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Informatica Economic vol. 16, no. 2/2012 27

Marriage in Honey Bees Optimization Algorithm for Flow-shop Problems

Pedro PALOMINOS1, Francisco TOLEDO1, Andrs VJAR2, Miguel ALFARO11 Industrial Engineering Department, University of Santiago of Chile, Santiago, Chile

2 INRA, UMR 782 Gnie Microbiologique et Alimentaire, AgroParisTech, INRA, 78850Thiverval-Grignon, France

{pedro.palominos, francisco.toledo,miguel.alfaro}@usach.cl, [email protected]

The objective of this work is to make a comparative study of the Marriage in Honeybees Op-

timization (MBO) metaheuristic for flow-shop scheduling problems. This paper is focused on

the design possibilities of the mating flight space shared by queens and drones. The proposed

algorithm uses a 2-dimensional torus as an explicit mating space instead of the simulated an-

nealing one in the original MBO. After testing different alternatives with benchmark datasets,

the results show that the modeled and implemented metaheuristic is effective to solve flow-

shop type problems, providing a new approach to solve other NP-Hard problems.

Keywords: Metaheuristics, Flow-Shop, Scheduling, Bio Inspired Optimization

IntroductionEvolutionary computation refers to the set

of methods that have a biological inspirationand allow solving complex problems. Thesemethods are based on the principle that evo-lution is an optimization process in which itis attempted to improve the skills of individ-uals and therefore achieve better adaptation

to their environment as a result of greateroverall survival. In particular, Swarm Intelli-gence (SI) is an area of artificial intelligencefocused on modeling the behavior of socialinsects like ants and bees [1]. Ant ColonyOptimization (ACO) is one of the betterknown models of the SI type in which the de-sign of optimization algorithms is inspired bythe decentralized and collective behavior ofant colonies [2], [3]. Another metaheuristicof the SI type is Marriage in Honeybees Op-

timization (MBO), which is inspired by themating flight of honey-making bees, and wasproposed by Abbass [4-6] to apply it to the

SAT problem, which is an NP-Completeproblem. The purpose of this work is to testthe efficiency of the MBO metaheuristic inthe permutational flow shop problem andcompare the results between the different de-sign alternatives of the mating flight space.

1.1 Marriage in Honey-Bees Optimization

The main stages of the MBO algorithm are: The mating flights of the queens with thedrones to choose the parents of the futurelarvae.

Creation of new offspring by the queens,which correspond to the crossing of genet-ic material from the queen bee with thedrones chosen in the mating flight.

Improvement of the health quality of theoffspring by the workers by mutation.

A schematic diagram is presented on Figure

1.

1


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28 Informatica Economic vol. 16, no. 2/2012

Fig. 1. Graphic representation of the MBO technique

The key stage of the metaheuristic is the mat-ing flight of the queens with the drones, be-cause in this stage the queen can choose tomate with a drone depending on the qualityof the drone found. The queen stores the ge-netic material of the chosen drones in itsspermatheca. As the flight takes place, thequeen loses energy and speed. The processends when the queen uses up the availableenergy or fills its spermatheca to capacity.The loss of speed affects the queens ability

to choose the drones; at first the queensmovements are long and allow it to chooseamong a large number of drones. As thequeens speed decreases, its movements be-come shorter, so it can only choose thedrones that are closest, within the range de-fined by its speed.The energy of a queen at the beginning of aflight is represented byE

0, 00, i{0,1,2},

0is the importance

factor of speed, 1

is the importance factor of

the individuals drones best position, p1, and

2

is the importance factor of the best drones

best position, p2. If

1>

2, more importance

is assigned to individual learning, in refer-ence to the best position found by the droneitself. On the other hand, if

2

>1

, more im-

portance is assigned to the learning of theswarm, in reference to the best position


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found by the set of drones. Every time aqueen chooses a drone for a possible cross-ing, the probability that the queen will storethe drones genetic material in its

spermatheca is given by:

P(q,d) = min{exp(-f/S),1} (3)

Where f is the absolute difference betweenthe fitness of the drone dand queen q, and Sis the speed of the queen. The probability ofmating is high when the distance betweentheirs fitness is short or when the queensspeed is high. After each transition state, thequeens energy and speed are updated ac-

cording to the following equations:

Ek+1

= gEk

(4)

Sk+1

= (1a)Sk

(5)

Where the initial speed S0

of the queen is

chosen uniformly, S0

~ U(0,1), and 0


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),( kJi

t : Time for processing taskJi

on ma-

chine k. n : Number of tasks.

m : Number of machines.To be able to represent the problem from theMBO perspective, the following steps weretaken:

2.1 StructureFirst, the genotypes of the drones, thequeens, and the larvae must be established.Each of these genotypes will be represented

by a permutational sequence of works of thesame length for all the individuals that will

depend on the number of tasks existing ineach problem. For example, Fig. 2 depictsrepresentations of a queen, a drone and a lar-va when eight jobs are available.

Fig. 2. Structure of the individuals' genotype

2.2 PopulationA queen bee population is generated, the bestof which is the one obtained by means ofPalmers Heuristic. The rest of the queen

bees will be obtained by a random procedure,to generate more diversity. The drone popu-lation will be obtained by a random proce-dure in which the tasks that make up their se-quence are assigned in such a way that each

drone generated is feasible. Also, the dronescannot have the same genetic material, i.e., indifferent flights different drones will be cre-ated, with the purpose of increasing the ex-

ploration of permutations.

2.3 Trajectory generationTo be able to generate the trajectories of thequeens and the drones they must first be po-sitioned randomly in the n-dimensionalspace, i.e., initial positions will be created bychance for each individual. This stage will bedivided into two: one corresponding to thequeen bee transition and the other to the

drones. The queen bees will move randomlyover a limited neighborhood, and the dronesaccording to (1) and (2).

2.4 Drone selectionAs already mentioned, the trajectories of thequeen bees will depend on chance. Once thathas happened, the probability that the droneshave of mating with a queen bee is calculated(3)-(5). Once the probability of mating with a

particular drone is obtained, the queen beechooses randomly if it will open itsspermatheca to the chosen drone. For that itwill generate a number (W) between 0 and 1that will represent the queen bees decision.

If that number is smaller than the dronesmating probability, the queen will open itsspermatheca to allow the entry of the dronessperm. This happens because the queens

predisposition to mate is greater than theprobability of the drone, so the queens deci-sion will be to mate with it.

2.5 Genetic crossingOnce the drone selection has been made, thenext step is processing that information, orthe genetic step, which has two operators, de-fined as order crossing and mutation. Thecrossing used is that proposed by Davis [8],called OX. In Fig. 3, five jobs are used to de-scribe a crossover between queen Q

1and

droneD1

to produce two larvae.

Fig. 3. OX crossing between queen bee and

drone


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Mutation involves the random alteration ofeach component in the sequence of jobs. Themutation used is illustrated in Fig. 4.

Fig. 4. Mutation for a larva

The mutation is triggered as follows: A ran-dom number between 0 and 1 is generated

for each larva and it is compared with its mu-tation probablility; if this number is smallerthan the mutation probability, the larva mu-tates. Otherwise the larvas genotype remainsintact.

2.6 Evaluation of the queen and the larvaThe evaluation function must determine if anindividual is apt or not, in the configurationspace, by means of a cost associated with it.This mode of implementation seeks to obtain

the best makespan for each individual.

2.7 Updating population

This procedure considers replacing the leastadapted queen bees by the best larvae ob-tained from the crossing of a queen bee and aselected drone.

2.8 Updating parametersIn each of their flights the queen bees looseenergy and speed, and they may accept somedrone for mating, which implies that theywill have less space in their spermatheca toaccept another mating. That is why the pa-rameters must be updated in each flight.

2.9 Stopping criteriaWhen the heuristic is iterated, it will always

voraciously reduce the target function. Inview of that, the criterion for making a moreexhaustive study of the efficiency of theMBO algorithm is the definition of a certainnumber of flights (iterations) initialized andentered at the start of the program. In thiscase the stopping criterion was 1000 itera-tions.Finally, Table 1 summarizes the values of themetaheuristics parameters for the flow-shop

problem. It should be noted that some wereobtained empirically and others were takenfrom the literature.

Table 1. Parameters for MBO ImplementationParameters Source Value

Number of queens Experiment 3Drone population Experiment 100Capacity of spermatheca (M) Experiment 100Initial speed of queens (S0) Experiment 10

Initial energy of queens (E

0

) Literature [6] U(0,1)

Energy reduction factor for queens (g) Literature [6] 0.1

3 Proposed variants for the MBO

metaheuristicThe original MBO algorithm is a hybridmetaheuristic of simulated annealing (SA),genetic algorithms (GA), and local search(LS), in view of the use of a mating functionsimilar to SA, and the advantages of GA andLS in the generation and improvement of the

larvae by the worker bees. However, theMBO metaheuristic has a large number of

particular characteristics that distinguish itfrom the rest[4-6].As described previously, the main MBO pro-cesses are the following: Mating in flight of the queen bee with the

drones. Creation of new queen bee larva. Larvae improvement by the workers.

Replacement of the less apt queens by thebest larva.


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The key process of interest for the presentwork is the flight mating process of thequeen bee where the drones are. With the

purpose of improving the pure MBO algo-

rithm, we will use the PSO model[7] in atoroidal space to select apt drones in eachswarm for mating with the queens.All the remaining processes will be kept con-stant and as defined in the MBO developed

by Abbas[5, 6]. Given the logic of the PSOalgorithm, the creation space of the matingflight is defined as an n-dimensional space,and as a way of interfering as little as possi-

ble with the drone generation space we willuse a toroidal space.

The surface of the 2-dimensional torus usedas flight space. In the torus, the coordinates

),(TTyx follow:

T

T

y

x=

)(mod

)(mod

y

x

ny

nx(9)

Where nx

and ny

are the length of each coor-

dinate axis and (x,y) are the coordinates ofthe Euclidean space.In PSO the drones fly through the problems

space, where each one keeps a record of itscoordinates in the problems space that areassociated with the best solution that it hasachieved so far. Each drone will be repre-sented as a vector that is updated at each cy-cle. Therefore, each position of the drones

will be represented by a tuple ),( TT yx in thetorus. The size of this space will be given ar-

bitrarily by the experiments made, and willdepend on the size of the population to be

generated. In this case the population was setat 100 drones and the space used was of theorder of )100,100(),( =yx nn . The distance

between two points in the torus was calculat-ed by the following formula:

),( vud = 21

)( yx + (10)

Where (x

,y) represents the shortest path

between one point and another for each axis:},min{ xxxxxx vunvu = (11)

},min{ yyyyyy vunvu = (12)

4 Experimental ResultsThe computational development for testingthe proposed heuristic was made on a com-

puter with an AMD Athlon 64 3000+ proces-sor, clock speed of 2.0 GHz, 1 GB RAM,with Linux, programmed in Python. The in-stances used were extracted from the data

base available in the OR-Library[9], where120 Taillards instances [10] were consid-ered, grouped as shown in Table 2, whereeach configuration has 10 test instances.

Table 2. Characteristics of used instancesConfiguration Tasks MachinesInstances

(N) (M)1 20 5 ta001-010

2 20 10 ta011-0203 20 20 ta021-0304 50 5 ta031-0405 50 10 ta041-0506 50 20 ta051-0607 100 5 ta061-0708 100 10 ta071-0809 100 20 ta081-090

10 200 10 ta091-10011 200 20 ta101-11012 500 20 ta111-120

The results obtained for each configurationevaluated are presented in Table 3, where thefirst column gives the number of the configu-

ration; the second gives the number tasks; thethird gives the number of machines; and thefourth gives the mean error of the original


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MBO metaheuristic measured as percentageof the best solution found and the best solu-tion from the data base, divided by the latter;the fifth gives the mean efficiency of the

modified MBO metaheuristic, measured aspercentage of the best solution found and thebest solution from the data base, divided bythe latter; and the sixth gives the difference

between the original MBO metaheuristic andthe proposed modified MBO metaheuristic.

In Table 3 it is seen that the overall averageerror for the 120 test instances is 4.93% forthe original MBO metaheuristic and 3.19%for the modified MBO metaheuristic, com-

pared with the best solution given in the OR-Libray database. These results are consideredsatisfactory for the proposed metaheuristic,since there are significant differences be-tween the two metaheuristics studied.

Table 3. ResultsConfiguration Tasks

(N)Machines

(M)Meanerror

Mean efficien-cy

Difference

1 20 5 3.6 1.4 -2.2

2 20 10 4.8 2.5 -2.33 20 20 4.7 2.3 -2.44 50 5 3.3 0.9 -2.45 50 10 7.3 4.4 -2.86 50 20 9.1 6.2 -2.97 100 5 2.3 0.8 -1.48 100 10 4.6 2.7 -1.99 100 20 9.1 6.6 -2.5

10 200 10 4.1 2.3 -1.811 200 20 8.9 6.6 -2.312 500 20 6.3 5.0 -1.3

Mean 4.93 3.19 -1.75Max 9.1 6.6Min 3.6 0.8

5 ConclusionThe main contribution of this work is the

proposal of a new metaheuristic based onhoneybee mating, proposed by Abbass [4-6]to tackle a scheduling problem of the flowshop type, achieving very good results for the120 tested instances. Also, a first applicationof the MBO technique to the scheduling

problem of the combinatorial flow-shop typeis presented. However, the proposed MBOmetaheuristic is a first approximation, so fu-ture work may take two routes: (1) Test theefficiency of the metaheuristic in other typesof combinatorial problems like that of thetraveling salesman, among others, and (2)study other larva improvement techniquesand carry out a parameterization study, inview of the large number of parameters used

by this technique.

AcknowledgmentThis project was supported by The Universi-ty of Santiago of Chile (060517PB DICYT-USACH).

References[1] E. Bonabeau, M. Dorigo, and

G. Theraulaz, Swarm intelligence: fromnatural to artificial systems. Oxford Uni-versity Press, USA, 1999.

[2] M. Dorigo and L. Gambardella, Ant col-ony system: A cooperative learning ap-

proach to the traveling salesman prob-lem,IEEE Transactions on EvolutionaryComputation, vol. 1, no. 1, pp. 5366,1997.

[3] M. Dorigo and G. Di Caro, Ant colonyoptimization: a new meta-heuristic, in

Proceedings of the 1999 Congress onEvolutionary Computation, vol. 2.1em


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plus 0.5em minus 0.4emIEEE, 1999, pp.14701477.

[4] H. Abbass, A Single Queen SingleWorker HoneyBees Approach to 3-

SAT, in Proceedings of the Genetic andEvolutionary Computation Conference

(GECCO-2001), San Francisco, USA,2001, pp. 807814.

[5] H. Abbass, MBO: Marriage in HoneyBees Optimization - A HaplometrosisPolygynous Swarming Approach, inProceedings of the 2001 Congress on

Evolutionary Computation, vol. 1.1emplus 0.5em minus 0.4emIEEE, 2001, pp.207214.

[6] H. Abbass, A monogenous MBO ap-proach to satisfiability, in Proceedings ofthe International Conference on Computa-

tional Intelligence for Modelling, Control

and Automation, CIMCA, Las Vegas, NV,USA., 2001.

[7] J. Kennedy and R. Eberhart, ParticleSwarm Optimization, in Proceeedings ofthe 1995 IEEE International Conference

on Neural Networks, vol. 4, Piscataway,

NJ, USA, 1995, pp. 19421948.[8] L. Davis, Job shop scheduling with ge-netic algorithms, in Proceedings of the1st International Conference on Genetic

Algorithms.1em plus 0.5em minus0.4emHillsdale, NJ, USA: L. Erlbaum As-sociates Inc., 1985, pp. 136140.

[9] J. E. Beasley, OR-Library: distributingtest problems by electronic mail,Journalof the Operational Research Society,vol. 41, no. 11, pp. 10691072, 1990.

[10] E. Taillard, Benchmarks for basicscheduling problems, European Journalof Operational Research, vol. 64, no. 2,

pp. 278285, January 1993.

Dr. Pedro Ivn PALOMINOS is currently Provost of the University ofSantiago of Chile. He holds a Bachelor Degree and a Professional Degree inIndustrial Engineering at the University of Santiago of Chile. He also holds aM.Sc. Degree in Engineering Production at the Federal University of Rio deJaneiro, Brazil and a Dr. Degree in Industrial Engineering at the Universityof Catalunya, Spain. Dr. Palominos has done teaching and research in Oper-ations Management and Metaheuristics.

Francisco J. TOLEDO received his BSc. in Industrial Engineering (2007)from the University of Santiago of Chile, and the M.Sc. in sciences of theengineering, mention engineering industrial (2008) from the University ofSantiago of Chile, and PhD student in Innovation (2010-2013) from the Uni-versity of Lorraine, ERPI, Nancy, 5400, France. His research interests in-clude the impacts of innovation and entrepreneurship, sustainable supply

chains, sustainable development innovation and models based on systemsmultiagents.

Miguel D. ALFARO (b. May 4, 1956) received his BSc. in Industrial Engi-neering (1985) from the University of Santiago of Chile, and the M.Sc. inEngineering Economy (1990) from The Catholic University of Chile, and theDoctor degree in Automatic Production (1998) from the University HenriPoincar, Nancy I, France. He is full-time professor at the Industrial Engi-neering Department, University of Santiago of Chile since 1998. His currentresearch interests include Non linear Systems Dynamics, Artificial Intelli-

gence in Manufacturing Systems and Agent-based Modeling Systems. He has authored over

30 articles in journals and proceedings, and participated in more than 50 conferences andworkshops. He has acted as chairperson at several conferences in Chile, member of scientificcommittees, and referee of various journals.
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