduling problem with variable job processing times. The objective is to minimize a linear combination of earliness, tardiness, the window location, window size, and weighted number of tardy jobs. We proposed a polynomial-time algorithm, respectively, for the learning effect and the deteriorating jobs. Obviously, if

Yu-Bin Wu and Ping Ji declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful to the anonymous referees for their helpful comments on earlier versions of this paper. This research was supported by the Science Research Foundation of Shenyang Aerospace University (Grant no. 201304Y) and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. PolyU 517011).