Expert Systems with Applications 38 (2011) 7243–7250
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
An improved adaptive genetic algorithm based on hormone modulation mechanism for job-shop scheduling problem Lei Wang a,⇑, Dun-bing Tang b a b
College of Mechanical and Automotive Engineering, Anhui Polytechnic University, Wuhu 241000, China College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
a r t i c l e
i n f o
Keywords: Job-shop scheduling problem (JSP) Hormone modulation mechanism Improved adaptive genetic algorithm (IAGA) Partheno-genetic operation (PGO)
a b s t r a c t An improved adaptive genetic algorithm (IAGA) for solving the minimum makespan problem of job-shop scheduling problem (JSP) is presented. Though the traditional genetic algorithm (GA) exhibits implicit parallelism and can retain useful redundant information about what is learned from previous searches by its representation in individuals in the population, yet GA may lose solutions and substructures due to the disruptive effects of genetic operators and is not easy to regulate GA’s convergence. The proposed IAGA is inspired from hormone modulation mechanism, and then the adaptive crossover probability and adaptive mutation probability are designed. The proposed IAGA is characterized by simplifying operations, high search precision, overcoming premature phenomenon and slow evolution. The proposed method by employing operation-based encoding is effectively applied to solve a dynamic job-shop scheduling problem (DJSP) and a complicated contrastive experiment of JSP in manufacturing system. Meanwhile, in order to ensure to create a feasible solution, a new method for crossover operation is adopted, named, partheno-genetic operation (PGO). The computational results validate the effectiveness of the proposed IAGA, which can not only find optimal or close-to-optimal solutions but can also obtain both better and more robust results than the existing genetic algorithms reported recently in the literature. By employing IAGA, machines can be used more efficiently, which means that tasks can be allocated appropriately, production efficiency can be improved, and the production cycle can be shortened efficiently. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Job-shop scheduling problem (JSP) is a typical combinatorial optimizing problem; it is an NP problem (Carey, Johnson, & Sethi, 1976) which satisfies task collocation and sequence requirement for resource allocation. Genetic algorithm (GA), which is as a stochastic optimization method, has shown a good performance for solving combinatorial optimization problems (Tamaki & Sannomiya, 1998); therefore it was used to optimize JSPs (Beatrice & Mario, 2004); however, it is not easy to regulate GA’s convergence so that GA often suffers from premature convergence (Leung, Gao, & Xu, 1997; Rudolph, 1994; Schmitt, Nehaniv, & Fujii, 1998; Yang & Donglas, 1998); therefore some improved algorithms (Masato, Kenichi, & Mitsuo, 2005; Wang, Wu, & Song, 2004; Wang & Zheng, 2001, 2002) were proposed to solve it. However, studying new methods is still an important task to some extent. As we know, hormone modulation mechanism is one of the major physiological systems and has some special modulation ⇑ Corresponding author. Address: College of Mechanical and Automotive Engineering, Anhui Polytechnic University, Beijing Road, Wuhu City of Anhui Province 241000, China E-mail address:
[email protected] (L. Wang). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.12.027
mechanism; therefore it has better self-adaptability and stability (Farhy, 2004). However, there are no reports on JSP based on hormone modulation mechanism. Some researchers have presented several simulation models of hormone release (Keenan, Licinio, & Veldhuis, 2001; Liu, Ding, & Wang, 2008; Liu, Ren, & Ding, 2005), which are used only in the medical field. For this reason, an improved adaptive genetic algorithm (IAGA) based on hormone modulation mechanism is applied to JSP, and it can overcome premature phenomenon which exists in the traditional GA and has a high search precision. The remainder of this paper is organized as follows: Section 2 describes JSP; Section 3 introduces the traditional GA; An IAGA based on hormone modulation mechanism is proposed to apply to JSP in Section 4. A DJSP and two complicated contrastive experiments of JSPs are given by using the proposed IAGA and other algorithms in Section 5. Concluding remarks and future research directions are presented in Section 6.
2. JSP model In general, JSP is described as follows (Gen & Cheng, 1997). There are n different jobs and m different machines to be
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scheduled. Each job is composed of a set of operations and the operation order on machines is prespecified. The required machine and the fixed processing time characterize each operation. There are several constraints on jobs and machines. A job does not visit the same machine twice. There are no precedence constraints among the operations of different jobs. Operations cannot be interrupted. Each machine can process only one job at a time. Neither release times nor due dates are specified. The problem is to determine the operation sequences on the machines in order to minimize the makespan—that is, the time required to complete all jobs. The objective scheduling problem based on minimizing the maximal makespan can be described as follows.
min f ðxÞ ¼ maxðcik Þ;
ð1Þ
s:t: cik t ik þ Mð1 aihk Þ cih ; cjk cik þ Mð1 xihk Þ tjk ;
ð2Þ ð3Þ
cik 0;
ð4Þ
aihk ¼ 0; 1;
ð5Þ
xijk ¼ 0; 1;
ð6Þ
where M is a very big positive number; cik denotes the finishing time of job i (i = 1, 2, . . . , n) on machine k (k = 1, 2, . . . , m); tik denotes the processing time of job i on machine k; aihk = 1, if job i is processed on machine h (h = 1, 2, . . . , m) before on machine k, or else aihk = 0; xijk = 1, if job i is processed on machine k before job j (j = 1, 2, . . . , n), or else xijk = 0. Eq. (1) is object function for scheduling; Eq. (2) presents job sequence restriction; and Eq. (3) presents machine restriction, namely, each machine can process only one job at a time. 3. Traditional GA GA is stochastic search techniques based on analogy to Darwinian natural selection. Individuals who fit the environment best should have a better chance to propagate their offspring. By the same reason, solutions that have the best ‘‘fitness’’ should receive
higher probability to search their ‘‘neighbors’’. The main advantage of GA lies in its powerful implicit parallelism. In Holland’s theory, a GA implicitly evaluates a number of patterns larger than population size without additional computational time and memory. The overall structure of GA can be described as follows: (1) Encoding: For any GA, a chromosome encoding is needed to describe each chromosome in the population. The encoding method determines how the problem is structured in the algorithm and the genetic operators that are used. Each chromosome is made up of a sequence of genes from certain alphabet which can consist of binary digits (0 and 1), floating-point numbers, integers, symbols (i.e., A, B, C, D), etc. Each chromosome represents a solution for the problem. (2) Initial population: An initial population of size P can be randomly generated. The length of each chromosome in a population should be the same. (3) Fitness evaluation: The fitness is computed for each chromosome in the current generation. (4) Selection: At each iteration, the best chromosomes are chosen for reproduction by one among three different methods, i.e., binary tournament, n-size tournament and linear ranking. (5) Offspring generation: The offspring generation is obtained by the processes of crossover and mutation. New individuals are generated until a fixed maximum number of individuals is reached. (6) Termination criterion: Fixed number of generations is reached. If the stop criterion is satisfied, the algorithm ends and the best chromosome, together with the corresponding result, is given as output. Otherwise, the algorithm iterates again steps 3–5. The basic flow chart of traditional GA is presented in Fig. 1. However, the traditional GA is not easy to regulate GA’s convergence so that GA often suffers from premature convergence. What’s more, its crossover probability and mutation probability are fixed, the self-adaptation is not enough good. Therefore, in order to improve the disadvantages of the traditional GAs, an IAGA based on hormone modulation mechanism is proposed to apply to JSP. To describe our IAGA, we describe the mechanism of hormone modulation in Section 4.1 firstly, then we detail in the following
Fig. 1. Basic flow chart of GA.
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