Due Date Single Machine Scheduling Problems with Nonlinear Deterioration and Learning Effects and Past Sequence Dependent Setup Times

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.


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][4][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  ≤ 0 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 2 Mathematical Problems in Engineering 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  ( > 0) and ( > 0) 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.

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 ,  = 1, 2, . . ., , is being scheduled in position  in a sequence, its actual processing time is  * .  is basic processing time of job scheduled in position . ( > 0) and ( > 0) are nonlinear deterioration effect, which is the amount of increase in the processing time of a job per unit delay in its starting time.
( < 0) is the learning index.  is starting time of job scheduling in position , and ( * −1 ) is the actual completion time of the job scheduled in position ( − 1).Thus, the actual processing time  * is formulized as follows: where ( * −1 ) is the actual completion time of the job scheduled in position ( − 1).
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 ps-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 ( psd  ) of job  if it is scheduled in the th position in a sequence is given as follows: where  ≥ 0 is a normalizing constant.The value of the normalizing constant  determines the actual lengths of the required setups and, when  = 0, 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  p-q-d .

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.

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  V , whose sequence is  + 1, such that   <  V . *  and  * V express completion times of   and  V scheduled at positions  and V (V =  + 1) 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  V , we obtain schedule   .The completion times of the jobs processed before jobs   and  V 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 ∑ (  ) − ∑ () ≥ 0, where   <  V , ( < 0), ( > 0), and ( > 0).Consequently, ∑ (  ) > ∑ ().The total lateness under  is strictly less than that under   .This completes the proof.
The maximum tardiness  max is defined as  max = {0,  max }.The obtained results can be transferred directly to the case of

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 ,   =  | ∑(  +   ) 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  V , whose sequence is  + 1, such that   <  V . *  and  * V express completion times of   and  V scheduled at positions  and V (V =  + 1) under the nonlinear deterioration and learning effect, respectively. is starting time of the schedule.
Earliness of job in position  can be calculated by