Advanced Materials Research Vols. 97-101 (2010) pp 2449-2454 © (2010) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.97-101.2449
An Improved Genetic Algorithm for Multi-objective Flexible Job-shop Scheduling Problem Chaoyong Zhanga, Xiaojuan Wangb and Liang Gaoc State Key Laboratory of Digital Manufacturing Equipment and Technology Department of Industrial & Manufacturing Systems Engineering, Huazhong Univ.of Sci. & Tech., Wuhan, 430074, China a
[email protected],
[email protected],
[email protected]
Keywords: Flexible job shop scheduling; multi-objective genetic algorithm; Pareto-optimality; immune and entropy principle
Abstract. Flexible job shop scheduling problem (FJSP) is an extended traditional job shop scheduling problem, which more approximates to real scheduling problems. This paper presents a multi-objective genetic algorithm (GA) based on immune and entropy principle to solve the multi-objective FJSP. In this improved multi-objective GA, the immune and entropy principle is used to keep the diversity of individuals and overcome the problem of premature convergence. Advanced crossover and mutation operators are proposed to adapt to this special chromosome structure. The proposed algorithm is evaluated on three representative instances and the computational results and comparison with some other approaches show that the proposed multi-objective algorithm is effective and potential. Introduction Flexible job-shop scheduling problem (FJSP) is an extended traditional job shop scheduling problem. It breakthroughs the restriction of unique resources and allows each operation to be processed by several different machines, thus making the job-shop scheduling problem accord with actual production situation more accurately. FJSP is more complicated than the job-shop scheduling problem (JSP), since that it need to assign each operation to a machine from a set of capable machines and then sequence the assigned operations on each machine[1]. Bucker and Schlie[2] intimately proposed the problem that one operation could be processed on several machines and have studied this problem deeply, it marks the beginning of study on FJSP. The methods of solving this kind of problem can be concluded into hierarchical approaches and integrated approaches. Hierarchical approach which firstly was proposed by Brandimarte[3] considered the assigning sub-problem and the sequencing sub-problem separately, and its basic idea is decomposing the complex problem into some sub-problems as to decrease the complexity. However the integrated approaches solve the assigning sub-problem and the sequencing sub-problem simultaneously. Recently, multi-objective FJSP has gained wide attention of many researches. The complexity of this problem lead to the appearance of much heuristic approaches, and the research is mainly concentrated on the hybrid and evolutionary algorithms. For example, Kacem et al.[4]proposed a Pareto approach based on the hybridization of fuzzy logic (FL) and evolutionary algorithms (EAs) to solve the flexible job-shop scheduling problem (FJSP). This hybrid approach exploits the knowledge representation capabilities of FL and the adaptive capabilities of EAs. Tanev et al.[5]presented an approach for scheduling of customers’ orders in factories of plastic injection machines (FPIM) as a case of real-world flexible job shop scheduling problem. Low et al.[6] made use of one of the multiple objective decision-making methods, a global criterion approach, to develop a multi-objective model for solving FMS scheduling problems with consideration of three performance measures, namely minimum mean job flow time, mean job tardiness, and minimum mean machine idle time, simultaneously. In addition, hybrid heuristics, which are a combination of two common local search methods, simulated annealing and tabu search, are also proposed for solving the addressed FMS
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scheduling problems. Xia and Wu[1] proposed a practical hierarchical solution approach for solving multi-objective FJSP. In this paper, an improved genetic algorithm (GA) based on immune and entropy principle are used to solve the multi-objective flexible job shop scheduling problem. In this improved multi-objective GA, the fitness scheme based on Pareto-optimality is applied. The improved GA utilizes the double-layer child forming chromosome structure in order to prevent the loss or destruction of elite solutions. Advanced crossover and mutation operators are proposed to adapt to this special chromosome structure. The immune and entropy principle are used to keep the diversity of individuals and overcome premature convergence. In Section 2, our model of multi-objective flexible job shop scheduling problem are given. Some concepts about muiti-objective optimization are given in Section 3.The immune and entropy principle are presented in Section 4. In Section 5, the procedures of GA for FJSP are elaborated. In Section 6 the computational results and the comparison with previous approaches are presented. The final conclusions are given in Section 7. Multi-objective FJSP Model The multi-objective flexible job shop scheduling problem is described as follows: there are n jobs which need to be processed on m machines. The Set of all the machines is denoted as M, M = {M 1 , M 2 ,..., M m } . Each job j consists of a sequence of n j operations O j ,1 , O j ,2 ,..., O j ,n j , any operation i ( O j ,i ) of job j is allowed to be executed by one machine out of a given set M j ,i . The task is assigning each operation O j ,i on the given machines and sequencing the operations on all the machines, with the aim of minimizing the two objectives as below: F1 : makespan (maximal completion time); F2 : the total workload of machines (the total working time of all machines); Multi-objective Optimization Basic Concepts. The general multi-objective optimization problem is described in this form: Minimize y = f ( x) = ( f1 ( x), f 2 ( x),..., f q ( x)) Where x = R p , and y ∈ R q The following two basic concepts are often used in multi-objective optimization case and are used in this paper. on-dominated Solutions: A solution a is said to dominate solution b if and only if: f i (a ) ≤ f i (b) ， ∀i ∈ {1, 2,..., q} f i (a ) < f i (b) ， ∃i ∈ {1, 2,..., q} Pareto-optimality: A feasible solution is called Pareto-Optimal when it is not dominated by any other solution in the feasible space. Pareto-Optimal Set which is also called the Efficient Set is the collection of all Pareto-optimal solutions and their corresponding images in the objective space are called the Pareto-optimal frontier. Fitness Assignment Scheme. In this paper, the fitness of each particle is determined by the factors of dominance and being dominated. It resembles the fitness scheme in SPEA2 which was firstly proposed by Zitzler [7].However, there are still some differences that it doesn’t consider the strength niche in our scheme, because the diversity strategy based on immunity has been applied already. For the non-dominated individuals, the fitness of individual i is defined as follows: fitness (i ) = ni /( + 1) (1) is the size of population, ni is the number of individuals that be dominated by individual i. For the dominated individuals, the fitness of individual j is defined as follows:
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fitness ( j ) = 1 +
∑
fitness (i )
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(2)
i∈DSet ,i f j
In this equation, DSet is the set of all the non-dominated individuals, i f j represents that the individual i dominates the individual j.
Immune and Entropy Principle Because of three effects: selection pressure, selection noise and operation disruption, EAs based on a finite population tend to converge a single solution[8].However, the goal of multi-objective optimization is finding a set of non-dominated solutions on or approximating the Pareto front of the problem, and not only a single solution. A non-niching strategy based on immune and entropy principle using special fitness calculation is proposed by Cui et al.[9]. The more similar individuals are located in the current population, the more reproduction probability of an individual is degraded. This strategy does not require any distance parameter and use exponential fitness rescaling mechanism based on genetic similarity between an individual and the rest of the population. Here antibody of immune system is taken as individual in MOEAs. Supposing that denotes population size, M is the length of antibody (fixed length) and S denotes the size of symbolic set. The strategy is described as follows: Information-theoretic Entropy of Antibody. If a random vector X denotes the status feature of an uncertain system (where X = {x1 , x2 ,..., xn } ) and the probability value of X is denoted by P (where P = { p1 , p2 ,..., pn } , 0 ≤ pi ≤ 1 ,i=1,2,…,n, and
∑
n i =1
pi = 1 ), the information-theoretic entropy of the
system is mathematically defined as: n H = −∑ k =1 pk ln( pk )
(3)
and the entropy of the mth locus of the individual is defined: S H m ( ) = −∑ k =1 pkm ln( pkm )
(4)
Where pkm denotes the probability that the kth symbol appears the mth locus. Similarity of Antibody. The similarity of antibody indicates similar extent between individual i and individual j: 1 Ai , j = (5) 1 + H i , j (2) Where H i , j (2) is the average entropy of individual i and individual j, and it can be calculated according to H m ( ) when the value of is 2: 1 M H i , j (2) = H K (2) ∑ k =1 M Density of Antibody. The density of antibody is denoted by Ci :
(6)
Ci = (number of antibodies in population whose antibody similarity to the individual i exceedsλ) / , where λ is similarity constant, and generally its range is 0.9 ≤ λ ≤ 1 . Aggregation Fitness. We define aggregation fitness of an individual as a tradeoff result of two evaluations: f i ' = f i × exp( K × Ci ) (7) Where f i is the initial (usual) fitness function of antibody i, and K is a plus regulative coefficient which is determined by the size of population and experience.
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Handing MFJSP with GA based on Immune Principle For the flexible job-shop problem, the representation of each solution consists of two parts in GA. The first part is used to determine the processing sequence for all the operations, and the other part is used to assign a suitable machine for each operation. By integrating the two parts of representation, a feasible solution can be generated. Crossover Operation. In our algorithm, crossover operation of the two parts of chromosomes is implemented separately, in which the crossover of the first part of chromosome that basing on operation utilizes POX( Precedence Operation Crossover) crossover operator[10], while the crossover of the second part that basing on the machine assigned utilizes MPX(Multi-point Preservative Crossover ) crossover operator[11]. IPOX is a modification basing on the POX, which only crosses the sequence of the operations and copies the assigned machines to the children chromosomes. For example, P1 and P2 are the parents chromosomes, C1 and C2 are their children chromosomes by crossing. The crossover operator of IPOX works as in Fig.1. C1 P1 P2 C2
1 ↑ 1 2 ↓ 2
2
3 ↑ 3 3
1 ↑ 1 1
1 2 ↑ 2 1 2 1 2 3 ↓ 1 3 1 2 1 J1 = { 1, 3 }, J2 = { 2 }
2 2 2 ↓ 2
3 ↑ 3 1
3 ↑ 3 3
3
3
Fig.1 IPOX crossover operation MPX(Multi-point Preservative Crossover ) only crosses the assigned machines and copies the sequence of the operations to the children chromosomes. For example, C1 and C2 are the children chromosomes of the parents chromosomes P1 and P2 by crossover. The crossover operator of MPX works as in Fig.2. C1
1
P1 Rand0_1 P2
1 0 4
C2
4
Job1 2 ↑ 4 1 2 ↓ 4
1
2
1 0 3
2 0 3
3
3
Job2 4 ↑ 3 1 4 ↓ 3
2 ↑ 5 1 2 ↓ 5
3 3 0 4 4
Job3 3 ↑ 2 1 3 ↓ 2
4 4 0 3 3
Fig.2 MPX crossover operation Mutation Operation. In order to improve the ability of local search and keep the diversity of the population, we implied mutation operation in our algorithm. For the first part of chromosome, the mutation operation is implemented as shown in Fig.3. P1 C1
2 2
1 1
3 2
1 3
2 1
3 3
2 2
1 1
3 3
Fig.3 Mutation of the operation sequence representation For the second part of chromosome, the mutation operation is implemented as shown in Fig.4. P1 C1
Job1 1 4 1 1 4 4
Job2 2 3 5 2 3 2
Job3 3 2 4 3 2 4
Fig.4 Mutation of the machine assignment representation Main Algorithm. The algorithm keeps a fixed size of population. The main algorithm is executed in the follow steps. Step 1: Initialize parameters and initial antibodies. Step 2: Calculated the density of antibody and aggregation fitness for each antibody. Step 3: Find all the non-dominated solutions from the current population.
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Step 4: Evaluate each individual by the aggregation fitness. Step 5: If the termination condition is met, then terminate search; else go to the next step. Step 6: Copying 1% individuals to the child individuals and select the child solutions according to the selecting strategy. Step 7: If the fitness of two parent solutions is unequal, perform crossover with probability pc and the two best individuals are selected as the child solutions. Then, perform mutation operations with probability pm . The children solutions are generated through this step. Step 8: If the terminating condition is satisfied, the algorithm ends, otherwise, go to step 2.
Simulation and Results To test the performance of algorithm, three representative instances (problem 8×8, problem 10×10 and problem 15×10) which are taken from Kacem et al. [12] have been taken into this experiment. In our algorithm, the population size is set 150, the maximal generation is set 300, the crossover probability is 0.8 and the mutation probability is 0.1. To illustrate the efficiency of our algorithm, we compare the results with Classic GA, Approach by localization, AL+CGA and PSO+SA which are taken from Xia and Wu[1]. The computational results and comparisons are given in Table 1, Table 2 and Table 3. F1 is makespan and F2 is the total workload of machines. Table 1 Comparison of results on problem 8×8
＋
Classic GA
AL
AL+CGA
PSO SA
GA
F1
16
16
15
16
15
16
16
15
F2
77
75
79
75
75
73
73
75
Table 2 Comparison of results on problem 10×10 Classic GA
AL
AL+CGA
PSO SA
＋
GA
F1
7
8
7
7
7
F2
53
46
45
44
43
Table 3 Comparison of results on problem 15×10 AL+CGA
F1
23
24
＋
PSO SA 12
GA 12
F2 95 91 91 91 For the problem 8×8 and 15×10, we can see that the solutions obtained from our algorithm are better than those from the other methods except PSO+SA and are the same with those from PSO+SA. For the problem 10×10, our algorithm performs better than all the other methods. In summary, these values show the efficiency of our algorithm as compared to those of the other algorithms.
Conclusions We have put forward an efficient modified multi-objective genetic algorithm, which is based on immune and entropy principle, for solving multi-objective flexible job-shop scheduling problems. Apart from the general GA, our algorithm applies a novel crossover operator for FJSP. Meanwhile, selection pressure of similar individuals can be decreased by combining the immune and entropy principle. The numerical experiments indicate the effectiveness of the proposed approach. However, there are still a number of extended research directions need to be considered in the future. For example, some other objectives can be considered for FJSP but not only two objectives.
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Furthermore, some other heuristic algorithms such as particle swarm optimization (PSO) and ant colony optimization (ACO) can substitute GA to generate more efficient multi-objective optimization algorithms based on immune and entropy principle.
Acknowledgment This project is supported by 863 High Technology Plan Foundation of China under Grant No. 2007AA04Z107, the National Natural Science Foundation of China No. 60973086, and the program for New Century Excellent Talents in University under Grant No. NCET-08-0232 and the National Natural Science Foundation of China under grant No. 50825503.
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