Multiple Heterogeneous Ant Colonies with Information Exchange Arash Arami, Bijan Rahmizadeh Rofoee, and Caro Lucas
Abstract— The method of Multiple Heterogeneous Ant Colonies with Information Exchange (MHACIE) is presented in this paper with emphasis on the speed of finding the optimal solution and the corresponding computational complexity. The proposed method which is inspired by biology and psychology has a structure composed of several ant colonies. These colonies participate in solving problems in a concurrently manner and also exchange information with each other in communicational steps. Each ant colony is considered as an intelligent agent with behavioral traits. These behavioral traits play a key role in the solving procedure, in interrelation circumstances and in installation of relations. Faster solutions have been achieved using different employments of agents in the algorithm structure. Experimental results show the superiority of Multiple Heterogeneous Ant Colonies algorithm in comparison to the standard ant colony system (ACS) and particle swarm optimization (PSO) algorithms on different benchmarks. A dynamic, control engineering benchmark is also provided in order to gain a more complete evaluation of the proposed algorithm.
I. INTRODUCTION
G
LOBAL Optimization methods are extensively used to solve different optimization problems such as task scheduling, pattern recognition and machine learning. There is an area of interest in designing and modifying Global Optimization algorithms with the aim of achieving faster solutions for the problems besides reducing computational cost and complexity. Different global optimization methods are inspired by observing the natural behaviors of biological systems such as gens, immune systems, proteins or by perceiving social behaviors of insects, birds and fishes [1-5]. Some other algorithms are inspired by social and political observations in human history such as Imperialist Competitive Algorithm [6]. Ant colony optimization methods are famous algorithms which are inspired by social behavior of ants in finding the shortest path between food and nest. This mechanism is based on pheromone deposition and its evaporation. These algorithms have gained great success in solving many of optimization problems and especially in combinational optimization problems, such as Travel Salesman Problem Arash Arami is with the Control and Intelligent Processing Center of Excellence (CIPCE), school of Electrical and Computer Engineering, University of Tehran, North Kargar Street, Tehran, Iran. (Phone:+98-21-61114313; e-mail:
[email protected] ) Bijan Rahmizadeh Rofoee is with Electrical and Computer Engineering Department, Tabriz University, 29 Bahman Street, Tabriz, Iran. E-mail :
[email protected] Caro Lucas is with the CIPCE, school of Electrical and Computer Engineering, University of Tehran, Tehran, PO Box 11365/4563, Iran. E-mail:
[email protected], and school of Cognitive Science, IPM, Tehran, PO Box 10395/5746, Iran. E-mail:
[email protected]
[7] and Graph coloration [8]. In this article, some distinct ant colonies are defined to work on the problem separately, but they have relation and cooperation with each other. Although they work concurrent, each colony has its own treatment confronting the problem and shows its own characteristics while contacting with other colonies. A set of structures are supposed for installing relationship between colonies. After random positioning of the colonies, an employed structure defines which colonies should exchange information. This paper is organized as follows: in section 2, ACS method is introduced briefly. In section 3, method of Multiple Heterogeneous Ant Colonies with Information Exchange is introduced. In section 4, the results of MHACIE, ACS, and PSO are compared with each other. Lastly the conclusion is presented. II. ANT COLONY SYSTEM (ACS) This algorithm was propounded by M. Dodrigo in 1997. In these methods, the pheromones of the edge on the paths are updated with 2 mechanisms: one of them is Local pheromone updating and the other one is Global pheromone updating. As soon as an ant passes an edge, the rule of local pheromone update is being utilized. It means that the total pheromone content of the edge diminishes which leads to probability decrease of choose of this edge by next ants. This mechanism obviously increases the chance of passing the other edges which yields to increase of diversity of searching. The Global pheromone update rule is utilized for the paths which belong to the best solution. On the other hand deposit of pheromone happens only in the edges of this path. The formulations of updating and state transition rules are cited below. State transition rule which defines the ants’ strategy for choosing next node is defined by formula 1.
Pijk (t ) = ⎧ 1 ⎪ ⎪ ⎪ 0 ⎨ ⎪ [τ ij (t )][ηij (t )]β ⎪ β ⎪ ∑k [τ iu (t )][ηiu (t )] u ∈ J ⎩ i
if {q ≤ q0 & j = j *} if {q ≤ q0 & j ≠ j *}
(1)
otherwise
Pijk is the probability of choosing the jth node after the ith node by the kth ant of a colony.
τ iu (t )
is the amount of pheromone in iu edge in time t.
The iu edge is the edge between ith and uth nodes.
ηiu (t ) encapsulates
the partial perception of ants about
the iu edge, for example, inverse of distance between ith and uth nodes. β is the relative importance parameter between τ and η . J(i) includes all the nodes that are linked to ith node. The update rules are presented in formulae (2), (3), (4), and (5).
j * = arg max u
([τ iu (t )][η iu (t )] β ) k
u∈J i
In the communication step, based on the configuration of information exchange, colonies are chosen randomly to interchange the information of their best discovered path with each other. Each colony apprises its counterpart of its best discovered path including the edges and pheromones ( Τ recieved ). The characteristics of counterpart colony have significant effect on its Start
(2)
Local pheromone updating rule:
τ ij (t + ε ) ← (1 − ρ )τ ij (t ) + ρτ 0
Colony 1
Global pheromone updating rule:
τ ij (t + 1) ← (1 − α ) τ ij (t ) + α Δτ ij (t )
Colony 2
Colony 3
Colony n
(3) Individual Attempts Step
(4)
Δτ ij (t) = ⎧⎪e × (Lgb ) −1 if edge(i, j) ∈ globalbesttour ⎨ ⎪⎩0 otherwise
Communication Step
(5) Colonies communicate using predefined configurations
n
And L gb is the length or cost of globally best tour from the beginning of the trial.
Pheromone update
Pheromone update
Pheromone update
n
III. PROPOSED METHOD In this method, some separated ant colonies cooperate to solve the problem. The incipient idea is inspired by a work done by Ellabib et al. [9]. Algorithm structure has two main steps; the first one is the individual attempts step and the next one is the communication step. In the first step, each colony attempts to solve the problem using a definite number of iterations; and in the second step, the colonies communicate with each other using preset configuration. The way colonies communicate each other is defined via the mentioned configuration. The information exchange is implemented using pheromone deposition which is described in details in section B. Flowchart of the MHACIE algorithm is presented in Fig. 1. By these assumptions, each colony can be considered as an intelligent agent that has its special treats, such as a conservative colony with a greater property of exploitation rather than exploration, or an explorer colony with more exploration property. A. Individual attempts step Colonies solve the problem separately in different iterations due to their characteristics. Decision making of each colony is based on its deposited pheromone. These colonies with different characteristics have different behavior confronting the problems. In other words, they have different exploration-exploitation proportion which is resulted from different iterations and different ways of deposition. For major exploration property, it is sufficient to increase the rate of pheromone evaporation in local update rule, or to let all the ants inside the colony to deposit pheromone in various volumes. To make a colony more hardworking, the number of exploration iterations should be increased. B. Communication step
Pheromone update
n No
Stop condition satisfied?
Yes Done
Fig. 1. Flowchart of the MHACIE
interpretation of received pheromonal information. The counterpart colony deposits pheromone on the recommended path, which is under effect of colony`s interpretation. This deposition mechanism leads to the biasing of solving procedure of the counterpart colony with an intensity proportional to γ colony depending on its interpretation. Formula 6 demonstrates the counterpart pheromone deposition.
{
' Τ recieved = τ ih' ,τ hk' , ...,τ mn
}
(6)
Τ← ⎯⎯ Τ + γ colony × Τ recieved Τ represents the amount of pheromones of the counterpart colony on the edges. Also, interchanges and cooperations which every colony has with others may be different from all because of its own characteristics (a stubborn colony or a colony with the lack of self confidence). Furthermore, how much the colony pays heed to its results or others results is important. Some examples for information exchange configuration are provided in Fig. 2. The connected colonies are capable of exchanging information.
a
b
c
d
Fig. 2. Structures and configurations of information exchange between colonies. (a) In presence of leader (b) chain form (c) cubic (d) one by one.
By utilizing various types of interchanges between colonies, better solutions might be accessible. An operator for controlling the way of searching of the colonies could be gained by considering the colony types and the correlation between them. Total computations of iterations are approximately equal to the sum of computations in partial iterations of each colony. IV. EXPERIMENTAL RESULTS In order to solve different sorts of problems using various types of ant colony optimizations (ACO), it is necessary to have the problems graph or constructions graph. For this purpose, a fully connected graph is used (Fig. 3.). Each variable is coded into definite number of nodes (bits variables). Two additional nodes are added to the bits variables, as the start and the end of the paths. Nodes are all accessible from the first node; also the final node is reachable from all other nodes. When an ant visits a node, it sets its bit to one and obviously unvisited nodes stay zero. For comparing the results of the MHACIE method with other global optimization algorithms, three functional benchmarks are utilized and described as follows.
Fig. 3. Fully connected construction graph
f1)
( x12 + x22 ) 0.25 (1 + sin 2 50 ( x12 + x22 ) 0.1 ) xi ∈ [−10, 10] f2) x12 + 2 x22 − 0.3 cos( 3πx1 ) − 0.4 cos( 4πx2 ) + 0.7
xi ∈ [−10, 10]
sub-iteration. In the other words, each iteration is equivalent to 30 ACS iterations. The result of standard ACS method and the proposed method are compared in Table 2 on iteration number. M.I.N. (Mean Iteration Number) is the average required iterations (equal iterations in the MHACIE) and S.D. is the Standard Deviation of min required iterations to obtain the best solution. The solution is valid when the difference between error cost function response and the main value is less than 0.0001 in predefined number of iterations for each method (2400 iterations for ACS, 3000 iterations for PSO, and 80 iterations for the proposed method). The multiplied number 30 in M.I.N. column of the proposed method shows the scaled measurement which provides a comparison of the proposed method and ACS (Table 2). Then two different methods, Ant Colony System (ACS) and Particle Swarm Optimization (PSO) are employed to solve benchmark problems. The results, which are obtained by using MATLAB 7.1 Profiler, of the proposed method, ACS and PSO are demonstrated in Table 3. In addition, evolution of cost of f1 problem and convergence of methods are demonstrated in Fig. 5, Fig. 6, and Fig. 7. The achieved results show that the proposed method has so rapid and powerful ability to solve optimization problems. Another analysis of the MHACIE is presented in Table 4. This table shows colonies characteristics in detail and the percentage of wins of each colony (finding global solution faster). Three first columns of the Table 4 show the prosperity percentage for each colony. Wining means the less time to achieve the global solution. Obviously, colony No. 1 has a great part in this method. The fifth column has three parts: Evaporation rate in global pheromone update; evaporation rate in local pheromone update; and the ants which deposit pheromone in each colony. The sixth column is determinant of e coefficient in global pheromone update
f3)
( x1 − x2 ) + ( x1 + x2 − 10 / 3) 2
2
xi ∈[ −20, 20] As an example, 3D plot of benchmark function f1 is depicted in Fig. 4.
Fig. 4. 3D plot of f1benchmark
In order to solve the problems, four ant colonies are generated (MHACIE: n = 4 ). The characteristics of each colony are stated in Table 4. Information exchange between colonies is performed by the d configuration. In order to find out which combination of colonies leads to the fastest solution, experiments are done using different sets of colonies. Each problem variable is coded with 10 bits (10 nodes for each variable). Condition of the experiment for each problem is 100 times of executing algorithms, running on a Pentium4 with 2.4 GHz CPU, 512 Mbytes of RAM and MATLAB program version 7.1. Results of different combinations are gathered in Table 1. It must be noticed that this table just contains the best solutions. Based on results from Table 1, the combination of all four different colonies lead to better solutions. In each iteration of the MHACIE method, the first and the second colonies execute their searches for 10 subiteration while the third and the fourth ones search in 5
TABEL I EMPLOYING DIFFERENT COMBINATIONS OF COLONIES F1 F2
F3
Combination of colonies
E.T.T
S.D.T.
E.T.T
S.D.T.
E.T.T
S.D.T.
1,2,3,4
1.124333
0.168999
1.4492
0.428768
4.613
1.829024
1,1,3,3
2.504125
1.627298
1.9135
0.497803
4.873
1.91601
2,2,4,4
1.4811
0.51695
3.0521
2.17995
5.528
2.67370
1,1,2,2
1.8274
1.378858
2.3404
0.87752
5.037
2.861545
1,1,2,3
1.6694
0.75231
1.6659
2.115663
4.934
3.326735
E.T.T. is the expected total time of achieving the global solution, S.D.T. is the Standard Deviation of procedure time.
TABEL II
Results of Functional Benchmark solutions (Iteration comparison) Index of bench marks
Conventional ACS
MHACIE
V.S.P.
M.I.N.
S.D.
V.S.P.
M.I.N.
S.D.
f1
63%
933.091
987.013
100%
4.28× 30 = 128.4
2.654 × 30 = 79.62
f2
92%
696.273
801.842
100%
4.773× 30 = 143.19
7.843 × 30 = 235.29
79%
345.444
439.761
97%
10.05 × 30 = 301.5
10.091 × 30 = 302.73
f3
Comparison between standard ACS and the MHACIE on different benchmarks V.S.P. is the percentage of valid solutions.
Index of benchmarks
TABLE III Results of Functional Benchmark solutions Conventional ACS
PSO
MHACIE
V.S.P
E.T.T
S.D.T.
V.S.P
E.T.T
S.D.T.
V.S.P
E.T.T
S.D.T.
f1
63%
16.241
0.812354
100%
15.803
0.414365
100%
1.1243
0.16899
f2
92%
7.8351
0.527082
100%
7.6571
0.273649
100%
1.4492
0.428768
f3
79%
6.9195
1.360384
95%
7.1901
0.228136
97%
4.613
1.829024
These results are achieved using MATLAB 7.1 Profiler. E.T.T. is the expected total time of achieving the global solution, S.D.T. is the Standard Deviation of procedure time, V.S.P. is the percentage of valid solutions..
Cost evolution
Cost Evolution
4
1.4
3.5
1.2
3
1 2.5
Cost
cost
0.8 2
0.6
1.5 1
0.4
0.5
0.2
0
0
500
1000
1500
2000
2500 3000 Iteration
3500
Fig. 5. Cost evolution for applying ACS to f1 problem
4000
4500
5000
0
1
1.5
2
2.5 Iteration
3
3.5
4
Fig. 7. Cost evolution for applying MHACIE to f1 problem
which is related to each colony characteristic. The seventh column shows the γ colony
(introduced in formula 6). The last column indicates the characteristics of each colony.
Fig. 6. Cost evolution of applying PSO to f1 problem
Application in tune of PID controller In tuning the PID controller, three parameters called Kp , Ki, and Kd must be calculated, so there are three kinds of nodes in problems graph. The cost function is defined as below: Cost = 3 × max_ overshoot + 2 × rise _ time + 6 × setteling _ time + IAE
2(+ S)( =
1 S2+
TABLE IV COLONY BEHAVIOR ANALYSIS Percent of wins colonies
2 sys
internal iterations Number per epoch f1
f2
f3
colony I
41%
44%
22%
colony II
32%
29%
26%
colony III
19%
16%
34%
colony IV
8%
11%
18%
Rate of Pheromone vaporization and use of ants information
Global : --Local : 0.1 all colony’s ants inf. Global : 0.1 10 Local : 0.2 all colony’s ants inf. Global : 0.2 Local : 0.1 5 The most the most successful ant of colony inf. Global : --Local : 0.3 5 all colony’s ants inf. Characteristics of each colony and their prosperity in solving propounded benchmarks 10
This cost function provides a better transient response for the system by decreasing the overshoot, settling time and rise time. Also, using IAE makes some improvements in permanent response. PID controllers are designed for 2 different systems below, and the results are compared with Ziegler-Nichols PID. The results are demonstrated below: (S + 2) sys1 = 3 ( S + 5S 2 + 5S + 5) Designed controllers are: 27.88 + 15 PIDzigler − nichols = 2.02 S + S 7.139 PID proposed _ a lg orithm = 4.292 S + + 4.907 S The step response of the main system and the step response after applying compensators are shown in Fig. 9. The response characteristics are brought to table 5.
PID:
e
Intensity of influencing by the others
Colony personality In society
2
0.01
hardworking opinionated
2
2
hardworking credulous
4
1
Not very hardworking Logical
2
0.5
Not very hardworking Recluse
TABLE V PERFORMANCE INDEXES Settling P.O. Time
Rise time
Ziegler4.28s 39.5% 0.34s Nichols Proposed 1.39s 1.65% 0.89s algorithm Comparison Between Ziegler-Nichols PID and tuned PID (system 1)
Designed controllers are:
PID zigler − nichols = 0.757 S + PID proposed _ a lg orithm = 4.59 S +
1 + 1.74 S 2.959 + 3.628 S
Step responses of the main system and the system after applying compensators are shown in Fig.10. Also the response characteristics are brought to table 6.
PID:
TABLE VI PERFORMANCE INDEXES Settling P.O. Time
Rise time
Ziegler11.7s 0% 6.4s Nichols Proposed 4.21s 1.58% 0.904s algorithm Comparison Between Ziegler-Nichols PID and tuned PID (system 2)
Fig. 9. Step response of system1(blue), with Ziegler-Nichols PID(green), with tuned PID(red)
[9] Issmail Ellabib, Paul Calamai, Otman Basir, “Exchange strategies for multiple Ant Colony System” Elsevier, Information Sciences 177 (2007) 1248–1264. [10] E. Alba, G. Leguizamon, G. Ordon˜ ez., “Parallel ant algorithms for the minimum tardy task problem”, in: CongresoArgentino de Ciencias de la Computacion, San Justo, Argentina, October 4–8, 2004, vol. CACIC’04, pp. 1835–1846.
Fig. 10. Step response of system2(blue), with Ziegler-Nichols PID(green), with tuned PID(red)
The results show the better performance of the MHACIE in short searching period.
V. Conclusion A modified optimization algorithm named Multiple Heterogeneous Ant Colonies Optimization with Information Exchanges is presented. In this method, different types of colonies are attempting to solve the problem with concurrent processes and information exchange. The way that colonies explore and contact with each other depends on behavioral traits and characteristics of each colony. For installing relations between colonies some configurations are presented and applied. Experimental results on different functional benchmarks show that the MHACIE method can reach to the optimum solution in less searching time and lower computation effort rather than ACS and PSO methods. Furthermore, the proposed method is applied to tune PID controllers. All the achieved results show that the MHACIE is very rapid and furnishes a powerful tool in solving global optimization problems. REFERENCES [1]
[2] [3] [4]
[5]
[6]
[7]
[8]
M. Dorigo, G. Di Caro, “The ant colony optimization metaheuristic”. In D. Corne, M. Dorigo, F. Glover (Eds.), new Ideals in optimization, McGraw-Hill 1999. R.L. Haupt and S. E. Haupt, “Practical Genetic Algorithms”, second edition, New Jersey: John Wiley & Sons, 2004. M. Melany, “An Introduction to Genetic Algorithms”, Massachusetts, MIT Press, 1999. Walter J. Gutjahr, “A Graph-based Ant System and its convergence”, Elsevier, Future Generation Computer Systems 16 (2000) 873–888. M. Dorigo, V. Manieezzo, and A. Colorni, “Ant system optimization by a colony of cooperating agents”, IEEE Trans. On System, Man, and Cybernetics-Part B: C bernetics, vol. 26, no. 1, pp. 29-41, February 1993. E. Atashpaz_Gargari, C. Lucas, “Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic competition”, Proc. IEEE Congress on Evolutionary Computing, 4661-4667, 2007. M. Dorigo, L.M. Gambardella, “”Ant Colony System: A cooperative learning approach to the Traveling Salesman Problem”, IEEE Transactions on Evolutionary Computation, Vol.1, No.1, 1997. D. Costa and A. Hertz, “Ants can color graph”, J. Op Res. Soc., vol. 48, no. 3, pp. 295-305, Mar. 1997.
