Study on Immunized Ant Colony Optimization Wei Gao Wuhan Polytechnic University, Wuhan, Hubei 430023 P. R. China E-mail: [email protected] Abstract Ant Colony Optimization (ACO) is a new natural computation method from mimic the behaviors of ant colony.It is a very good combination optimization method. To extend the ant colony optimization, some continuous Ant Colony Optimizations have been proposed. To improve the searching performance, the principles of evolutionary algorithm and artificial immune algorithm have been combined with the typical continuous Ant Colony Optimization, and one new Immunized Ant Colony Optimization is proposed here. In this new algorithm, the ant individual is transformed by adaptive Cauchi mutation and thickness selection. To verify the new algorithm, the typical functions, such as Schaffer function and “Needle-in-a-haystack” function, are all used. And then, the results of Immunized Ant Colony Optimization are compared with that of continuous Ant Colony Optimization. The results show that, the convergent speed and computing precision of new algorithm are all very good.

1. Introduction Ant Colony Optimization (ACO) is a new natural computation algorithm from mimic the behaviors of ants colony, and proposed by Italy scholar M. Dorigo in 1990’s [1-3]. The original intention of ACO is to solve the complicated combination optimization problems, such as TSP, so the traditional ACO is a very good combination optimization method. Its basic biologic principle is briefly introduced as follows. As one kind of social insect, the behavior of ant is very simple, but the ant colony can represent very complicated behavior, and can complete very complicated task. The scientists have noticed this phenomenon for a long time. From a lot of researches, they found that, the information is delivered among ant colony by one kind of hormone. Through this kind of information exchange, the ant can cooperate and complete very complicated task. As the ant moves, the hormone is released on its path. The follow ants can

Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007

perceive this hormone and recognize its density. And have a larger probability to move along the path that has the greater hormone density. So, this is a kind of information positive feedback: the more ants move along a road, the more probability of follow ants also move along this road. Based on above principles, the ant colony optimization can solve very complicated combination optimization problems, such as TSP, Job-Shop problem, et al [4-5].

2. Immunized Ant Colony Optimization 2.1. Continuous ant colony optimization Initialization Extrinsic cycle start, k=1 Random distribution ant colony Fitness computing, record the best fitness Internal cycle start, s=1 Probability move of ant individul s=s+1 s
Yes

Hormone update to each ant k=k+1 k
Yes

Output result

Figure 1. Basic flow chart of continuous ant colony optimization For imitation information exchange among ant colony, the ACO can solve the complicated combination optimization problem, and its effect is

better than that of traditional methods. So according to the information cooperation of ant colony, the continuous optimization method based on principles of ant colony should be feasible. Based on above thought, some continuous Ant Colony Optimizations have been proposed [6-8]. To study simply and effectively, here a new continuous Ant Colony Optimization is proposed, which flow chart is as follow Figure 1.

2.2. Immunized Ant Colony Optimization From the above flow chart of continuous Ant Colony Optimization, we can see that, in the internal cycle, there is only one operation, which is the probability move of ant individual. So, the ant individual cannot explore effectively in internal cycle, the efficiency and precision of whole algorithm will be damaged largely. To improve continuous Ant Colony Optimization, here the mature methods used in evolutionary algorithm and artificial immune system are introduced into continuous Ant Colony Optimization, and a new Immunized Ant Colony Optimization (IACO) is proposed for the first time. The detailed modification is that, in internal cycle, after the probability move is finished, the self-adaptive mutation operation and immunized selection operation are used. After this kind of modification, the ant individual can move more purposively, so the search effect can be improved very high. The detailed process of IACO is as follows. 2.2.1. Creation of initial ant colony. In the solution space of optimization problem, the ant colony is created randomly, so the ant initial population can be gotten. That is to say, the ant colony is randomly distributed in the solution space. The site of one ant individual is corresponding to one point of solution space, which is one solution vector. 2.2.2. Fitness function of ant individual. Fitness result is the criterion to evaluate the ant site, and also is one of the criterions to control the progress of algorithm. The fitness function of individual is generally a kind of transformation of its objective function. The purpose of this transformation is not only to guarantee the fitness value of each individual positive, but also to maintain the diversity among individuals. 2.2.3. Probability move of ant individual. The probability move of ant individual is the key operation of algorithm. Its move probability can be expressed as follows.

Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007

Pij =

[τ j ]α [η ij ] β

∑ [τ

α

k]

[η ik ] β

k

where, τ j is hormone intensity of ant individual. At initial stage, it is a constant, which is τ 0 = c . In this study, we take c = 0.001 . τ j is related with f(i) through ∆τ . η ij = f i − f j , which express the modification quantity of objective function after the ant individual moves. When f i = f j is occurred, the movement should be repeated, until they are not equal. The α and β are two variables, which ranges are as follows, 1 ≤ α ≤ 5 , 1 ≤ β ≤ 5 . Here, we take, α = 1 and β = 5 . 2.2.4. Mutation operation. Here, the mutation operation is a kind of adaptive Cauchi mutation [9], which form is as follows. Suppose one individual of the population is X = (a1 ,..., a4 ) , the new mutated individual is X ′ = (a1′ ,..., a4′ ) , then the component of the new individual can be described as follows.

ai′ = ai +σi ⋅ T ⋅ Ci (0,1)

where, σi = 1.0

(i=1~4)

βi F( X ) + γ i

(i=1~4) where, σ i is the standard deviation of the parameters; Ci(0,1) is the Cauchi random number; F(X) is the fitness of individual; β i , γ i are two parameters, there β i = 1 , γ i = 0 ; T is a adaptive parameter, which description is as follows,

T=

T0 Fmax − Favr

where, T0 is the initial parameter, which value is 2.5; Fmax, Fmin are the maximum and minimum fitness of the current population. The adaptive mutation can make the disturbing extent of mutation adaptively changeable through the computation iterative extent, so the search performance of the whole algorithm can be improved. 2.2.5. Selection operation. According to the stimulative reaction and restraining reaction in immune system, one kind of thickness factor is introduced into selection operation to adjust the score of individual [9]. If the thickness of individual is high, the premature is easy to appear, so the individual whose thickness is high should be restrained and the selection probability

of individual whose fitness is large must be high. The detailed method is as follows, the adjustive modification based on thickness and fitness of individual is added to the score of individual, which is as follows. p′(i).scores = p(i).scores+ α ⋅ C ⋅ (1 − +β ⋅

F (i) ) ⋅ p(i).scores Fmax

F (i) ⋅ p(i).scores Fmax

In order to analysis the optimal capability of Immunized Ant Colony Optimization, two standard testing functions are used, and also the results are compared with the continuous Ant Colony Optimization.

3.1. Simulation experiment 1

where α , β are parameters which are in the range of [0~1], and here α = β = 0.5 ; C expresses the thickness of individual, whose definition is the ratio of number of individuals whose fitness is highest or near highest to total number of population, and here it can be expressed as follows, C=

3. Simulation experiments

t ⋅ (0.8 ⋅ Fmax → Fmax ) N

where, numerator is the summation of individuals whose fitness is between Fmax and 0.8*Fmax; F(i) is the fitness value of individual whose sequence number is i; Fmax if the maximum fitness of the population. From the score formula equation, we can see that, as to the individuals whose thickness is high and total number is t, if their fitness is high, their scores are little (this can be seen from second term of score formula). While if the thickness of individual is low, the score of individuals whose fitness is high must be higher due to the third term of score formula.

In this simulation experiment, the optimization function is Schaffer function[10]. Its expression is as follows. f ( x1 , x2 ) = 0.5 −

sin2 x12 + x22 − 0.5 [1.0 + 0.001( x12 + x22 )]2

The search range is −100 ≤ x1, x2 ≤ 100 . This function has infinite extremums, hut only one (0,0) is global maximum point, where the function value is 1. The character of the function is that there exists a circular raphe around the global maximum point, where the function value is 0.990284. So, when optimizing this function, the optimal method is often trapped to local maximum. The generally used optimal method is often invalidated to this function. Therefore, this function has become a standard problem to test the performance of evolutionary algorithm, and has been used in many studies. The distributing map of this function is showed in Figure 2.

2.2.6. Hormone update operation. After the above operations, the hormone information of new ant colony must be updated. The method of hormone update can be expressed as follows.

τ new = ρ ⋅ τ old j j +

∑ ∆τ

k j

k

where, ρ is volatile rate of hormone, which can be taken as about 0.3. ∆τ j is residual quantity of hormone. ∆τ j can be expressed as follows.

 Q ant moved along pathwayij ∆τ =   0 otherwise where, Q is a constant. At start, the quantity of hormone is taken as a constant, which is c. The optimal combination of parameters, which are Q, c, α , β , ρ , generally can be confirmed by trial method [1]. Apparently, these parameters can be confirmed by other optimization methods, such as evolutionary algorithm.

Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007

Figure 2. The distributing map of testing function Another character of this function is that the range of parameters is very large. As the previous study showing, the range of parameters has strongly affected to the performance of evolutionary algorithm. So, the large range of parameters of Schaffer function is a large difficulty to many evolutionary algorithms. In order to illuminate the problem and make the computing simple, here the results of thirty independent computation is used and the threshold value of evolutionary algorithm is 0.999, that is to say, when the result of evolutionary algorithm reaches this value, we think this algorithm convergent. Firstly, the comparison study between Immunized Ant Colony Optimization and continuous Ant Colony Optimization is maken.

In computation, the number of population is taken as 50, the ranges of the parameters are as [-10,10], and maximum iterative number is 100. The results are as follows. For Immunized Ant Colony Optimization, all thirty times of computation are convergent to threshold value, and the average evolutionary generation is 12.52. While at the same condition, using continuous Ant Colony Optimization, there are 28 times which are convergent to threshold value in thirty times computation, and their average evolutionary generation is 26.64. So, we can see that, the Immunized Ant Colony Optimization can certainly improve the evolutionary speed very well, that is to say that the evolutionary generation of continuous Ant Colony Optimization is the 2.13 times of Immunized Ant Colony Optimization. And then, we can draw the follow conclusions, the evolutionary mutation and immune selection operations can improve the algorithm performance very largely, and the new algorithm is a very good optimization method. Secondly, to analyze the computational capability of new algorithm, the effect of population number is studied. In this computation, the results of ten independent computation is used and the threshold value of evolutionary algorithm is also 0.999. The ranges of the parameters are taken as [-10,10], and maximum iterative number is 100. To compare, the population number is taken as four kinds, such as 100, 50, 25 and 20. And then, the computing results are as follows. With four population numbers, ten times of computation are all convergent to threshold value, that is to say, the computing stability of Immunized Ant Colony Optimization is very good, and can not be affected by population number largely. But their computing speed are different very large. When the population number is 100, the average convergent speed is 3.7. When the population number is 50, the average convergent speed is 10.2. When the population number is 25, the average convergent speed is 21.5. And when the population number is 20, the average convergent speed is 22.6. So, we can see that, the convergent speed under population number 100 is 3 times than that of population number 50, and then is 6 times than that of population number 25, and 7 times than that of population number 20. So, the population number is a very important factor for convergent speed.

3.2. Simulation experiment 2 In this experiment, the “Needle-in-a-haystack” function [10] is used, which expression is as follows.

Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007

f ( x1 , x 2 ) = 3

2

[0.05 + ( x12 + x 22 )]

+ ( x12 + x 22 ) 2

where, the search range is −5.12 ≤ x1 , x 2 ≤ 5.12 . There are four local extremums for this function, which are (+5.12, +5.12), (-5.12, -5.12), (+5.12, -5.12) and (-5.12, +5.12), and their function values are all 2748.78. And for this function, there are only one (0,0) is global maximum point, where the function value is 3600. For its complicated performance, this function has become a very typical test function for optimization method, and be used in many studies, which distributing map is showed in Figure 3.

Figure 3. The distributing map of testing function As the above study, the comparison study between Immunized Ant Colony Optimization and continuous Ant Colony Optimization is maken also in this experiment. In computation, the number of population is taken as 100, the ranges of the parameters are as [-5.12,5.12], the convergent precision is 0.00001, and maximum iterative number is 100. Here the results of 10 independent computation is used. The results are as follows. For Immunized Ant Colony Optimization, 9 times of computation are convergent to convergent precision, and the average evolutionary generation is 19.33. While at the same condition, using continuous Ant Colony Optimization, there are also 9 times which are convergent to convergent precision in 10 times computation, and their average evolutionary generation is 27.89. So, we can see that, the computing stability of two algorithms is uniform, but the speed of new algorithm is increased about two times. To compare the performance of new algorithm for two functions, the number of population is taken as 50, he convergent precision is 0.0001, and maximum iterative number is 100. The results are as follows. For Immunized Ant Colony Optimization, 5 times of computation are convergent to convergent precision, and the average

iterative number is 32. While using continuous Ant Colony Optimization, there are 6 times which are convergent to convergent precision in 10 times computation, and their average iterative number is 38. So, we can see that, the computing performance of two algorithms is about similar, but the synthetical performance of new algorithm is better. For the first function, the computing speed of new algorithm can increase two times, but for second function, the computing performance of two algorithms is about similar. So, the “Needle-in-a-haystack” function is harder for optimization methods. From the results of above simulation experiments, we can draw the follow conclusions. The new Immunized Ant Colony Optimization is a very good method for continuous optimization problems, and should be applied in complicated optimization problems widely.

4. Conclusions Ant colony optimization has been become a very useful method for combination optimization problems. Based on close connections between combination optimization and continuous optimization, nowadays some scholars have studied to apply ant colony optimization to continuous optimization problems, and proposed some continuous ant colony optimizations. To improve the performance of those continuous ant colony optimizations, here the principles of evolutionary algorithm and artificial immune algorithm have been combined with the typical continuous Ant Colony Optimization, and the adaptive Cauchi mutation and thickness selection are used to operate the ant individual, so a new Immunized Ant Colony Optimization is proposed. To verify the new algorithm, the typical functions, such as Schaffer function and “Needle-in-a-haystack” function, are used. The results show that, the convergent speed and computing precision of new algorithm are all very good. So, the new Immunized Ant Colony Optimization is a very good optimization method, and is suitable to be used in very complicated optimization problems.

5. References [1] M. Dorigo, V. Maniezzo, and A. Colorni, “Ant System: Optimization by a colony of cooperaing agents,” IEEE Trans. on SMC, 1996, pp. 29-41. [2] Daniel Angus, “Ant Colony Optimisation: From Biological Inspiration to an Algorithmic Framework,” Technical Report No. TR013, Swinburne University of Technology, Melbourne, Australia, 2006.

Third International Conference on Natural Computation (ICNC 2007) 0-7695-2875-9/07 $25.00 © 2007

[3] M. Dorigo, and C. Blumb, “Ant colony optimization theory: A survey,” Theoretical Computer Science, vol. 344, 2005, pp. 243-278. [4] M. Dorigo, and T. Stutzle, Ant Colony Optimization, Cambridge: MIT Press, 2004. [5] E. Bonabeau, M. Dorigo, and G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems, Oxford: Oxford University Press, 1999. [6] K. Socha, “ACO for Continuous and MixedVariable Optimization”, In M. Dorigo, M. Birattari, C. Blum, L. M. Gambardella, F. Mondada, and T. Stuetzle, (eds.), Ant Colony Optimization and Swarm Intelligence. Fourth International Workshop, ANTS 2004, LNCS 3172, Springer Verlag, Berlin, Germany, 2004, pp. 25-36. [7] S. H. Pourtakdoust and H. Nobahari, “An Extension of Ant Colony System to Continuous Optimization Problems”, In M. Dorigo, M. Birattari, C. Blum, L. M. Gambardella, F. Mondada, and T. Stuetzle, (eds.), Ant Colony Optimization and Swarm Intelligence. Fourth International Workshop, ANTS 2004, LNCS 3172, Springer Verlag, Berlin, Germany, 2004, pp. 294-301. [8] J. Dréo,and P. Siarry, “Continuous interacting ant colony algorithm based on dense heterarchy”, Future Generation Computer Systems, vol. 20, 2004, pp. 841856. [9] W. Gao, “Fast Immunized Evolutionary Programming”, CEC2004, 2004, pp. 666-670. [10] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, New York: Springer Press, 1996.