978-1-4673-6469-0/13/$31.00 © 2013 186

Constraint Handling in Firefly Algorithm

Aditya M. Deshpande, Gaurav Mohan Phatnani Optimization and Agent Technology (OAT) Research Lab

Maharashtra Institute of Technology, 124 Paud Road Pune 411038, India

{amdeshpande, gmphatnani}@oatresearch.org

Anand J. Kulkarni Optimization and Agent Technology (OAT) Research Lab

Maharashtra Institute of Technology, 124 Paud Road Pune 411038, India

ajkulkarni@oatresearch.org; kulk0003@ntu.edu.sg

Abstract—Most of the contemporary nature-/bio-inspired techniques are unconstrained algorithms. Their performance may get affected when dealing with the constrained problems. There are number of constraint handling techniques developed for these algorithms. This paper intends to compare the performance of the emerging metaheuristic swarm optimization technique of Firefly Algorithm when incorporated with the generalized constrained handling techniques such as penalty function method, feasibility-based rule and the combination of both, i.e. combined approach. Seven well known test problems have been solved. The results obtained using the three constraint handling techniques are compared and discussed with regard to the robustness, computational cost, rate of convergence, etc. The associated strengths, weaknesses and future research directions are also discussed.

Keywords—Metaheuristic, swarm optimization technique, firefly algorithm, penalty function, feasibility-based rule

I. INTRODUCTION The performance of most of the unconstrained optimization

algorithms such as Evolutionary Algorithms (EAs), Genetic Algorithms (GAs), Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), Simulated Annealing (SA), Probability Collectives (PC), etc. may significantly get affected when applied for solving constrained problems [1-6]. The most adopted approach for handling constraints is the penalty function approach which can be referred to as a generalized constraint handling method because of its simplicity and ability to handle nonlinear constraints and it can be used with most of the unconstrained optimization methods [4, 7, 8]. This approach is highly sensitive to the choice of the associated penalty parameter and its updating scheme, which may require several associated preliminary trials as well [5, 7]. Another approach is the use of feasibility-based rule which allows the objective function and the constraint information to be considered separately [5, 9, 10]. According to [9-11], this approach is very effective to make the solution jump out of possible local minima; however, it may result into slower convergence and increased computational cost.

The Firefly Algorithm (FA) is an emerging metaheuristic swarm optimization technique based on the idealized behavior of the �ashing characteristics of �re�ies [12-17]. According to [12, 16], the FA may converge very quickly as the fireflies perform independent and parallel search in the problem solution space. The FA with penalty function approach has

been tested solving a constrained Economic Dispatch (ED) problem [16], as well as constrained FA (CFA) with feasibility-based rule [15] solving well known test problems [15, 18-22]. This paper intends to compare and investigate the performance of FA using three constraint handling approaches, mainly, penalty function approach, a variation of the feasibility-based rule originally presented in [11] and in association with FA in [15], and a combination of the penalty function and feasibility-based rule referred to as combined approach. The FA with these approaches was applied solving seven (G04, G06, G07, G08, G09, G11, G12) well studied constrained problems with equality and inequality constraints [15, 18-20]. These problems include polynomial, quadratic, nonlinear and cubic objective as well as constraint functions. The results highlighted clear distinction between the three constrained FA approaches with respect to the convergence rate, robustness and associated computational cost.

The paper is organized as follows. Section II in detail describes the FA. The associated three constraint handling techniques are described in Section III. The discussion on the performance of the FA with the three constraint handling techniques for solving seven test problems is provided in Section IV. Some concluding remarks and future work are at the end in Section V.

II. FIREFLY ALGORITHM As mentioned before, the FA is an emerging metaheuristic

swarm optimization technique based on the natural behavior of fireflies. The natural behavior of fireflies is based on bioluminescence phenomenon [12, 16]. They produce short and rhythmic flashes to communicate with other fireflies and attract potential prey. The light intensity/brightness I of the flash at a distance r obeys inverse square law, i.e. 21I r∝ in addition to the light absorption by surrounding air. This makes most of the fireflies visible only till a limited distance, usually several hundred meters at night, which is enough to communicate. The flashing light of fireflies can be formulated in such a way that it is associated with the objective function to be optimized, which makes it possible to formulate optimization algorithms [12, 16].

According to [12-17], three idealized rules for the basic structure of FA are as follows: 1) all fireflies are unisex and therefore one firefly will be attracted to other fireflies regardless of their sex; 2) attractiveness is proportional to their brightness, thus for any two flashing fireflies, the less brighter

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one will move towards the more brighter one. Also as the attractiveness is proportional to the brightness, they both decrease as their distance increases. If there is no brighter one than a particular firefly, it will move randomly; 3) the light intensity iI of any firefly i is determined according to the nature of an objective function, i.e. if the problem is to maximize the objective function then the firefly having the larger value for objective function is considered brighter and for the minimization problem, the firefly having lower value of objective function is considered brighter. Simplest form for variation of light intensity I along with the distance between the source and the observer is given by

20 /I I r= (1)

where 0I is the intensity of light at the source and r is the distance between the observer and the source; however, to avoid the indeterminate condition when the distance between the source and the observer becomes zero, i.e. I = ∞ and also taking into account the effect of light absorption due to the surroundings the following Gaussian form is used

2

0rI I e γ−= (2)

Furthermore, the attractiveness β of any firefly is proportional to its light intensity seen by the other fireflies adjacent to it. It is given by

2

0re γβ β −= (3)

where, r is the distance between two fireflies, 0β is attractiveness at 0r = , and γ is light absorption coefficient. The value of 0β and γ can be chosen from within ( ]0,1 and

[ ]0,1 , respectively.

The distance between fireflies i and j at ix and jx , respectively, is given by

( )2

, ,1

d

ij i j i k j kk

r x x x x=

� �= − = −� �� �� (4)

where d is the number of dimensions and k represents the dimension under consideration. The movement of any firefly i towards more attractive firefly j is determined by

2

0 ( ) ( 0.5)ijri i j ix x e x x randγβ α−= + − + − (5)

where α is randomization coefficient and rand is a random number generator uniformly distributed in [0,1]. The choice of the value of α essentially decides rate of convergence. A high value of α may lead to premature convergence while a low value may lead to delayed convergence.

III. CONSTRAINT HANDLING IN FIREFLY ALGORITHM In accordance with the FA, three types of constrained

handling methods, mainly, penalty function method, feasibility-based rule and the combined approach are discussed here.

A. Penalty Function Method In this method, the constrained problem is transformed into unconstrained one by forming a pseudo-objective function as follows

2 2

1 1( , ) ( ) [ ( )] [ ( )]

p m

i ii i

r f r h gφ +

= =

= + +� �

�� �x x x x (6)

where ( ) max(0, ( ))i ig x g x+ = , ( )ig x represents the inequality constrains where ( ) 0ig x ≤ and ( )ih x represents the equality constrains and r is the scalar penalty parameter.

B. Feasibility-based Rule In every iteration, a variation of the feasibility-based rule [4,

11, 15] was applied to compare the solution associated with every individual firefly i with every other firefly j . The rule is given below.

1. If both fireflies are at feasible positions and firefly j is at better position than firefly i then firefly i moves towards firefly j .

2. If firefly i is at an infeasible position and firefly j is at a feasible position then i moves to firefly j .

3. If positions of firefly i and firefly j are infeasible and number of constrains satisfied by firefly j are more than that of firefly i then firefly i moves to firefly j .

4. Once the position of the firefly is updated using above rules 1 to 3, if the updated position of the firefly i presents improved solution over the solution associated with its previous iteration position, then firefly i accepts its current solution, else retains its previous iteration solution.

Firefly Algorithm using Feasibility-based Rule Objective function ( )f x , ( )1,...,

Tdx x=x

Assign random initial position ix ( )1,...,i n= to each firefly i

Light intensity iI at ix determined by ( )if x Define light absorption coefficient γ while ( )_t max generations<

for 1:i n= for 1:j n= Compare fireflies i and j as per feasibility-based rule; Obtain attractiveness β of fireflies i and j ;

Attractiveness varies with distance r via 2exp rγ� �−� �

Evaluate new solutions; Compare the old and the new solutions given by firefly i using feasibility-based rule; Move firefly i towards firefly j in d dimensions; Evaluate new solutions and update light intensity β end for j end for j Rank the fireflies and find the current best end while Post process and results evaluation

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TABLE I SUMMARY OF MAIN PROPERTIES OF THE BENCHMARK PROBLEMS

Problem Nature Problem type No. of variables Optimum values LI NI NE a F S G4 Minimization Quadratic 5 -30665.5 9 0 0 2 52.123% G6 Minimization Cubic 2 -6961.814 0 2 0 2 0.006% G7 Minimization Quadratic 10 24.306 3 5 0 6 0.000% G8 Maximization Non-linear 2 0.095825 0 2 0 0 0.856% G9 Minimization Polynomial 7 680.63 0 4 0 2 0.512%

G11 Minimization Quadratic 2 0.75 0 0 1 1 0.000% G12 Minimization Quadratic 3 1 0 9 0 0 4.779%

NE : Nonlinear Equality, NI : Nonlinear Inequality, LI : Linear Inequality, a : number of active constraints at optimum

TABLE II SUMMARY OF THE BEST, MEAN AND WORST FA RESULTS

Problem Penalty Function Feasibility-based Rule Combined Best Mean Worst Best Mean Worst Best Mean Worst

G04 -30664.905 -30657.036 -30636.6 -30665.694 -30664.862 -30664.2 -30665.035 -30664.671 -30663.9 G06 -6960.3151 -6936.5239 -6916.64 -6960.6717 -6955.8041 -6950.63 -6960.5123 -6956.6334 -6953.47 G07 24.685374 25.28593 25.95709 24.61202 24.64413 24.81805 24.380464 24.47053 24.60237 G08 0.095825 0.0958243 0.095822 -0.095825 0.0958244 -0.09582 -0.095825 0.0958246 0.095824 G09 680.7697 681.48073 682.3398 680.67764 680.9101 681.3614 680.84633 681.04151 681.2603 G11 0.7490034 0.7490136 0.749018 0.7490003 0.7490027 0.74901 0.7490001 0.7490024 0.749012 G12 0.9999485 0.9999469 0.999946 1 1 1 0.999948 0.9999466 0.999946

TABLE III PERFORMANCE OF FA

Problem Penalty Function Feasibility-based Rule Combined FE SD Time (sec) FE SD Time (sec) FE SD Time (sec)

G04 8400 11.74036 1.04 12000 0.396542 37.70 12000 0.475541 41.78 G06 18000 12.40036 3.64 12000 3.287116 38.54 12000 2.192753 42.24 G07 30000 0.452786 6.42 30000 0.10338 50.26 50000 0.059658 79.84 G08 24000 1.02E-06 1.78 12000 6.74E-06 15.34 14000 2.88E-06 18.08 G09 12000 0.54283 2.32 12000 0.21128 26.83 12000 0.15336 27.06 G11 12000 1.31E-05 3.58 12000 2.91E-06 35.68 12000 3.42E-06 38.45 G12 20000 9.22E-07 21.76 12000 5.07E-09 305.74 12000 7.78E-07 314.50

TABLE IV COMPARISON OF THE BEST SOLUTIONS OF FA WITH OTHER CONTEMPORARY APPROACHES

Problem GA [18] SAF [19] PSO [20] CFA [15] FA

Penalty Function Feasibility-based Rule Combined G04 -30665.5386 -30665.5 -30665.5 - -30664.905450 -30665.69408 -30665.03468 G06 -6961.8139 -6961.8 -6943.5 -6961.813 -6960.315063 -6960.671654 -6960.51225 G07 24.3294 24.48 24.6269 24.308 24.68537401 24.61201961 24.38046395 G08 0.095825 0.095825 0.09583 0.095825 0.095825015 -0.095825027 -0.095825 G09 680.6303 680.64 680.837 680.63 680.7696984 680.6776396 680.8463346 G11 0.75 0.75 0.75 0.749 0.749003433 0.749000308 0.749000097 G12 - - 1 1 0.999948524 1 0.999948

C. Combined Penalty Function and Feasibility-based Rule In this method, in every iteration, the pseudo-objective function ( , )rφ x in (4) is minimized and the acceptance or rejection of the solution associated with the position of the individual firefly i , is decided based on the feasibility-based rule discussed above. This approach may help any firefly at infeasible position to quickly move towards the improved and feasible solution.

IV. EXPERIMENTS AND RESULTS The FA incorporated with the above discussed constraint

handling techniques was tested solving seven constrained test problems [21, 22]. These problems include polynomial, quadratic, nonlinear and cubic objective as well as constraint functions. The characteristics of these problems along with the ratio of the size of the feasible search space F to the size of the whole search space S (which represent the degree of

difficulty) are presented in Table 1. These problems are well studied in the literature [15, 18-20]. The approach of GA [18] was incorporated with the repair approach utilizing the gradient information derived from the constraints set. This approach may become tedious when the number of constraints increases. In addition, clear problem specific information may be necessary to decide the magnitude by which the solution requires to be pushed towards the feasibility. In Self-adaptive Fitness formulation (SAF) [19] the dimensionality of the problem is reduced by representing the constraint violations by a single infeasibility measure. The approach is applied in three stages. First, each individual is assigned infeasibility, second, the bounding solutions of the search space are identified, and finally, the infeasible solutions are penalized. The convergence rate of the SAF essentially depends on the selection of the associated penalty parameter. The PSO approach [20] is driven essentially by the optimal particle movements being generated in every iteration of the algorithm. Similar to the approach presented in this paper, the

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Constrained FA (CFA) [15] is also based on the feasibility-based rule [11]; however, while handling constraints, it considers the position of other fireflies only. On the other hand, FA with the feasibility-based rule implemented here considers the position of every firefly and its associated feasibility even after its movement to a new position. This may reduce the number of function evaluations required considerably.

Every Problem was solved 30 times using the FA three constraint handling techniques discussed in Section III. The firefly population size n , randomization coefficient α and absorption coefficient γ were chosen to be 20, 0.2 and 1.0, respectively. The penalty parameter r was chosen as 1015 and kept constant throughout the run of the algorithm. The algorithm was coded in MATLAB 7.7.0 (R2008B) on Windows platform with Intel i5, 2.8GHz processor speed and 4GB RAM. The best, mean and worst solutions of the FA associated with the penalty function approach, feasibility-based rule and the combined approach is presented in Table 2. The standard deviation (SD), number of function evaluations (FE) and the computational time are presented in Table 2, and the solution comparison with the contemporary optimization techniques such as GA using gradient based method [18], Self-Adaptive Fitness (SAF) approach [19], PSO [20], and CFA [15] is presented in Table 3.

Since the standard deviations for feasibility-based rule and the combined approach are lower, as shown in Table 3, it can be concluded that the feasibility-based rule and the combined approach are more robust than the penalty function approach solving these problems. This may be because in case of FA with the feasibility-based rule and combined approach, no firefly moves towards a firefly which is infeasible, whereas, in case of the FA with penalty function approach, each firefly is attracted towards every other firefly, with a better pseudo-objective function value, even though the other firefly may be infeasible. In addition, in case of the FA with feasibility-based approach delayed convergence with increased computational cost was observed. The computational time for the FA with penalty function approach was marginally shorter as compared with the FA with feasibility-based rule. In addition, further delayed convergence for the FA with the combined approach was observed. The comparison of the best FA solutions using the three constrained handling techniques incorporated into it with the other contemporary algorithms solving the test problems is presented in Table 4. It could be concluded that the performance of FA was competitive, if not better.

V. CONCLUSIONS AND FUTURE WORK The paper discussed the performance of FA with three

constraint handling techniques, mainly, penalty function approach, feasibility-based rule and the combination of the penalty function and feasibility-based rule, i.e. combined approach incorporated into it. The comparison was carried out by solving a variety of constrained test problems. A variation of the feasibility-based rule originally proposed in [11] and further incorporated into FA [15] as well as its combination with penalty function approach for handling constraints in FA was also successfully presented.

Similar to other algorithms and in agreement with the no-free-lunch theorem [23] several parameters associated with the FA such as firefly population size n , randomization coefficient α and absorption coefficient γ affect the rate of convergence as well the quality of the solution. It also necessitated some preliminary trials choosing these parameters. This may necessitate a self adaptive scheme to fine-tune these parameters. Furthermore, the robust and competitive FA solutions underscored its future potential solving more realistic problems such as machine shop scheduling and urban traffic control [24], structural optimization [9,25] as well as problems involving uncertainty [26] etc. In addition, the constraint handling technique may be further improved/developed using a multi-criteria optimization approach [1-3, 27], and can further be applied in fault-tolerant systems [28]. The authors also intend modify the FA to solve the problems involving logarithmic constraints.

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