#### Abstract

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.

#### 1. Introduction

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 [1], the car repair or maintenance by a worker, and the patient diagnosis and treatment by a doctor [2]. Thus, the processing time of a job is shorter if it is scheduled later, rather than in the sequence. Mosheiov [3] determined that this phenomenon is known in the literature as a “learning effect.” Biskup [4] was the first to investigate the learning effect in the scheduling problems. He assumed that the production time as a function of the number of repetitions of the production of a single item with a learning effect decreases. A number of researchers [3–5] studied E/T problem with learning effect, which is with a single machine and no deterioration effect. They solved unrestricted common due date problem as an assignment problem. In this work, we used Biskup’s learning effect (see [4]). Consider where is basic processing time of job performed at position when is learning index.

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 [6] pointed out another case searching for an object under worsening weather or growing darkness. Although J. N. D. Gupta and S. K. Gupta [7] provided an example of steel rolling miles where the temperate of an ingot, while waiting to enter the rolling machine, drops below a certain level, which required the ingot to be reheated before rolling, Kunnathur and Gupta [8] gave a fire fighting example where the time and effort required to control a fire increase if there is a delay in the start of the fire-fighting effort. Alidaee and Womer [9] classified deteriorating jobs models into three different types: linear, piecewise linear, and nonlinear. In this paper, we are interested in nonlinear deterioration effect. One has where is starting time of and and are nonlinear deterioration effect, which is the amount of increase in the processing time of a job per unit delay in its starting time.

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 [10] studied single machine scheduling problem with deteriorating jobs and learning effects to minimize the makespan when Wang [11] solved polynomially single machine scheduling problems with deteriorating jobs and learning effects. Toksari and Güner [12] proposed mixed nonlinear integer model for parallel machine earliness/tardiness scheduling problem with sequence dependent setup time and the effects of learning and deterioration. In another paper, they [13] proposed that the optimum solution of the same problem is V-shaped. Toksari and Güner [12] consider a parallel machine earliness/tardiness (ET) scheduling problem with different penalties under the effects of position based learning and linear and nonlinear deteriorations. The same authors [13] also worked on some scheduling problems with the nonlinear effects of learning and deterioration. Cheng et al. [1] derived polynomial-time optimal solutions for some scheduling problems with deteriorating jobs and learning effect. Linear deterioration effect was used in all studies on scheduling problems effects of learning and deterioration simultaneously. Güner and Toksari [14] studied the same scheduling problems with position based learning effect and linear deterioration simultaneously. We extended this study using nonlinear deterioration effect for the same problems.

#### 2. Problem Formulation

Güner and Toksari [14] considered the position dependent learning effect of a job and linear deterioration effect for the same processing time. In this study, these two effects are combined as follows: There are jobs to be scheduled on single machine. If job , , is being scheduled in position in a sequence, its actual processing time is . is basic processing time of job scheduled in position . and are nonlinear deterioration effect, which is the amount of increase in the processing time of a job per unit delay in its starting time. is the learning index. is starting time of job scheduling in position , and is the actual completion time of the job scheduled in position . Thus, the actual processing time is formulized as follows: where is the actual completion time of the job scheduled in position .

We also take setup times into consideration in the scheduling model by adopting the notion of Koulamas and Kyparisis [15] that setup times are past sequence dependent. Koulamas and Kyparisis [15] motivate the assumption of p-s-d setup times by the manufacturing of integrated circuit (IC) boards. Specifically, an IC board consists of a number of electronic components mounted on it and the processing of any electronic component will have an adverse effect on the “readiness” of other components on the board due to the passage of electricity through the board. Consequently, each component prior to processing requires a setup operation to restore it to “full-readiness” status and the setup time depends on the component’s degree of “unreadiness,” which is proportional to the actual processing times of the already processed components. Thus, the p-s-d setup time of job if it is scheduled in the th position in a sequence is given as follows: where is a normalizing constant. The value of the normalizing constant determines the actual lengths of the required setups and, when , there is no need for any p-s-d setups (see [15]). We denote the effects of the learning and deterioration in (3) by learning effect and nonlinear deterioration effect and denote the p-s-d setup given in (5) by .

#### 3. Some Single Machine Scheduling Problems with Individual Due Dates under Nonlinear Effects of Learning and Deterioration

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

Lemma 1. *Consider
**
where
**
when is starting time of the schedule.*

*Proof. *One has
then, we can obtain the following:
where
This completes the proof.

Let , , , and represent the total lateness, total tardiness, maximum lateness, and maximum tardiness of a given permutation, respectively. These problems with past sequence dependent setup times were studied by Biskup and Herrmann [16] and Güner and Toksari [14] studied the same scheduling problems with position based learning effect and linear deterioration simultaneously. We used the same approach with Güner and Toksari [14] to prove the theorems.

Theorem 2. *The problem can be solved optimally by sequencing jobs in nondecreasing order of their processing times (SPT rule).*

*Proof. *Consider an optimal schedule . Assume, under , there are two adjacent jobs, job , whose sequence is , followed by job , whose sequence is , such that . and express completion times of and scheduled at positions and under the nonlinear deterioration and learning effect, respectively. is starting time of the schedule.

Lateness of job in position can be calculated by
then, from Lemma 1,
where explains the completion time of when it is in position at schedule . If we assume that is total lateness up to position , then
Performing a pairwise interchange on jobs and , we obtain schedule . The completion times of the jobs processed before jobs and are not affected by interchange; then,
If we assume that is total lateness up to position , then
The difference between the values of and is
, where , , , and .

Consequently, .

The total lateness under is strictly less than that under . This completes the proof.

*Example 3. *
Let , , , , when learning rate , , and . Then, sequence leads to , , , , , , and . The SPT sequence leads to , , , , , , and .

Theorem 4. *The problem can be solved optimally by sequencing jobs in nondecreasing order of their processing times (SPT rule).*

*Proof. *Tardiness is defined as . The results of Theorem 2 can be transferred directly to the case of . This completes the proof.

Theorem 5. *The problem both and can be solved optimally by sequencing jobs in nondecreasing order of their processing times (SPT rule) (or equivalently earliness due date (EDD) rule) if the due dates and the processing times are agreeable; that is, if implies .*

*Proof. *We would not duplicate , , , and due to knowing them from Theorem 4.

The difference between the values of and is
, where , , , and .

Consequently, .

The difference between the values of and is
, where , , and .

Consequently, .

The total lateness under is strictly less than that under .

The maximum tardiness is defined as . The obtained results can be transferred directly to the case of .

This completes the proof.

*Example 6. *
Let , , , , when learning rate , , and . Then, the SPT sequence leads to , , , , , , and . Sequence leads to , , , , , , and .

#### 4. Common Due Date Problem

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

Theorem 7. *Consider V-shape: jobs in set of jobs that are completed on or before the due date are sequenced in nonincreasing order of the ; jobs in set of jobs that are completed after the due date are sequenced in nondecreasing order of the .*

*Proof. *Consider an optimal schedule . Assume, under , there are two adjacent jobs, job , whose sequence is , followed by job , whose sequence is , such that . and express completion times of and scheduled at positions and under the nonlinear deterioration and learning effect, respectively. is starting time of the schedule.

Earliness of job in position can be calculated by
then, from Lemma 1,
If we assume that is total earliness up to position , then
Performing a pairwise interchange on jobs and , we obtain schedule . The completion times of the jobs processed before jobs and are not affected by interchange; then,
If we assume that is total earliness up to position , then
The difference between the values of and is
, where , , and .

Consequently, ; it means that jobs in set of jobs that are completed before the due date are sequenced in nonincreasing order of the .

If a similar argument follows for the jobs that are started after the common due date , we will obtain the following:
where , , and .

Consequently, ; it means that jobs in set of jobs that are completed after the due date are sequenced in nondecreasing order of the .

#### 5. Conclusions

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.

#### Conflict of Interests

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