A new hybrid genetic algorithm for job shop scheduling problem
Introduction
Scheduling is one of the most critical issues in the planning and managing of manufacturing processes. The difficulty of finding the optimal schedule depends on the shop environment, the process constraints and the performance indicator. One of the most difficult problems in this area is the job shop scheduling problem (JSSP), which has been proved to be an NP-complete problem [1]. Since the benchmarks of JSSP were presented by Fisher and Thomson in 1963 [2], JSSP has been studied by a great number of researchers, and many exact methods and approximation algorithms have been proposed [3], [4], [5], [6], [7]. Exact methods, such as branch and bound, linear programming and Lagrangian relaxation guarantee global convergence and have been successful in solving small instances. However, they require a very high computing time as the size of problem increases. So a lot of researchers paid their attention to meta-heuristic and intelligent hybrid search optimization strategies for solving JSSP. Among these, the shifting bottleneck approach, particle swarm optimization, ant colony optimization, simulated annealing, Tabu search, genetic algorithm, neural network, immune algorithm are the typical examples. These intelligent optimization algorithms are relatively easy to implement and could conveniently be adapted to different kinds of scheduling problems, and especially, it has been proven by experiments that they can find high-quality solutions within reasonable computational time. These have made the research on intelligent hybrid algorithm for JSSP increasingly popular in the recent years.
Genetic algorithms were proposed by Holland [8] and have been successfully used in a variety of practical problems. Davis [9] first applied a genetic algorithm to the JSSP in 1985 successfully, and now genetic algorithms have been proved to be an effective approach for the JSSP. There are many such works, e.g., a genetic algorithm with search area adaptation was proposed by Someya and Yamamura [10]. Zhou and Feng [11] proposed a hybrid heuristics GA for JSSP, where the scheduling rules, such as shortest processing time (SPT) and most work remaining (MWKR), were integrated into the process of genetic evolution. Park et al. developed an efficient method based on genetic algorithm to address JSSP. The scheduling method based on parallel genetic algorithm was designed in [12]. Mattfeld and Bierwirth [13] considered a multi-objective job shop scheduling problem with release and due dates, as well as tardiness as objectives. Watanabe et al. [14] proposed a modified genetic algorithm with search area adaptation for solving the job shop scheduling problem. Li presented a two-row chromosome structured new genetic algorithm based on working procedure and machine distribution [15]. The relevant crossover and mutation operations were also designed in [15].
However, the existing genetic algorithms for the JSSP are usually with a slow convergence speed and easy to trap into local optimal solutions. In order to enhance the convergence speed, many researchers focused their attention on combining the genetic algorithm with local search schemes to develop some hybrid optimization strategies for JSSP. Wang and Zheng [16] developed a hybrid optimization strategy for JSSP by combining the simulated annealing algorithm and the genetic algorithm. Gonçalves et al. [17] presented a hybrid genetic algorithm for the JSSP, in which the schedules were constructed by using a priority rule and the priorities were defined by the genetic algorithm, and then a local search heuristic was applied to improve the solution. Zhang proposed a genetic simulated algorithm to solve the JSSP by combining the GA and SA [18]. Liu et al. [19] combined the traditional genetic algorithm with the Taguchi method and the proposed algorithm possessed the merits of global exploration and robustness. Xu proposed an immune genetic algorithm by combining the immune theory and the genetic algorithm [20]. Vilcot proposed a fast and elitist genetic algorithm based on NSGA-II for solving the multi-objective SSSP [21]. Zhang combined the TS algorithm and the genetic algorithm to develop a hybrid algorithm for JSSP [22]. Zhang proposed a hybrid simulated annealing algorithm based on a novel immune mechanism for the JSSP [23].
However, there are the following shortcomings for the aforementioned algorithms: (1) these algorithms have not taken into account the diversity of population. This can easily lead to the “premature” convergence because of the population being easily filled with more similar individuals. (2) Most of the crossover operators and mutation operators have not made use of the characteristics of the JSSP structure itself. Many researchers designed these operators according to the structure of code and only changed the form of code. Thus, these operators usually cannot guide the search to move to the better solutions, and are also difficult to integrate the merit of the parent individuals. (3) The local search ability is not satisfactory.
In order to overcome these drawbacks of the algorithms, in this paper, a new crossover operator and mutation operator were designed by sufficiently making use of the information and structure of the JSSP, and these operators are efficient for JSSP. Moreover, in order to enhance the diversity of the population and avoid trapping into the local optimal solutions, a specifically designed mixed selection scheme is presented. Furthermore, to improve the speed of the algorithm, an efficient local search approach is proposed and integrated into the proposed genetic algorithm. The detailed contributions are as follows:
- (1)
Clear definitions of the similarity and the concentration were given. And then a mixed selection operator based on the fitness value and the concentration value was proposed. This operator increased the diversity of the population and could prevent the “premature” convergence to some extent.
- (2)
New kinds of crossover operator based on the machine and mutation operator based on the critical path were specifically designed according to the graph model of JSSP. To calculate the critical path, we presented a new algorithm for finding it from schedule.
- (3)
A local search operator was designed so as to improve the local search ability of GA.
- (4)
Based on these genetic operators, we proposed a hybrid genetic algorithm (HGA) and proved its convergence. Finally, the efficiency of the proposed algorithm was verified by computer simulations on some typical scheduling problems.
The remainder of the paper is organized as follows. In Section 2, the models of the JSSP are set up. In Section 3, a hybrid genetic algorithm to the JSSP is presented and its convergence is proved. Section 4 presents the experimental results. The conclusions are made in Section 5.
Section snippets
Modeling the job shop scheduling problem
The JSSP with which we are concerned can be described as follows [24]: There are n different jobs to be processed on m different machines. Each job needs m operations and each operation needs to be processed without preemption for a fixed processing time on a given machine. There are several constraints on jobs and machines:
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A job can visit a machine once and only once.
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There are no precedence constraints among the operations of different jobs.
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Preemption of operations is not allowed.
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Each machine
Encoding and decoding
Defining an appropriate representation of individual is the most important and critical issue for constructing an efficient GA. As mentioned above, determining a schedule is equivalent to determine a graph G(N,A,E′), where , and is a directed connected path which traverses all vertices associated with Ej, that is, if we can make sure the sequence of the operations processed on each machine, we can determine a schedule. Thus, the following encoding scheme proposed by
Simulation results
In order to verify the good performance of the proposed hybrid genetic algorithm, we use 43 instances from two classes of standard JSSP test problems: Fischer and Thompson [2] instances FT06, FT10, FT20 and Lawrence [29] instances LA01–LA40. The HGA was compared with some algorithms reported in literature [16], [30], [31], [32], [33], [34], [35] in recent years.
The parameters used in simulations are as follows: the population size N=100, the crossover probability pc=0.7, the mutation
Conclusions
To solve the JSSP more effectively, a mixed selection operator based on the fitness value and the concentration was designed in order to increase the diversity of the population. Crossover operator based on the machine, and mutation operator based on the critical path were designed according to the graph model of JSSP. To calculate the critical path, a new algorithm was presented. A local search operator was designed in order to improve the quality of the solutions. Based on these, a hybrid
Acknowledgment
The work is supported by National Natural Science Foundation of China (60873099) and Ph.D. Programs Foundation of Ministry of Education of China (20090203110005).
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