Numerical Research on Distributed Genetic Algorithm with Redundant

Numerical Research on Distributed Genetic Algorithm with Redundant Binary Number

1

Sayori Seto,

Akinori Kanasugi

1,2

Graduate School of Engineering, Tokyo Denki University, Japan [email protected], [email protected]

Abstract

Genetic algorithm (GA) is one of optimization algorithm based on an idea for evolution of life. GA can be applied various combination optimization problem. This paper proposes a novel distributed genetic algorithm (DGA) with redundant binary number. Since a redundant binary number has redundancy, solution expression becomes variegated. For this reason, it is expected the algorithm easily find the optimized solution, and the error rates decrease. The effectiveness of the proposed algorithm was confirmed by computer simulations.

Keywords : Genetic Algorithm, Distributed Ga, Redundant Binary Number

1. Introduction

Genetic algorithm (GA) is one of optimization algorithm based on an idea for evolution of life [1,2].

GA can be applied various problems such as combination optimization problem, machine learning and so on. A distributed genetic algorithm (DGA) divides a solution group into some solution groups

“island”, and performs genetic operation in each island [3]. In DGA, in order to exchange the solution among each island, “migration” operation is performed. Since DGA has few numbers of individuals per island, premature convergence takes place easily. However, since diversity is also maintainable by migration, compared with conventional GA, effective solution search is expectable.

This paper proposes a novel DGA with redundant binary number, while conventional DGA expresses chromosomes in binary number [4,5]. Since a redundant binary number has redundancy, solution expression becomes variegated. For this reason, it is expected the algorithm easily find the optimized solution, and the error rates decrease [6]. In the proposed algorithm, different numerical systems are used on each island. Therefore, since diversity is further maintainable, the further improvement in performance is expectable. The effectiveness of the proposed algorithm was confirmed by computer simulations.

2. Genetic Algorithm

Genetic algorithm (GA) is proposed in 1975 by Prof. John Holland. The algorithm is based on Darwin's evolutionary theory and likens solution to gene. The flow chart of GA is shown in figure 1.

Figure 1. Flow Chart of GA

The procedure of GA is as follows.

(1) Initialization: The first process decides initial genotype, namely value and genetic length. For example, if we assume values are ‘0’ or ‘1’, and length is eight, an example of chromosome is shown in figure 2.

(2) Evaluation: The second process calculates the fitness for each individual with the target function.

The evaluation depends on each problem.

(3) Termination Judgment: If the process satisfies the termination condition, the operation finishes and outputs the individual with the best fitness as the optimized solution.

(4) Selection: To generate the children, this process chooses parents from individuals. For example, if we assume parents the first generation, children become the second generation. The children generate the next children again. The children inherited the characteristic of parents are generated in this way.

(5) Crossover: This process crosses individuals chosen by selection operation and generates the individuals of the next generation. An example of crossover operation is shown in figure 3.

(6) Mutation: This process mutates the chromosome of new generation (figure 4). The mutation operation is effective to escape from a local optimum solution.

Figure 2. Chromosome

Figure 3. Crossover Operation

Figure 4. Mutation Operation A. Distributed Genetic Algorithm

A distributed genetic algorithm (DGA) divides a solution group into some solution groups “island”, and performs genetic operation in each island. In DGA, in order to exchange the solution among each island, “migration” operation is performed (figure 5). Since DGA has few numbers of individuals per island, premature convergence takes place easily. However, since diversity is also maintainable by migration, compared with conventional GA, effective solution search is expectable. Figure 6 shows a simple flow chart of DGA.

Figure 5. Migration Operation

Figure 6. Flow Chart of DGA

B. Redundant Binary Number

In the proposed algorithm, redundant binary number system is utilized. The advantage of GA with redundant binary number is increase of total expression number of optimized solution. From this advantage, improvement in searching speed and decrease of error rate are expected.

The redundant binary number uses values ‘0’, ‘1’ and ‘-1’. However, because conventional digital circuits cannot deal the value ‘-1’, we express each genetic information in two bits. We assume that ‘0’

is assigned to “00” or “11”, ‘1’ is assigned to “01”, ‘-1’ is assigned to “10”. These correspondences are summarized in table 1.

Table 1. Expression of Redundant Binary Number

Bit

String

For example, we express decimal number “seven” in binary number of the four bits precision and the corresponding redundant binary numbers, as shown in figure 7. The chromosomes of the redundant binary number become longer in comparison with the normal binary number. However, there are many expression way. For example, there are nine expression ways in the case of figure 7. Since a redundant binary number has redundancy, solution expression becomes variegated. For this reason, it is expected the algorithm easily find the optimized solution, and the error rates decrease.

Figure 7. Comparison between Binary Number and Redundant Binary Number

The GA based on redundant binary number is almost the same as conventional GA. However, decoding from redundant binary number to binary number is required. In the proposed algorithm, we separate chromosomes into odd number bit and even number bit. Then we subtract even number bit from odd number bit. An example is shown in figure 8.

Figure 8. Decoding from Redundant Binary Number to Binary Number

3. Proposed DGA

The concept of DGA proposed in this paper is shown in figure 9. In this figure, GA_B, GA_G, and GA_RB express GA using binary number, GA using Gray code, and GA using redundant binary number, respectively. A part of solutions are exchanged through the migration unit. In the migration, code conversions are performed.

Figure 10 shows the migration pattern. The random number ‘r’ chooses the migration pattern. The number of the chromosomes which emigrate is one.

Figure 9. Concept of proposed DGA

Figure 10. Migration Pattern

Although search results depend on type of solution code, stable good results are expected to many problems by the proposed DGA. In figure 9, although there are two sets of GA with binary number, one set of GA with Gray code, and one set of GA with redundant binary number, this is just an example. Of course, various combinations are possible.

The authors are planning to develop the processor for this algorithm. Then, in order to suppress the circuit scale, the composition of figure 9 was illustrated. Namely, since the scale is small, two sets of circuits treating a binary number are used, and since the scale is large, as for the circuit treating a gray code or a redundant binary number, only one set is used.

4. Simulation

The comparison of simulation results were performed in four types of GA (binary number, Gray code, redundant binary and proposed DGA). The performance of each GA was evaluated by solving following three functions. The corresponding graphs are illustrated in figure 11, 12 and 13, respectively.

(Eq. 1) f ( x )  x

 20000 (Solution: x  141 )

Figure 11. Graph of Equation 1

(Eq. 2) f ( x )  x

 2000000000 (Solution: x  44721 )

Figure 12. Graph of Equation 2

(Eq. 3) f ( x )  ( x  100 )( x  40000 )

(Solution: x  40000 )

Figure 13. Graph of Equation 3

The simulation program was implemented in C language and executed on HP Z400 workstation.

The error rates of three functions are summarized in figure 14. Each result is the average of 500 times of trial. In calculation of error rate, only the case where a solution is completely same as the optimal solution is judged as a correct solution. Therefore, if a solution is not in agreement with the optimal solution, even if very close to the optimal solution, it has judged as an error. Figure 14 shows that good results were obtained by the proposed DGA in various problems.

The parameters are summarized in table 2. Although the solution is denoted by 16 bits in GA with binary number and Gray code, the solution is denoted by 32 bits in GA with redundant binary number.

Figure 14. Error rate

Table 2. GA Parameters

Generation

Population

Selection

Crossover

Crossover rate

Mutation rate

5. Conclusion

In this paper, a novel distribute genetic algorithm with redundant binary number was proposed and discussed. The proposed algorithm showed stable and good performance by combining the various coding system. It is confirmed that the proposed algorithm was effective for improvement of error rate by computer simulation results.

The future works are evaluation for other difficult problems and hardware implementation.

6. Acknowledgment

This work was supported by Tokyo Denki University Science Promotion Fund (Q12J-03).

7. References

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2009