Single-Machine Scheduling with Upper Bounded Maintenance Time under the Deteriorating Effect

Abstract

We consider a single-machine scheduling problem with upper bounded actual processing time and upper bounded maintenance time under deteriorating effect. The actual processing time of a job is a position-dependent power function. If the actual processing time of a job exceeds the upper bound, tardiness penalty of the job should be paid. And if the maintenance time exceeds the corresponding upper bound, tardiness penalty of the maintenance should also be paid. The maintenance duration studied in the paper is a position-dependent exponential function. The objective is to find jointly the optimal maintenance frequency and the optimal job sequence to minimize the total cost, which is a linear function of the makespan and the total tardiness. We show that the studied scheduling problem can be transformed as a classic assignment problem to solve. There is also shown that a special case of the scheduling problem can be optimally solved by a lower order algorithm.

1. Introduction

In recent years, scheduling problems with the deteriorating effect have attracted increasing attention. In case of the deteriorating effect, the actual processing time of a job will be longer if it is scheduled later in a sequence. Browne and Yechiali [1] initiated research on scheduling problem with the deteriorating effect, where the actual processing time of a job is a linear nondecreasing start-time-dependent function. A time-dependent deteriorating model was proposed by Rudek [2], where the actual time required to perform a job is a function of the sum of the normal processing time of jobs already processed. For extensive surveys related to time-dependent processing time, the reader can refer to the papers [3–7]. Hsu et al. [8] studied single-machine scheduling and due date assignment problems with position-dependent processing time. They showed that the problems are polynomial time solvable. Mosheiov [9] investigated the scheduling problem with general, nondecreasing, job-dependent, and position-dependent deterioration function under the setting of parallel identical machines to minimize the total load. Rustogi and Strusevich [10] presented polynomial-time algorithms for single-machine problems with generalized positional deterioration effects under machine maintenance. They assumed that the decisions should be taken regarding possible sequences of jobs and on the number of maintenance activities to be included into a schedule to minimize the overall makespan. More recent papers which have considered position-dependent job processing time could be seen in [11–14].

Researchers have studied a variety of scheduling problems with job completion time due window. Jobs should be finished as close as possible to their due dates to cope with global competition and improve customer demand. A job will have to be stored in inventory when it is finished before its due date, which may lead to an earliness penalty. Contrarily, a job will get a tardiness penalty when it is finished after its due date because it violates the contractual obligation with the customer. For extensive surveys related to scheduling problems with the job completion time due window, reader can refer to the papers [15–19]. In this paper, we set the upper bound for the actual processing time of each job. The actual processing time of a job is required to be within a given interval; otherwise tardiness penalty should be paid. For example, in brick manufacturing processes, the actual processing time cannot exceed a given upper bound; otherwise the brick may have quality flaws.

On the other hand, it is reasonable and necessary to perform maintenance in manufacturing processes, because it can help improve the production efficiency. Some scheduling problems with deteriorating effect and machine maintenance have been studied. A single-machine scheduling problem with a cyclic process of deteriorating effect and maintenance activities was addressed by Kuo and Yang [20]. For the problem, they provided polynomial algorithms to minimize the makespan. Zhao and Tang [21] extended the model of Kuo and Yang [20]. The position-dependent deteriorating effect they considered is described by a general exponential function. They claimed that the problem can be transformed as a classic assignment problem to solve. Chen [22] studied a single-machine scheduling problem with periodic maintenance activities and nonresumable jobs to minimize the number of tardy jobs. S. J. Yang and D. L. Yang [12] considered a single-machine scheduling problem with position-dependent deteriorating effect under variable maintenance activities to minimize the makespan of all jobs. It is necessary to maintain the machine, but the maintenance time should be completed within a time interval, otherwise it will affect the machine efficiency (see, e.g., Lee and Chen [23] and Kubzin and Strusevich [24]). Thus, in this paper, we set the upper bound for the maintenance time. Once the maintenance time exceeds the upper bound, the tardiness penalty of the maintenance should also be paid.

However, to the best of our knowledge, research on scheduling simultaneously with upper bounded actual processing time of a job and upper bounded maintenance time under deteriorating effect considerations has rarely been studied. Motivated by these points, this paper investigates a scheduling problem with upper bounded actual processing time of a job and upper bounded maintenance time under deteriorating effect. If the actual processing time of a job exceeds the upper bound, tardiness penalty of the job should be paid. And if the maintenance time exceeds the corresponding upper bound, tardiness penalty of the maintenance should also be paid since it will affect the machine efficiency. We assume that the machine may be subject to several maintenance activities during the scheduling horizon and the maintenance duration is a variable function. The objective is to minimize the total cost, which is assumed to conclude production fee and total tardiness costs, through exploring jointly the optimal maintenance frequency, the optimal maintenance position, and the optimal job sequences. We show that the studied problem in the scheduling problem remains polynomially solvable.

The remaining part of this paper is structured as follows. We formally introduce the notation and terminology used throughout the rest of this paper in the next section. In Section 3, we propose the main results of this paper. In Section 4, we conclude with a summary of the results and suggest directions for future research.

2. Notations and Problem Formulation

Consider a single machine to process a set of 𝑛 independent jobs, which are all available for processing at time zero. The machine can handle one job at a time. In manufacturing processes, the job preemption is not allowed. To improve the production efficiency, maintenance activities may be performed on the machine. During maintenance the machine is stopped, and the machine will revert to its initial state after the maintenance. We assume that the actual processing time of a job will be longer when it is scheduled later in a sequence due to the deteriorating effect of the machine. And the maintenance duration is a function of the maintenance position of the machine. The jobs will be processed from a group consecutively. Thus, the schedule can be denoted as 𝜎=[𝐺₁,𝑀₁,𝐺₂,𝑀₂,…,𝐺_𝑘,𝑀_𝑘,𝐺_𝑘+1],0≤𝑘≤(𝑛−1), where 𝐺_𝑖, 1≤𝑖≤𝑘+1, denotes the 𝑖th group and 𝑀_𝑖, 1≤𝑖≤𝑘, denotes the 𝑖th maintenance. 𝐶_[𝑙,𝑟] is the completion time of the job scheduled in 𝑟th position of the 𝑙th group. The following a positional deterioration model of the actual processing time of job 𝐽_𝑗 is discussed. The actual processing time of job 𝐽_𝑗, if scheduled in position 𝑟 of group 𝐺_𝑖, is given by 𝑝^𝑟_[𝑖,𝑗]=𝑝_[𝑖,𝑗]𝑟^{𝑎[𝑖,𝑗]},for𝑖=1,2,…,𝑘+1,𝑗,𝑟=1,2,…,𝑛_𝑖, (1) where 𝑝_[𝑖,𝑗] is the normal processing time of job 𝐽_𝑗 and 𝑎_[𝑖,𝑗] is the deteriorating factor of job 𝐽_𝑗. The number of jobs of group 𝐺_𝑖 is denoted as 𝑛_𝑖.

In this study, we examine a model of the maintenance duration which concerns the position-dependent deteriorating effect. If the maintenance is the 𝑖th maintenance in the sequence, its actual maintenance duration is defined by 𝑚_𝑖=𝑡₀𝑏^(𝑖−1),for𝑖=1,2,…,𝑘, (2) where 𝑡₀>0 denotes the basic maintenance time and 𝑏>1 is the deteriorating factor of the maintenance. If the maintenance is arranged later in the sequence, the actual maintenance duration will be longer in this model due to the deteriorating effect.

Observing from (1), we find no matter what the group is, the actual processing time of job 𝐽_𝑗 is only dependent on its position in a group. For convenience, we reformulate (1) as follows: 𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,for𝑗=1,2,…,𝑛,𝑟=1,2,…,𝑛_𝑖,𝑖=1,2,…,𝑘+1, (3) where 𝑝_𝑗 and 𝑎_𝑗>0 are the normal processing time and the deteriorating factor of job 𝐽_𝑗, respectively.

Let 𝑝_𝑗𝑏₀ denote the upper bound of the actual processing time of job 𝐽_𝑗, where 𝑏₀>1 is a constant number. The tardiness of job 𝐽_𝑗 is denoted as 𝑇_𝑗; that is, 𝑇_𝑗=max{0,𝑝^𝑟_𝑗−𝑝_𝑗𝑏₀}. Then it can be obtained that the total tardiness of all jobs is ∑^𝑛_𝑗=1𝑇_𝑗. Let 𝑡₀𝑢 denote the upper bound of the maintenance time, where 𝑢>1 is a constant number. The tardiness of the 𝑖th maintenance is denoted as 𝑇^′_𝑖, that is, 𝑇^′_𝑖=max{0,𝑡₀𝑏^(𝑖−1)−𝑡₀𝑢}. Then the total tardiness of all maintenances is denoted by ∑^𝑘_𝑖=1𝑇^′_𝑖. Let 𝐶_max denote the makespan; that is, 𝐶_max=max{𝐶_𝑗∣𝑗=1,…,𝑛}.

In manufacturing processes, the length of working time determines the production fee. The tardiness penalties are assumed to be linear relationship with the total tardiness of all jobs and all maintenances, respectively. Thus, in the case of setting the upper bounds for the processing time of jobs and maintenance time of the machine simultaneously, we define the total cost as follows: 𝑇𝐶=𝛼𝐶_max+𝛽^𝑛_𝑗=1𝑇_𝑗+𝛾^𝑘_𝑖=1𝑇^′_𝑖, (4) where 𝛼,𝛽, and 𝛾 are the unit production fee, the unit tardiness cost of all jobs, and the unit tardiness cost of all maintenances, respectively. 𝛼, 𝛽, and 𝛾 should be positive numbers, that is, 𝛼>0, 𝛽>0, and 𝛾>0. The objective of this study is to minimize the total cost through exploring jointly the optimal maintenance frequency, the optimal maintenance positions, and the optimal job sequences.

3. Total Cost Minimization

Using the three-field notation 𝛼/𝛽/𝛾 of Graham et al. [25], we denote our problem as 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶, where 𝑀 and 𝑘 denote the maintenance and the maintenance frequency, respectively. We set the upper bounds for the actual processing time of each job and the maintenance time of the machine simultaneously. If the actual processing time of a job exceeds the upper bound, the tardiness penalty should be paid. And if the maintenance time also exceeds the corresponding upper bound, tardiness penalty of the maintenance should also be paid. The associated objective of the problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶 is given by 𝑇𝐶=𝛼𝐶_max+𝛽^𝑛_𝑗=1𝑇_𝑗+𝛾^𝑘_𝑖=1𝑇^′_𝑖. (5)

A group balance principle was presented by Kuo and Yang [20]. In the next part, we will prove that the group balance principle remains valid for the problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶. Assume that there are 𝑛 independent jobs to be assigned. If the machine is maintained 𝑘 times in a schedule, then the jobs are divided into (𝑘+1) groups. Application of the group balance principle ensures that the number of jobs in groups is as close as possible.

3.1. Group Balance Principle

Assume that the machine is maintained 𝑘 times in a schedule and the jobs are divided into (𝑘+1) groups. The number of the jobs in every group is ⌈𝑛/(𝑘+1)⌉−1 or ⌈𝑛/(𝑘+1)⌉, that is, ⌈𝑛/(𝑘+1)⌉−1≤𝑛_𝑖≤⌈𝑛/(𝑘+1)⌉.

Lemma 1. For the problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶, there exists such an optimal schedule that the number of jobs in groups satisfies the group balance principle.

Proof. Using the similar proof of Lemma 2 in Zhao and Tang [21], we assume that an optimal schedule 𝜎 consisting of 𝑛 independent jobs and 𝑘 maintenance activities does not satisfy the group balance principle. The maintenance and group sequence 𝜎 can be described as 𝜎=(𝐺₁,𝑀₁,𝐺₂,𝑀₂,…,𝐺_𝑘,𝑀_𝑘,𝐺_𝑘+1). Then somewhere in 𝜎 there must exist at least two groups 𝐺_𝑖 and 𝐺_𝑗, in which the difference in the number of jobs is greater than one. We assume that 𝑛_𝑖>𝑛_𝑗, then 𝑛_𝑖−𝑛_𝑗>1, where 𝑛_𝑖 and 𝑛_𝑗 denote the number of jobs in the 𝐺_𝑖 and 𝐺_𝑗, respectively. Let 𝜋₁, 𝜋₂, and 𝜋₃ denote the partial schedules of the 𝜎, then 𝜎=(𝜋₁,𝐺_𝑖,𝑀_𝑖,𝜋₂,𝐺_𝑗,𝑀_𝑗,𝜋₃).
Move the last job of group 𝐺_𝑖 to the last position of group 𝐺_𝑗, then we obtain a new schedule 𝜎^′=(𝜋₁,𝐺^′_𝑖,𝑀_𝑖,𝜋₂,𝐺^′_𝑗,𝑀_𝑗,𝜋₃). The moving of the job 𝐽_{[𝑖,𝑛𝑖]} is illustrated by Figure 1. For simplicity, we let the job 𝐽_{[𝑖,𝑛𝑖]} be the job 𝐽_𝑗. In schedule 𝜎 and 𝜎^′, the production cost of the other jobs remains unchanged since the positions of them remain unchanged. Let 𝑇𝐶(𝑝_𝑗) and 𝑇𝐶^′(𝑝_𝑗) denote the contribution of 𝑝_𝑗 to the total cost in the schedule 𝜎 and 𝜎^′, respectively. Since the maintenance duration is only dependent on its position in the schedule, moving the last job of group 𝐺_𝑖 to the last position of group 𝐺_𝑗 can not change the maintenance time. Then in the schedules 𝜎 and 𝜎^′, 𝛾∑^𝑘_𝑖=1𝑇^′_𝑖 remains unchanged.
In schedule 𝜎, the contribution of 𝑝_𝑗 to the total cost is given by 𝑇𝐶𝑝_𝑗=𝛼𝑝_𝑗𝑛^𝑎𝑗_𝑖+𝛽max0,𝑝_𝑗𝑛^𝑎𝑗_𝑖−𝑝_𝑗𝑏₀, (6) where 𝑎_𝑗 is the deteriorating factor of job 𝐽_𝑗.
In schedule 𝜎^′, the contribution of 𝑝_𝑗 to the total cost is given by 𝑇𝐶^′𝑝_𝑗=𝛼𝑝_𝑗𝑛_𝑗+1^𝑎𝑗+𝛽max0,𝑝_𝑗𝑛_𝑗+1^𝑎𝑗−𝑝_𝑗𝑏₀. (7) Combining (6) and (7), we get the following equality: 𝑇𝐶𝑝_𝑗−𝑇𝐶^′𝑝_𝑗=𝛼𝑝_𝑗𝑛^𝑎𝑗_𝑖−𝑛_𝑗+1^𝑎𝑗+𝛽max0,𝑝_𝑗𝑛^𝑎𝑗_𝑖−𝑝_𝑗𝑏₀−max0,𝑝_𝑗𝑛_𝑗+1^𝑎𝑗−𝑝_𝑗𝑏₀. (8) Since 𝛼>0,𝛽>0,𝑛_𝑖−𝑛_𝑗>1, and 𝑎_𝑗>0, we can obtain that 𝑇𝐶(𝑝_𝑗)−𝑇𝐶^′(𝑝_𝑗)>0. Hence, we can obtain that the total cost of schedule 𝜎^′ is less than that of schedule 𝜎, which contradicts the optimality of schedule 𝜎. Lemma 1 is proved.

Figure 1 

The illustration of the moving of job 𝐽[𝑖,𝑛𝑖].

In the following, we show that the problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶 remains polynomially solvable and can be solved in 𝑂(𝑛⁴) time. The associated total cost is given by 𝑇𝐶=𝛼𝐶_max+𝛽Σ^𝑛_𝑗=1𝑇_𝑗+𝛾Σ^𝑘_𝑖=1𝑇^′_𝑖=𝛼^𝑘+1𝑛_𝑖=1^𝑖_𝑟=1𝑝_[𝑖,𝑟]𝑟^{𝑎[𝑖,𝑟]}+^𝑘_𝑖=1𝑡₀𝑏^(𝑖−1)+𝛽^𝑘+1𝑛_𝑖=1^𝑖_𝑟=1max0,𝑝_[𝑖,𝑟]𝑟^{𝑎[𝑖,𝑟]}−𝑝_[𝑖,𝑟]𝑏₀+𝛾^𝑘_𝑖=1max0,𝑡₀𝑏^(𝑖−1)−𝑡₀𝑢=^𝑘+1𝑛_𝑖=1^𝑖_𝑟=1𝛼𝑟^{𝑎[𝑖,𝑟]}+𝛽max0,𝑟^{𝑎[𝑖,𝑟]}−𝑏₀𝑝_[𝑖,𝑟]+^𝑘_𝑖=1𝛼𝑡₀𝑏^(𝑖−1)+𝛾max0,𝑡₀𝑏^(𝑖−1)−𝑡₀𝑢. (9) Then, it can be seen whatever the group is, the contribution of a job to the total cost only depends on its position in a group, and for the given 𝑘, ∑^𝑘_𝑖=1(𝛼𝑡₀𝑏^(𝑖−1)+𝛾max{0,𝑡₀𝑏^(𝑖−1)−𝑡₀𝑢}) is a constant. We explore to find a polynomial to minimize the total cost. The problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶 can be reformulated as a standard assignment problem, which can be described as follows: Minimize^𝑛_𝑗=1^𝑘+1𝑛_𝑖=1^𝑖_𝑟=1𝑤_𝑗𝑖𝑟𝑥_𝑗𝑖𝑟+^𝑘_𝑖=1𝛼𝑡₀𝑏^(𝑖−1)+𝛾max0,𝑡₀𝑏^(𝑖−1)−𝑡₀𝑢 (10) subject to ^𝑛_𝑗=1𝑥_𝑗𝑖𝑟=1,𝑖=1,2,…,𝑘+1,𝑟=1,2,…,𝑛_𝑖,_{𝑘+1𝑛𝑖=1𝑖𝑟=1}𝑥_𝑗𝑖𝑟𝑥=1,𝑗=1,2,…,𝑛,_𝑗𝑖𝑟=0or1,𝑗=1,2,…,𝑛,𝑖=1,2,…,𝑘+1,𝑟=1,2,…,𝑛_𝑖, (11) where 𝑤_𝑗𝑖𝑟=(𝛼𝑟^𝑎𝑗+𝛽max{0,𝑟^𝑎𝑗−𝑏₀})𝑝_𝑗. If job 𝐽_𝑗 is scheduled in the 𝑟th position in group 𝐺_𝑖, 𝑥_𝑗𝑖𝑟=1, otherwise 𝑥_𝑗𝑖𝑟=0. Constraint sets (11) can ensure that each job is scheduled exactly once and each position is taken by one job. A special case should be noted as follows. In the case of 𝑘=0, there is no maintenance in the schedule, and the objective of the assignment problem is not ∑^𝑛_𝑗=1∑^𝑘+1_𝑖=1∑^𝑛𝑖_𝑟=1𝑤_𝑗𝑖𝑟𝑥_𝑗𝑖𝑟+∑^𝑘_𝑖=1(𝛼𝑡₀𝑏^(𝑖−1)+𝛾max{0,𝑡₀𝑏^(𝑖−1)−𝑡₀𝑢}), but ∑^𝑛_𝑗=1∑^𝑘+1_𝑖=1∑^𝑛𝑖_𝑟=1𝑤_𝑗𝑖𝑟𝑥_𝑗𝑖𝑟.

It is known that the assignment problem can be optimally solved in 𝑂(𝑛³) time by the classic Hungarian algorithm. In order to minimize the total cost, we propose a polynomial time algorithm to determine jointly the optimal 𝑘 and the optimal job sequence.

Algorithm 2.  
Step 1. For each 𝑘(𝑘=0,1,…,𝑛−1), solve the assignment problem (10)-(11), and let the corresponding objective value be 𝑇𝐶(𝑘).
Step 2. Let (𝑇𝐶(𝑘))^∗=min(𝑇𝐶(𝑘),(𝑘=0,1,…,𝑛−1)), and the corresponding schedule is the result schedule.

Theorem 3. The problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶 can be optimally solved by Algorithm 2 in 𝑂(𝑛⁴) time.

Proof. For a fixed maintenance frequency 𝑘, we can obtain the optimal maintenance positions and the number of jobs in each group by Lemma 1. The problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶 can be optimally solved via the assignment problem (10)-(11) in 𝑂(𝑛³) time. Note that 𝑘 has 𝑛 possible values. Then, (𝑇𝐶(𝑘))^∗=min(𝑇𝐶(𝑘),(𝑘=0,1,…,𝑛−1)) is the optimal objective value for the considered problem. Therefore, to solve the problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶, the computational complexity is 𝑂(𝑛⁴).

Using the similar method of Theorem 3, the following corollary can be easily obtained.

Corollary 4. For the scheduling problem of only setting the upper bound for the actual processing time of a job, it can be optimally solved in 𝑂(𝑛⁴) time.

In the following, we investigate a special case of the problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎𝑗,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶. Let the deteriorating factor 𝑎_𝑗=𝑎, where 𝑎 is a common deteriorating factor. We denote the special case of the problem as 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶 and explore to find a more efficient algorithm.

First, we give a lemma which is useful for the following results.

Lemma 5. If sequence 𝑥₁,𝑥₂,…,𝑥_𝑛 is ordered nondecreasingly and sequence 𝑦₁,𝑦₂,…,𝑦_𝑛 is ordered nonincreasingly, the sum ∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖 of products of the corresponding elements is minimized [26].

Theorem 6. The problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶 can be optimally solved by scheduling the jobs in a nonincreasing order of their normal processing time 𝑝_𝑗 and then arranging the jobs one by one into each group in turn. The time complexity of the problem is 𝑂(𝑛log𝑛).

Proof. For a given maintenance frequency 𝑘=𝑘₀, let ℎ be the remainder of 𝑛 divided by (𝑘₀+1); that is, ℎ=mod(𝑛,𝑘₀+1). If ℎ≠0, without loss of generality, we assume that there are 𝑑 jobs in each of the first ℎ groups and (𝑑−1) jobs in each of the other groups. Let 𝑤_[𝑖,𝑟]=𝛼𝑟^𝑎+𝛽max{0,𝑟^𝑎−𝑏₀}, where 𝑤_[𝑖,𝑟] is the positional weight of the corresponding job. Then the associated total cost is given as follows: 𝑇𝐶=𝛼𝐶_max+𝛽^𝑛_𝑗=1𝑇_𝑗+𝛾^𝑘0_𝑖=1𝑇^′_𝑖=^ℎ𝑑_𝑖=1𝑟=1𝑤_[𝑖,𝑟]𝑝_[𝑖,𝑟]+^𝑘0+1_𝑖=ℎ+1^𝑑−1_𝑟=1𝑤_[𝑖,𝑟]𝑝_[𝑖,𝑟]+^𝑘0_𝑖=1𝛼𝑏^(𝑖−1)+𝛾max0,𝑏^(𝑖−1)−𝑢𝑡₀. (12)(13) Since 𝛼,𝑡₀,𝑏,𝛾, and 𝑢 are constant numbers, for the given 𝑘₀, ∑^𝑘0_𝑖=1(𝛼𝑏^(𝑖−1)+𝛾max{0,𝑏^(𝑖−1)−𝑢})𝑡₀ is a constant number. From (13), it can be seen that 𝛼+𝛽max0,1−𝑏₀=𝑤_[1,1]=𝑤_[2,1]=⋯=𝑤_[𝑘0+1,1]<𝛼2^𝑎+𝛽max0,2^𝑎−𝑏₀=𝑤_[1,2]=𝑤_[2,2]=⋯=𝑤_[𝑘0+1,2]<⋯<𝛼(𝑑−1)^𝑎+𝛽max0,(𝑑−1)^𝑎−𝑏₀=𝑤_[1,𝑑−1]=𝑤_[2,𝑑−1]=⋯=𝑤_{[𝑘0+1,𝑑−1]}<𝛼𝑑^𝑎+𝛽max0,𝑑^𝑎−𝑏₀=𝑤_[1,𝑑]=𝑤_[2,𝑑]=⋯=𝑤_[ℎ,𝑑]. (14) Hence, if 𝑝_[1,1]≥𝑝_[2,1]≥⋯≥𝑝_[𝑘0+1,1]≥𝑝_[1,2]≥𝑝_[2,2]≥⋯≥𝑝_[𝑘0+1,2]≥⋯≥𝑝_[1,𝑑−1]≥𝑝_[2,𝑑−1]≥⋯≥𝑝_{[𝑘0+1,𝑑−1]}≥𝑝_[1,𝑑]≥𝑝_[2,𝑑]≥⋯≥𝑝_[ℎ,𝑑], (15) then, by Lemma 5, the total cost is the least one. Therefore, there exists an optimal schedule in which jobs are scheduled in nonincreasing order of their normal processing time. Then, schedule the job in the first position of each group one by one. If the first position of each group is filled, then schedule the remaining job in the second position of each group one by one. If all the second positions are filled, fill the third position, and so on, until all jobs are scheduled. The time complexity of arranging the jobs in a nonincreasing order of their normal processing time is 𝑂(𝑛log𝑛). The time complexity of assigning 𝑛 jobs one by one to each group in turn in a nonincreasing order of their normal processing time is 𝑂(1). Thus, the problem 1/𝑝^𝑟_𝑗=𝑝_𝑗𝑟^𝑎,𝑀=𝑘,𝑚_𝑖=𝑡₀𝑏^(𝑖−1)/𝑇𝐶 can be optimally solved in 𝑂(𝑛log𝑛) time.

We demonstrate the results of Theorem 6 in the following example.

Example 7. Data: 𝑛=5,𝑝₁=3,𝑝₂=5,𝑝₃=5,𝑝₄=8,𝑝₅=11,𝑎=0.2,𝛼=2,𝛽=25, 𝛾=100,𝑡₀=4,𝑏=1.1,𝑢=1.2,and𝑏₀=1.3. The values of the number of jobs in each group, the positional weights, the optimal schedule, and the total cost are given in Table 1.
Observing from Table 1, it can be seen that the case of 𝑘=1 is optimal. The jobs should be divided into 2 groups, where 𝑛₁=3,𝑛₂=2. The optimal schedule is (11, 5, 3, 8, 5). Then 𝑇𝐶^∗=70.47.

Table 1 

The number of jobs in each group, the positional weights, the optimal schedule, and the total cost.

4. Conclusions

The paper investigated a single-machine scheduling problem with upper bounded actual processing time and upper bounded maintenance time under deteriorating effect. The maintenance duration studied in the paper is a position-dependent exponential function. The objective is to minimize the total cost that is a linear function of the makespan and the tardiness penalties. We proved that the problem considered can be optimally solved in 𝑂(𝑛⁴) time. Moreover, for a special case that the deteriorating factor of the job processing time is assumed as a constant, we showed that the total cost minimization problem with deteriorating effect can be solved in 𝑂(𝑛log𝑛) time. We provided a numerical example for the special case, where the optimal solutions can be easily obtained. Future research may focus on the scheduling problem with upper bounded actual position-dependent processing time and upper bounded maintenance time under deteriorating effect in the context of parallel machine scheduling problems or job-shop scheduling problems.

Acknowledgments

The authors wish to thank editors and anonymous referees for their helpful comments. This work is supported by National Nature Science Foundation Project of China (71171046) and National Science-technology Support Plan Project (2012BAH69F03).