Minimizing the Makespan for Scheduling Problems with General Deterioration Effects

Abstract

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 single-machine 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.

1. 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-time-based 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 𝐽_𝑖 is 𝑝_𝑖𝑟=𝑝_𝑖𝑟^𝑎 (1) if job 𝐽_𝑖 is scheduled in position 𝑟 of the job sequence, where 𝑝_𝑖 and 𝑎 (𝑎<0) are the normal processing time and the learning rate of job 𝐽_𝑖, respectively. The model in Biskup [1] was extended to be 𝑝_𝑖𝑟=𝑝_𝑖𝑟^𝑎𝑖, (2) 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 𝐽_𝑗 is 𝑝^𝐴_𝑗=𝑝_𝑗+𝑎_𝑗𝑡, (3) where 𝑝_𝑗, 𝑎_𝑗, and 𝑡 are the normal processing time, deterioration rate, and starting time of job 𝐽_𝑖, respectively. Cheng and Wang [9] introduced an actual processing time model of 𝐽_𝑖 as follows: 𝑝_𝑗=𝑎_𝑖−𝑣_𝑖min𝑛_𝑖,𝑛_0𝑖,𝑖=1,2,…,𝑛, (4) where 𝑝_𝑗 and 𝑣_𝑖 are the normal processing time of job 𝐽_𝑖 and the learning effect coefficient indicating the relative degree of learning, respectively. The notation 𝑛_𝑖 is a nonnegative integer such that 0≤𝑛_𝑖≤𝑛−1 which indicates the number of jobs processed before job 𝐽_𝑖 in the schedule (i.e., 𝑛_𝑖+1 is the position of job 𝐽_𝑖 in the schedule), and the notation 𝑛_0𝑖 is a nonnegative integer such that 𝑛_0𝑖≤𝑛−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 𝐽_𝑗 as follows: 𝑝_𝑗𝑟(𝑡)=𝑝_𝑗+𝛼𝑡^𝑎, (5) where 𝑎>0 and 𝛼>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: 𝑝_𝑗𝑟(𝑡)=𝛼_𝑗𝑡𝑟^𝑎,𝑝_𝑗𝑟(𝑡)=𝑝₀+𝛼_𝑗𝑡𝑟^𝑎, (6) where 𝛼_𝑗 and 𝑡 are the deterioration rate of job 𝐽_𝑗 and the starting time of job 𝐽_𝑗, 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 𝐽_𝑗 scheduled in position 𝑟 of a sequence is 𝑝_𝑖𝑟=𝑝_𝑖∑1+^{𝑟−1𝑘=1}𝑝_[𝑘]∑^𝑛_𝑘=1𝑝_𝑘^𝑎1𝑟^𝑎2, (7) where 𝑎₁<0 and 𝑎₂<0 denote the learning rates for sum-of-processing-times-based and position-based learning, respectively. In the model, the subscript [𝑘] denotes the job in position 𝑘 of a sequence, and 𝑝_[𝑖] denotes the normal processing time of job 𝑖. 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 𝐽_𝑗 scheduled in position 𝑟 of a sequence is 𝑝_𝑖𝑟=𝑝_𝑖𝑓^𝑟−1_𝑘=1𝑝_[𝑘]𝑔(𝑟), (8) where ∑⁰_𝑘=1𝑝_[𝑘]=0,𝑓∶[0,+∞)→(0,1] is a differentiable nonincreasing function where 𝑓 is nondecreasing on [0,+∞), 𝑓(0)=1, and 𝑔∶[0,+∞)→(0,1] is a nonincreasing function with 𝑔(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 𝑝_𝑖𝑟=𝑝_𝑖𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]𝑔(𝑟), (9) where ∑⁰_𝑙=1𝛽_𝑙𝑝_[𝑙]∶=0,𝑓∶[0,+∞)→(0,1] is a differentiable nonincreasing function where 𝑓^′ is nondecreasing on [0,+∞), 𝑔∶[1,+∞)→(0,1] is a nonincreasing function with 𝑔(1)=1, and 𝛽_𝑙 is the weight of position 𝑟. 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.

2. Single-Machine Scheduling

The formulation of the considered single-machine scheduling problems can be described as follows. At time zero, there are 𝑛 independent jobs ready to be processed on a single-machine. Preemption is not allowed and a machine is only able to process one job at a time. Job 𝑗 has a normal processing time 𝑝_𝑗(𝑗=1,2,…,𝑛).

In this paper, we propose two general deterioration models with respect to both position-based effect and sum-of-processing-time-based effect. By the first model, if job 𝑗 is scheduled in the 𝑟 position of the sequence, the actual processing time of job 𝑗 is 𝑝_𝑗𝑟=𝑝_𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]+𝑔(𝑟), (10) where ∑⁰_𝑙=1𝛽_𝑙𝑝_[𝑖,𝑙]∶=0,𝑓∶[0,+∞)→[1,+∞) is a differentiable nondecreasing function and 𝑓(0)=1, 𝑔∶[1,+∞)→[1,+∞) is a nondecreasing function with 𝑔(1)=1, 𝛽_𝑙 is the weight of position 𝑙, 𝑝_[𝑙] denotes the normal processing time of job 𝑖, and the subscript [𝑙] denotes the job in position 𝑙 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 0≤𝛽₁≤𝛽₂≤⋯𝛽_𝑛.

By the second model, if job 𝑗 is scheduled in the 𝑟 position, its actual processing time is 𝑝_𝑗𝑟=𝑝_𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]𝑔(𝑟), (11) 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 single-machine scheduling problem with respect to the first model can be denoted as 1∣𝑝_𝑗𝑟=𝑝_𝑗[𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙])+𝑔(𝑟)],DE∣𝐶_max, where DE denotes the deterioration effect. Similarly, the single-machine scheduling problem with respect to the second model can be denoted as 1∣𝑝_𝑗𝑟=𝑝_𝑗𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙])𝑔(𝑟),DE∣𝐶_max.

In the following, we first state two lemmas that will be used in the proofs of the following theorems in this section.

Lemma 1. If 𝑓∶[0,+∞)→[1,+∞) is nondecreasing and its derived function 𝑓^′ is nonincreasing on [0,+∞),𝜆≥1,𝛿₁≥1, and 𝑥≥0, then (1−𝜆)(𝑓(𝐴)𝛿₂+1) + 𝜆(𝑓(𝐴+𝑥)𝛿₂+𝛿₁) - (𝑓(𝑎+𝜆𝑥)𝛿₂+𝛿₁)≥0.

Proof. We let ℎ(𝑥)=(1−𝜆)(𝑓(𝐴)𝛿₂+1)+𝜆(𝑓(𝐴+𝑥)𝛿₂+𝛿₁)−(𝑓(𝐴+𝜆𝑥)𝛿₂+𝛿₁). The first derivation of ℎ(𝑥) with respect to 𝑥 can be obtained that ℎ^′(𝑥)=𝜆𝛿₂[𝑓^′(𝐴+𝑥)−𝑓^′(𝐴+𝜆𝑥)]≥0 under the condition that 𝑓∶[0,+∞)→[1,+∞) is a differentiable nondecreasing function with 𝑓^′ being nonincreasing on [0,+∞),𝜆≥1, and 𝛿₁≥1. Then, it is obvious that ℎ(𝑥) is an increasing function on 𝑥. Therefore, we have ℎ(𝑥)≥ℎ(0)=(1−𝜆)(1−𝛿₁)≥0 for 𝜆≥1,𝛿₁≥1.

With a similar differentiation method, it is easy to obtain the following lemma.

Lemma 2. If 𝑓∶[0,+∞)→[1,+∞) is nondecreasing and its derived function 𝑓^′ is nonincreasing on [0,+∞),𝜆≥1,𝛿≥1, and 𝑥≥0, then (1−𝜆)𝑓(𝐴)+𝜆𝛿𝑓(𝐴+𝑥)−𝛿𝑓(𝐴+𝜆𝑥)≥0.

Theorem 3. The problem 1∣𝑝_𝑗𝑟=𝑝_𝑗(𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙])+𝑔(𝑟)),𝐷𝐸∣𝐶_max can be optimally solved by the Longest Processing Time first rule of 𝑝_𝑗 (the LPT rule) if 𝑓^′ is nonincreasing on [0,+∞).

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 𝜎 be [𝜎₁,𝐽_𝑗,𝐽_𝑘,𝜎_𝑖2,𝜎₂] and let the schedule 𝜎^′ be [𝜎₁,𝐽_𝑘,𝐽_𝑗,𝜎₂], where 𝜎₁ and 𝜎₂ are partial job sequences, and the difference between 𝜎 and 𝜎^′ is a pairwise interchange of two adjacent jobs 𝐽_𝑗 and 𝐽_𝑘 in the schedule. Concretely, 𝐽_𝑗 and 𝐽_𝑘 are assumed to be assigned in the 𝑟th and 𝑟+1th position, respectively, in the schedule 𝜎. In 𝜎^′, 𝐽_𝑗 and 𝐽_𝑘 are assigned in the 𝑟+1th and 𝑟th position, respectively. Without loss of generality, the normal processing times of 𝐽_𝑗 and 𝐽_𝑘 are assumed to satisfy 𝑝_𝑗≤𝑝_𝑘. To show that 𝜎^′ dominates 𝜎, it suffices to show that 𝐶_𝑘(𝜎)≥𝐶_𝑗(𝜎^′) and 𝐶_𝑢(𝜎)≥𝐶_𝑢(𝜎^′) for any job 𝐽_𝑢 in 𝜎₂. Let the completion time of the last job in partial schedule 𝜎₁ be 𝐶_𝜎1. In the schedule 𝜎, the completion times of 𝐽_𝑗 and 𝐽_𝑘 are 𝐶_𝑗(𝜎)=𝐶_𝜎1+𝑝_𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙](𝑟,𝐶+𝑔)_𝑘(𝜎)=𝐶_𝜎1+𝑝_𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]+𝑔(𝑟)+𝑝_𝑘𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]+𝛽_𝑟𝑝_𝑗+𝑔(𝑟+1), (12)(13) respectively. In the schedule 𝜎^′, the completion times of 𝐽_𝑘 and 𝐽_𝑗 are 𝐶_𝑘𝜎^′=𝐶_𝜎1+𝑝_𝑘𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙](𝑟𝐶+𝑔)_𝑗𝜎^′=𝐶_𝜎1+𝑝_𝑘𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]+𝑔(𝑟)+𝑝_𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]+𝛽_𝑟𝑝_𝑘+𝑔(𝑟+1), (14)(15) respectively. Combining (13) and (15), we have 𝐶_𝑘(𝜎)−𝐶_𝑗𝜎^′=𝑝_𝑗−𝑝_𝑘𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]+𝑔(𝑟)+𝑝_𝑘𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]+𝛽_𝑟𝑝_𝑗+𝑔(𝑟+1)−𝑝_𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]+𝛽_𝑟𝑝_𝑘(×𝑝+𝑔𝑟+1)_𝑘𝑝_𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]1𝑔(𝑟)+1=𝑝_𝑗𝑔(𝑟𝑝)1−^𝑘𝑝_𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]1𝑔(𝑟)+𝑝+1_𝑘𝑝_𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]+𝛽_𝑟𝑝_𝑗×1𝑔(𝑟)+𝑔(𝑟+1)𝑔(𝑟)−𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]+𝛽_𝑟𝑝_𝑗𝑝_𝑘𝑝_𝑗×1𝑔(𝑟)+𝑔(𝑟+1)𝑔(𝑟). (16) Let 𝐴=∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙],𝜆=𝑝_𝑘/𝑝_𝑗,𝛿₁=(𝑔(𝑟+1)/𝑔(𝑟)),𝛿₂=(1/𝑔(𝑟)), and 𝑥=𝛽_𝑟𝑝_𝑗. Based on the formulation of the considered problem, it is easy to obtain that 𝐴≥0,𝜆≥1,𝛿₁≥1,0<𝛿₂≤1, and 𝑥≥0. Based on Lemma 1 and (16), we have 𝐶_𝑘(𝜎)−𝐶_𝑗𝜎^′=𝑝_𝑗𝑔(𝑟)(1−𝜆)𝑓(𝐴)𝛿₂𝑓()𝛿+1+𝜆𝐴+𝑥₂+𝛿₁−𝑓(𝑎+𝜆𝑥)𝛿₂+𝛿₁≥0. (17) Assuming that 𝐽_𝑠 is the job scheduled in the 𝑟+2 position of the job sequence, then the completion times of 𝐽_𝑠 in 𝜎 and 𝜎^′ are 𝐶_𝑠(𝜎)=𝐶_𝑘(𝜎)+𝑝_𝑠𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖,𝑙]+𝛽_𝑟𝑝_𝑗+𝛽_𝑟+1𝑝_𝑘(,𝐶+𝑔𝑟+2)_𝑠𝜎^′=𝐶_𝑗𝜎^′+𝑝_𝑠𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖,𝑙]+𝛽_𝑟𝑝_𝑘+𝛽_𝑟+1𝑝_𝑗+𝑔(𝑟+2), (18)(19) respectively. Since 𝑝_𝑗≤𝑝_𝑘 and 𝛽_𝑟≤𝛽_𝑟+1, it can be obtained that 𝛽_𝑟𝑝_𝑗+𝛽_𝑟+1𝑝_𝑘−𝛽_𝑟𝑝_𝑘+𝛽_𝑟+1𝑝_𝑗=𝛽_𝑟−𝛽_𝑟+1𝑝_𝑗−𝑝_𝑘≥0. (20) Since 𝑓 is a nondecreasing function, (20) is equivalent to 𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖,𝑙]+𝛽_𝑟𝑝_𝑗+𝛽_𝑟+1𝑝_𝑘≥𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖,𝑙]+𝛽_𝑟𝑝_𝑘+𝛽_𝑟+1𝑝_𝑗. (21)
Let 𝐶_𝑠(𝜎) and 𝐶_𝑠(𝜎^′) denote the completion time of job 𝐽_𝑠 in the schedule 𝜎 and 𝜎^′, respectively. From (17)–(19) and (21), we have 𝐶_𝑠(𝜎)≥𝐶_𝑠(𝜎^′). Thus, we show that 𝐽_𝑠 starts earlier in 𝜎^′ than in 𝜎 and completes earlier in 𝜎^′ than in 𝜎. With a similar proof method, it can be shown that 𝐶_𝑢(𝜎)≥𝐶_𝑢(𝜎^′) for any job 𝐽_𝑢 in 𝜎₂. 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.

Theorem 4. The problem 1∣𝑝_𝑗𝑟=𝑝_𝑗(𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙])𝑔(𝑟)),𝐷𝐸∣𝐶_max can be optimally solved by the Longest Processing Time first rule (the LPT rule) if 𝑓^′ is nonincreasing on [0,+∞).
If 𝑔(𝑟) is unchanged with respect to 𝑟, then two actual processing time models can be both simplified as 𝑝_𝑗𝑟=𝑝_𝑗𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]). Similarly, if 𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙]) is unchanged with respect to ∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑙], then two actual processing time models can be both simplified as 𝑝_𝑗𝑟=𝑝_𝑗𝑔(𝑟). It is obvious that the single-machine problem with the above special models can be also optimally solved by the LPT rule.

3. Flow Shop Scheduling

The formulation of the flow shop scheduling problem can be described as follows. There are 𝑛 independent jobs 𝐽₁,𝐽₂,…,𝐽_𝑛 ready to be processed on 𝑚 machines 𝑀₁,𝑀₂,…,𝑀_𝑚 at time zero. Job 𝐽_𝑗 consists of 𝑚 operations 𝑂_1𝑗,𝑂_2𝑗,…,𝑂_𝑚𝑗. 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 𝑂_𝑖+1,𝑗 may start only after operation 𝑂_𝑖𝑗 has been completed. Operation 𝑂_𝑖𝑗 has a normal processing time 𝑝_𝑖𝑗(𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛), and the actual processing time of operation 𝑂_𝑖𝑗 is 𝑝_𝑖𝑗𝑟 if it is scheduled in position 𝑟 (𝑟=1,2,…,𝑛) in a sequence. In this paper, we propose two general deterioration models. By the first model, if operation 𝑂_𝑖𝑗 is scheduled in the 𝑟 position of the sequence, the actual processing time of operation 𝑂_𝑖𝑗 is 𝑝_𝑖𝑗𝑟=𝑝_𝑖𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙]+𝑔(𝑟), (22) where ∑⁰_𝑙=1𝛽_𝑙𝑝_[𝑖,l]∶=0,𝑓∶[0,+∞)→[1,+∞) is a differentiable nondecreasing function, and 𝑓(0)=1,𝑔∶[1,+∞)→[1,+∞) is a nonincreasing function with 𝑔(1)=1, and 𝛽_𝑙 is the weight of position 𝑙; 𝑝_[𝑖𝑙] denotes the normal processing time of the operation scheduled in the 𝑟th position on machine 𝑀_𝑖, where the subscript [𝑖,𝑙] denotes the job in position 𝑙 of the sequence on 𝑀_𝑖. 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≤𝛽₁≤𝛽₂⋯𝛽_𝑛.

By the second model, if job 𝑗 is scheduled on the 𝑟 position of machine 𝑖, its actual processing time is 𝑝_𝑖𝑗𝑟=𝑝_𝑖𝑗𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙]𝑔(𝑟), (23) 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 𝑗 on the 𝑟 position of machine 𝑖 is 𝑝_𝑖𝑗𝑟=𝐻𝑝_𝑖𝑗,𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙],𝑔(𝑟), (24) where 𝑝_𝑖𝑗1=𝐻(𝑝_𝑖𝑗,𝑓(0),𝑔(1))=𝑝_𝑖𝑗 and 𝐻 is nondecreasing in 𝑝_𝑖𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙]), and 𝑔(𝑟). Obviously, model (22) and model (23) are both special cases of new model (24).

Lemma 5. The problem 𝐹∣𝑝_𝑖𝑗𝑟=𝐻[𝑝_𝑖𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙]),𝑔(𝑟)],𝑝_2𝑗=𝑝₂,𝐷𝐸∣𝐶_max has the same optimal schedule as the problem 1∣𝑝_1𝑗𝑟=𝐻[𝑝_1𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1𝑙]),𝑔(𝑟)],𝐷𝐸∣𝐶_max if 𝑝_1𝑗≥𝑝₂.

Proof. Let 𝐶_[𝑗] denote the completion time of the job scheduled in 𝑗th position of the job sequence. For 𝑗=1, we have 𝐶_[1]=𝑝_[11]+𝑝₂.
Because 𝑝_2𝑗=𝑝₂ and 𝑝_1𝑗≥𝑝₂, it can be obtained that 𝑝_1𝑗𝑟=𝐻[𝑝_1𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1𝑙]),𝑔(𝑟)]≥𝑝₂ for 𝑗=1,2,…,𝑛 and 𝑟=1,2,…,𝑛. The schedule is illustrated in Figure 1.
Then, we have the completion times for jobs 𝑗=2,3,…,𝑛 in the following: 𝐶_[2]=max𝐶_[1],𝑝_[11]+𝐻𝑝_[12],𝑓𝛽₁𝑝_[1,𝑙],𝑔(2)𝑝_[11]+𝑝₂=𝑝_[11]+𝐻𝑝_[12],𝑓𝛽₁𝑝_[1,𝑙],𝑔(2)+𝑝₂,⋮𝐶_[𝑘+1]=max𝐶_[𝑘],^𝑘_𝑟=1𝐻𝑝_[1𝑟],𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1,𝑙],𝑔(𝑟)×𝑝_[11]+𝑝₂=^𝑘_𝑟=1𝐻𝑝_[1𝑟],𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1,𝑙],𝑔(𝑟)+𝑝₂,⋮𝐶_max=𝐶_[𝑛]=max𝐶_[𝑘],^𝑘_𝑟=1𝐻𝑝_[1𝑟],𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖,𝑙],𝑔(𝑟)×𝑝_[11]+𝑝₂=^𝑛_𝑟=1𝐻𝑝_[1𝑟],𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1,𝑙],𝑔(𝑟)+𝑝₂. (25) Since 𝑝₂ is constant, an optimal schedule of minimizing the makespan for the problem 𝐹2∣𝑝_1𝑗𝑟=𝐻[𝑝_1𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1𝑙],𝑔(𝑟)],DE∣𝐶_max must be the optimal schedule of minimizing for the problem 𝐹2∣𝑝_𝑖𝑗𝑟=𝐻[𝑝_𝑖𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1𝑙]),𝑔(𝑟)],𝑝_2𝑗=𝑝₂,DE∣𝐶_max.

Figure 1 

The illustration of the problem 𝐹∣𝑝𝑖𝑗𝑟 = 𝐻(𝑝𝑖𝑗,𝑓(∑𝑟−1𝑙=1𝛽𝑙𝑝[𝑖𝑙]),𝑔(𝑟)),𝑝2𝑗 = 𝑝2,DE∣𝐶max.

With a similar proof method of Lemma 5, we can obtain the following theorem.

Theorem 6. The problem 𝐹𝑚∣𝑝_𝑖𝑗𝑟 = 𝐻[𝑝_𝑖𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙]),𝑔(𝑟)],𝑝_2𝑗𝑟 = 𝑝_3𝑗𝑟 = ⋯=𝑝_𝑚𝑗𝑟 = 𝑝,𝐷𝐸∣𝐶_max has the same optimal schedule as the problem 1∣𝑝_1𝑗𝑟 = 𝐻(𝑝_1𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1𝑙],𝑔(𝑟)),𝐷𝐸∣𝐶_max if 𝑝_1𝑗≥𝑝.

Combining Theorems 3 and 6, we have the following corollary.

Corollary 7. The problem 𝐹𝑚∣𝑝_𝑖𝑗𝑟=𝑝_𝑖𝑗[𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙])+𝑔(𝑟)],𝑝_2𝑗𝑟=𝑝_3𝑗𝑟=⋯=𝑝_𝑚𝑗𝑟=𝑝,𝐷𝐸∣𝐶_max can be optimally solved by the Longest Processing Time first rule of 𝑝_𝑗 (the LPT rule) if 𝑓^′ is nonincreasing on [0,+∞) and 𝑝_1𝑗≥𝑝.

Similarly, based on Theorems 4 and 6, it is easy to get the following result.

Corollary 8. The problem 𝐹𝑚∣𝑝_𝑖𝑗𝑟=𝑝_𝑖𝑗𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙])𝑔(𝑟),𝑝_2𝑗𝑟=𝑝_3𝑗𝑟=⋯=𝑝_𝑚𝑗𝑟=𝑝,𝐷𝐸∣𝐶_max can be optimally solved by the Longest Processing Time first rule of 𝑝_𝑗 (the LPT rule) if 𝑓^′ is nonincreasing on [0,+∞) and 𝑝_1𝑗≥𝑝.

Theorem 9. If a schedule simultaneously minimizes 𝐻(𝑝_[2𝑛],𝑓(∑^𝑛−1_𝑙=1𝛽_𝑙𝑝_[2,𝑙]),𝑔(𝑛)) and the makespan of the problem 1∣𝑝_1𝑗𝑟=𝐻(𝑝_1𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1𝑙],𝑔(𝑟)),𝐷𝐸∣𝐶_max, it must be the optimal schedule of the problem 𝐹2∣𝑝_𝑖𝑗𝑟=𝐻(𝑝_𝑖𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙],𝑔(𝑟)),min(𝑝_1⋅)≥max(𝑝_2⋅),𝐷𝐸∣𝐶_max.

Proof. By using the same notation as that in the proof of Theorem 6, we let 𝐶_[𝑗] denote the completion time of the job scheduled in 𝑗th position of the job sequence. For 𝑗=1, we have 𝐶_[1]=𝑝_[11]+𝑝₂.
Because min(𝑝_1⋅)≥max(𝑝_2⋅), it can be obtained that 𝐻𝑝_1𝑗,𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1𝑙](𝑟𝑝,𝑔)≥𝐻_2𝑗,𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[2𝑙],𝑔(𝑟) (26) for 𝑗=1,2,…,𝑛 and 𝑟=1,2,…,𝑛. Then, we have the completion times for jobs 𝑗=2,3,…,𝑛 in the following: 𝐶_[2]=max𝐶_[1],𝑝_[11]+𝐻𝑝_[12],𝑓𝛽₁𝑝_[1,𝑙](2𝑝,𝑔)+𝐻_[22],𝑓𝛽₁𝑝_[2,1],𝑔(2)=𝑝_[11]+𝐻𝑝_[12],𝑓𝛽₁𝑝_[1,𝑙](2)𝑝,𝑔+𝐻_[22],𝑓𝛽₁𝑝_[2,1](2),⋮𝐶,𝑔_[𝑘+1]=max𝐶_[𝑘],^𝑘+1_𝑟=1𝐻𝑝_[1𝑟],𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1,𝑙](𝑟𝑝,𝑔)+𝐻_[2,𝑘+1],𝑓^𝑘_𝑙=1𝛽_𝑙𝑝_[2,𝑙]()=,𝑔𝑘+1_𝑘+1𝑟=1𝐻𝑝_[1𝑟],𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1,𝑙](𝑟)𝑝,𝑔+𝐻_[2,𝑘+1],𝑓^𝑘_𝑙=1𝛽_𝑙𝑝_[2,𝑙](),⋮𝐶,𝑔𝑘+1_max=𝐶_[𝑛]=max𝐶_[𝑛−1],_𝑛𝑟=1𝐻𝑝_[1𝑟],𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1,𝑙](𝑟)𝑝,𝑔+𝐻_[2𝑛],𝑓^𝑛−1_𝑙=1𝛽_𝑙𝑝_[2,𝑙](𝑛)=,𝑔_𝑛𝑟=1𝐻𝑝_[1𝑟],𝑓^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1,𝑙](𝑟)𝑝,𝑔+𝐻_[2𝑛],𝑓^𝑛−1_𝑙=1𝛽_𝑙𝑝_[2,𝑙],𝑔(𝑛). (27)
Since ∑^𝑛_𝑟=1𝐻(𝑝_[1𝑟],𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1,𝑙]),𝑔(𝑟)) can be explained as the makespan of the problem 1∣𝑝_1𝑗𝑟=𝐻(𝑝_1𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1𝑙],𝑔(𝑟)),𝐷𝐸∣𝐶_max, the schedule minimizes 𝐻(𝑝_[2𝑛],𝑓(∑^𝑛−1_𝑙=1𝛽_𝑙𝑝_[2,𝑙]),𝑔(𝑛)) simultaneously and the makespan of the problem 𝐹2∣𝑝_1𝑗𝑟 = 𝐻(𝑝_1𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[1𝑙],𝑔(𝑟)),𝐷𝐸∣𝐶_max is the optimal schedule of the problem 𝐹2∣𝑝_𝑖𝑗𝑟 = 𝐻(𝑝_𝑖𝑗,𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙],𝑔(𝑟)),min(𝑝_1⋅) ≥ max(𝑝_2⋅),𝐷𝐸∣𝐶_max.

Combining Theorems 3 and 9, we have the following corollary.

Corollary 10. The problem 𝐹2∣𝑝_𝑖𝑗𝑟=𝑝_𝑖𝑗[𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙])+𝑔(𝑟)],min(𝑝_1⋅)≥max(𝑝_2⋅),𝐷𝐸∣𝐶_max can be optimally solved by the Longest Processing Time first rule of 𝑝_𝑗 (the LPT rule) if 𝑓^′ is nonincreasing on [0,+∞).

Similarly, combining Theorems 4 and 9, we can obtain the following result.

Corollary 11. The problem 𝐹2∣𝑝_𝑖𝑗𝑟=𝑝_𝑖𝑗𝑓(∑^𝑟−1_𝑙=1𝛽_𝑙𝑝_[𝑖𝑙])𝑔(𝑟),min(𝑝_1⋅)≥max(𝑝_2⋅),𝐷𝐸∣𝐶_max can be optimally solved by the Longest Processing Time first rule of 𝑝_𝑗 (the LPT rule) if 𝑓^′ is nonincreasing on [0,+∞).

4. 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 single-machine 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.

Acknowledgments

The authors wish to thank the editors and the anonymous referees for their helpful comments on the earlier versions of this paper. This work is supported by the National Nature Science Foundation of China (71171046), the National Science and Engineering Research Council of Canada (Discovery Grant CISE-539050), the College Graduate Research and Innovation Foundation of Jiangsu Province (CXLX 0162), and the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1239).