International Congress Series 1291 (2006) 1 &9~ 17':.

ELSEVIER \vww_ics~el sevic::r.com

Genetic algorithm with modified reproduction strategy based on self-organizing map and

usable schema

Ryosuke Kubota a. *, Keiichi Horio b' Takeshi Yamakawa b

a Gradua1e School ~~( C omplllcr Science and 5\sh>ms Engineering. Kyuslw lnstiwtr (~l Ter1molog_l-; ]~4 1/ihikino. Wakamatsu~ku. Kitakyush11, Fukrmk.1. 80/o:~0/9(,, Japan

b Graduate School of L{fe Science and 5)·stems f:'ngim:ering, Kyushu lnHitute of Technology. Japan

Abstract In this paper, v.:e propose a new updating method considering usability of each element of

inputs and apply it to the reproduction of the GA to achieve more effective search than the traditional reproduction. In the proposed updating method, the order of updating elements is decided by ave-raging the corresponding elements multiplied by fitn~ss values of the chromosomes. To verify the effectiveness of the proposed updating method, the GA with the modi lied reproduction is applied to 0 -- 1 knapsack problem. © 2006 Published by Elsevier B.V

Ke_vword~ · Genetic a lgorithms; Self-organizing map; Binary: Genetic diversity: Updating rule: Order of inverting dcm~nts

I. Introduction

Genetic algori thm (GA) is a search algorithm based on the mechanics of natural selection and natural genetics [I]. Searching in GA is achieved by iterating reproduction, crossover and mutation strategies. In these strategies, the reproduction strategy is the most imponan t for decid ing the searching points of the next generation. However. tradi tional reprod uction operators may lose a genetic diversity of population in an early stage, because they cannot generate new chromosomes which are d ifferent from the present chromosomes. A decrease of the genetic diversity leads to ineffect iveness of searchi ng [2].

• Corresponding author. T~l.: +81 93 695 6!23; fax: +Sl 93 695 6133. E-mail address: kubota-I")'Osuk..;!,~cdu.brain.k~l.lk..;h.:"h.'.jp (R. Kubota).

0531-5131 / :tj 2006 Published by Elsevier B.V. Jui: 10.1016/j ICS.200o.OH\02

170 R. Kubota er u/. / lntemational Cm1grcss Series 119/ (2006) /69---- 172

To cope with this problem, we have proposed the reproduction strategy by utilizing an approximation ability of a binaty version of se l f~organizing map (BSOM) [3] to maintain the genetic di versity of the population 1"41. This BSOM-based reproduction can generate many kinds of chromosomes with high 1ltness values. However, an order of updating elements of a binary weight vector in the reproduction based on the previous BSOM [4) influences the searching pcrfonnancc of the GA.

In tillS paper, we propose a modified BSOM-bascd reproduction strategy with new updating method ofbina1y weight vectors consideting usability of each element to achieve more effective search than the traditional BSOM-based reproduction.

2, Genetic algorithm with reproduction strategy based on binary self-organizing map

Fig. 1 shows the outline of the GA with the BSOM-based reproduction, where the elements of the input and weight vectors are represented by binmy numbers. The chromosomes of the present generation are used as the input vectors of the BSOM, and the weight vectors after teaming are employed as the chromosomes of the next generation. In other words, the new chromosomes are reproduced by learning the BSOM. In the updating phase of the BSOM, the updating of weight vectors is realized by adjusting elements which are randomly selected in the weight vectors, when the corresponding elements of the input and the weight vector are different. The number of adjusting elements CL(t) is a leaming rate and decided by coefticients with respect to fitness values of the input and weight vectors as following equation:

(1)

where, l J, N,t;,j:._.,,, and d represent Gauss ' notation. the parameter characterized by the number of clements, fitness values of input and weight vector before updating, and Eucl idean distance bel:\veen the best matching unit and the unit to be updated, respectively. h(d,(,) represents the neighbourhood fiJnction and is defined by following Gaussian function:

(2)

The same procedure was also employed in the traditional BSOM-based reproduction (4). This updating can generate new chromosomes with high fitness values. Moreover, their

genetic diversity is preserved. Thus. BSOM-based reproduction operator can achieve the

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Fig. I. Outline of GA \Vith BSOl\·1-based reproduction.

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R. Kubv r.1 l'l a /. / fnrcm arional C vngrcss Saies I '!91 (2006) /69 17 2 Iii

effective search . llowevcr. the order of updating elements of binaty weight vector innuences a search ing performance of the GA. Therefore, the order of updating elements in the updating phase must be considered to realize more effective search than the traditional BSOM reproduction.

3. Proposed updating method based on usable schema

In this paper, we propose a new updating method considering the order of updating elements and apply it to the BSOM-based reproduction. In the proposed updating method, the order of updating elements is decided by a usability of the schema. The usable schema is proper for one element which may contribu te to the high fitness value in a set of the input vectors. Specifically. the usability of ith element is decided by S; as following equation:

0~5,~1. (3)

where K represents th~ number ot' input vectors. In Eq. (3), S; represents the propo11ion of the binary values in a column of the input vectors which have high fitness values. Note that the usability is higher, when S; is closer to "o·· or''!''. On the contrary. the usability is lower. when S, is closer to ''0.5'' . In the proposed updating, each element of weight vector is adjusted to input vector based on th~ order of the usability. when elements of input and weight vectors are di ffercnt.

4. Simulation results

In order to verifY the effecti veness of the BSOM-based reproduction employing the proposed updating method, it is applied to the 0-1 knapsack problem (one of the primaty combinatorial optimization problem). The searching performance of the BSOM-based reproduction employing the proposed updating method is compared with Roulette Wheel Selection (RWS) and that employing the previous updating method. In this simulation , population size, crossover probability and mutation probability are 25. 0.3 and 0.0 I , respectively. These methods ar~ compared trom the view points of the number of generations, and CPU time until the quasi-optimal solution is acquired. Tabk l shows the simulation results. Ten nms are executed and the average value is calculated to suppress dependencies on all stochastic effects. These results show that the GA with the BSOM-

Tahtc I Number of ge-nerat ion and CPU rime

Number of subjects

~0

50

Methods

RWS

Prt:vious [ 4\ Proposed RWS

Previous 14) Proposed

Number of generations

1~80

3 18

164 7089

107 76

CPU time (s)

0.76 0.63 0.44

2.R4 0.62 0.42

17~ R. Kuhota et ,,,_ / lntemational Congrt'S.\' S(;:rie,\· I !Y 1 (2 006) /69 172

based reproduction operator employing the proposed updating method can realize the faster search in smaller number of generations comparing with that employing the traditional reproduction operators .

5. Conclusion

[n thi s paper, we proposed the new updating method of weight vectors to achieve the more effective search of the GA employing BSOM-based reproduction comparing with that employing the traditional BSOM. The ineffective searching using the traditional updating method is attributed to the problem which the elements to be updated arc selected randomly. ln the proposed updating method. the order of updating elements is considered. The order is decided by the usability which is defined by averaging the corresponding elements multiplied by fitness values of the input vectors. The GA employing the BSOM­based reproducti on with the proposed updating method can achieve the more effective search comparing with that employing the traditional and the previous BSOM-based reproductions. The etTectiveness and validity of proposed updating method were shown by solving the combinatorial optimization problem.

Acknowledgements

This work was partially supported be the :? 1st Century COE Program in Kyushu Institute of Technology, entitled "'World of Brain CompLlting Interwoven out of Animals and Robots,"' and was also supported by the Ministry of Education, Science. Sports and Culture, Grant-in-Aid for Scientific Research (A), 2002. 14205038.

References

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algorithm perfom1<mce in dynamic l!nvironment.s, Ad,·anced in Infom1ation Systl!ms LeciUre Note in

Compurcr Science 2457 (200c) 3!4 - 323. [3 ] T. Kohonen. Self-organization :md Associat ive Memory. 1984. [~] R. Kubota. K. Horio. T. Yamak:l\va. Binary SOM Bastd on Significance of Lnputs anJ its Applicatum to

Rcprouucrion or GA. In : Proc. or WS0:-.1"05 (2005) 21 1-218.

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