Single-Machine Scheduling with Accelerating Learning Effects

Abstract

Scheduling with learning effects has been widely studied. However, there are situations where the learning effect might accelerate. In this paper, we propose a new model where the learning effect accelerates as time goes by. We derive the optimal solutions for the single-machine problems to minimize the makespan, total completion time, total weighted completion time, maximum lateness, maximum tardiness, and total tardiness.

1. Introduction

In classical scheduling, the processing times of the jobs are assumed to be constant over the entire planning horizon [1]. However, this assumption may not be appropriate for all the cases [2–5]. In many realistic situations, because the employees do the same job repeatedly, they learn how to do similar jobs more efficiently. As a result, the actual processing time of a job is shorter if it is scheduled later, rather than earlier in the sequence. This phenomenon is known as the “learning effect” and has been applied in scheduling problems. Dondeti and Mohanty [6], Biskup [7], Cheng and Wang [8], and Gawiejnowicz [9] are among the pioneers that incorporate the learning effect into scheduling problems. Dondeti and Mohanty [6] consider a single-machine scheduling problem in which the machine experiences learning as it continues to work. Biskup [7] points out that the learning effect in scheduling may arise in firms that produce similar jobs. Cheng and Wang [8] study a single-machine scheduling problem in which the job processing times will decrease as a result of learning. Since then, a number of researchers have considered the effects of learning in different scheduling settings, including single-machine scheduling, two-machine flowshop scheduling, and parallel machine scheduling. Cheng et al. [10], Gawiejnowicz [11], Biskup [12], and Rustogi and Strusevich [13] provide comprehensive reviews of scheduling models with time-dependent processing times.

Recently, Janiak and Rudek [14] introduce a new learning effect model into the scheduling field where learning is generalized in two ways. First, they allow each job to have a different learning effect to the operator. Second, they assume that the job processing time is a nonincreasing 𝑘-stepwise function, which in general is not restricted to a certain learning curve; hence, it can accurately fit every possible shape of a learning function. Wang et al. [15] consider some single-machine problems with a time-dependent learning effect. They find the optimal solutions or derive the worst-case error bounds for some proposed heuristic algorithms. Janiak and Rudek [16] consider and analyze a learning effect model in which the learning curve is S-shaped. They provide the NP-hard proofs for two cases of the problem to minimize the makespan. Lee and Wu [17] observe that robots with neural networks are used in computers, motor vehicles, and many assembly lines. The actions of a robot are constantly modified through self-learning in processing jobs. On the other hand, the operators in the control centre learn how to give commands efficiently through work experience. This motivates the consideration of both “position-based” and “sum-of-processing-time-based” learning effects simultaneously. Combining the models of Biskup [7] and Koulamas and Kyparisis [18], Cheng et al. [19] introduce a model with position-based and sum-of-processing-timed-based learning effects in which the actual processing time of a job is a function of the total normal processing times of the jobs already processed and of the job position in a schedule. Wu and Lee [20] study the problem with learning effects to minimize the total completion time in a multiple machine permutation flowshop. Janiak and Rudek [21] suggest a new approach, called multiability learning, which generalizes the existing ones and models more precisely real-life settings. They focus on the problem with the proposed learning model to minimize the makespan and provide optimal polynomial time algorithms for some special cases. Cheng et al. [22] develop a learning effect model where the actual job processing time is a logarithm function of the normal processing time of jobs already processed. They provide the optimal solutions for some single-machine problems. Lee et al. [23] consider a single-machine bi-criterion problem with learning effects and job release times. Wang et al. [24] study some single-machine problems with past-sequence-dependent setup times. Lee et al. [25] investigate a single-machine problem with the learning effect and release times to minimize the makespan. Ji and Cheng [26] consider a scheduling problem with job-dependent learning effects and multiple rate-modifying activities. They show that the problem to minimize the total completion time remains polynomially solvable. Wu and Lee [27] and Yin et al. [28] further generalize the model proposed by Cheng et al. [19]. Zhang and Yan [29] modify the models of Cheng et al. [19] and Wu and Lee [27] and provide the optimal solutions for some single-machine and flowshop problems. Wang et al. [30] and Wang and Wang [31] provide the optimal solutions for some single-machine problems with an exponential sum-of-actual-processing-time-based learning effect. Okołowski and Gawiejnowicz [32] consider a parallel-machine scheduling problem with DeJong’s learning effect. For the makespan problem, they propose a sequential branch-and-bound algorithm and a parallel branch-and-bound algorithm. Wang et al. [33] consider resource allocation scheduling with learning effects where the processing time of a job is a function of the job’s position in a sequence and its resource allocation. They provide a polynomial algorithm to find the optimal job sequence and resource allocation. Hsu et al. [34] and Kuo et al. [35] study the total completion time problem on unrelated parallel machines with past-sequence-dependent setup times and learning effects. Zhu et al. [36] investigate some single-machine group scheduling problems with resource allocation and learning effects. Rudek [37] provide the computational complexity and solution algorithms for flowshop scheduling problems with the learning effect. Lai and Lee [38] propose a learning effect model where the actual processing time of a job is a general function of the normal processing times of the jobs already processed and the job’s own scheduled position.

Biskup [12] classifies the existing models into two different approaches, namely, position-based learning effect and sum-of-processing-time-based learning effect models. He points out that position-based learning corresponds to situations where learning arises from processing-time independent operations like setting up machines. This is a realistic assumption for the case because the actual processing times of the jobs are mainly machine-driven. The sum-of-processing-time-based approach takes into account the experience that workers gain from producing the jobs. This might, for example, be the case for offset printing, where running the press itself is a highly complicated and error-prone process. In some precision manufacturing environments, such as the aircraft manufacturing industry, new employees must spend long time to become skilled workers. In this case, the learning effect is not evident in the early stage because of the low learning rate at that stage, but the learning rate will gradually increase as employees become skilled. In other words, the learning rate accelerates over time. This phenomenon also exists in labour-intensive industries, such as hand-made umbrellas, fans, and embroidery, in which craftsmen need to learn and perfect their crafts over a long period of time practising their skills. In this study we propose a scheduling model with the learning effect that might accelerate.

The remainder of this paper is organized as follows. We present the solution procedures for the single-machine problems to minimize the makespan, total completion time, total weighted completion time, maximum lateness, maximum tardiness, and total tardiness in Section 2. We conclude the paper in Section 3.

2. Some Single-Machine Problems

There are 𝑛 jobs ready to be processed on a single machine. For each job 𝑗, there is a normal processing time 𝑝_𝑗, a due date 𝑑_𝑗, and a weight 𝑤_𝑗. Due to the learning effect as in [39], the actual processing time of job 𝑗 if it is scheduled in the 𝑟th position of a sequence is 𝑝_𝑗[𝑟]=𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎, (1) for 𝑟=1,2,…,𝑛, where 𝑎<0, 𝑝_[𝑘] denotes the normal processing time of the job scheduled in the 𝑘th position of the sequence and 𝛼_𝑟,𝑖≤𝛼_𝑟,𝑗 if 𝑖<𝑗 and 𝛼_𝑟,𝑗≥𝛼_𝑟′,𝑗≥0 if 𝑟>𝑟^′. Throughout the paper, let 𝐶_𝑗, 𝐿_𝑗=𝐶_𝑗−𝑑_𝑗, and 𝑇_𝑗=max{0,𝐶_𝑗−𝑑_𝑗} denote the completion time, lateness, and tardiness of job 𝑗, respectively. In addition, we let 𝐶_max=max{𝐶_𝑗}, 𝐿_max=max{𝐿_𝑗}, and 𝑇_max=max{𝑇_𝑗}. Using the well-known three-field notation for describing scheduling problems, we denote the problems under study as 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|𝐶_max, 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|∑𝐶_𝑖, 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|∑𝑤_𝑖𝐶_𝑖, 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|𝐿_max, 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|𝑇_max, and 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|∑𝑇_𝑖, respectively.

Before presenting the main results, we first state some lemmas that will be used in the proofs in the sequel.

Lemma 1. (𝜃−1)(1+𝑐+𝑢)^𝑎−𝜃(1+𝑢+𝜆𝑡)^𝑎+(1+𝑢+𝜆𝜃𝑡)^𝑎≥0 for 𝑎<0, 𝜃≥1, 𝜆≥0, 𝑐<0, 𝑡≥0, 𝑢>0, and 1+𝑐+𝑢≥0.

Proof. Let 𝐹(𝑡)=(𝜃−1)(1+𝑐+𝑢)^𝑎−𝜃(1+𝑢+𝜆𝑡)^𝑎+(1+𝑢+𝜆𝜃𝑡)^𝑎. Taking the first derivative of 𝐹(𝑡) with respect to 𝑡, we have 𝐹^′(𝑡)=𝑎𝜆𝜃(1+𝑢+𝜆𝜃𝑡)^𝑎−1−(1+𝑢+𝜆𝑡)^𝑎−1≥0. (2) Since 𝑎<0 and 𝜃≥1, 𝐹(𝑡) is a nondecreasing function so 𝐹(𝑡)≥𝐹(0)≥0. This completes the proof.

Lemma 2. (1+𝑢+𝑐)^𝑎(1−𝜃)+𝜆₂𝜃(1+𝑢+𝜆𝑡)^𝑎−𝜆₁(1+𝑢+𝜆𝜃𝑡)^𝑎≤0 for 𝑎<0, 𝜃≥1, 𝜆≥0, 𝑐<0, 𝑡≥0, 0≤𝜆₂≤𝜆₁≤1, 𝑢>0, and 1+𝑐+𝑢≥0.

Proof. Let 𝐹(𝜃)=(𝜃−1)(1+𝑢+𝑐)^𝑎+𝜆₁(1+𝑢+𝜃𝑡)^𝑎−𝜆₂𝜃(1+𝑢+𝑡)^𝑎. To show that 𝐹(𝜃) is nonnegative, it suffices to show that 𝐹^′(𝜃)≥0 because 𝐹(1)=(1+𝑢+𝑡)^𝑎(𝜆₁−𝜆₂)≥0. Taking the first derivative of 𝐹(𝜃) with respect to 𝜃, we have 𝐹^′(𝜃)=(1+𝑢+𝑐)^𝑎+𝑎𝜆₁𝑡(1+𝑢+𝜃𝑡)^𝑎−1−𝜆₂(1+𝑢+𝑡)^𝑎. (3) Since 𝑎<0, 𝑢>0, 𝑡≥0, and 𝜆₁≥0, we have 𝐹^′(𝜃)≥(1+𝑢+𝑐)^𝑎+𝑎𝜆₁𝑡(1+𝑢+𝑡)^𝑎−1−𝜆₂(1+𝑢+𝑡)^𝑎=𝐹^′(1). (4) To show that 𝐹^′(1)≥0, we define the following function: 𝐺𝜉,𝜂=(1+𝑢+𝑐)^𝑎+𝑎𝑡𝜂(1+𝑢+𝑡)^𝑎−1−𝜉(1+𝑢+𝑡)^𝑎. (5) First, we claim that 𝐺(1,1)≥0. We have 𝐺(1,1)=(1+𝑢+𝑐)^𝑎+𝑎𝑡(1+𝑢+𝑡)^𝑎−1−(1+𝑢+𝑡)^𝑎=(1+𝑢+𝑐)^𝑎−(1+𝑢)^𝑎+(1+𝑢)^𝑎−(1+𝑢+𝑡)^𝑎+𝑎𝑡(1+𝑢+𝑡)^𝑎−1. (6) By the mean value theorem, there exists a 𝜎 with 0≤𝜎≤1 such that (1+𝑢)^𝑎−(1+𝑢+𝑡)^𝑎=−𝑡𝑎(1+𝑢+𝜎𝑡)^𝑎−1. (7) It follows that 𝐺(1,1)=(1+𝑢+𝑐)^𝑎−(1+𝑢)^𝑎−𝑡𝑎(1+𝑢+𝜎𝑡)^𝑎−1+𝑎𝑡(1+𝑢+𝑡)^𝑎−1. (8) Since 𝑎<0, we have (1+𝑢+𝑐)^𝑎≥(1+𝑢)^𝑎 and (1+𝑢+𝜎𝑡)^𝑎−1≥(1+𝑢+𝑡)^𝑎−1 for 0≤𝜎≤1. Thus, 𝐺(1,1)≥0 and the claim is proved.
Since 𝐺(𝜉,𝜂) is nonincreasing with respect to 𝜉 and 𝜂, we have 𝐺(𝜉,𝜂)≥𝐺(1,1)≥0. This implies that 𝐹^′(1)=𝐺(𝜆₂,𝜆₁) is nonnegative. Thus, we have 𝐹^′(𝜃)≥𝐹^′(1)≥0 and the proof is completed.

We prove the following properties using the pairwise job interchange technique. Suppose that 𝑆 and 𝑆^′ are two job schedules and the difference between 𝑆 and 𝑆^′ is a pairwise interchange of two adjacent jobs 𝑖 and 𝑗. That is, 𝑆=(𝜋,𝑖,𝑗,𝜋^′) and 𝑆^′=(𝜋,𝑗,𝑖,𝜋^′), where 𝜋 and 𝜋^′ each denote a partial sequence. Furthermore, we assume that there are 𝑟−1 jobs in 𝜋. In addition, let 𝐴 denote the completion time of the last job in 𝜋. Under the proposed model, the completion times of jobs 𝑖 and 𝑗 in 𝑆 and 𝑆^′ are 𝐶_𝑖(𝑆)=𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎,𝐶_𝑗(𝑆)=𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑖^𝑎,𝐶_𝑗𝑆^′=𝐴+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎,𝐶_𝑖𝑆^′=𝐴+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗^𝑎. (9)(10)(11)(12)

Property 1. An optimal schedule for the 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|𝐶_max problem is obtained by sequencing the jobs in the shortest processing time (SPT) order.

Proof. Suppose that 𝑝_𝑗≥𝑝_𝑖. To show that 𝑆 dominates 𝑆^′, it suffices to show that 𝐶_𝑗(𝑆)≤𝐶_𝑖(𝑆^′). Taking the difference between (10) and (12), we have 𝐶_𝑖𝑆^′−𝐶_𝑗(𝑆)=𝑝_𝑗−𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗^𝑎−𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑖^𝑎. (13) Substituting 𝑢=∑^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘], 𝜃=𝑝_𝑗/𝑝_𝑖, 𝑐=∑^𝑟−1_𝑘=1(𝛼_𝑟,𝑘−𝛼_𝑟+1,𝑘)𝑝_[𝑘], 𝑡=𝑝_𝑖, and 𝜆=𝛼_𝑟+1,𝑟 into (13), we have from Lemma 1 that 𝐶_𝑗(𝑆^′)≥𝐶_𝑖(𝑆). This completes the proof.

Similar to the proof of Property 1, we have the following result.

Property 2. An optimal schedule for the problem 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|∑𝐶_𝑖 is obtained by sequencing the jobs in the SPT order.

Rudek [40] shows that the total weighted completion time problem with sum-of-processing-time-based learning effects is at least NP-hard. Since it is a special case of our proposed model, the problem to minimize the total weighted completion time under the proposed model is at least NP-hard. Thus, the weighted shortest processing time first (WSPT) rule does not yield the optimal solution under the proposed model. However, we show in the next property that the WSPT rule provides an optimal solution for the total weighted completion time problem if the normal processing times and the weights are agreeable; that is, 𝑝_𝑖≤𝑝_𝑗 implies that 𝑤_𝑖≥𝑤_𝑗 for all jobs 𝑖 and 𝑗.

Property 3. An optimal schedule for the 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|∑𝑤_𝑖𝐶_𝑖 problem is obtained by sequencing the jobs in the WSPT order if the normal processing times and the weights are agreeable.

Proof. Suppose that 𝑝_𝑖≤𝑝_𝑗. From Property 1, we have 𝐶_𝑗(𝑆)≤𝐶_𝑖(𝑆^′). Thus, to show that 𝑆 dominates 𝑆^′, it suffices to show that 𝑤_𝑖𝐶_𝑖(𝑆)+𝑤_𝑗𝐶_𝑗(𝑆)≤𝑤_𝑗𝐶_𝑗𝑆^′+𝑤_𝑖𝐶_𝑖𝑆^′. (14) From (9) to (12), we have 𝑤_𝑖𝐶_𝑖(𝑆)+𝑤_𝑗𝐶_𝑗(𝑆)−𝑤_𝑗𝐶_𝑗𝑆^′+𝑤_𝑖𝐶_𝑖𝑆^′=𝑤_𝑖𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎−𝑤_𝑗𝐴+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑤_𝑗𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑖^𝑎−𝑤_𝑖𝐴+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗^𝑎=𝑤_𝑖+𝑤_𝑗𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎𝑝1−^𝑗𝑝_𝑖+𝑤_𝑗𝑤_𝑖+𝑤_𝑗𝑝_𝑗𝑝_𝑖×1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑖^𝑎−𝑤_𝑖𝑤_𝑖+𝑤_𝑗×1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗^𝑎. (15) Substituting 𝜃=𝑝_𝑗/𝑝_𝑖, 𝜆₁=𝑤_𝑖/(𝑤_𝑖+𝑤_𝑗), 𝜆₂=𝑤_𝑗/(𝑤_𝑖+𝑤_𝑗), 𝑢=∑^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘], 𝑡=𝑝_𝑖, 𝑐=∑^𝑟−1_𝑘=1(𝛼_𝑟,𝑘−𝛼_𝑟+1,𝑘)𝑝_[𝑘], and 𝜆=𝛼_𝑟+1,𝑟 into (15), we have 𝑤_𝑖𝐶_𝑖(𝑆)+𝑤_𝑗𝐶_𝑗(𝑆)−𝑤_𝑖𝐶_𝑖𝑆^′+𝑤_𝑗𝐶_𝑗𝑆^′=𝑤_𝑖+𝑤_𝑗𝑝_𝑖(1+𝑢+𝑐)^𝑎(1−𝜃)+𝜆₂𝜃(1+𝑢+𝜆𝑡)^𝑎−𝜆₁(1+𝑢+𝜆𝜃𝑡)^𝑎. (16) From Lemma 2, we have 𝑤_𝑖𝐶_𝑖(𝑆)+𝑤_𝑗𝐶_𝑗(𝑆)−[𝑤_𝑖𝐶_𝑖(𝑆^′)+𝑤_𝑗𝐶_𝑗(𝑆^′)]≤0. Therefore, repeating this job interchange argument for all the jobs not sequenced in the WSPT order completes the proof for the 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|∑𝑤_𝑖𝐶_𝑖 problem.

Next, we show that the earliest due date (EDD) rule yields an optimal solution under the proposed models if the normal processing times and due dates are agreeable; that is, 𝑑_𝑖≤𝑑_𝑗 implies that 𝑝_𝑖≤𝑝_𝑗 for all jobs 𝑖 and 𝑗.

Property 4. An optimal schedule for the 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|𝐿_max problem is obtained by sequencing the jobs in the EDD order if the normal processing times and due dates are agreeable.

Proof. Suppose that 𝑑_𝑖≤𝑑_𝑗, which implies that 𝑝_𝑖≤𝑝_𝑗. The maximum lateness among the first 𝑟−1 jobs is the same since they are processed in the same order. Since the makespan is minimized by the SPT rule (Property 1), the maximum lateness in the partial sequence 𝜋^′ of 𝑆 will not be greater than that in the partial sequence 𝜋^′ of 𝑆^′. Thus, to prove that the maximum lateness of 𝑆 is less than or equal to that of 𝑆^′, it suffices to show that max{𝐿_𝑖(𝑆),𝐿_𝑗(𝑆)}≤𝐿_𝑖(𝑆^′).
From (9) to (12), we can derive that the lateness of jobs 𝑖 and 𝑗 in 𝑆 is 𝐿_𝑖(𝑆)=𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎−𝑑_𝑖,𝐿_𝑗(𝑆)=𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑖^𝑎−𝑑_𝑗. (17) Similarly, the lateness of jobs 𝑗 and 𝑖 in 𝑆^′ is 𝐿_𝑗𝑆^′=𝐴+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎−𝑑_𝑗,𝐿_𝑖𝑆^′=𝐴+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗^𝑎−𝑑_𝑖. (18) Thus, 𝐿_𝑖(𝑆)≤𝐿_𝑖(𝑆^′) since 𝑝_𝑖≤𝑝_𝑗 and 𝑝_𝑖(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗)^𝑎≥0. From Property 1 and 𝑑_𝑖≤𝑑_𝑗, we see that 𝐿_𝑗(𝑆)≤𝐿_𝑖(𝑆^′). Thus, 𝑆 dominates 𝑆^′. Therefore, repeating this job interchange argument for all the jobs not sequenced in the EDD order completes the proof of the result for the 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|𝐿_max problem.

Similar to the proof of Property 4, we have the following result.

Property 5. An optimal schedule for the 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|𝑇_max problem is obtained by sequencing the jobs in the EDD order if the normal processing times and due dates are agreeable.

Property 6. An optimal schedule for the 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|∑𝑇_𝑖 problem is obtained by sequencing the jobs in the EDD order if the normal processing times and due dates are agreeable.

Proof. Suppose that 𝑑_𝑖≤𝑑_𝑗, which implies 𝑝_𝑖≤𝑝_𝑗. The total tardiness of the first 𝑟−1 jobs is the same since they are processed in the same order. Since the makespan is minimized by the SPT rule (Property 1), the total tardiness of the partial sequence 𝜋^′ in 𝑆 will not be greater than that of the partial sequence 𝜋^′ in 𝑆^′. Thus, to prove that the total tardiness of 𝑆 is less than or equal to that of 𝑆^′, it suffices to show that 𝑇_𝑖(𝑆)+𝑇_𝑗(𝑆)≤𝑇_𝑗𝑆^′+𝑇_𝑖𝑆^′. (19) From (9) to (12), we can derive that the tardiness of jobs 𝑖 and 𝑗 in 𝑆 is 𝑇_𝑖(𝑆)=max𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎−𝑑_𝑖,𝑇,0_𝑗(𝑆)=max𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑖^𝑎−𝑑_𝑗,0. (20) Similarly, the tardiness of jobs 𝑗 and 𝑖 in 𝑆^′ is 𝑇_𝑗𝑆^′=max𝐴+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎−𝑑_𝑗,𝑇,0_𝑖𝑆^′=max𝐴+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗^𝑎−𝑑_𝑖,0. (21)
To compare the total tardiness of jobs 𝑖 and 𝑗 in 𝑆 and 𝑆^′, we consider two cases. In the first case where 𝐴+𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎≤𝑑_𝑗, the total tardiness of jobs 𝑖 and 𝑗 in 𝑆 and in 𝑆^′ is 𝑇_𝑖(𝑆)+𝑇_𝑗(𝑆)=max𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎−𝑑_𝑖,0+max𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑖^𝑎−𝑑_𝑗,𝑇,0_𝑗𝑆^′+𝑇_𝑖𝑆^′=max𝐴+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗^𝑎−𝑑_𝑖,0. (22)
Suppose that neither 𝑇_𝑖(𝑆) nor 𝑇_𝑗(𝑆) is zero. Note that this is the most restrictive case since it comprises the case where either one or both 𝑇_𝑖(𝑆) and 𝑇_𝑗(𝑆) are zero. From Property 1 and 𝑑_𝑖≤𝑑_𝑗, we have 𝑇_𝑗𝑆^′+𝑇_𝑖𝑆^′−𝑇_𝑖(𝑆)+𝑇_𝑗(𝑆=𝑝_𝑗)−𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗^𝑎−𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑖^𝑎+𝑑_𝑗−𝐴−𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎≥0. (23) Thus, {𝑇_𝑗(𝑆^′)+𝑇_𝑖(𝑆^′)}−{𝑇_𝑖(𝑆)+𝑇_𝑗(𝑆)}≥0 in the first case. In the second case where 𝐴+𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎>𝑑_𝑗, the total tardiness of jobs 𝑖 and 𝑗 in 𝑆 and 𝑆^′ is 𝑇_𝑖(𝑆)+𝑇_𝑗(𝑆)=max𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎−𝑑_𝑖,0×max𝐴+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑖^𝑎−𝑑_𝑗,𝑇,0_𝑗𝑆^′+𝑇_𝑖𝑆^′=2𝐴+2𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗^𝑎−𝑑_𝑖−𝑑_𝑗. (24)
Suppose that neither 𝑇_𝑖(𝑆) nor 𝑇_𝑗(𝑆) is zero. From Property 1, 𝑑_𝑖≤𝑑_𝑗 and 𝑝_𝑖≤𝑝_𝑗, we have 𝑇_𝑗𝑆^′+𝑇_𝑖𝑆^′−𝑇_𝑖(𝑆)+𝑇_𝑗(𝑆𝑝)=2_𝑗−𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]^𝑎+𝑝_𝑖1+^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑗^𝑎−𝑝_𝑗1+^𝑟−1_𝑘=1𝛼_𝑟+1,𝑘𝑝_[𝑘]+𝛼_𝑟+1,𝑟𝑝_𝑖^𝑎≥0. (25) Thus, {𝑇_𝑗(𝑆^′)+𝑇_𝑖(𝑆^′)}−{𝑇_𝑖(𝑆)+𝑇_𝑗(𝑆)}≥0 in the second case. Thus, 𝑆 dominates 𝑆^′. Therefore, repeating this job interchange argument for all the jobs not sequenced in the EDD order completes the proof of the result for the 1|𝑝_𝑗[𝑟]=𝑝_𝑗(1+∑^𝑟−1_𝑘=1𝛼_𝑟,𝑘𝑝_[𝑘])^𝑎|∑𝑇_𝑖 problem.

3. Conclusions

In this paper, we propose a new scheduling model where the learning effect accelerates. We derive the optimal solutions for some single-machine problems. For the makespan and the total completion time problems, we show that the SPT rule yields the optimal schedules. For the total weighted completion time problem, we show that the WSPT rule yields the optimal schedule if the normal processing times and weights are agreeable. Furthermore, we prove that the EDD rule yields the optimal schedules for the maximum lateness, maximum tardiness, and the total tardiness problems if the normal processing times and due dates are agreeable.

Acknowledgments

The authors are grateful to the referees for their comments. This work is supported by the National Science Council of Taiwan, under Grant no. NSC 1000-2221-E-035-029-MY3.