We present some problems against due dates with nonlinear learning and deterioration effects and past sequence dependent setup times. In this study, two effects (learning and deterioration) are used for the same processing time. The processing time of a job is shorter if it is scheduled later, rather than in the sequence. This phenomenon is known in the literature as a “learning effect.” On the other hand, in many realistic scheduling settings, a job processed later consumes more time than the same job processed earlier—this is known as scheduling with deteriorating jobs. In the past sequence dependent setup times approach, the setup time of a job is proportionate to the sum of processing times of the jobs already scheduled. In this study, we demonstrated that some problems with due dates remain polynomially solvable. However, for some other problems, we concentrated on finding polynomially solves under their special cases.

This paper addresses several problems against due dates with nonlinear learning and deterioration effects and past sequence dependent setup times. A scheduling problem is very important for a manufacturing system. So, numerous scheduling problems have been studied by researchers. In classical scheduling theory, the processing times of jobs are assumed to be known and fixed. However, in many realistic settings, workstations improve continuously as a result of repeating the same or similar activities such as the production of glass crafts by a skilled craftsman [

There is a growing interest in the literature to study scheduling problems of deterioration jobs; that is, jobs whose processing times are increasing functions of their starting times. Mosheiov [

In the scheduling literature, many researchers have worked different combinations of these three special cases to solve single machine, parallel machine, and flow shop scheduling problems. In this paper, we introduce firstly position dependent learning effect, nonlinear deterioration effect, and past sequence dependent setup time to solve due date problems in the single machine scheduling environment.

In the literature, there are a few studies on scheduling problems with effects of learning and deterioration simultaneously. Wang and Cheng [

Güner and Toksari [

We also take setup times into consideration in the scheduling model by adopting the notion of Koulamas and Kyparisis [

Before presenting the main results, we first present a lemma, which will be used in the proofs of the theorems in the sequel.

Consider

One has

Let

The problem can be solved optimally by sequencing jobs in nondecreasing order of their processing times (SPT rule).

Consider an optimal schedule

Lateness of job in position

Consequently,

The total lateness under

Let

The problem

Tardiness

The problem both

We would not duplicate

The difference between the values of

Consequently,

The difference between the values of

Consequently,

The total lateness under

The maximum tardiness

This completes the proof.

Let

In this section, we tackle common due date problem with past sequence dependent setup times under learning effect and nonlinear deterioration effect. An excellent introduction to common due date problems is given as

Consider

Consider an optimal schedule

Earliness of job in position

Consequently,

If a similar argument follows for the jobs that are started after the common due date

Consequently,

In this paper, we introduced some problems against due dates with past sequence dependent setup times under learning effect and nonlinear deterioration effect. In this study, effects of learning and deterioration are considered simultaneously. We present that some problems remain polynomially solvable. These problems are minimizing total lateness, total tardiness, maximum lateness (with agreeable due dates), maximum tardiness (with agreeable due dates), and earliness/tardiness problem with common due date.

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