We investigate a common due-date assignment scheduling problem with a variable maintenance on a single machine. The goal is to minimize the total earliness, tardiness, and due-date cost. We derive some properties on an optimal solution for our problem. For a special case with identical jobs we propose an optimal polynomial time algorithm followed by a numerical example.

Recently, as a competitive strategy to provide high quality service for customer demand, just-in-time (JIT) production has received considerable attention from the manufacturing enterprises [

On the other hand, to prevent production disruption caused by machine breakdown, machine maintenance needs to be performed to perserve production efficiency. Since 1996, researchers begun to take maintenance into consideration in scheduling (see Lee [

In the papers by Kubzin and Strusevich [

Another popular topic in recent years is that of scheduling with simultaneous considerations of due-date assignment and maintenance. Mosheiov and Oron [

In this paper we introduce a new scheduling model which combines the due-date assignment and the machine maintenance. We assume that the duration of maintenance is variable and the maintenance must be started prior to a given deadline.

As a practical example for the proposed model, we may consider the steel-making process in the steel plant [

In the second section we provide the notation and formulation on our model. The third section derives some important properties on an optimal solution. In Section

Our problem can be described as follows. There are

The classical due-date assignment scheduling problem (without maintenance) was introduced by Panwalkar et al. [

In order to solve our problem, we first derive some properties on an optimal solution. We also use the small perturbations technique.

There exists an optimal solution in which the schedule starts at time zero and contains no idle time among the jobs, and the maintenance is scheduled between the two consecutive jobs without idle time.

First we show that there is no idle time between the jobs.

Assume that there exists idle time between the jobs

Next, we show that the maintenance is scheduled between the two consecutive jobs without idle time.

Assume that there exists idle time between the job

Assume that there exists idle time between the maintenance and the job

Clearly, we know

By the above analysis, we can treat all the jobs and the maintenance as a consecutive whole without idle time.

Finally, we show that the schedule starts at time zero. Assume that there exists a solution which does not start at time zero. Then we move the whole to the left by some times to assure that the new schedule starts at time zero and reset a smaller common due-date than the original due-date to obtain a new solution, which does not increase the objective value.

With the above argument, we conclude Lemma

Idle time between jobs

Idle time between job

Idle time between the maintenance VM and job

The optimal common due-date is the completion time of the job in position

First we show that in an optimal solution the common due-date

Then

With the above discussion, we conclude that in an optimal solution the optimal common due-date

Now, we assume that the common due-date

The case

In an optimal solution, the maintenance is scheduled either at time 0, or after the common due-date.

Suppose that there exists a solution in which the maintenance starts at time

Now we construct a new solution as follows. Starting the maintenance at time zero and scheduling all the jobs according to their original order just after the maintenance. Setting the common due-date to the new completion time of job

The duration of the maintenance decreases as it starts earlier.

The earliness of jobs

The common due-date

The maintenance starts at time

The maintenance starts at time 0.

In this section we consider a special case for our problem. We assume all the jobs are identical; that is,

Recall that the due-date is the completion time of job in the

With the above analysis, we propose our algorithm as follows.

If

Compute

Output the schedule with the minimal objective value from all the constructed schedules

From the properties on an optimal solution as shown in Lemmas

The

The job processing times are identical with

Applying Algorithm H, we first compute the parameters as follows:

Their corresponding objective values are

When comparing the costs in

In this paper we consider the common due-date assignment scheduling problem with a variable maintenance on a single machine. The goal is to minimize the total earliness, tardiness, and due-date cost. We derive some properties on an optimal solution for our problem. For a special case with identical jobs we propose an optimal polynomial time algorithm running in

For the general case with nonuniform processing times of jobs, whether problem is NP-hard or not is open and deserves the further research.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author is grateful to the editor and two anonymous referees for their helpful comments and suggestions.