This paper focuses on a preventive maintenance plan and production scheduling problem under reentrant Job Shop in semiconductor production. Previous researches discussed production scheduling and preventive maintenance plan independently, especially on reentrant Job Shop. Due to reentrancy, reentrant Job Shop scheduling is more complex than the standard Job Shop which belongs to NPhard problems. Reentrancy is a typical characteristic of semiconductor production. What is more, the equipment of semiconductor production is very expensive. Equipment failure will affect the normal production plan. It is necessary to maintain it regularly. So, we establish an integrated and optimal mathematical model. In this paper, we use the hybrid particle swarm optimization algorithm to solve the problem for it is highly nonlinear and discrete. The proposed model is evaluated through some simple simulation experiments and the results show that the model works better than the independent decisionmaking model in terms of minimizing maximum completion time.
In 1993, Kumar [
RJSP is more complex than JSP because the same job will visit the same machine more than once at different stages. The main solving method is the heuristic algorithm. An approach of artificial neural network is proposed to solve multidecision scheduling problems of semiconductor wafer fabrication [
Although many achievements have been made in semiconductor manufacturing production scheduling, they are studied only from the perspective of the production department to optimize production scheduling. They ignore the fact that the semiconductor manufacturing workshop equipment maintenance department arranges a large number of preventive maintenance tasks so that they will affect the equipment production. What is more, the random failure of machines will interrupt production scheduling. It will generate a production scheduling conflict when we discuss production scheduling and equipment maintenance separately. Obviously, the related theoretical study of production scheduling does not reflect the actual status of the reentrant shop job scheduling very well at present.
Equipment preventive maintenance is also important to improve production efficiency. Ma et al. [
Although joint optimization is rarely studied on semiconductor production, many researches have been carried out in traditional production scheduling and equipment maintenance. Cassady and Kutanoglu [
Judging from the current research results, although joint optimization of production and maintenance has aroused more and more attention, studies on solving joint research of reentrant manufacturing production and equipment maintenance operations are very rare. In this paper, we combine production scheduling with preventive maintenance planning and establish an integrated optimization model of scheduling and preventive maintenance with the goal of minimizing completion time.
The paper is organized as follows. Section
Standard JSP contains two subproblems: the machine selection problem and the operation sequencing problem. During the production, each job can only be processed on different machines only once. And each machine can process a job at the same time. The processing diagram is shown as Figure
Processing diagram of JSP.
Processing diagram of RJSP.
RJSP and equipment preventive maintenance planning problem can be represented as follows: there are
The processing time for each operation in a particular machine is defined.
A machine can only process one operation at a certain time (resources constraint).
Each job must be processed on one machine at a given time, so jumping the queue is not allowed.
All the jobs have the same probability to be scheduled at the beginning, which means different jobs have the same priority.
Each operation of each job is not allowed to be interrupted in processing time; that is, preventive maintenance operations must be before or after production.
All machines are new at the beginning of the production process.
Each machine recovers as new after the implementation of preventive maintenance.
The time of small repair and machine setup is ignored during the processing.
The indices and decision variables used in the model are as follows.
In the process of production, equipment maintenance is a basic work. Timely maintenance can restore equipment performance, fix the trouble, and effectively prolong the service life of equipment. So, it is one of the most important ways to ensure smooth production. For preventive maintenance, it can be roughly divided into two categories. One is preventive maintenance which is based on machine reliability, and the other is periodic preventive maintenance.
For preventive maintenance based on the reliability of the machine, when the reliability of the machine reaches the threshold set, it is time for equipment maintenance. Some scholars suppose the machines will return to the new state after maintenance in their studies. In fact, equipment performance is gradually degraded with decreasing machine age. Under the same maintenance conditions and maintenance time, the equipment degree of maintenance decreases with the increase of the maintenance times. So, it will take less and less time to get the reliability threshold. So, lots of scholars assume that the reliability threshold will reduce after maintenance. Periodic preventive maintenance is the most common method in industrial production. It means that when each machine achieves the maximum maintenance interval, it must be maintained in time. When there is small equipment failure, we assume that the maintenance time is negligible. So, we do not consider the failure rate of the machine. Thus, this article mainly uses the periodic preventive maintenance strategy.
In practice, many companies perform PM at a fixed interval. This kind of fixed interval is according to the processing time when starting production. If we calculate time according to processing time, the machines will be maintained excessively because some machines may not process any job during the maintenance cycle. So, we adopt other methods to determine PM intervals. When we calculate time, we focus on the age of the machine. This method is used in many researches [
Job processing time on machines for the example.
Machine 1  Machine 2  

Job 1  Operation 
10  5 
Operation 
5  10 
If we calculate time according to machine age, each machine just needs only onetime maintenance as depicted in Figure
Maintenance based on machine age.
Maintenance based on processing time.
The main goal in this paper is to minimize the makespan (
The starting production time of the
completion time of previous jobs on jth machine:
completion time of the previous operation of the ith job on another machine
time for implementing PM. In our model, PM is regarded as perfect, which means the machine will be renewed to an asgoodasnew status. We assume that the initial age of the machines is zero. Then, according to the machines’ age and fixed maintenance cycle, we can decide whether PM should be conducted. Hence, the completion time of the current job can be calculated as follows:
The completion production time of the
Due to reentrancy, the same kind of job in different stages will wait to be produced on one machine. So, we should decide which one will be produced firstly. The calculation is as follows:
With high nonlinearity and discreteness, the reentrant Job Shop scheduling problem is a strongly NPhard problem. It is very hard to be solved by traditional optimization methods within an acceptable time. In recent years, many scholars have been committed to using intelligent optimization algorithms to solve this problem, such as genetic algorithm, ant colony algorithm, and particle swarm optimization algorithm. General particle swarm optimization algorithm is to find the optimum value by tracking individual best value and group best value [
Workflow of the proposed HPSOA is shown in Figure
The workflow of HPOSA.
Initialize the number of particles, the maximum number of iterations, learning factors, and so on.
Initialize the particle’s initial position and velocity and calculate the fitness value.
Initialize the particle’s personal best and global best.
Calculate the value of inertia weight
Calculate the fitness value of particles and then sort order according to the fitness value.
Improve particles by crossover and mutation operation.
Recalculate the fitness value.
Obtain updated personal best and global best.
If the termination condition is met, output the optimal solution. Otherwise, execute the loop body Steps
In our encoding, the particle is composed of twodimensional vectors. The job sequence in the first dimension decides the job’s processing sequence. The second dimension is a set of random numbers which can decide particles’ velocity and position. The range of those random numbers is [
If the particle has reached the condition of crossover, we choose the particles to cross with the best personal particle and best group particle. The method of crossover is integer crossover. At first, we choose the position of crossover randomly. And then the selected one crosses with the personal best or group best. For example, if we choose the second position and fourth position to cross, the operation will be depicted as follows.
2
1
1
1
2
2
0.1576
0.2785
0.5469
0.9575
0.9649
0.9706
2
2
1
1
2
1
0.1419
0.4218
0.4854
0.8003
0.9157
0.9572
2
2
1
1
2
2
0.1576
0.4218
0.4854
0.8003
0.9649
0.9706
After crossing, the amount of processing of some jobs will increase or decrease. In this situation, we will take necessary actions to make sure the amount of processing meets the conditions. In the crossover process, we will choose the best individual to update the particle swarm according to the fitness value.
In the mutation operation, two positions are changed with each other in the same individual. At first, we choose the mutation position, position 1 and position 2, and then change the mutual position. We just change the position of the job not the particle’s position. For example, if we change the second position and fourth position, it will be described as follows: 2 2 1 1 2 1 0.1419 0.4218 0.4854 0.8003 0.915 0.9572
2
1
1
2
2
1
0.1419
0.4218
0.4854
0.8003
0.915
0.9572
In real semiconductor manufacturing scheduling, the production process is very complex. Each job needs to go through hundreds of operations. In order to validate the superiority of joint optimization model and the effectiveness of hybrid particle swarm algorithm, we compare it with the independent decisionmaking model. In the example, there are 4 jobs that are processed on 5 machines in the system. Some machines will produce the same job more than once. In order to reflect the competition of similar products at different stages on the same machine, we assume that job 1 and job 2 are the same products. Job operation information is depicted in Table
Job processing time on each machine (time units).












(1,20)  (3,10)  (2,39)  (5,24)  (4,33)  (2,17)  (3,26)  (1,22)  (2,19) 

(1,20)  (3,10)  (2,39)  (5,24)  (4,33)  (2,17)  (3,26)  (1,22)  (2,19) 

(1,18)  (2,37)  (3,36)  (4,45)  (2,30)  (3,18)  (4,31)  (1,24)  (5,33) 

(3,21)  (2,33)  (1,40)  (4,35)  (3,15)  (2,34)  (1,18)  (5,25)  (4,37) 
Maintenance has degraded effect in the whole production process. Kubzin and Strusevich proposed a linear degradation function to calculate PM time [
By using hybrid particle swarm optimization algorithm, the optimal solution is 392.5. The Gantt chart of integrated optimization model is shown in Figure
The Gantt chart of integrated optimization model.
In actual production, the production department is responsible for the production scheduling, and the equipment maintenance department just makes equipment maintenance plan. So, their work is separated. The maintenance department calculates the best maintenance cycle, and the production department calculates the best production scheduling, and then the production department arranges a production plan. In the study, we assume that the production sequence is fixed. And if the machine’s cumulative processing time is more than maintenance cycle, the machine needs to be maintained right now. Used by the same algorithm, the optimal solution is 411.5. Its scheduling Gantt chart is shown in Figure
The Gantt chart of independent decisionmaking.
Compared with independent decisionmaking, the result is improved by 4.62%. From the Gantt chart, we also can get that the maintenance time of joint decisionmaking is less than of independent decisionmaking. So, this will decrease the maintenance cost in some ways. In order to prove the superiority of the combined optimization model, we simulate a few more complex numerical experiments, and the simulation data can be checked in Supplementary Materials available online at
Comparison of independent decision making and integrated optimization.
Jobs  Machines  Reentrancy times  independent decision making  integrated optimization model  Improvement 

4  5  3  411.5  390.5  4.62% 
4  5  5  669  646  3.44% 
5  7  3  696  650.5  6.54% 
5  7  5  920.5  876.5  4.78% 
Genetic algorithm (GA) is a stochastic search and optimization method which is based on natural selection and genetic mechanism [
To verify the validity of HPSOA, we make some tests on different examples compared with GA. In the testing, we make comparison with GA which can also be used to solve this problem. And the results indicated that HPSOA is better than GA to solve this problem.
At first, we analyze the example of 4 jobs and 5 machines we talked about above. The final result is 403.5 which is worse than HPSOA. Its scheduling Gantt chart is shown in Figure
Job processing and machine maintenance by GA.
The machines’ reentrant times will affect dispatching, so reasonable arrangements for these machines play a big role in improving production efficiency. So, we give two more complex examples to verify the superiority of our method by increasing reentrancy times. One is 5 jobs/7 machines with 3 times reentrancy. The other is 5 jobs/7 machines with 5 times reentrancy. They are all solved using GA and HPSOA. The results are shown in Table
Comparison of GA and HPSOA for complex problems.
Jobs  Machines  Reentrancy times  GA  HPSOA  Improvement 

4  5  3  403.5  390.5  3.22% 
4  5  5  700.5  646  7.78% 
5  7  3  711.5  650.5  8.57% 
5  7  5  1065.5  876.5  18.69% 
In actual production, production scheduling and preventive maintenance are closely linked and interactive. In this paper, we build an optimal model of reentrant manufacturing scheduling and equipment maintenance. We adopt fixed maintenance cycle with penalty function strategy and solve the problem effectively by the hybrid particle swarm optimization algorithm. By comparison with GA, this method is preferable to solve the combined optimization problem.
We make lots of assumptions to simplify the complex problem for our study. However, they will not exist in practical production. Firstly, equipment failure is difficult to avoid, so the machines’ minor repair time should not be ignored. What is more, equipment age is changeable after maintenance in reality. At last, we focus on only one single objective in the study. However, cost is an important factor to consider which includes production cost, preventive maintenance cost, minimal repair cost for unexpected failures, and tardiness cost. As a consequence, future efforts are still needed to solve this composite problem in reentrant Job Shop production scheduling.
The authors declare that they have no competing interests.
This work is supported by the National Natural Science Foundation of China (no. 71472125).