Flowshop Scheduling Problems with a Position-Dependent Exponential Learning Effect

We consider a permutation flowshop scheduling problemwith a position-dependent exponential learning effect.The objective is to minimize the performance criteria of makespan and the total flow time. For the two-machine flow shop scheduling case, we show that Johnson’s rule is not an optimal algorithm for minimizing themakespan given the exponential learning effect. Furthermore, by using the shortest total processing times first (STPT) rule, we construct the worst-case performance ratios for both criteria. Finally, a polynomial-time algorithm is proposed for special cases of the studied problem.


Introduction
Many researchers have studied flowshop scheduling problems under various assumptions and with different objective functions.Minimizing the makespan of the classical flowshop is known to be an NP-hard problem except for Johnson's [1] two-machine case.Dannenbring [2], Gonzalez and Sahni [3], Smutnicki [4], and Cepek et al. [5] have developed approximation algorithms for some special cases of machine flowshop problems.
In traditional machine scheduling theory, the processing time of a job is independent of its processed position.However, due to workers' learning ability, working attitude, and their continuously improved skills with the passage of time, the processing time of a job is shorter if it is scheduled later in the production sequence.This phenomenon is known as a learning effect, which has been employed in management science since its discovery by Wright [6].Although the learning theory was first applied to industry more than 70 years ago, it remains an interesting important topic in scheduling research.Biskup [7] and Cheng and Wang [8] investigated the effect of learning in the framework of scheduling.Since then, scheduling with learning effect has received growing attention.Mosheiov [9] investigated Biskup's learning effect model in various scheduling problems and through several examples showed that the optimal schedule is very different from that of the classical versions.Mosheiov and Sidney [10] considered a single-machine scheduling problem with jobdependent learning effects with objectives such as makespan and total flow time, which are proved to be polynomialtime solvable.Bachman and Janiak [11,12] investigated several single machine scheduling problems with positiondependent processing time.
Many researchers have extended several kinds of learning effect model in machine scheduling problems.Wang [13] studied flowshop scheduling problems with job processing times dependent on their positions and suggested Johnson's rule as a heuristic algorithm to analyze the worst-case ratio of the makespan and special cases of -machine flowshop.Wang and Xia [14] and Xu et al. [15] extended Biskup's learning effect to minimize makespan and the total completion time in a flowshop setting.Biskup [16] provided an extensive review of scheduling with learning effects.Janiak and Rudek [17] considered a learning effect model in which the learning curve is S-shaped and provided NP-hard proofs for two cases of the problem to minimize the makespan.S.-J.Yang and D.-L.Yang [18] investigated a single-machine scheduling with a position-dependent aging effect described by a power function and variable maintenance duration.Li and Hsu [19] considered the case of two agents competing for a common single machine with learning effect.Lee et al. [20] studied a uniform parallel machine problem to jointly find an optimal assignment of operators to machines and an optimal schedule to minimize the makespan.Jiang et al. [21] introduced an actual time-dependent and job-dependent learning effect into single-machine scheduling problems.Wu et al. [22] proposed two truncated learning models in singlemachine scheduling problems.The related literature also includes Koulamas and Kyparisis [23], Mosheiov [24], Janiak et al. [25], Lee and Wu [26], Huang et al. [27], Zhang and Yan [28], Koulamas [29], J.-B.Wang and J.-J.Wang [30], and Kuo [31] and further references.
We know that the model   =     is proposed by Biskup [16], the effect for the given job processed in different positions is not stable; that is, if job   is processed at th and ( + 1)th position, respectively, then  + /  = (( + 1)/)  , and the processing time decreases quickly if  < 0 and far from zero.However, in many realistic settings, the learning process of workers should be stable, which is of utmost importance to guarantee the quality of the product.Therefore, we propose a position-dependent exponential learning effect model   =    −1 , where  ∈ (0, 1] is a learning index.In such model, the processed position effect for a given job in some schedule is a constant because of  + /  = , and the processing time decreases slowly if  is near 1, and the proposed learning effect model can reflect reasonably the stability of the manufacturing process. In this paper, we study permutation flowshop scheduling problems with a position-dependent exponential learning effect to minimize one of the following two regular performance criteria: makespan and the total flow time.An example is constructed to show that the classical Johnson rule is not optimal for the two-machine case with minimizing makespan under such a position-dependent exponential learning effect.We use the shortest total processing time first (STPT) rule (Dannenbring [2]) to solve our problem and obtain the same worst-case performance ratio for both criteria and prove this performance ratio to be tight.Furthermore, a polynomialtime algorithm is proposed for two special cases: identical processing times on all the machines for any given jobs and the flowshop scheduling problem with dominant machines.
The rest of the paper is organized as follows.In Section 2, we formalize the problem.We develop the worst-case performance ratios of the STPT algorithm and use it to solve our problem in Section 3. In Section 4, we analyze several special cases and prove that all the problems are solvable in polynomial time.Finally, concluding remarks are given in Section 5.

Formulation of the Problem
( 1 ,  2 , . . .,   ) jobs are to be processed on an -machine flowshop.We assume that the normal processing time of job   on machine   is denoted as  , and the actual processing time of job   on machine   at the th position is denoted as  ,, .Namely, the actual processing time of a job is characterized by a position-dependent exponential function:  ,, =  ,  −1 ,  = 1, 2, . . ., ; ,  = 1, 2, . . ., , where  ∈ (0, 1] denotes the common learning index for all jobs.The aim in this paper is to find a schedule so that  jobs should be processed on the  machines to minimize a given objective function. For a given schedule ,   () represents the completion time of job   ,  max () = max{  () |  = 1, 2, . . ., } denotes the makespan, and  = ([1], [2], . . ., []) denotes a schedule, where [] denotes a job that occupies the th position in .In the remaining part of this paper, all problems considered are denoted by the three-field notation scheme || introduced by Graham et al. [32]; that is,  |  ,, =  ,  −1 | (), where () ∈ { max , ∑   }.
Johnson [1] proved that 2 ‖  max can be optimally solved by Johnson's rule.However, in the following example, we show that this policy is not optimal for problem 2 |  ,, =  ,  −1 |  max , where 0 <  ≤ 1 is a constant learning index.
Before analyzing the worst-case ratio of the problems, we provide two lemmas.By simple interchange technique, the following results can be easily obtained.

Lemma 2.
For problem 1 |  , =    −1 |  max , an optimal schedule can be obtained by shortest processing time first (SPT) rule.
In the following, we turn our attention to obtain quasioptimal schedule.For the  ‖ ∑   problem, Dannenbring [2] gave the approximation algorithm shortest total processing time first (STPT) (a nondecreasing order of {  } (  = ∑  =1  , )) and showed that it has worst-case performance ratio .

Special Cases Solvable in Polynomial Time
We now consider the special case where a job has the same processing times on all machines for any given job; that is,  , =   .We know that for the  |  , =   |  max problem, the completion time of ,  [2] , . . .,  [] } (Pinedo [34]), where  denotes a schedule and [] denotes the job that occupies the th position in .Hence, for the (2) Theorem 6.For problem  |  ,, =    −1 |  max , an optimal schedule can be obtained by the SPT rule.
Proof.Consider a schedule .Suppose that job  [] occupy the th position in  and that  [] is defined accordingly.The schedule   is obtained from the schedule  by swapping jobs  [] and  [𝑖+1] .Assume that the schedule , in which , is optimal.The difference in makespan for the two schedules is given as follows: Since we have max That is,  max () >  max (  ), which contradicts our assumption that  is an optimal schedule.Therefore, the schedule obtained by using the SPT rule is optimal.
Proof.The proof is similar to that of Theorem 6 and is omitted.
From ( 6), we have where the first term is only dependent on job  [1] , while the second term is the total flow-time of all the jobs on machine   and is minimized by the SPT rule (Lemma 3).Therefore, an optimal schedule can be constructed for the problem  |  ,, =  ,  −1 | ∑   by Algorithm A. Proof.The proof is similar to that of Theorem 8 and is omitted.

Conclusion
In order to overcome the lack of stability in Biskups model in a manufacturing setting, we focus on flowshop scheduling problems with position-dependent exponential learning effect to minimize the makespan and the total flow time.For the two-machine case, we show that Johnson's algorithm does not obtain optimal result.We thus develop the worstcase ratios of the STPT algorithm to minimize the makespan and the total flow time, respectively, and empirically prove that both bounds are tight by several examples.Finally, for special cases, we show that the problems remain polynomially solvable.
Notice that the complexity of the studied problem remains unreciprocated and the worst-case ratios we developed heavily depend on the number of machines.The dependency of machine number in the flowshop is a limitation of our algorithm as it may create notable discrepancy between the optimal solution and that derived by our method.Therefore, our future research includes proving the complexity of the flowshop scheduling problem with position-dependent exponential learning effect and developing generalizable and effective algorithms for diverse problems.