Minimizing Total Weighted Completion Time on Single Machine with Past-Sequence-Dependent Setup Times and Exponential Time-Dependent and Position-Dependent Learning Effects

Abstract

This paper addresses a single-machine problem in which the past-sequence-dependent (p-s-d) setup times and exponential time-dependent and position-dependent learning effects are considered. By the exponential time-dependent learning effect, it means that the processing time of a job is defined by an exponent function of the total actual processing time of the already processed jobs. The setup times are proportional to the length of the already processed jobs. The aim is to minimize the total weighted completion time, this is an NP-hard problem. Under certain conditions, it is shown that the classical WSPT rule is optimal for the problem.

1. Introduction

In classical scheduling problems, the processing time of a job is assumed to be a constant. However, in many realistic problems of operations management, both machines and workers can improve their performance by repeating the production operations. Therefore, the actual processing time of a job is shorter if it is scheduled later in a sequence. This phenomenon is known as the “learning effect” in literature (see [1]). During the last few years, learning effect has attracted growing attention from the scheduling research community on account of its significance. Concerning the learning effects, two different approaches in scheduling environments have been suggested. The first one can be best described as position-dependent learning, which was first introduced into the field of scheduling by Biskup in [2] and Cheng and Wang in [3], meaning that learning is affected by the pure number of jobs being processed. Many references have studied such a learning effect model thereafter, a sample of these papers can be seen in [4–13]. However, the position-dependent learning effects neglect the processing times of the jobs already processed. If human interactions have a significant impact during the processing of jobs, the processing time will add to the employees' experience and cause learning effects. For situations like this, Kuo and Yang in [14] introduced a time-dependent learning effect, that is, the sum of processing time dependent learning, and showed that the shortest processing time (SPT) rule is optimal for the objective of minimizing the total completion time. For more papers about this time-dependent learning model, the reader is referred to the papers [15–17]. Besides, Koulamas and Kyparisis in [18] introduced a general sum of processing time-dependent learning effect model in which employees learn more if they perform a job with a longer processing time. Recently, Wu and Lee in [19] proposed a new learning model where the actual job processing time not only depends on its scheduled position but also depends on the sum of the processing times of jobs already processed. Let 𝑝_𝑗𝑟 denote the actual processing time job 𝐽_𝑗 scheduled in 𝑟th position, then 𝑝_𝑗𝑟=(1+∑^𝑟−1_𝑘=1𝑝_[𝑘]/∑^𝑛_𝑘=1𝑝_𝑘)^𝑎1𝑟^𝑎2, where 𝑝_[𝑘] is the normal processing time of the 𝑘th job in a sequence and 𝑎₁≤0,𝑎₂≤0. Under this model they showed that the SPT sequence is optimal for minimizing makespan and total completion time on single machine under the proposed learning model. In addition, they showed that the WSPT sequence is optimal for minimizing the sum of the weighted completion times if jobs have agreeable weights. Cheng et al. in [20] consider another learning model, in which the learning effect is given as 𝑝_𝑗𝑟=(1−∑^𝑟−1_𝑘=1𝑝_[𝑘]/∑^𝑛_𝑘=1𝑝_𝑘)^𝑎1𝑟^𝑎2, where 𝑎₁≥1,𝑎₂≤0. Under this learning model, they obtained the same results as that in [19]. Further, they showed that the EDD sequence is optimal for the maximum lateness if jobs have agreeable due dates. For a recent state-of-the-art review on scheduling models and problems with learning effects, the readers are referred to [21].

On the other hand, it is reasonable and necessary to consider scheduling problems with setup times (see [22]). There are two types of setup time or setup cost, namely, sequence-independent and sequence-dependent. In the first case, the setup time/cost depends solely on the task to be processed, regardless of its preceding task. While in the sequence-dependent type, setup time/cost depends on both the task and its preceding task. For recent results and trends in scheduling problems with setup times or costs, the reader may refer to the recent review paper [23]. Koulamas and Kyparisis in [24] first introduced a scheduling problem with past-sequence-dependent (p-s-d) setup times, that is, the setup time is dependent on all already scheduled jobs. They showed that the standard single-machine scheduling problems with p-s-d setup times to minimize makespan, total completion time and the total absolute differences in completion times are polynomially solvable. They also extended their results to nonlinear p-s-d setup times. Subsequently, Kuo and Yang in [25], Wang in [26], Biskup and Herrmann [27], Wang et al. in [28] studied different single-machine scheduling problems with time-dependent learning effect and setup times considerations.

From the learning models mentioned earlier, it is not hard to see that the actual processing time of, say, the 𝑟th job is affected by the sum of the normal processing times of the preceding 𝑟-1 jobs. This learning effect model is based on the assumption that the number of operating processes in a job depends on the repetitions of the same operation. However, in many situations, the operating processes of a job are different, for example, car repair or maintenance and patient diagnosis and treatment. The conditions of cars or patients are different. Hence, there are no identical repetitions of operating processes in the job. Nevertheless, there still exists a certain learning effect after operating the job. In such situations, the learning effect is due to the experiences of operating jobs, that is, the total actual processing time of jobs. To the best of our knowledge, apart from the recent paper [29], it has not been investigated the scheduling models considering the actual time-dependent learning effect. In this paper, we consider a single machine total completion time minimization problem with past-sequence-dependent setup times and learning effect where the actual processing time of a job is not only an exponential function of the total actual processing times of the jobs already processed but also a function of the job's scheduled position.

2. Problem Formulation

The model is described as follows. There are given a single machine and 𝑛 independent and nonpreemptive jobs that are immediately available for processing. The normal processing time of job 𝐽_𝑗 is 𝑝_𝑗 (1≤𝑗≤𝑛). Associated with each job 𝐽_𝑗(1≤𝑗≤𝑛) is a weight 𝑤_𝑗 and a due-date 𝑑_𝑗. Let 𝑝_[𝑘] and 𝑝^𝐴_[𝑘] be the normal processing time and the actual processing time of a job when it is scheduled in the 𝑘th position in a sequence, respectively. Let 𝑝^𝐴_𝑗𝑟 be the actual processing time of job 𝐽_𝑗 when it is scheduled in position 𝑟 in a sequence. Then

𝑝^𝐴_𝑗𝑟=𝑝_𝑗𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏, (2.1) where 𝑝^𝐴_[1]=𝑝_[1], 𝑝^𝐴_[𝑠]=𝑝_[𝑠]𝛼^{−𝛽(∑𝑠−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑠^𝜏 (1<𝑠≤𝑟−1), 𝛼≥1, 𝛽≥0, and 𝜏≤0. For convenience, we denote this actual time-dependent and job-position dependent learning effect defined in (2.1) by 𝐿𝐸_{𝑎𝑡-𝑝}.

As in [24, 26], we assume that the p-s-d setup time of a job if it is scheduled in position 𝑟 is given by

𝑠_[1]=0,𝑠_[𝑟]=^𝑟−1_𝑘=1𝑏𝑝^𝐴_[𝑘], (2.2) where 𝑏≥0 is a normalizing constant, and we denote the p-s-d setup given in (2.2) by 𝑠_𝑝𝑠𝑑.

For a given schedule 𝑆, let 𝐶_𝑗(𝑆) denote the completion time of job 𝐽_𝑗 and 𝐶_[𝑟](𝑆) represent the complete time of the job scheduled in the 𝑟th position in 𝜋. For single-machine scheduling with past-sequence-dependent setup times and time-dependent learning effect, Wang in [26] showed that the completion time of the 𝑟th job in a given sequence 𝑆 is given by

𝐶_[𝑟](𝑆)=^𝑟_𝑘=1(𝑏(𝑟−𝑘)+1)𝑝^𝐴_[𝑘]. (2.3) Similarly, the completion time of the 𝑟th job in a given sequence 𝑆 under (2.1) is

𝐶_[𝑟](𝑆)=^𝑟_𝑘=1(𝑏(𝑟−𝑘)+1)𝑝_[𝑘]𝛼^{−𝛽(∑𝑘−1𝑙=1𝑝𝐴[𝑙]/∑𝑛𝑙=1𝑝[𝑙])}𝑘^𝜏. (2.4)

Let 𝐶_max denote the makespan, where 𝐶_max=max{𝐶₁,𝐶₂,…,𝐶_𝑛}. Let ∑𝑤_𝑗𝐶_𝑗 represent the weighted sum of completion times. In the remaining part of the paper, the problem will be denoted using the three-field notation scheme 1/𝐿𝐸_{𝑎𝑡-𝑝},𝑠_𝑝𝑠𝑑/∑𝑤_𝑗𝐶_𝑗 introduced by Graham et al. in [30].

3. Total Completion Time Minimization Problem

In this section, we consider the total completion time minimization problems with past-sequence-dependent setup times defined in (2.2) and the learning effect defined in (2.1).

First, we give three lemmas, which are useful for the following theorem.

Lemma 3.1. It holds that, 𝑡^𝜆((𝑟+1)/𝑟)^𝜏−𝜆𝑡((𝑟+1)/𝑟)^𝜏+𝜆−1>0for𝜆>1,0≤𝑡≤1,𝜏≤0,and𝑟=1,2,…,𝑛−1.

Proof. Let 𝐹(𝑡)=𝑡^𝜆(𝑟+1𝑟)^𝜏−𝜆𝑡(𝑟+1𝑟)^𝜏+𝜆−1. (3.1) Taking the first derivative of 𝐹(𝑡) with respect to 𝑡, we have 𝐹^′(𝑡)=𝜆𝑡^𝜆−1(𝑟+1𝑟)^𝜏−𝜆(𝑟+1𝑟)^𝜏=𝜆(𝑟+1𝑟)^𝜏(𝑡^𝜆−1−1). (3.2) Since 𝜆>1 and 0≤𝑡≤1, we have 𝐹^′(𝑡)≤0. This implies that 𝐹(𝑡) is nonincreasing on 0≤𝑡≤1. Hence, 𝐹(𝑡)≥𝐹(1)=(𝑟+1𝑟)^𝜏−𝜆(𝑟+1𝑟)^𝜏+𝜆−1=(𝜆−1)(1−(𝑟+1𝑟)^𝜏)>0 (3.3) for 𝜆>1, 𝜏≤0 and 𝑟=1,2,…,𝑛−1. This completes the proof.

Lemma 3.2. It holds that, 1−𝜆₁𝑡((𝑟+1)/𝑟)^𝜏+𝜆₂𝑡ln𝑡((𝑟+1)/𝑟)^𝜏≥0for0<𝑡≤1,𝜏≤0,0≤𝜆₁≤𝜆₂≤1,and𝑟=1,2,…,𝑛−1.

Proof. Let 𝐹(𝑡,𝜆₁,𝜆₂)=1−𝜆₁𝑡(𝑟+1𝑟)^𝜏+𝜆₂𝑡ln𝑡(𝑟+1𝑟)^𝜏. (3.4) Then 𝜕𝐹/𝜕𝜆₁=−𝑡((𝑟+1)/𝑟)^𝜏<0,𝜕𝐹/𝜕𝜆₂=𝑡ln𝑡((𝑟+1)/𝑟)^𝜏≤0, thus we have 𝐹(𝑡,𝜆₁,𝜆₂)≥𝐹(𝑡,1,1)=1−𝑡(𝑟+1𝑟)^𝜏+𝑡ln𝑡(𝑟+1𝑟)^𝜏. (3.5) Let 𝑓(𝑡)=1−𝑡(𝑟+1𝑟)^𝜏+𝑡ln𝑡(𝑟+1𝑟)^𝜏. (3.6) Then 𝑓^′(𝑡)=((𝑟+1)/𝑟)^𝜏ln𝑡≤0, hence 𝑓(𝑡) is nonijncreasing on 𝑡 and 𝑓(𝑡)≥𝑓(1)=1−((𝑟+1)/𝑟)^𝜏≥0 for 0<𝑡≤1. Therefore, 𝐹(𝑡,𝜆₁,𝜆₂)≥0 for 0<𝑡≤1, 𝜏≤0, 0≤𝜆₁≤𝜆₂≤1, and 𝑟=1,2,…,𝑛−1. This completes the proof.

Lemma 3.3. It holds that, 𝜇[1−𝜆₁𝑡((𝑟+1)/𝑟)^𝜏]−[1−𝜆₂𝑡^𝜇((𝑟+1)/𝑟)^𝜏]≥0 for𝜇≥1,0<𝑡≤1,𝜏≤1, 0≤𝜆₁≤𝜆₂≤1, and 𝑟=1,2,…,𝑛−1.

Proof. Let ℎ(𝜇)=𝜇[1−𝜆₁𝑡(𝑟+1𝑟)^𝜏]−[1−𝜆₂𝑡^𝜇(𝑟+1𝑟)^𝜏]. (3.7) To take the first and second derivatives of (3.7) with respect to 𝜇, we have ℎ^′(𝜇)=1−𝜆₁𝑡(𝑟+1𝑟)^𝜏+𝜆₂𝑡^𝜇ln𝑡(𝑟+1𝑟)^𝜏,ℎ^′′′(𝜇)=𝜆₂𝑡^𝜇ln²𝑡(𝑟+1𝑟)^𝜏≥0. (3.8) Hence, ℎ^′(𝜇) is nondecreasing on 𝜇≥1, that is, ℎ^′(𝜇)≥ℎ^′(1) for 𝜇≥1. By Lemma 3.2, we have ℎ^′(1)=1−𝜆₁𝑡(𝑟+1𝑟)^𝜏+𝜆₂𝑡ln𝑡(𝑟+1𝑟)^𝜏≥0. (3.9) Therefore, ℎ^′(𝜇)≥ℎ^′(1)≥0. Hence, ℎ(𝜇) is nondecreasing on 𝜇, that is, ℎ(𝜇)≥ℎ(1)=(𝜆₂−𝜆₁)𝑡((𝑟+1)/𝑟)^𝜏≥0 for 𝑡≥0,𝜏≤0, 0≤𝜆₁≤𝜆₂≤1 and 𝑟=1,2,…,𝑛−1. This completes the proof.

Theorem 3.4. For the total completion time minimization problem 1/𝐿𝐸_{𝑎𝑡-𝑝},𝑠_𝑝𝑠𝑑/∑𝑤_𝑗𝐶_𝑗 when 𝑝^𝐴_𝑗𝑟 is given by (2.1), an optimal schedule can be obtained by sequencing jobs in the WSPT rule if 𝑝_𝑙≤((𝑏+1)/3^𝜏𝛽ln𝛼)∑^𝑛_𝑘=1𝑝_𝑘(𝑙=1,…,𝑛) and jobs have reversely agreeable weights, that is, 𝑝_𝑖<𝑝_𝑗 implies 𝑤_𝑖≥𝑤_𝑗 for all jobs 𝐽_𝑖 and 𝐽_𝑗.

Proof. Consider an optimal schedule 𝑆 that does not follow the WSPT rule. Then in this schedule there must be at least two adjacent jobs, say job 𝐽_𝑗 followed by job 𝐽_𝑖 such that 𝑝_𝑗/𝑤_𝑗>𝑝_𝑖/𝑤_𝑖 which implies that 𝑝_𝑗≥𝑝_𝑖 by reversely agreeable weights. If 𝑝_𝑗>𝑝_𝑖, then 𝑤_𝑖≥𝑤_𝑗; if 𝑝_𝑖=𝑝_𝑗, then 𝑤_𝑖>𝑤_𝑗. Assume that job 𝐽_𝑗 is scheduled in position 𝑟. By performing an adjacent pair-wise interchange of jobs 𝐽_𝑗 and 𝐽_𝑖 and remaining all the other jobs in their original positions, we can obtain a new schedule, denoted by 𝑆^′. Whereas under the original schedule 𝐽_𝑗 is scheduled in position 𝑟 and 𝐽_𝑖 is scheduled in position 𝑟+1, under 𝑆^′, 𝐽_𝑖 is scheduled in position 𝑟, and 𝐽_𝑗 is scheduled in position 𝑟+1. Denote 𝑆=(𝜎₁,𝐽_𝑗,𝐽_𝑖,𝜎₂), let 𝐶_𝜎1 be the completion time of the last job in 𝜎₁, and let 𝐽_𝑘 be the first job in 𝜎₂.
The completion times of the jobs processed before jobs 𝐽_𝑗 and 𝐽_𝑖 are not affected by the interchange. In the following we will show that 𝑤_𝑖𝐶_𝑖(𝑆^′)+𝑤_𝑗𝐶_𝑗(𝑆^′)≤𝑤_𝑖𝐶_𝑖(𝑆)+𝑤_𝑗𝐶_𝑗(𝑆) and the completion times of the jobs processed after jobs 𝐽_𝑗 and 𝐽_𝑖 are not increased by the interchanging argument. Under 𝑆, the completion times of jobs 𝐽_𝑗 and 𝐽_𝑖 are 𝐶_𝑗(𝑆)=^𝑟−1_𝑘=1(𝑏(𝑟−𝑘)+1)𝑝^𝐴_[𝑘]+𝑝_𝑗𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏,𝐶_𝑖(𝑆)=^𝑟−1_𝑘=1(𝑏(𝑟+1−𝑘)+1)𝑝^𝐴_[𝑘]+(𝑏+1)𝑝_𝑗𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])𝑟𝜏}+𝑝_𝑖𝛼^{−𝛽((∑𝑟−1𝑘=1𝑝𝐴[𝑘]+𝑝𝐴𝑗𝑟)/∑𝑛𝑘=1𝑝[𝑘])}(𝑟+1)^𝜏. (3.10) whereas under 𝑆^′, they are 𝐶_𝑖(𝑆^′)=^𝑟−1_𝑘=1(𝑏(𝑟−𝑘)+1)𝑝^𝐴_[𝑘]+𝑝_𝑖𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏,𝐶_𝑗(𝑆^′)=∑^𝑟−1_𝑘=1(𝑏(𝑟+1−𝑘)+1)𝑝^𝐴_[𝑘]+(𝑏+1)𝑝_𝑖𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏+𝑝_𝑗𝛼^{−𝛽((∑𝑟−1𝑘=1𝑝𝐴[𝑘]+𝑝𝐴𝑗𝑟)/∑𝑛𝑘=1𝑝[𝑘])}(𝑟+1)^𝜏. (3.11) So we have 𝑤_𝑖𝐶_𝑖(𝑆)+𝑤_𝑗𝐶_𝑗(𝑆)−𝑤_𝑖𝐶_𝑖(𝑆^′)−𝑤_𝑗𝐶_𝑗(𝑆^′)=𝑏(𝑤_𝑖−𝑤_𝑗)^𝑟−1_𝑘=1𝑝^𝐴_[𝑘]+𝑏(𝑤_𝑖𝑝_𝑗−𝑤_𝑗𝑝_𝑖)𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏+(𝑤_𝑖+𝑤_𝑗)(𝑝_𝑗−𝑝_𝑖)𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏+𝑤_𝑖𝑝_𝑖𝛼^{−𝛽((∑𝑟−1𝑘=1𝑝𝐴[𝑘]+𝑝𝐴𝑗𝑟)/∑𝑛𝑘=1𝑝[𝑘])}(𝑟+1)^𝜏−𝑤_𝑗𝑝_𝑗𝛼^{−𝛽((∑𝑟−1𝑘=1𝑝𝐴[𝑘]+𝑝𝐴𝑖𝑟)/∑𝑛𝑘=1𝑝[𝑘])}(𝑟+1)^𝜏. (3.12) Let 𝑥=𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}, 𝜆₁=𝑤_𝑗/(𝑤_𝑖+𝑤_𝑗), 𝜆₂=𝑤_𝑖/(𝑤_𝑖+𝑤_𝑗), 𝜇=𝑝_𝑗/𝑝_𝑖, and 𝑡=𝛼^{−𝛽(𝑝𝐴𝑖𝑟/∑𝑛𝑘=1𝑝𝑘)}. Clearly 𝜇≥1, 0≤𝜆₁≤𝜆₂≤1, and 𝑡>0. Then the last three terms in (3.12) can be rewritten as 𝑤_𝑖𝐶_𝑖(𝑆)+𝑤_𝑗𝐶_𝑗(𝑆)−𝑤_𝑖𝐶_𝑖(𝑆^′)−𝑤_𝑗𝐶_𝑗(𝑆^′)=𝑝_𝑖(𝑤_𝑖+𝑤_𝑗)𝑟^𝜏𝑥{𝜇[1−𝜆₁𝑡(𝑟+1𝑟)^𝜏]−[1−𝜆₂𝑡^𝜇(𝑟+1𝑟)^𝜏]}≥0(byLemma3.3). (3.13) From 𝑝_𝑗/𝑤_𝑗>𝑝_𝑖/𝑤_𝑖, we have 𝑏(𝑤_𝑖𝑝_𝑗−𝑤_𝑗𝑝_𝑖)𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏>0. In addition, from 𝑝_𝑗≥𝑝_𝑖, which implies that 𝑤_𝑖≥𝑤_𝑗, we have 𝑏(𝑤_𝑖−𝑤_𝑗)∑^𝑟−1_𝑘=1𝑝^𝐴_[𝑘]≥0. Hence, 𝑤_𝑖𝐶_𝑖(𝑆)+𝑤_𝑗𝐶_𝑗(𝑆)>𝑤_𝑖𝐶_𝑖(𝑆^′)+𝑤_𝑗𝐶_𝑗(𝑆^′). (3.14) Next, we will show that the completion times of the jobs in 𝜎₂ are not increased by the interchanging argument. Firstly, we show that 𝐶_𝑗(𝑆^′)≤𝐶_𝑖(𝑆) after interchanging. Since 𝐶_𝑗(𝑆^′)−𝐶_𝑖(𝑆)𝑝_𝑖𝑥𝑟^𝜏𝑝=(𝑏+1)(1−^𝑗𝑝_𝑖𝑝)+^𝑗𝑝_𝑖𝛼^{−𝛽(𝑝𝐴𝑖𝑟/∑𝑛𝑘=1𝑝[𝑘])}(𝑟+1𝑟)^𝜏−𝛼^{−𝛽(𝑝𝐴𝑗𝑟/∑𝑛𝑘=1𝑝[𝑘])}(𝑟+1𝑟)^𝜏, (3.15) then let 𝑡=𝛼^{−𝛽(𝑝𝐴𝑖𝑟/∑𝑛𝑘=1𝑝𝑘)} and 𝜆=𝑝_𝑗/𝑝_𝑖. It is clearly that 𝑡≥0 and 𝜆>1. By substituting 𝑡 and 𝜆 into (3.15), it is simplified to 𝐶_𝑗(𝑆^′)−𝐶_𝑖(𝑆)𝑝_𝑖𝑥𝑟^𝜏={(𝑏+1)(1−𝜆)+𝜆𝑡(𝑟+1𝑟)^𝜏−𝑡^𝜆(𝑟+1𝑟)^𝜏}. (3.16) By Lemma 3.1, we have 𝐶_𝑗(𝑆^′)<𝐶_𝑖(𝑆).
Secondly, we show that the completion times of the jobs in 𝜎₂ are not increased. Note that 𝐽_𝑘 is the first job in 𝜎₂, it is scheduled in (𝑟+2)th position, let Λ₁=𝑝_𝑖𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏+𝑝_𝑗𝛼^{−𝛽(∑𝑟𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}(𝑟+1)^𝜏,Λ₂=𝑝_𝑗𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏+𝑝_𝑖𝛼^{−𝛽(∑𝑟𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}(𝑟+1)^𝜏. (3.17) Then we have Λ₁−Λ₂𝑝_𝑖𝑥𝑟^𝜏=(1−𝜆)+𝜆𝑡(𝑟+1𝑟)^𝜏−𝑡^𝜆(𝑟+1𝑟)^𝜏. (3.18) By Lemma 3.1 we have Λ₁<Λ₂. Since 𝐶_𝑘(𝑆^′)=𝐶_𝑗(𝑆^′)+𝑏(^𝑟−1_𝑙=1𝑝^𝐴_[𝑙]+𝑝^𝐴_𝑖𝑟+𝑝^𝐴_𝑗𝑟+1)+𝑝_𝑘𝛼^{−𝛽((∑𝑟−1𝑙=1𝑝𝐴[𝑙]+𝑝𝐴𝑖𝑟+𝑝𝐴𝑗𝑟+1)/∑𝑛𝑙=1𝑝[𝑙])}(𝑟+2)^𝜏=^𝑟−1_𝑘=1(𝑏(𝑟+2−𝑘)+1)𝑝^𝐴_[𝑘]+(2𝑏+1)𝑝_𝑖𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏+(𝑏+1)𝑝_𝑗𝛼^{−𝛽((∑𝑟−1𝑘=1𝑝𝐴[𝑘]+𝑝𝐴𝑖𝑟)/∑𝑛𝑘=1𝑝[𝑘])}(𝑟+1)^𝜏+𝑝_𝑘𝛼^{−𝛽((∑𝑟−1𝑙=1𝑝𝐴[𝑙]+𝑝𝐴𝑖𝑟+𝑝𝐴𝑗𝑟+1)/∑𝑛𝑙=1𝑝[𝑙])}(𝑟+2)^𝜏,𝐶_𝑘(𝑆)=𝐶_𝑖(𝑆)+𝑏(^𝑟−1_𝑙=1𝑝^𝐴_[𝑙]+𝑝^𝐴_𝑗𝑟+𝑝^𝐴_𝑖𝑟+1)+𝑝_𝑘𝛼^{−𝛽((∑𝑟−1𝑙=1𝑝𝐴[𝑙]+𝑝𝐴𝑗𝑟+𝑝𝐴𝑖𝑟+1)/∑𝑛𝑙=1𝑝[𝑙])}(𝑟+2)^𝜏=^𝑟−1_𝑘=1(𝑏(𝑟+2−𝑘)+1)𝑝^𝐴_[𝑘]+(2𝑏+1)𝑝_𝑗𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏+(𝑏+1)𝑝_𝑖𝛼^{−𝛽((∑𝑟−1𝑘=1𝑝𝐴[𝑘]+𝑝𝐴𝑗𝑟)/∑𝑛𝑘=1𝑝[𝑘])}(𝑟+1)^𝜏+𝑝_𝑘𝛼^{−𝛽((∑𝑟−1𝑙=1𝑝𝐴[𝑙]+𝑝𝐴𝑗𝑟+𝑝𝐴𝑖𝑟+1)/∑𝑛𝑙=1𝑝[𝑙])}(𝑟+2)^𝜏. (3.19) Then we have 𝐶_𝑘(𝑆^′)−𝐶_𝑘(𝑆)=𝑏(𝑝_𝑖−𝑝_𝑗)𝛼^{−𝛽(∑𝑟−1𝑘=1𝑝𝐴[𝑘]/∑𝑛𝑘=1𝑝[𝑘])}𝑟^𝜏+(𝑏+1)(Λ₁−Λ₂)+𝑝_𝑘(𝑟+2)^𝜏(𝛼^{−𝛽((Λ+Λ1)/𝑃)}−𝛼^{−𝛽((Λ+Λ2)/𝑃)})<(𝑏+1)(Λ₁−Λ₂)+𝑝_𝑘(𝑟+2)^𝜏(−𝛽ln𝛼𝑃𝛼^{−𝛽((Λ+𝜉)/𝑃)})(Λ₁−Λ₂)=(Λ₂−Λ₁)[𝑝_𝑘(𝑟+2)^𝜏(𝛽ln𝛼𝑃𝛼^{−𝛽((Λ+𝜉)/𝑃)})−(𝑏+1)]<[(𝛽3^𝜏𝑝ln𝛼)^𝑘𝑃−(𝑏+1)](Λ₂−Λ₁). (3.20) For any job 𝐽_𝑙, by assumption 𝑝_𝑙≤(𝑏+1)/3^𝜏𝛽ln𝛼∑^𝑛_𝑘=1𝑝_𝑘 for 𝛼≥1 and 𝛽≥0. Then (3.20) is not larger than zero, that is, 𝐶_𝑘(𝑆^′)≤𝐶_𝑘(𝑆). Similarly, we have 𝐶_ℎ(𝑆^′)≤𝐶_ℎ(𝑆) for any 𝐽_ℎ in 𝜎₂.
Repeating this interchange argument for all jobs not sequenced in the WSPT rule yields the theorem.

Corollary 3.5. For the problem1/𝐿𝐸_{𝑎𝑡-𝑝},𝑠_𝑝𝑠𝑑/𝐶_max when 𝑝^𝐴_𝑗𝑟 is given by (2.1), if 𝑝_𝑙≤∑^𝑛_𝑘=1𝑝_𝑘/3^𝜏𝛽ln𝛼(𝑙=1,…,𝑛), then there exists an optimal schedule in which the job sequence is determined by the SPT rule.

Proof. it can be directly from the proof of Theorem 3.4.

Corollary 3.6. For the problem 1/𝐿𝐸_{𝑎𝑡-𝑝},𝑠_𝑝𝑠𝑑/∑𝐶_𝑗 when 𝑝^𝐴_𝑗𝑟 is given by (2.1), if 𝑝_𝑙≤∑^𝑛_𝑘=1𝑝_𝑘/3^𝜏𝛽ln𝛼(𝑙=1,…,𝑛), then there exists an optimal schedule in which the job sequence is determined by the SPT rule.

Corollary 3.7. For the problem 1/𝐿𝐸_{𝑎𝑡-𝑝},𝑠_𝑝𝑠𝑑/∑𝐶^𝑘_𝑗 when 𝑝^𝐴_𝑗𝑟 is given by (2.1), if 𝑝_𝑙≤∑^𝑛_𝑘=1𝑝_𝑘/3^𝜏𝛽ln𝛼(𝑙=1,…,𝑛), then there exists an optimal schedule in which the job sequence is determined by the SPT rule.

Proof. Here, we still use the same notations as in the proof of Theorem 3.4. In order to show 𝑆^′ dominates 𝑆, it suffices to show that (i) 𝐶_𝑗(𝑆^′)≤𝐶_𝑖(𝑆), (ii) 𝐶_𝑙(𝑆^′)≤𝐶_𝑙(𝑆) for any job 𝐽_𝑙 (𝑙≠𝑖,𝑗) and (iii) 𝐶^𝑘_𝑖(𝑆^′)+𝐶^𝑘_𝑗(𝑆^′)≤𝐶^𝑘_𝑖(𝑆)+𝐶^𝑘_𝑗(𝑆).
The proof of part (i) and part (ii) are given in Theorem 3.4. In addition, from 𝑝_𝑖≤𝑝_𝑗, we have 𝐶_𝑖(𝑆^′)≤𝐶_𝑗(𝑆). Since 𝑘 is a positive real number, we have 𝐶^𝑘_𝑖(𝑆^′)+𝐶^𝑘_𝑗(𝑆^′)≤𝐶^𝑘_𝑖(𝑆)+𝐶^𝑘_𝑗(𝑆). This completes the proof.

Corollary 3.8. For the problem 1/𝐿𝐸_{𝑎𝑡-𝑝},𝑠_𝑝𝑠𝑑,𝑝_𝑗=𝑝/∑𝑤_𝑗𝐶_𝑗 when 𝑝_𝑗𝑟 is given by (2.1), an optimal schedule can be obtained by sequencing jobs in non-increasing order of 𝑤_𝑗.

Corollary 3.9. For the problem 1/𝐿𝐸_{𝑎𝑡-𝑝},𝑠_𝑝𝑠𝑑,𝑤_𝑗=𝑘𝑝_𝑗/∑𝑤_𝑗𝐶_𝑗when 𝑝_𝑗𝑟 is given by (2.1), if 𝑝_𝑙≤((𝑏+1)/3^𝜏𝛽ln𝛼)∑^𝑛_𝑘=1𝑝_𝑘(𝑙=1,…,𝑛), then an optimal schedule can be obtained by sequencing jobs in SPT rule.

Remark 3.10. In this paper, the learning effect is defined in (2.1) and p-s-d setup times is defined in (2.2), where the 𝛼≥1, 𝛽≥0, 𝜏≤0, and 𝑏≥0 are controlled parameters. In practises, even human interactions have a significant impact during the processing of jobs, the impact should be limited. In general, half of the decreasing on the processing time is impossible. So the parameters should have practical meaning and properly given. If there parameters satisfy that ((𝑏+1)/3^𝜏𝛽ln𝛼)<1, then for any job 𝑝_𝑙(𝑙=1,2,…,𝑛) we have 𝑝_𝑙≤((𝑏+1)/3^𝜏𝛽ln𝛼)∑^𝑛_𝑘=1𝑝_𝑘. Thus, Theorem 3.4 holds under agreeable condition .

4. Conclusions

In this paper we investigated the single machine total weighted completion time minimization problem with p-s-d setup times and learning effect where the actual processing time of a job is not only an exponential function of the total actual processing times of the jobs already processed but also a function of the job's scheduled position. We show that the classical WSPT rule is optimal under certain conditions. It is suggested for future research to investigate other efficient heuristic algorithm to the problem or extend this kind of learning into the context of other more complicated scheduling settings, or propose more general and practical models.

Acknowledgment

This research is supported by the National Natural Science Foundation of China (60774049).