IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 5, MAY 2013 2085

A Multiobjective Firefly Approach Using Beta ProbabilityDistribution for Electromagnetic Optimization Problems

Leandro dos Santos Coelho , Teodoro Cardoso Bora , Felipe Schauenburg , and Piergiorgio Alotto

Industrial and Systems Eng. Graduate Program (PPGEPS),Pontifical Catholic University of Parana, Curitiba, PR, BrazilDepartment of Electrical Engineering, Federal University of Parana (UFPR), Curitiba, PR, Brazil

Dip. Ingegneria Industriale, Università di Padova, Italy

Current research on optimization methods is increasingly focused on biology-inspired metaheuristics as efficient tools for the solutionof many electromagnetic optimization problems. The firefly algorithm (FA) is an algorithm of this class, and is based on the idealizedbehavior of the flashing characteristics of fireflies. In FA, the flashing light can be represented in such a way that it is associated withthe objective function to be optimized, which makes it possible to formulate a biology-inspired algorithm. This paper briefly introducesthe basics of FA and its multiobjective version (MOFA) and proposes a novel multiobjective variant which uses the beta probabilitydistribution (MOBFA) in the tuning of control parameters, which is useful to maintain the diversity of solutions, as well as the use ofcrowding-based archiving of the Pareto solutions. Numerical results refer to a simple analytical benchmark as well as a multiobjectiveconstrained brushless dc motor design problem, both showing that the resulting MOBFA algorithm outperforms the standard one.

Index Terms—Brushless dc motor design, firefly algorithm (FA), multiobjective optimization, optimization.

I. INTRODUCTION

C URRENT research on optimization methods is stronglyfocused on biology-inspired techniques for the solution

of engineering design problems. The firefly algorithm (FA) [1],[2] is a recent nature-inspired optimization algorithm that sim-ulates the flashing pattern and behavioral characteristics of fire-flies. FA can be considered as a typical swarm-based approachfor optimization, in which the search algorithm is inspired bythe social behavior of fireflies and the phenomenon of biolu-minescent communication. Bioluminescent flashing in fireflies,while primarily a sexual signal, is known to deter some preda-tors while attracting others.FA has already shown interesting performance for single-ob-

jective continuous and mixed integer/continuous constrainedoptimization problems [3]–[8]. Furthermore, very recent liter-ature shows that FA lends itself to be naturally extended to thesolution of multiobjective optimization problems (MOFA) [9]through Pareto approaches, i.e. techniques based on finding theset of non-dominated solutions.However, the proposed modified MOFA (MOBFA) departs

from the original scheme and uses the constraint-handling tech-nique from NSGA-II (Non-dominated Sorting Genetic Algo-rithm-II) [10], the beta probability distribution [11] in the tuningof the randomization parameter of the algorithm, and an ex-ternal archive of nondominated solutions.This paper uses both a simple analytical benchmark as

well as an application-oriented brushless direct current (DC)wheel motor benchmark problem [12], [13] to investigate theperformance of the standard MOFA and the proposed MOBFAapproaches.

Manuscript received November 04, 2012; revised December 20, 2012; ac-cepted January 04, 2013. Date of current version May 07, 2013. Correspondingauthor: P. Alotto (e-mail: alotto@die.unipd.it).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMAG.2013.2238902

The remainder of the manuscript is structured as follows: InSection II, the fundamentals of the FA are presented, while Sec-tion III is devoted to the multiobjective algorithms MOFA andMOBFA. Section IV presents the results obtained byMOFA andMOBFA on an analytical benchmark. Section V contains the de-scription of the brushless dc motor design problem and the dis-cussion of optimization results. Conclusions are briefly drawnin Section VI.

II. FUNDAMENTALS OF THE FIREFLY ALGORITHM

FA is based on the behavior of social insects (fireflies). Insocial insect colonies, each individual seems to have its ownagenda and yet the group as a whole appears to be highly orga-nized. FA, as a population-based metaheuristic, works on fol-lowing principles [1], [2]. i) Fireflies are unisex, so that onefirefly will be attracted to all other fireflies. ii) Attractivenessis proportional to the brightness of the flashing, and for any twofireflies, the less bright one will be attracted to the brighter one;however, the perceived brightness decreases with distance, andif no firefly is brighter than a given firefly, this will move ran-domly. iii) The brightness of a firefly is associated with the ob-jective function. Since the attractiveness is proportional to thelight intensity seen by adjacent fireflies, the attractiveness of afirefly is defined in terms of Cartesian distance between firefliesand . In this case, the movement of a firefly attracted to abrighter one is determined by

(1)

where the second term is due to the attraction (with the absorp-tion coefficient in this paper, and the attractiveness at

) and the third term is due to randomization ( being arandomization parameter). The random number of (1) isusually a uniformly distributed random number in [0, 1].It can be observed that if the attractiveness does not de-

pend on the distance between fireflies so that all will be attractedto the current global optimum. In this case the behavior of the

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2086 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 5, MAY 2013

Fig. 1. Pseudo code of the adopted standard FA.

Fig. 2. Pseudo code of the standard MOFA.

algorithm becomes very similar to the one of Particle SwarmOptimization (PSO), which also includes such a term. On theother hand if the movement is purely random. The FAalgorithm will therefore behave in a way which is between thesetwo extremes depending on the value of which is therefore avery critical parameter. In spite of the interesting performanceof the method reported in literature, the performance of MOFAgreatly depends on its control parameters and it may suffer frombeing trapped in local optima. The authors’ experience with FAis that the algorithm, without ad-hoc tuning, is slightly less effi-cient than PSO or Differential Evolution (DE) while being a bitmore robust. Furthermore, FA has intrinsic clustering featureswhich may be appealing in the multimodal case and in the caseof extension to multiobjective problems.The pseudo code for implementing FA is shown in Fig. 1.

III. MULTIOBJECTIVE FIREFLY ALGORITHM

The previously described algorithm can be extended quitenaturally to multiobjective problems, thus resulting in MOFA,with the scheme showed in Fig. 2.In the implemented standard, MOFA was generated with

uniform distribution in range [0, 1], while was adoptedbased on comments in [14].In order to improve the performance of MOFA we propose

following modifications to the standard algorithm: First of all

Fig. 3. Pseudo code of the proposed MOBFA.

the use of the beta probability distribution [11] in the tuning ofthe randomization parameter can be useful to preserve diver-sity and helps to explore hidden areas in the search space, espe-cially in multiobjective optimization approaches. Furthermorein MOBFAwe adopt the selection mechanism of NSGA-II [10],a classic and effective multiobjective algorithm, which has aspromising features his elitist characteristics which are achievedwithout major loss of diversification. Nondominated solutionsare collected in an external archive of nondominated solutions,and ranked according to front index and crowding within thefront with same index (see [10] for details).The algorithm of the proposed MOBFA is shown in Fig. 3.Before arriving to the proposed version of MOBFA, exten-

sive testing of algorithmic options have been considered andtested on the analytical example of Section IV as well as onother standard benchmark functions. In particular, 76 alterna-tive implementations have been tested, all based on archivingand NSGA-II type criteria, including:1) always with2) linearly increasing and decreasing during iteration al-ways with

3) proportional to various combinations of max., min., avg.values of the objectives always with

4) linearly increasing and decreasing during iteration withproportional to various combinations of max., min., avg.

values of the objectives5) and/or obtained from beta distributionAmong these variants the overall most successful one, used

in the next sections used a scaled beta distribution for and ,where the scaling factor was given by the extension of parameterspace in each dimension.

IV. ANALYTICAL BENCHMARK

The analytical benchmark refers to a simple multiobjectiveproblem related to the electrical circuit shown in Fig. 4. Imaginethat a sinusoidal voltage generator with open-circuit-voltageand internal impedance is connected in serieswith a load impedance . The power transferred tothe load is while the effi-ciency is . The goals of maximizingthe transferred power and the efficiency are conflicting since itis not possible to contemporarily optimize both objectives by

DOS SANTOS COELHO et al.: MULTIOBJECTIVE FIREFLY APPROACH USING BETA PROBABILITY DISTRIBUTION 2087

TABLE IOPTIMIZATION VARIABLES AND RANGE

Fig. 4. Simple circuit related to the analytical test problem.

Fig. 5. Results of MOFA and MOBFA on the analytical test case.

operating on and . It can be easily shown that the Paretofront for this problem is given by . In thefollowing we assume for the sake of simplicity that

, while and .Fig. 5 shows the results obtained by MOFA and MOBFA on

this test problem for two sample runs (similar results are ob-tained over multiple runs, but only two are shown for the sakeof clarity of the picture). The population consisted in both casesof 20 fireflies and the maximum number of generations was 30,i.e. a total of 600 function evaluations. The archive size was setto 100 but the archive was not filled up during the runs. It canbe easily seen that the solutions obtained by MOBFA are nearerto the true front and that they are much more well-dispersed.

V. BRUSHLESS DC MOTOR DESIGN

The usefulness of stochastic optimization methods cannot beassessed by analytical benchmarks only but domain-specific testcases are equally important to evaluate their suitability for spe-cific classes of problems, which usually feature specific issues.

TABLE IISPACING AND EUCLIDEAN DISTANCES INDICES (30 RUNS)

this index uses the data of the Pareto front filtered out of 30 runs

In the context of electromagnetics device optimization, abrushless dc wheel motor benchmark was presented in [12] andseveral optimization results obtained with various algorithmsare available in literature. Furthermore, the Matlab code forcomputing the objective function is publicly available [13],thus making the comparison independent of differences in thecalculation of the objective function. These features make itideal for comparing the performances of different techniques.The original single-objective problem is characterized by five

continuous design variables (Table I summarizes the degrees offreedom and their respective ranges), and the efficiency of themotor is to be maximized (which is equivalent to minimizingthe motor losses). Furthermore, the problem is subject to six in-equality constraints, which are related to technological and op-erational and considerations regarding the specific wheel motor.The single-objective problem has been optimized with deter-ministic as well as with several different stochastic algorithmsand the best result obtained so far is with a max-imum allowed weight which was an imposedconstraint of the benchmark.A multiobjective version of the above described brushless dc

motor benchmark is used here to compare the performances ofMOFA and MOBFA. The considered objectives are the mini-mization of , where is the efficiency of the machine,and , where is the total mass of the motor, whichhas the constraint .Both algorithms were run 30 times in order to collect mean-

ingful statistics with a population size of 40 fireflies, an externalarchive of 200 solutions, and a stopping criterion of 2,500 max-imum function evaluations. A summary of results is reportedin Table II, while Fig. 6 shows the obtained Pareto fronts (fil-tered out of the 30 runs). A visual inspection of the fronts showsthe superiority of MOBFA especially in the region ,while Table II gives quantitative information on this claim.It can be observed that the proposed MOBFA obtains the

highest number of solutions on the combined Pareto front, i.e.49 vs. 43 and furthermore such solutions are better in terms of

2088 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 5, MAY 2013

Fig. 6. Pareto front (filtered of 30 runs) using MOFA and MOBFA.

TABLE IIIOPTIMIZATION RESULTS (30 RUNS)

both normalized Euclidean distance and Normalized harmonicmean.Table III shows the best result obtained by a deterministic op-

timizer on the single-objective problem compared with resultsobtained by MOBFA on the multiobjective problem. It is inter-esting to note that the best result obtained by MOBFA on thePareto front in terms of efficiency is not far from the result ofthe single-objective problem, thus showing that the true Paretofront in the neighbourhood of is very well ap-proximated by the MOBFA optimizer.

VI. CONCLUSION

In this paper, the performances of a standard (MOFA) and animproved (MOBFA) multiobjective firefly algorithms are com-pared on a significant test problem. Both techniques are capableof solving the brushless dc motor benchmark problem, with the

proposed improved MOBFA technique providing better resultsin terms of mean values of spacing, normalized Euclidean dis-tance and normalized harmonic mean indices. Future researchwill focus on enhancing MOBFA with opposition mechanisms[15].

ACKNOWLEDGMENT

This work was supported by the National Council of Scien-tific and Technologic Development of Brazil—CNPq—underGrants 476235/2011-1/PQ and 304785/2011-0/PQ and by theUniversity of Padova PRAT2011 grant CPDA115285.

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