Joint optimization of preventive maintenance and production scheduling for parallel machines system

Abstract

This paper deals with the joint problem of production scheduling and maintenance planning as it becomes one important base for intelligent manufacturing. Due to the dual requirements both on cost and delivery date, a multi-objective optimization approach is proposed for parallel machines to allow decision makers to find compromise solution between production scheduling and maintenance planning: minimizing both maximum completion time and total maintenance cost. The new optimization approach is one multiple nondominated improved NSGA-II algorithm based on greedy idea and pre-distribution thinking, which can quickly and effectively seek Pareto optimal solution to solve the joint problem. As well as the introduced idea of averaging machine utilization, the job processing time model is constructed by considering machine degradation to meet real manufacturing situation. Moreover, a penalty function to maintenance (delay or advance) is proposed. Finally, the experimental analysis demonstrates the effectiveness and efficiency of this pre-distributed NSGA-II optimization approach, which could help solve the joint decision-making problem of production scheduling and preventive maintenance for parallel machines system.

Keywords

Production scheduling preventive maintenance NSGA-II algorithm parallel machines integer programming

1 Introduction

In real manufacturing, one machine may fail due to its degradation and usage. The consequent repair and replacement will make machine unavailable, which disorders production scheduling. Hence, how to schedule maintenance planning to keep machines in good operation condition and high reliability and further make production scheduling based on machine maintenance has become a vital issue for the achievement of intelligent manufacturing [1, 8]. Recently, a few researchers have studied on this joint problem of production scheduling and maintenance planning. For example, Pan et al. [16] proposed a scheduling model for single machine system incorporating production scheduling and machine maintenance, so as to maximize machine’s availability. Wong et al. [20] proposed a method for joint production scheduling problem by considering multi-resources and preventive maintenance, and proved its validity. Lee and Ni [9] determined maintenance and product dispatching policies based on condition-monitoring information and the relationship between machine degradation and associated product quality. They used a Markov decision process for long-term decision making and integer programming for short-term decision making with a multi-product and multi-station system. Cui et al. [4] used genetic algorithm to solve the joint problem for a single machine, and joint decision-making policy could obtain better result than independent policy. Mirabedini and Iranmanesh [14] integrated flow shop scheduling problem with preventive maintenance activities in order to minimize the total completion time, and proposed two meta-heuristics based on simulated annealing and genetic algorithm. Bajestani et al. [2] addressed the problem of integrated maintenance and production scheduling in a deteriorating multi-machine production system over multiple periods. They formulated a Markov decision process model to determine the maintenance plan and develop sufficient conditions guaranteeing its monotonicity in both machine condition and demand. Huang et al. [7] studied on a dual objectives problem and improved genetic algorithm based on mixed-coding. Sun and Li [17] proposed a heuristic method based on first-fit decreasing (FFD) to minimize maintenance cost, under an assumption of fixed maintenance cycle. According to the literatures abovementioned, as these scheduling problems are strongly NP-hard, it is obvious that heuristic method such as genetic algorithm [15] has become an important approach which is widely used.

Hence, this paper studies on identical parallel machines to search an optimal policy to balance production scheduling and preventive maintenance planning to minimize maximum completion time and total maintenance cost. As this kind of problem is proved to be NP-Hard problem [10, 18], this paper proposes a multiple nondominated improved NSGA-II algorithm based on greedy idea and pre-distribution thinking. This optimization approach first introduces a pre-distribution idea to average machine utilization, which can help seek Pareto optimal solution quickly and effectively to solve the joint problem. In addition, the existing literatures often ignore the influence of machine degradation to production scheduling, and only few researchers studied job processing time model based on machine’s usage [4, 13]. However, machine’s usage cannot describe its real health condition, thus job processing time model should be improved. In order to meet practical manufacturing situation, this paper constructs job processing time model based on machine’s health condition. Moreover, a penalty function for maintenance (delay/advance) is proposed so as to estimate real job’s completion time.

2 Problem description

This paper studied on a joint problem of production scheduling and machine maintenance planning for identical parallel machines in order to meet dual requirements of minimizing maximum completion time and total maintenance cost. Through constructing the optimization model, an improved scheduling algorithm is developed to search optimal solution to plan job processing and maintenance operation (seen in Fig. 1).

There are some assumptions given as below:

At the initial time, n jobs simultaneously arrive on m machines for processing. The job set is j ={ j_i|i = 1, 2, 3, …, n } and the machine set is M ={ M_i|i = 1, 2, 3, …, m }.

Only one job is processed by one machine at one time, and its processing cannot be interrupted.

The processing speed of each machine is different due to different machine’s condition.

The initial health condition of each machine is new.

The maintenance cycle of preventive maintenance is flexible.

The preventive maintenance operation is performed in maintenance window, and can restore the machine to be “as good as new”.

3 Model construction

In order to meet real manufacturing process, the job processing time should be adjusted according to machine’s health condition, hence this paper builds job processing time model by considering machine’s health condition. And based on this, an integrated optimization model with the aim of minimizing maximum completion time and total maintenance cost is constructed.

3.1 Flexible maintenance window

In order to maintain machine in its best operational condition, preventive maintenance operation is assumed to be performed in flexible maintenance window [u, v], where u is the earliest maintenance time and v is the latest maintenance time. Once machine’s heath index H_k is H_safe, the machine needs to be performed preventive maintenance operation and restores to be as good as new (seen in Fig. 2). Due to machine degradation, maintenance cycle RML_k (k is the maintenance cycle index) will be shorter along with increased maintenance times, which means that the machine should be performed preventive maintenance operation earlier (i.e. maintenance window is flexible). Once machine’s heath index is H_fail in the N^th maintenance cycle, the machine needs to be replaced, which is practical in manufacturing process. Assume the adjustment factor of maintenance window is δ_k, it is obtained the maintenance window for different maintenance cycle is

$\begin{matrix} [u_{k}, v_{k}] = δ_{k} \times [u_{k - 1}, v_{k - 1}] \\ k \in (0, 1, 2, \dots, N) \end{matrix}$ (1)

3.2 Job processing time model

This paper introduces heath index H_k to describe machine’s health condition. As the prediction models of machine’s health index usually take ARMA (auto-regressive and moving average) structure because of its basic architecture (see as [6, 11], this paper adopts ARMA structure for time sequence data. However, because the prediction model is not simply according to time, it is essential to make simple function approximation to better predict machine’s health index. Through the principle of least square method, the prediction model can be fit by polynomial curve. Hence, the form of prediction model is shown as $H_{k} = a_{1} x_{k}^{n} + a_{2} x_{k}^{n - 1} + \dots + a_{n} x_{k}^{1} + a_{0}$ (2) where x_k is machine’s usage of the k^th maintenance cycle ({k = 0, 1, … , N }). After preventive maintenance, the machine will be renewed, thus x_k will be zero. Here, n is set through verifying the fit factor R². While R² is closer to 1, the smaller the residual is, which proves the better fitting result of regression curve. The formula of R² is $R^{2} = \frac{S_{R}^{2}}{S_{T}^{2}} = \frac{\sum_{i = 1}^{n} {(\hat{y_{i}} - \bar{y})}^{2}}{\sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}}$ (3) where m denotes the number of samples, $\hat{y_{i}}$ denotes the predicted value, y_i denotes the actual value and $\bar{y}$ denotes the average value of y_i. As the recession of machine’s health index will accelerate along with the increased maintenance cycle, machine’s health index should be adjusted (i.e. machine’s health index according to its usage is different in different maintenance cycle). According to the adjusted maintenance window given in Section 3.1, assume b_k is the adjustment factor of machine’s usage in the k^th maintenance cycle ({k = 0, 1, … , N }), thus machine’s usage x_k could be adjusted as

$\begin{matrix} x_{k}^{'} & = & b_{k} \times b_{k - 1} \times \dots \times b_{0} \times x_{k} \\ k \in (0, 1, 2, \dots, N) \end{matrix}$ (4)

With the consideration of same maintenance threshold (i.e. machine’s health index at the latest maintenance time), it is obvious that $b_{k} = \frac{1}{δ_{k}}$ (5)

Hence, the improved prediction model of machine’s health index is

$\begin{matrix} H_{k}^{'} & = & a_{1} x_{k}^{' n} + a_{2} x_{k}^{' n - 1} + \dots + a_{n} x_{k}^{' 1} + a_{0} \\ k \in (0, 1, 2, \dots, N) \end{matrix}$ (6)

Due to machine’s deterioration effect, job’s processing time will increase while machine health gets worse. Although a few research have studied on this, they only consider job’s processing time linearly increase along with the increased machine’s usage. However, machine’s usage is unable to describe machine’s health condition directly. Machine’s health index which is key factor for machine’s condition is a direct influence factor for job’s actual processing time. Thus, the actual processing time of job i based on machine’s health index is shown as $p_{i} = p_{io} + α_{i} \times (1 - H_{k}^{'})$ (7) where p_i denotes the actual processing time of job i, p_io denotes job’s basic processing time and α_i denotes the factor of machine degradation [4] which is only related to job’s basic processing time and p_io/α_i = p_jo/α_j.

3.3 Multi-objective integrated optimization model

The time to failure for this machine is subjected to Weibull distribution with scale parameter θ and shape parameter ɛ (ɛ > 1) [4]. When the machine fails, there are minimal repair time t_r and minimal repair cost c_r. a_i denotes machine’s usage before processing the i^th job and b_i denotes machine’s usage after processing the i^th job. While performing preventive maintenance operation, preventive maintenance time is t_pm and preventive maintenance cost is c_pm. Once preventive maintenance operation is performed, the machine will restore to be as good as new and its usage becomes zero. Hence, the expected value of the completion time and total maintenance cost of the i^th job is $\begin{matrix} E (t_{i}) & = & E (t_{i - 1}) + p_{i} + t_{r} E (O_{i}) + t_{pm} w_{i} \\ = & \sum_{i = 1}^{n} (t_{pm} w_{i} + p_{i} + t_{r} E (O_{i})) \end{matrix}$ (8) $\begin{matrix} E (c_{i}) & = & E (c_{i - 1}) + c_{r} E (O_{i}) + c_{pm} w_{i} \\ = & \sum_{i = 1}^{n} (c_{pm} w_{i} + c_{r} E (O_{i})) \end{matrix}$ (9) where w_i is a decision variable. It is shown that $w_{i} = {\begin{matrix} 1, & if maintenance is performed prior \\ to the i^{th} job \\ 0, & otherwise \end{matrix}$ (10)

Here, E (O_i) denotes the expected times of minimal repair, and there is $E (O_{i}) = \int_{a_{i}}^{b_{i}} \frac{ɛ}{θ} {(\frac{t}{θ})}^{ɛ - 1} dt = {(\frac{b_{i}}{θ})}^{ɛ} - {(\frac{a_{i}}{θ})}^{ɛ}$ (11)

In Equation (11), a_i and b_i are given as $\begin{matrix} a_{i} & = & b_{i - 1} (1 - w_{i}) \\ i & = & 1, 2, \dots, n \end{matrix}$ (12) $\begin{matrix} b_{i} & = & a_{i - 1} + p_{i} \\ i & = & 1, 2, \dots, n \end{matrix}$ (13)

Hence, the objective functions of minimizing maximum completion time and total maintenance cost are given as $min f_{1} : T = \max (T_{1}, \dots, T_{j}, \dots, T_{m})$ (14) $min f_{2} : C = \sum_{j = 1}^{m} C_{j}$ (15) where T_j denotes the completion time of the j^th machine and C_j denotes the maintenance cost of the j^th machine.

4 Model solution

This paper proposes a pre-distribution NSGA-II algorithm based on greedy idea and pre-distribution thinking to search the optimal solution. First, pre-distributing all jobs to the machines, then arranging the pre-distributed jobs by this optimization algorithm. This proposed algorithm can help quickly and effectively seek Pareto optimal solution for this joint scheduling problem. Its solving process is presented as below.

4.1 Build integer programming model

As one multi-objective scheduling problem for parallel machines is proved to be NP-hard problem, the involvement of machine maintenance planning is even increasing the complexity and expanding its solution space. Along with large quantity of jobs in scheduling problem, the problem complexity would increase, thus it is essential to find a better method to effectively obtain the satisfied solution. In this paper, the pre-distribution thinking can help reduce the problem complexity. It is thus useful for the scheduling problem in real manufacturing. If max(T₁, …, T_m) ⪢ min(T₁, …, T_m), there would occur much machine idleness. This is a kind of resource waste. Hence, based on greedy idea, this paper first introduces the consideration of averaging machine utilization into scheduling algorithm. The first step is sorting all jobs for different machines according to pre-distribution thinking in which machine utilization is the same or similar. The second step is arranging the sorted jobs for different machine with optimal Pareto solution. The method of pre-distribution for n jobs on m machines is given by $min T = \sum_{j = 2}^{m} (T_{1 o} - T_{jo})$ (16) $s . t . T_{jo} = \sum_{i = 1}^{n} p_{io} \times z_{ij}$ (17) $T_{1 o} \geq T_{jo}$ (18) $\sum_{i = 1}^{n} z_{ij} \geq [\frac{n}{m}]$ (19) $\sum_{j = 1}^{m} z_{ij} = 1$ (20) $z_{ij} = {\begin{matrix} 1, & job i is processed by machine j \\ 0, & job i is not processed by \\ machine j \end{matrix}$ (21) $i \in (1, 2, \dots, n)$ (22) $j \in (1, 2, \dots, m)$ (23)

Equation (17) denotes the total basic completion time of sorted jobs processed by machine j. Equation (18) assumes the total basic completion time of sorted jobs processed by machine 1 (i.e. T_1o) is maximum. Equation (19) denotes the number of assigned jobs on each machine is almost the same. Equation (20) denotes each job could only be processed by one machine. z_ij in Equation (21) is a 0-1 decision variable.

4.2 Prove pre-distribution thinking

In general, the dominance relation of Pareto can be represented by ≻_p [3, 5]. It means: In a v-dimensional decision space, there are p objective function values solved by two feasible solutions X and Y. If f_i (X) ≤ f_i (Y) (i = 1, 2, …, p) and f_i (X) < f_i (Y) by at least one objective function value, X dominates for a multi-objective minimization problem, which can be represented as f (X) ≻ _pf (Y).

As unbalanced job assignment will easily lead to large variance between actual completion time by different machine (i.e. T_max ⪢ T_min), it is obvious that this kind of solution is not a suitable solution with the aim of minimizing maximum completion time. Assume the current solution set is Set 1 and the total maintenance cost is C. For balanced job assignment, the solution set is Set 2 (i.e. $T_{\max}^{'} \approx T_{\min}^{'}$ ). It can be proved that both Set 2 and Set 1 at least belong to the same Pareto frontier level for minimizing both maximum completion time and total maintenance cost, even Set 2 dominates Set 1. The brief proof is given as follows.

First, assume preventive maintenance operation is performed in flexible maintenance window [u_k, v_k], thus it is known that $T_{\max} > T_{\max}^{'}$ (24)

Then, the average actual completion time of m machines in Set 2 is $\bar{T_{m}^{'}} = \frac{\sum_{j = 1}^{m} T_{j}^{'}}{m} \approx T_{\max}^{'} \approx T_{\min}^{'}$ (25)

In Set 1 (T_max ⪢ T_min), assume the set of $T_{j} > \bar{T_{m}^{'}}$ is T_a (a = 1, 2, . . , m_a) and the set of $T_{j} < \bar{T_{m}^{'}}$ is T_b (b = 1, 2, . . , m_b). Due to machine degradation, the longer one machine’s processing time is, the greater influence on job’s actual processing time. There is $\sum_{j = 1}^{m_{a}} (T_{j} - \bar{T_{m}^{'}}) \geq \sum_{j = 1}^{m_{b}} (\bar{T_{m}^{'}} - T_{j})$ (26)

Then, it can be deduced that $\sum_{j = 1}^{m} T_{j} > \sum_{j = 1}^{m} T_{j}^{'}$ (27)

While preventive maintenance operation is performed in flexible maintenance window [u_k, v_k], it is considered that the longer the total actual completion time is, the more the preventive maintenance times is. There is $N \geq N^{'}$ (28)

According to Equations (9) and (27), there is $C \geq C^{'}$ (29)

Finally, based on Equations (14)–(15), it can be known that $Set 2 ≻_{p} Set 1$ (30)

Therefore, the pre-distribution thinking is proved to be rational. Moreover, this paper discusses Pareto solution solved in previous literatures to show its effectiveness. For example, Wang and Chang [19] used a hybrid particle bee colony algorithm to search optimal Pareto solution. In this literature, the number of machines is m ={ 2, 4, 6 } and the number of jobs is n ={ 20, 50, 100, 200, 300 }. Its results are given in Table 1.

It can be seen in Table 1 that T_max and T_min are almost the same. That is to say, the optimal or near-optimal solution belongs to the solution set solved by pre-distribution thinking (i.e. the optimal or near-optimal solution should satisfy $T_{\max}^{'} \sim T_{\min}^{'}$ ), which well proves the effectiveness and rationality of this proposed pre-distribution thinking.

4.3 Develop nondominated genetic algorithm

In this paper, a pre-distribution NSGA-II algorithm is introduced to solve the integrated model (i.e. deal with the sorted jobs by pre-distribution thinking). Its brief steps are given as below.

Fitness function: it is used to minimize the objective functions of minimizing maximum completion time and total maintenance cost (i.e. f₁ and f₂). In order to accelerate the algorithm convergence speed and ensure the effectiveness of the solution, when preventive maintenance operation is not performed in a maintenance window, the solution should be punished. Based on previous research [21], this paper considers: if preventive maintenance operation is not performed in maintenance window [u_k, v_k], the completion time b_last of the last job in k^th maintenance cycle would be punished with a penalty coefficient β, and minimal repair cost could be calculated based on new b_last. The penalty function is $b_{last} = b_{last} + Δ b_{last} = b_{last} + Δ t \times β$ (31)

where Δb_last is the penalty term and Δt = b_last - v_k or Δt = u_k - b_last.

Gene encoding: processing sequence is encoded by real number and preventive maintenance operation in encoded by 0-1 number. If machine is performed preventive maintenance operation after processing job i, it should be encoded by 1, otherwise, it is encoded by 0. For example, a chromosomal gene is encoded as below. It means job’s processing sequence is 2-4-5-7-8, and one preventive maintenance operation must be performed after processing job 5.

Dominated sorting: according to multi-objective domination principle with penalty function, sort dominant parent and offspring hierarchically.

Selection of individual for crossover: as the scheduling problem of multi-jobs processed by multi-machine can be simplified to be a scheduling problem of multi-jobs processed by single machine based on pre-distribution thinking, the length of chromosome is shorten significantly. In order to avoid trapping into the local optimum, two individuals for crossover are selected randomly.

Gene crossover: the two-point crossover method is adopted to randomly select two positions for crossover of parent gene, with a certain probability, so as to generate offspring individual to ensure the variety of the solution (shown in Fig. 3).

Gene mutation: gene encoded by real number is exchanged its position randomly and gene encoded by 0-1 number is changed itself with a certain mutation probability.

Selection of species: in order to avoid trapping into the local optimum early, the championship selection method is adopted. First, the chromosome of parent and offspring are combined. Then, the dominant ones are sorted hierarchically and the congestion of each individual layer is calculated. Later, two individuals are selected and compared by levels. If two levels are different, the individual with lower level is selected. If two levels are the same, the congestion is compared, and the individual with larger distance is selected. Finally, father species for next generation are chosen until the number of individuals is same as the number of initial species. The congestion L [i] _d can be calculates by $L {[i]}_{d} = \sum_{m = 1}^{2} L {[i + 1]}_{m} - L {[i - 1]}_{m}$ (32)

5 Numerical example

Suppose there are 32 jobs to be processed by 4 machines, and their basic processing time p_io is randomly generated between 5 and 20. According to the prediction model for machine’s health index in Liao’ research [11], the predicted value determined every 0.5 hours is selected to be initial value. Based on Equation (3), this proposed prediction model could be solved. Its computation results are presented in Table 2.

In Table 2, it can be seen that from n = 4, R² changes slightly and its value is close to 1. n = 4 is thus to be chosen. The corresponding coefficient can be calculated to be $[- 0.0026, 0.1478, - 2.9197, 14.2816, 986.6367]$

Based on this proposed prediction model, it can be obtained that machine health index reaches the threshold 0.202 after 36.4 hours operation, which is almost the same as the result (0.206, 36.75 hrs) obtained by Liao [11] (seen in Fig. 4). Hence, it can be proved this proposed prediction model of machine’s health index is accurate to determine whether performing maintenance operation in maintenance window.

Assume $δ_{k} = \frac{4 \times k + 9}{5 \times k + 9}$ and maintenance threshold is the same [11]. Thus, the adjustment factor of machine’s usage in the k^th maintenance cycle is

$\begin{matrix} b_{k} = \frac{1}{δ_{k}} = \frac{5 \times k + 9}{4 \times k + 9} \\ k \in (0, 1, 2, \dots, N) \end{matrix}$ (33)

Hence, it can be deduced that $x_{k}^{'}$ is

$\begin{matrix} x_{k}^{'} & = & (\frac{5 \times k + 9}{4 \times k + 9} \times \frac{5 \times (k - 1) + 9}{4 \times (k - 1) + 9} \\ \times \dots \times \frac{5 \times 0 + 9}{4 \times 0 + 9}) x_{k} \end{matrix}$ (34)

Based on Equation (6), the prediction model of machine health index can be obtained as

$\begin{matrix} H_{k}^{'} & = & 0.001 \times (- 0.0026 x_{k}^{' 4} + 0.1478 x_{k}^{' 3} \\ - 2.9197 x_{k}^{' 2} + 14.2816 x_{k}^{'} + 986.6367) \\ k \in (0, 1, 2, \dots, N) \end{matrix}$ (35)

According to the proposed integer programming model with pre-distribution thinking, the initial solution is given as: j_i1, j_i2, j_i3 and j_i4. They present the jobs to be processed by machine 1, 2, 3 and 4 (see in Table 3).

Based on the abovementioned literatures, the parameters are adjusted appropriately. The parameters are set as follows:

Assume there are 32 jobs to be processed by 4 machines, and their basic processing time p_io is randomly generated between 5 and 20.

As the machine reaches the maintenance threshold 0.2 after 36.8 hours, the 0^th maintenance window is supposed to be [32.8, 36.8].

As the scale parameter of Weibull distribution θ is related to machine’s life, it is supposed that θ is 50 and the shape parameter ɛ is 2.

Assume preventive maintenance time t_pm is 4 hours, preventive maintenance cost C_pm is 300, minimal repair time t_r is 15 hours, minimal repair cost c_r is 500.

Suppose pio / - αi = pjo / - αj = 1.1, the penalty coefficient β is 0.5. hence, the machine should be replaced after 10 preventive maintenance cycles.

For this proposed NSGA-II algorithm, it is assumed that the number of initial population is 20, the maximum number of generations is 200, the crossover probability is 0.9 and the mutation probability is 0.2.

According to the optimization approach, the optimal solution for each machine can be obtained. In Fig. 5, the Pareto frontier solution for machine 1 is presented.

It can be seen that there is plenty of solutions for machine 1, and the distribution is uniform and concentrated. Figure 6 shows a policy of production scheduling and preventive maintenance planning by assuming the weight of time and cost is the same (the completion time is almost 121 hrs).

As the completion time of each machine is almost the same (i.e. T_max ∼ T_min), this solution can be viewed as a global solution. In addition, according to the computation results about completion time and cost shown in Table 4, it can be found that in different maintenance cycles, t_ik of the last job is in the relative maintenance window, which can prove preventive maintenance operation is indeed performed in flexible maintenance window. Thus, it is a feasible solution.

In addition, this proposed method is compared with standard NSGA-II method. There are {n|n = 16, 32, 64} and {m|m = 2, 4, 8}. The computation results are listed below. It can be found that in Table 5, the solution obtained by pre-distributed NSGA-II is much better than that obtained by standard NSGA-II. Moreover, the average processing time T_aver obtained by this proposed method is more stable, and the average processing cost C_aver is towards linear relationship with increased number of jobs n. while using standard NSGA-II, the method would easily trap into local optimization due to increased complexity, which makes the average processing time T_aver follow increasing tendency and the average processing cost C_aver is towards nonlinear relationship with increased number of jobs n. Hence, the thinking that multi-machines scheduling problem transforms to be single machine scheduling problem can greatly reduce the search space of solutions, and also avoid trapping into local optimization. The computation results can well demonstrate the effectiveness and efficiency of this proposed method.

6 Conclusion and future works

In this paper, due to the dual requirements both on cost and delivery date, one multi-objective optimization approach is proposed for parallel machines to find compromise solution between production scheduling and maintenance planning. It aims to minimize both maximum completion time and total maintenance cost. In order to obtain effective and efficient solution, a multiple nondominated pre-distributed NSGA-II algorithm is proposed based on greedy idea and pre-distribution thinking. This paper firstly introduces the idea of averaging machine utilization to simplify and accelerate the searching of Pareto optimal solution from solution space about multi-machine scheduling problem, which can transform multi-machine scheduling problem to be single machine scheduling problem. Hence, this proposed method can greatly reduce the search space of solutions and avoid trapping into local optimization. Secondly, this paper constructs job processing time model considering machine degradation to meet practical manufacturing situation. A penalty function to maintenance (delay or advance) is also proposed. Finally, through a numerical example, the computation results can well demonstrate the effectiveness and efficiency of this multi-objective optimization approach, which can help solve the joint decision-making problem of production scheduling and maintenance planning for parallel machines.

In summary, this proposed method is feasible in the field of joint scheduling problem and the results are reliable. Later, in order to meet more practical situation, the future work will focus on the extension of this optimization method. For example, due data could be taken into account as a constraint. Thus, this pre-distributed NSGA-II algorithm can be improved for real manufacturing process.

Footnotes

Acknowledgments

The authors would like to thank anonymous referees for their remarkable comments and great support by National Natural Science Foundation of China (71301176) and Research Fund for the Doctoral Program of Higher Education of China (No. 20130191120001).

References

Baban

, Baban

and Suteu

, Using a fuzzy logic approach for the predictive maintenance of textile machines, Journal of Intelligent and Fuzzy Systems 30(2) (2016), 999–1006.

Bajestani

M.A.

, Banjevic

and Beck

J.C.

, Integrated maintenance planning and production scheduling with Markovian deteriorating machine conditions, International Journal of Production Research 52(24) (2014), 7377–7400.

Bandyopadhyay

and Bhattacharya

, Solving Multi-objective Parallel Machine scheduling Problem by a Modified NSGA-II, Applied Mathematical Modelling 37 (2013), 6718–6729.

Cui

, Lu

and Pan

, Integration research of production scheduling and equipment maintenance based on multi-objective optimization, Computer Integrated Manufacturing System 6(17) (2014), 1398–1404.

Deb

, Pratap

and Agarwal

, A fast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transaction on Evolutionary Computation 6(2) (2002), 182–197.

, Xie

and Wang

, Optimal setup adjustment and control of a process under ARMA disturbances, IIE Transactions 47(3) (2015), 230–244.

Huang

, Lu

and Cui

, Joint optimization of production scheduling and maintenance plan to parallel machine system, Industrial Engineering and Management 18(4) (2013), 49–55.

Hussain

, Fuzzy information system for condition based maintenance of gearbox, Journal of Intelligent and Fuzzy Systems 28(6) (2015), 2509–2518.

Lee

and Ni

, Joint decision making for maintenance and production scheduling of production systems, The International Journal of Advanced Manufacturing Technology 66(5) (2013), 1135–1146.

10.

Lenstra

, Rinnooy

and Brucker

, Complexity of Machine Scheduling problems, Annals of Discrete Mathematics 1 (1997), 343–362.

11.

Liao

, Prospective Maintenance and Production Scheduling Integration of Equipment Based on Decline Mechanism[D]. Shanghai Jiaotong University, 2011.

12.

Liao

, Pan

, Wang

and Xi

, Application of Statistical pattern recognition and autoregressive moving average model to the residual life prediction, Journal of Shanghai Jiaotong University 45(7) (2011), 1000–1004.

13.

, Xu

and Zhou

, Study on scheduling problem of degradation single-machine system based on flexible maintenance, Computer Engineering and Application 51(10) (2015), 227–231.

14.

Mirabedini

and Iranmanesh

, A scheduling model for serial jobs on parallel machines with different preventive maintenance (PM), The International Journal of Advanced Manufacturing Technology 70(9) (2014), 1579–1589.

15.

Pajk

, Genetic - Fuzzy system of power units maintenance schedules generation, Journal of Intelligent and Fuzzy Systems 28(4) (2015), 1577–1589.

16.

Pan

E.S.

, Liao

W.Z.

and Xi

L.F.

, A single machine-based scheduling optimisation model integrated with preventive maintenance policy for maximising the availability, International Journal of Industrial and Systems Engineering 10(4) (2012), 451–469.

17.

Sun

and Li

, Scheduling problem with multiple maintenance activities and non-preemptive jobs on two identical parallel machines, International Journal of Production Economics 124(1) (2010), 151–158.

18.

Vlachos

, Ant Colony System algorithm solving a Thermal Generator Maintenance Scheduling Problem, Journal of Intelligent and Fuzzy Systems 24(4) (2013), 713–723.

19.

Wang

and Chang

, Study on Batch Scheduling of Parallel Machine System[D]. China University of Mining and Technology, 2014.

20.

Wong

C.S.

, Chan

F.T.S.

and Chung

S.H.

, A joint production scheduling approach considering multiple resources and preventive maintenance tasks, International Journal of Production Research 51(3) (2013), 883–896.

21.

Zhang

, Zhai

P.C.

and Zhang

B.Y.

, Penalty function method in the application of genetic algorithm to deal with constraints[J], Journal of Wuhan University of Technology 24(2) (2002), 56–59.