AN IDEA BASED ON HONEY BEE SWARM FOR NUMERICAL OPTIMIZATION (TECHNICAL REPORT-TR06, OCTOBER, 2005) Dervis KARABOGA

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1 AN IDEA BASED ON HONEY BEE SWARM FOR NUMERICAL OPTIMIZATION (TECHNICAL REPORT-TR06, OCTOBER, 2005) Devis KARABOGA Eciyes Univesity, Engineeing Faculty Compute Engineeing Depatment Kaysei/Tükiye I. INTRODUCTION Swam intelligence has become a eseach inteest to many eseach scientists of elated fields in ecent yeas. Bonabeau has defined the swam intelligence as any attempt to design algoithms o distibuted poblem-solving devices inspied by the collective behaviou of social insect colonies and othe animal societies [1]. Bonabeau et al. focused thei viewpoint on social insects alone such as temites, bees, wasps as well as othe diffeent ant species. Howeve, the tem swam is used in a geneal manne to efe to any estained collection of inteacting agents o individuals. The classical example of a swam is bees swaming aound thei hive; nevetheless the metapho can easily be extended to othe systems with a simila achitectue. An ant colony can be thought of as a swam whose individual agents ae ants. Similaly a flock of bids is a swam of bids. An immune system [2] is a swam of cells and molecules as well as a cowd is a swam of people [3]. Paticle Swam Optimization (PSO) Algoithm models the social behaviou of bid flocking o fish schooling [4].

2 Two fundamental concepts, self-oganization and division of labou, ae necessay and sufficient popeties to obtain swam intelligent behaviou such as distibuted poblemsolving systems that self-oganize and adapt to the given envionment: a) Self-oganization can be defined as a set of dynamical mechanisms, which esult in stuctues at the global level of a system by means of inteactions among its low-level components. These mechanisms establish basic ules fo the inteactions between the components of the system. The ules ensue that the inteactions ae executed on the basis of puely local infomation without any elation to the global patten. Bonabeau et al. have chaacteized fou basic popeties on which self oganization elies: Positive feedback, negative feedback, fluctuations and multiple inteactions [1]: i) Positive feedback is a simple behavioual ules of thumb that pomotes the ceation of convenient stuctues. Recuitment and einfocement such as tail laying and following in some ant species o dances in bees can be shown as the examples of positive feedback. ii) Negative feedback countebalances positive feedback and helps to stabilize the collective patten. In ode to avoid the satuation which might occu in tems of available foages, food souce exhaustion, cowding o competition at the food souces, a negative feedback mechanism is needed. iii) Fluctuations such as andom walks, eos, andom task switching among swam individuals ae vital fo ceativity and innovation. Randomness is often cucial fo emegent stuctues since it enables the discovey of new solutions. iv) In geneal, self oganization equies a minimal density of mutually toleant individuals, enabling them to make use of the esults fom thei own activities as well as othes.

3 b) Inside a swam, thee ae diffeent tasks, which ae pefomed simultaneously by specialized individuals. This kind of phenomenon is called division of labou. Simultaneous task pefomance by coopeating specialized individuals is believed to be moe efficient than the sequential task pefomance by unspecialized individuals [2,5-7]. Division of labou also enables the swam to espond to changed conditions in the seach space. Two fundamental concepts fo the collective pefomance of a swam pesented above, self-oganization and division of labou ae necessay and sufficient popeties to obtain swam intelligent behaviou such as distibuted poblem-solving systems that self-oganize and -adapt to the given envionment. II. BEHAVIOUR OF HONEY BEE SWARM The minimal model of foage selection that leads to the emegence of collective intelligence of honey bee swams consists of thee essential components: food souces, employed foages and unemployed foages and the model defines two leading modes of the behaviou: the ecuitment to a necta souce and the abandonment of a souce. i) Food Souces: The value of a food souce depends on many factos such as its poximity to the nest, its ichness o concentation of its enegy, and the ease of extacting this enegy. Fo the sake of simplicity, the pofitability of a food souce can be epesented with a single quantity [8]. ii) Employed foages: They ae associated with a paticula food souce which they ae cuently exploiting o ae employed at. They cay with them infomation about this paticula souce, its distance and diection fom the nest, the pofitability of the souce and shae this infomation with a cetain pobability. iii) Unemployed foages: They ae continually at look out fo a food souce to exploit. Thee ae two types of unemployed foages: scouts, seaching the envionment suounding the nest fo new food souces and onlookes waiting in the nest and

4 establishing a food souce though the infomation shaed by employed foages. The mean numbe of scouts aveaged ove conditions is about 5-10% [8]. The exchange of infomation among bees is the most impotant occuence in the fomation of the collective knowledge. While examining the entie hive it is possible to distinguish between some pats that commonly exist in all hives. The most impotant pat of the hive with espect to exchanging infomation is the dancing aea. Communication among bees elated to the quality of food souces takes place in the dancing aea. This dance is called a waggle dance. Since infomation about all the cuent ich souces is available to an onlooke on the dance floo, pobably she can watch numeous dances and decides to employ heself at the most pofitable souce. Thee is a geate pobability of onlookes choosing moe pofitable souces since moe infomation is ciculated about the moe pofitable souces. Employed foages shae thei infomation with a pobability popotional to the pofitability of the food souce, and the shaing of this infomation though waggle dancing is longe in duation. Hence, the ecuitment is popotional to the pofitability of the food souce [9]. In ode to undestand the basic behaviou chaacteistics of foages bette, let us examine Figue 1. Assume that thee ae two discoveed food souces: A and B. At the vey beginning, a potential foage will stat as unemployed foage. That bee will have no knowledge about the food souces aound the nest. Thee ae two possible options fo such a bee: i) It can be a scout and stats seaching aound the nest spontaneously fo a food due to some intenal motivation o possible extenal clue (S on Figue 1). ii) It can be a ecuit afte watching the waggle dances and stats seaching fo a food souce (R on Figue 1). Afte locating the food souce, the bee utilizes its own capability to memoize the location and then immediately stats exploiting it. Hence, the bee will become an employed foage. The

5 foaging bee takes a load of necta fom the souce and etuns to the hive and unloads the necta to a food stoe. Afte unloading the food, the bee has the following thee options: i) It becomes an uncommitted followe afte abandoning the food souce (UF). ii) It dances and then ecuits nest mates befoe etuning to the same food souce (EF1) iii) It continues to foage at the food souce without ecuiting othe bees (EF2). It is impotant to note that not all bees stat foaging simultaneously. The expeiments confimed that new bees begin foaging at a ate popotional to the diffeence between the eventual total numbe of bees and the numbe of pesent foaging. Figue 1. The behaviou of honey bee foaging fo necta In the case of honey bees, the basic popeties on which self oganization elies ae as follows: i) Positive feedback: As the necta amount of food souces inceases, the numbe of onlookes visiting them inceases, too. ii) Negative feedback: The exploation pocess of a food souce abandoned by bees is stopped. iii) Fluctuations: The scouts cay out a andom seach pocess fo discoveing new food souces. iv) Multiple inteactions: Bees shae thei infomation about food souce positions with thei nest mates on the dance aea.

6 III. PROPOSED APPROACH In this wok, a paticula intelligent behaviou of a honey bee swam, foaging behaviou, is consideed and a new atificial bee colony (ABC) algoithm simulating this behaviou of eal honey bees is descibed fo solving multidimensional and multimodal optimisation poblems. In the model, the colony of atificial bees consists of thee goups of bees: employed bees, onlookes and scouts. The fist half of the colony consists of the employed atificial bees and the second half includes the onlookes. Fo evey food souce, thee is only one employed bee. In othe wods, the numbe of employed bees is equal to the numbe of food souces aound the hive. The employed bee whose food souce has been exhausted by the bees becomes a scout. The main steps of the algoithm ae given below: Send the scouts onto the initial food souces REPEAT Send the employed bees onto the food souces and detemine thei necta amounts Calculate the pobability value of the souces with which they ae pefeed by the onlooke bees Stop the exploitation pocess of the souces abandoned by the bees Send the scouts into the seach aea fo discoveing new food souces, andomly Memoize the best food souce found so fa UNTIL (equiements ae met) Each cycle of the seach consists of thee steps: moving the employed and onlooke bees onto the food souces and calculating thei necta amounts; and detemining the scout bees and diecting them onto possible food souces. A food souce position epesents a possible solution to the poblem to be optimized. The amount of necta of a food souce coesponds to the quality of the solution epesented by that food souce. Onlookes ae placed on the food souces by using a pobability based selection pocess. As the necta amount of a food souce inceases, the pobability value with which the food souce is pefeed by onlookes inceases, too. Evey bee colony has scouts that ae the colony s exploes [10]. The exploes do not have any guidance while looking fo food. They ae pimaily concened with finding any kind of food souce. As a esult of such behaviou, the scouts ae chaacteized by low

7 seach costs and a low aveage in food souce quality. Occasionally, the scouts can accidentally discove ich, entiely unknown food souces. In the case of atificial bees, the atificial scouts could have the fast discovey of the goup of feasible solutions as a task. In this wok, one of the employed bees is selected and classified as the scout bee. The selection is contolled by a contol paamete called "limit". If a solution epesenting a food souce is not impoved by a pedetemined numbe of tials, then that food souce is abandoned by its employed bee and the employed bee is conveted to a scout. The numbe of tials fo eleasing a food souce is equal to the value of "limit" which is an impotant contol paamete of ABC. In a obust seach pocess exploation and exploitation pocesses must be caied out togethe. In the ABC algoithm, while onlookes and employed bees cay out the exploitation pocess in the seach space, the scouts contol the exploation pocess. In the case of eal honey bees, the ecuitment ate epesents a measue of how quickly the bee swam locates and exploits the newly discoveed food souce. Atificial ecuiting pocess could similaly epesent the measuement of the speed with which the feasible solutions o the optimal solutions of the difficult optimization poblems can be discoveed. The suvival and pogess of the eal bee swam depended upon the apid discovey and efficient utilization of the best food esouces. Similaly the optimal solution of difficult engineeing poblems is connected to the elatively fast discovey of good solutions especially fo the poblems that need to be solved in eal time. IV. SIMULATION RESULTS In the simulation studies, Atificial Bee Colony (ABC) Algoithm was applied fo finding the global minimum of the well-known thee test functions. One of the functions is Sphee function that is continuous, convex and unimodal function. x is in the inteval [-100, 100]. Global minimum value fo this function is 0 and the optimum solution is = x, x,..., x ) (0,0,...,0). The second function is a well known classic optimization x opt ( = poblem: Rosenbock valley. The global optimum is inside a long, naow, paabolic shaped flat valley. Theefoe, it is vey difficult to convege the global optimum. Vaiables of the function ae stongly dependent, and the gadients geneally do not point towads the optimum. x is in the inteval [-2.048, 2.048], the global minimum value is 0; and the optimum solution is = x, x ) (1,1 ). The global optimum of the function is the only x opt ( 1 2 =

8 optimum and the function is unimodal. The thid function is Rastigin function which is based on Sphee function with the addition of cosine modulation to poduce many local minima. x is in the inteval [-600, 600] and the minimum value is 0. The optimum solution fo this function is = x, x,..., x ) (0,0,...,0) x opt ( = Table 1: Benchmak functions tested by the ABC Algoithm Functions Ranges Minimum Value x i 100 f ( x = f 1 (0) = 0 1 ) x i i= f 2 ( x) = 100(x2 x1 ) + (x1 1 ) x f ( 1 ) = xi 600 f 3(x) = (xi 10cos(2πx i ) + 10) f 2 (0) = 0 i= 1 i 2 In the ABC algoithm, the maximum numbe of cycles was taken as The pecentages of onlooke bees and employed bees wee %50 of the colony and the numbe of scout bees was selected to be one. The incease in the numbe of scouts encouages the exploation pocess while the incease of onlookes on a food souce encouages the exploitation pocess. Paametes adopted fo the ABC algoithm ae given in Table 2. Table 2: Contol paametes adopted fo the ABC algoithm Contol paametes of ABC Algoithm swamsize 20 limit Numbe of onlooke bees *Dim. numbe of onlookes 50% of the swam numbe of employed bees 50% of the swam numbe of scouts 1 Each of the expeiments was epeated 30 times with diffeent andom seeds and the aveage function values of the best solutions wee ecoded. The mean and the standad deviations of the function values obtained by ABC algoithm fo the same conditions ae given in Table 3.

9 Table 3 The esults obtained by ABC algoithms. f ( ) 1 x f 2 ( x ) f 3 ( x) Functions Mean Std (5D Sphee) 4.45E E-17 (2D Rosenbock) (10D Rastigin) 4.68E E-17 V. CONCLUSION In this wok, a new optimization algoithm based on the intelligent behaviou of honey bee swam has been descibed. The new swam algoithm is vey simple and vey flexible when compaed to the existing swam based algoithms. It is also vey obust, at least fo the test poblems consideed in this wok. Fom the simulation esults, it is concluded that the poposed algoithm can be used fo solving unimodal and multi-modal numeical optimization poblems. In this wok, the algoithm was tested on a vey limited set of test poblems. The simulation study must be caied out on a lage set of test functions and the pefomance of the algoithm must be examined in detail. VI. REFERENCES 1. E. Bonabeau, M. Doigo, G. Theaulaz, Swam Intelligence: Fom Natual to Atificial Systems, New Yok, NY: Oxfod Univesity Pess, L.N. De Casto, F.J. Von Zuben, Atificial Immune Systems. Pat I. Basic Theoy And Applications, Technical Repot No. Rt Dca 01/99, Feec/Unicamp, Bazil, J. Vestestøm, J. Riget, Paticle Swams Extensions fo impoved local, multi-modal, and dynamic seach in numeical optimization, MSc.Thesis, May J. Kennedy, R. C. Ebehat, Paticle swam optimization, In Poceedings of the 1995 IEEE Intenational Confeence on Neual Netwoks, Vol. 4, pp , V. Teeshko, Reaction-diffusion model of a honeybee colony s foaging behaviou, M. Schoenaue, et al, Eds., Paallel Poblem Solving fom Natue VI, Lectue Notes in Compute Science, vol. 1917, Spinge-Velag: Belin, p , 2000.

10 6. R. L. Jeanne, "The Evolution of the Oganization of Wok in Social Insects" Monit. Zool. Ital. 20, , G. Oste, E. O. Wilson, "Castes and Ecology in the Social Insects", Pinceton, NJ: Pinceton Univesity Pess, (1978). 8. T. D. Seeley, The Wisdom of the Hive (Havad Univesity Pess, Cambidge, MA, 1995). 9. V. Teeshko, A. Loengaov, Collective Decision-Making in Honey Bee Foaging Dynamics, Computing and Infomation Systems Jounal, ISSN , vol. 9, No 3, Octobe T. D. Seeley, Vissche P.K., Assessing the benefits of coopeation in honeybee foaging: seach costs, foage quality, and competitive ability, Behav. Ecol. Sociobiol., 22: , 1988.