Minimizing the Makespan for Scheduling Problems with General Deterioration Effects

This paper investigates the scheduling problems with general deterioration models. By the deterioration models, the actual processing time functions of jobs depend not only on the scheduled position in the job sequence but also on the total weighted normal processing times of the jobs already processed. In this paper, the objective is to minimize the makespan. For the singlemachine scheduling problems with general deterioration effects, we show that the considered problems are polynomially solvable. For the flow shop scheduling problems with general deterioration effects, we also show that the problems can be optimally solved in polynomial time under the proposed conditions.


Introduction
In the last decade, there is a growing interest in the literatures to study the scheduling problems where the actual processing time of a job may be subject to change due to various possible changes of its starting time and/or its position in the scheduling sequence.In general, there are two main types of effects on actual processing time of jobs in the scheduling literature: position-based effect and sum-of-processing-timebased effect.
Many studies modeled the actual processing times of jobs as specific exponential functions of the scheduled positions of jobs.Biskup [1] considered a scheduling problem with an exponential model of position-dependent effect where the actual processing time of job   is if job   is scheduled in position  of the job sequence, where   and  ( < 0) are the normal processing time and the learning rate of job   , respectively.The model in Biskup [1] was extended to be where the learning rates of the jobs are different; that is, the learning effects are job dependent in Mosheiov and Sidney [2].Some recent studies (see, e.g., Kuo and Yang [3], Cheng et al. [4], Chang et al. [5], Yang et al. [6], and Browne and Yechiali [7]) have investigated this issue with exponential processing time models under different machine environments.Some other studies proposed different versions of linear processing time models of jobs.Zhao and Tang [8] considered a linear deterioration model in which the actual processing time of job   is where   ,   , and  are the normal processing time, deterioration rate, and starting time of job   , respectively.Cheng and Mathematical Problems in Engineering Wang [9] introduced an actual processing time model of   as follows: where   and V  are the normal processing time of job   and the learning effect coefficient indicating the relative degree of learning, respectively.The notation   is a nonnegative integer such that 0 ≤   ≤  − 1 which indicates the number of jobs processed before job   in the schedule (i.e.,   + 1 is the position of job   in the schedule), and the notation  0 is a nonnegative integer such that  0 ≤  − 1 indicating a threshold value.The scheduling problems with linear actual processing times of jobs were considered in many recent articles, which include Mosheiov [10], Bachman and Janiak [11], Li et al. [12], Cheng et al. [13], and Yin et al. [14].
In the last decade, many researchers have devoted themselves to address different combinations of nonlinear models and linear models for actual processing times of jobs in scheduling studies.Alidaee and Womer [15] proposed a simple processing time model of job   as follows: where  > 0 and  > 0 are both parameters of the nonlinear deterioration effect.Lee et al. [16] considered the bicriterion single-machine scheduling problem with two job processing time models as follows: where   and  are the deterioration rate of job   and the starting time of job   , respectively.Several researchers addressed the combined effect of position-based effect and sum-of-processing-time-based effect into various scheduling problems.S.-J.Yang and D.-L.Yang [17] proposed a model in which the actual processing time of job   scheduled in position  of a sequence is where  1 < 0 and  2 < 0 denote the learning rates for sumof-processing-times-based and position-based learning, respectively.In the model, the subscript [] denotes the job in position  of a sequence, and  [] denotes the normal processing time of job .Yin et al. [18] extended the model in S.-J.Yang and D.-L.Yang [17] and proposed another learning model in which the actual processing time of job   scheduled in position  of a sequence is where ∑ 0 =1  [] = 0,  : [0, +∞) → (0, 1] is a differentiable nonincreasing function where  is nondecreasing on [0, +∞), (0) = 1, and  : [0, +∞) → (0, 1] is a nonincreasing function with (1) = 1.Many recent studies (see, e.g., Wang [19], Toksarı and Güner [20], Yang and Kuo [21], Cheng et al. [22,23], Cheng et al. [24], Kuo [25], Yang [26], and Lee and Wu [27]) have been conducted to address the combined processing time models in scheduling problems.
In a recent paper, J.-B.Wang and J.-J.Wang [28] proposed a new general learning effect model for both position-based effect and sum-of-processing-time-based effect, which can be described as where ∑ 0 =1    [] := 0,  : [0, +∞) → (0, 1] is a differentiable nonincreasing function where   is nondecreasing on [0, +∞),  : [1, +∞) → (0, 1] is a nonincreasing function with (1) = 1, and   is the weight of position .They showed that even with the introduction of a general learning effect to job processing times, some single-machine scheduling problems can be optimally solved in polynomial time.They also proved that some special cases of the flow shop scheduling problems are polynomially solvable under the proposed model.
In this paper, we explore to consider scheduling problems with considerations of two general deterioration effects whose actual processing times functions of jobs are modified from the models in Lee and Wu [27] and J.-B.Wang and J.-J.Wang [28] with respect to deterioration effects.The motivation for our study stems from the hardware production line that cuts a class of products to proper sizes and shapes.The actual processing time of a product depends on the quality of the cutting tool in the manufacturing process.Due to tool blunting, the actual processing time of a product increases with respect to the sum of the processing time of products already processed by the cutting tool and the number of products already processed.The objective of our study is to find the optimal job sequence to minimize the makespan of all the jobs.
The remaining part of this paper is organized as follows.In Section 2, we consider two single-machine scheduling problems.In Section 3, we investigate the flow shop scheduling problems under proper given conditions.We conclude this paper and describe the possible extensions in the last section.

Single-Machine Scheduling
The formulation of the considered single-machine scheduling problems can be described as follows.At time zero, there are  independent jobs ready to be processed on a singlemachine.Preemption is not allowed and a machine is only able to process one job at a time.Job  has a normal processing time   ( = 1, 2, . . ., ).
In this paper, we propose two general deterioration models with respect to both position-based effect and sumof-processing-time-based effect.By the first model, if job  is scheduled in the  position of the sequence, the actual processing time of job  is where ∑ 0 =1    [,] := 0,  : [0, +∞) → [1, +∞) is a differentiable nondecreasing function and (0) = 1,  : [1, +∞) → [1, +∞) is a nondecreasing function with (1) = 1,   is the weight of position ,  [] denotes the normal processing time of job , and the subscript [] denotes the job in position  of a sequence.In some actual processes, different jobs always have different complexities which lead to different experience to the worker in production line; thus, we assume that By the second model, if job  is scheduled in the  position, its actual processing time is where all the notations have the same means as those of the first model.Note that the function used in the model (11) is the same as the function used by J.-B.Wang and J.-J.Wang [28].The difference between this model and the model in J.-B.Wang and J.-J.Wang [28] is that the former aims to study the deterioration effect while the latter (J.-B.Wang and J.-J.Wang [28]) considers learning effect.By using the three-field notation scheme for the scheduling problem introduced by Graham et al. [29], the singlemachine scheduling problem with respect to the first model can be denoted as ) + ()], DE |  max , where DE denotes the deterioration effect.Similarly, the single-machine scheduling problem with respect to the second model can be denoted as In the following, we first state two lemmas that will be used in the proofs of the following theorems in this section.
Proof.We let ℎ() = (1 − )(() 2 + 1) + (( + ) 2 +  1 ) − (( + ) 2 +  1 ).The first derivation of ℎ() with respect to  can be obtained that ℎ  () =  2 [  ( + ) −   ( + )] ≥ 0 under the condition that  : [0, +∞) → [1, +∞) is a differentiable nondecreasing function with   being nonincreasing on [0, +∞),  ≥ 1, and  1 ≥ 1.Then, it is obvious that ℎ() is an increasing function on .Therefore, we have ℎ With a similar differentiation method, it is easy to obtain the following lemma.Proof.To prove this theorem, it is sufficient to prove that a schedule can be enhanced by scheduling the job with longer processing time into the earlier position of the schedule under the given condition in the theorem.Let the schedule  be [ 1 ,   ,   ,  2 ,  2 ] and let the schedule where  1 and  2 are partial job sequences, and the difference between  and   is a pairwise interchange of two adjacent jobs   and   in the schedule.Concretely,   and   are assumed to be assigned in the th and +1th position, respectively, in the schedule .In   ,   and   are assigned in the + 1th and th position, respectively.Without loss of generality, the normal processing times of   and   are assumed to satisfy   ≤   .To show that   dominates , it suffices to show that   () ≥   (  ) and   () ≥   (  ) for any job   in  2 .Let the completion time of the last job in partial schedule  1 be   1 .In the schedule , the completion times of   and   are respectively.In the schedule   , the completion times of   and   are respectively.Combining ( 13) and ( 15), we have Let  = ∑ −1 =1    [] ,  =   /  ,  1 = (( + 1)/()),  2 = (1/()), and  =     .Based on the formulation of the considered problem, it is easy to obtain that  ≥ 0,  ≥ 1,  1 ≥ 1, 0 <  2 ≤ 1, and  ≥ 0. Based on Lemma 1 and ( 16), we have Assuming that   is the job scheduled in the  + 2 position of the job sequence, then the completion times of   in  and   are respectively.Since   ≤   and   ≤  +1 , it can be obtained that Since  is a nondecreasing function, ( 20) is equivalent to Let   () and   (  ) denote the completion time of job   in the schedule  and   , respectively.From ( 17)-( 19) and ( 21), we have   () ≥   (  ).Thus, we show that   starts earlier in   than in  and completes earlier in   than in .With a similar proof method, it can be shown that   () ≥   (  ) for any job   in  2 .By repeating the above argument, it is obvious to lead to the conclusion that a schedule can be enhanced by scheduling the job with longer processing time into the earlier position of the job sequence.
Based on Lemma 2, the following theorem can be obtained by using a similar proof method to that of Theorem 3. If () is unchanged with respect to , then two actual processing time models can be both simplified as ) is unchanged with respect to ∑ −1 =1    [] , then two actual processing time models can be both simplified as   =   ().It is obvious that the single-machine problem with the above special models can be also optimally solved by the LPT rule.

Flow Shop Scheduling
The formulation of the flow shop scheduling problem can be described as follows.There are  independent jobs  1 ,  2 , . . .,   ready to be processed on  machines  1 ,  2 , . . .,   at time zero.Job   consists of  operations  1 ,  2 , . . .,   .Preemption is not allowed and a machine is only able to process one operation at a time.All machines should process the operations of the jobs in the same order.Operation  +1, may start only after operation   has been completed.Operation   has a normal processing time   ( = 1, 2, . . ., ;  = 1, 2, . . ., ), and the actual processing time of operation   is   if it is scheduled in position  ( = 1, 2, . . ., ) in a sequence.In this paper, we propose two general deterioration models.By the first model, if operation   is scheduled in the  position of the sequence, the actual processing time of operation   is where ∑ 0 =1    [,l] := 0,  : [0, +∞) → [1, +∞) is a differentiable nondecreasing function, and p [11] H(p [12] , f( where all the notations have the same means as those of the first model.
In the following, we first introduce a new general deterioration model.The actual processing time of job  on the  position of machine  is where ), and ().Obviously, model ( 22) and model ( 23) are both special cases of new model (24).
Proof.Let  [] denote the completion time of the job scheduled in th position of the job sequence.For  = 1, we have  [1] =  [11] +  2 .
the Longest Processing Time first rule of   (the LPT rule) if   is nonincreasing on [0, +∞).

Conclusions
This paper considered three scheduling problems with general deterioration models, where the actual processing time function of a job depends not only on the total weighted normal processing times of jobs already processed but also on the scheduled position in the job sequence.Firstly, for two single-machine scheduling problems with general deterioration models associated with both the total weighted normal processing times of the jobs and the scheduled position in the job sequence, we proved that two singlemachine scheduling problems are both polynomially solvable under the condition that the derived actual processing time functions are nonincreasing, respectively.Secondly, based on the results of the single-machine problem, we also showed that four flow shop scheduling problems can be optimally solved in polynomial time under proposed conditions.
Future works may focus on developing randomized algorithms for the scheduling problem without pregiven conditions.The variability of tool changes or maintenance activities is also worth considering in scheduling problems with general deterioration effects.
1) = 1, and   is the weight of position ; []denotes the normal processing time of the operation scheduled in the th position on machine   , where the subscript [, ] denotes the job in position  of the sequence on   .In many real life scenarios, positions are characterized by a different grade of complexity; thus, each position can provide different complexity to the manufacturing process; hence, we assume that 0 ≤  1 ≤  2 ⋅ ⋅ ⋅   .By the second model, if job  is scheduled on the  position of machine , its actual processing time is