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Abstract—Flexible job-shop scheduling problem (FJSP) is a well-known difficult combinatorial optimization problem. Many algorithms have been proposed for solving FJSP in the last few decades. In this paper, we present a genetic algorithm for FJSP. The algorithm encodes the individual with parallel machine process sequence based code, integrates the Most Work Remaining, the Most Operation Remaining and random selection strategies for generating the initial population, and integrates the binary tournament selection and the linear ranking selection strategies to reproduce new individuals. Computational result shows that the integration of more strategies in a genetic framework leads to better results than the traditional genetic algorithms.

I. INTRODUCTION olution to a flexible job-shop scheduling problem (FJSP) involves an assignment of a set of jobs to be processed on machines in alternative routes. Its complexity of the large

solution space and its multi-objective function make the problem difficult. This problem is NP-Hard that cannot be optimally solved in a reasonable amount of computational time. For this reason, there are two research areas for FJSP. One is focus on heuristic methods to find a near-optimal solution, such as local searching algorithm [1], tabu search [2], priority dispatching rules [3], expert systems [4], branch-and-bound algorithm [5], shifting bottleneck algorithm [6], simulated annealing [7] and neural network [8]. The other is focus on methods to find optimal solutions, such as genetic algorithms [9], ant colony optimization [10] and artificial immune genetic algorithms [11]. In addition to the above methods, a hybrid method can be also proposed such as local search and simulated annealing algorithms [12], hybrid genetic algorithms [13].

In this paper, an innovative integrated GA for FJSP is presented. The rest of the paper is organized as follows. After introduction, section II depicts FJSP and presents the mathematical model of it, section III presents an integrated genetic algorithm for FJSP and section IV provides experimental results and analysis. Finally, conclusions are drawn in section V.

II. FLEXIBLE JOB-SHOP SCHEDULING PROBLEM There are m jobs Ji (1�i�m). Each job Ji consists of a chain

of li operations Oij (1�j�li), which have to be scheduled in a predetermined order. Oij is the jth operation of job Ji which

Ming Wan and Jianguo Nan are with the Air force Engineering Institute,

the Chinese Air Force Engineering University of Xi’an, Shannxi (email:afwanming@163.com and njg16@163.com).

Xiaohui Xu is with the Chinese AVIC Qingan Group CO. LTD. of Xi’an, Shannxi(email:xiaoxiaohui@sina.com.cn).

has to be processed on a machine. These operations must to be processed on n machines Mk (1�k�n). The classical static FJSP considers that:

� Each machine can perform only one operation at a time;

� Each operation can be processed by only one machine without interruption and preemption;

� For the operations of a job, the jth operation must be processed before the j’th operation if 1�j<j’�li;

� Each operation has determinate and constant processing time �ijk for machine Mk;

� All jobs and machines are available for processing at time zero;

� There are no limiting resources other than machines.

FJSP is concerned with finding a sequential allocation of competing resources that optimizes a particular objective function such as makespan, just-in time, cost, machine load or computational complexity. Although many objective functions can be considered in FJSP, makespan is the principal criterion for research and is able to capture the fundamental computational difficulty which exists implicitly in determining an optimal schedule.

The term makespan refers to the cumulative time to complete all operations on all machines. It is a measure of the time period from the starting time of the first operation to the ending time of the last operation. makespan minimization problem is well defined and is simple to understand. It is used in our work as a proving ground for the proposed algorithm for job shop scheduling problems.

Let xijk=1 if Oij is processed on machine Mk, 0 otherwise. �ij is the specific machine that Oij requires according to its process route. Sij and Cij are the start time and the completion time of Oij respectively. Ak is the earliest available time of machine Mk. The mathematical model is described as follows:

min Cmax = max Cij (1�i�m, 1�j�li) (1)

Subject to:

jixn

k ijk ,11

��� � (1.1)

)(,, kjiCS ijijijkij ���� � (1.2)

)'1(',,' iijij ljjjjiCS ��� (1.3)

),'(',',, '''' ijjiijjiij iijijiCSS �� � � (1.4)

Flexible Job-shop Scheduling with Integrated Genetic Algorithm

Ming Wan, Xiaohui Xu, and Jianguo Nan

S

13

Fourth International Workshop on Advanced Computational Intelligence Wuhan, Hubei, China; October 19-21, 2011

978-1-61284-375-9/11/$26.00 @2011 IEEE

jiSij ,0 �� (1.5)

kAk �� 0 (1.6)

Equation (1) presents the objective function, which aims to reduce the makespan of all jobs. Constraint (1.1) forces one operation being scheduled on a unique machine. Constraint (1.2) means that all operations can not be interrupt or preempt. Constraint (1.3) forces the operation scheduling sequence of a job meeting the requirement of the job. Constraint (1.4) means that a machine can and only can process one operation at a time. Constraint (1.5) ensures all operations start after time zero. Constraint (1.6) means all machines are available at time zero.

III. INTEGRATED GA FOR FJSP GA is a stochastic heuristics, which encompass

semi-random search method whose mechanism is based on the simplifications of evolutionary process observed in nature [14]. GA is an optimization method of searching based on evolutionary process. GA assigns a value to each individual in the population according to a problem-specific objective function. A survival-of-the-fittest step selects individuals from the old population. A reproduction step applies operators include selection, crossover and mutation to those individuals to produce a new population that is fitter than the previous one.

A. Chromosome Representation A chromosome of individual describes the assignment of

operations to the machines, and the order in which they appear in the chromosome describes the sequence of operations. Each chromosome represents a solution for the problem. A chromosome representation method should considers the complexity, the validity, the feasibility, the integrality and the Lamarkian characteristic of chromosomes, also includes space characteristic of results and memory requirement.

In the case of our problem, the chromosome is represented by a set of machines put in parallel and each machine is a vector which contains its assignment operations. These operations are represented by a job-operation pair with two terms. The first is the number of the job; the second is the order number of the operation in its operating sequence.

Three jobs with eight operations and three machines are shown in Table I. For this instance, a chromosome and the corresponding scheduling Gantt chart are shown in Figure 1.

The chromosome representation method possesses some advantage, as it gives directly a feasible schedule. The obtained solution contains operations assignment and sequence information. This allows the user to best manage park machines, and thereby the production.

TABLE I. A FLEXIBLE JOB-SHOP SCHEDULING INSTANCE

Processing time (�ijk) for machine (Mk) Job (Ji)

Operation(Oij) M1 M2 M3 O11 7 6 4 O12 4 8 5 J1 O13 9 5 4 O21 2 5 1 O22 4 6 8 J2 O23 9 7 2 O31 8 6 3 J3 O32 3 5 8

Fig. 1. Chromosome representation sample

B. Population Initialization In our GA, the initial population is obtained by a mix of

three known dispatching rules [13]:

(a) Randomly select a job and machine (Random);

(b) Most Work Remaining (MWR);

(c) Most Operations Remaining (MOR).

For example, 60% of initial chromosomes can be generated by rule (a), 20% by rule (b) and 20% by rule (c). Again, finding a robust tuning for the mix of these three rules for different classes of problem instances is one of the goals of our research. Again, finding a robust tuning for the mix of these three rules for different classes of problem instances is one of the goals of our research.

C. Selection Operator The selection operator is to choose the chromosomes for

reproduction. Individuals are chosen from the population and the best of them is selected for survival. In our approach, the criterion used to select the chromosomes to be included in the mating pool can be chosen among two selection methods well known in the GA literature: binary tournament selection and linear ranking selection.

(a) Binary tournament selection: Two individuals are randomly chosen from the population and the best of them is selected for reproduction;

(b) Linear ranking selection: Individuals are sorted according to their fitness and a rank },...,2,1{ Nri � is assigned to each, where N is the population size. The best individual gets rank N while the worst gets rank 1. Then,

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),...,2,1(,)1(

2)( NiNN

rip i ��

� (2)

is the probability of choosing the ith individual in the rank ordering.

Once the chromosomes for reproduction have been selected, the crossover and mutation genetic operators are applied to produce the offspring. Crossover operator applies to pairs of chromosomes, while mutation operator applies to single individuals. In both cases the operations to be exchanged are chosen arbitrarily.

D. Crossover Operator The crossover operator generates the offspring by

exchanging the assignment of a subset of operations between the two parents. For arbitrary Parent1 and Parent2, the crossover algorithm is:

Step 1. Randomly get a integer r (1�r�k);

Step 2. Exchange the machine vectors of Mr of two parents;

Step 3. Make chromosomes feasible;

Step 3.1. Delete all redundant operations in other machine vectors;

Step 3.2. For all operations not included in the vectors of the offspring, randomly select a machine vector except the machine vector of Mr and randomly insert it into a feasible position.

Fig. 2. Crossover process sample

An example is shown in Fig. 2. For Parent1 and Parent2, the machine vector of M2 is selected by crossover operator. For Offspring1, job-operation pair (1, 2), (2, 3) and (3, 1) are redundant obviously and will be deleted from the corresponding vectors. Then job-operation pair (2, 1) and (2, 2) are randomly inserted into the feasible position of the machine vector of M1 or M3. For Offspring2, job-operation pair (2, 1) and (2, 2) are redundant obviously and will be

deleted from the corresponding vectors. Then job-operation pair (1, 2), (2, 3) and (3, 1) are randomly inserted into the feasible position of the machine vector of M1 or M3.

E. Mutation Operator The mutation operator is critical to the success of the GA

since it diversifies the search directions and avoids convergence to local optimization. Not like the crossover operator, the mutation operator only exchanges the assignment of a single operation in an individual. For an arbitrary individual, the mutation algorithm is:

Step 1. Select a machine vector of Mk and a job-operation pair (Ji,Oij) randomly;

Step 2. Delete (Ji,Oij) from the machine vector of Mk;

Step 3. Randomly select a machine vector of Mk’ (k’� k) and insert (Ji,Oij) into the machine vector of it.

F. Integrated Genetic Algorithm Framework Step 1. Initialization population is generated randomly

with size Np, and it is feasible schedule;

Step 2. The fitness is defined by objective of FJSP model, and individual adaptive value is evaluated. Let iteration i=1;

Step 3. The crossover is operated in the population according to probability of crossover Pc;

Step 4. The individual is selected randomly according to probability of mutation Pm;

Step 5. The new individual adaptive value is calculated, parent and offspring are taken part in survival competition together;

Step 6. Modify i by i=i+1. Stop reproduction if i�Nr, going back to Step 3 otherwise;

Step 7. Evaluate the Cij and Cmax of all individuals and get all best solutions.

IV. EXPERIMENTAL EVALUATION First, we implement the traditional GA, proposed by

Yamada [15], for solving 14 small to medium sized benchmark problems proposed by Lawrence [1]. These problems are listed in Table II with their sizes, in terms of number of jobs and number of machines, where the first number represents the number of jobs and the second number represents the number of machines. As for example, 10×5 means 10 jobs and 5 machines.

In our experiment, we tested different values for a list of algorithm parameters, and computational experience proves that the random selection and the following values are more effective:

� Population size Np: 5000;

� Number of generations Nr: 1000;

� Rate of initial sequences with Random rule: 60%;

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� Rate of initial sequences with MWR rule: 20%;

� Rate of initial sequences with MOR rule: 20%;

� Crossover probability Pc: 90%;

� Mutation probability Pm: 1%.

In Table II, the result found using this approach is reported in the column 3 along with the column 2 that is the best found solution in the literature.

TABLE II. RESULT COMPARISON OF THE TRADITIONAL GA AND THE INTEGRATED GA

Problem Known

Best Solutions

Traditional GA

Solutions

Integrated GA Solutions

% Deviation

La01(10×5) La02(10×5) La03(10×5) La04(10×5) La05(10×5) La06(15×5) La07(15×5) La08(15×5) La09(15×5) La10(15×5) La11(20×5) La12(20×5) La13(20×5) La14(20×5)

666 655 597 590 593 926 890 863 951 958

1222 1039 1150 1292

794 686 699 662 593987 962 963 1071 1011 1340 1210 1362 1351

666655622597 5939268908639519581222103911501316

0.00% 0.00% 0.00% 1.19% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 1.86%

Average Percentage Deviation 0.218%

We solve the same set of benchmark problems with similar parameters, using the proposed integrated GA. The result obtained from this approach is presented in column 4 of Table II. Note that all the results reported in Table II are the best of all best fitness values in 10 independent runs.

Finally, the percentage deviations are presented in the last column which is the % gap between the results obtained from our integrated GA and known best in column 2. Our integrated approach obtained optimal (/known best) solution for 12 problems out of 14 benchmark problems solved by us. On the other hand, the traditional GA (column 3) failed to find equal or better solution other than one problem (La05).

V. CONCLUSION AND FUTURE RESEARCH In this paper we present an integrated genetic algorithm for

the FJSP. An extensive computational study shows that our algorithm outperforms the traditional GA for the same problem, and gives results comparable with the best algorithm known so far. In the future, it will be interesting to investigate on the following issues:

� Use of a mix of different selection criteria for choosing the best individual of a given generation of chromosomes;

� Improve the intelligent mutation operator by selecting a critical operation, instead of any operation, on the most critical machine.

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Scheduling. theory, algorithms and systems

Discrete Applied Mathematics,“”

“ ”International Journal of Production Research,

Applied mathematics and physics Department of Kyoto University,