The Effect of Updating the Local Pheromone on ACS Performance using Fuzzy Logic

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 7, No. 4, August 2017, pp. 2161~2168
ISSN: 2088-8708, DOI: 10.11591/ijece.v7i4.pp2161-2168  2161
Journal homepage: http://iaesjournal.com/online/index.php/IJECE
The Effect of Updating the Local Pheromone on ACS
Performance using Fuzzy Logic
Abdellatif El Afia, Safae Bouzbita, Rdouan Faizi
National School of Computer Science and Systems Analysis (ENSIAS), Mohammed V Universityin Rabat, Morocco
Article Info ABSTRACT
Article history:
Received Feb 21, 2017
Revised May 13, 2017
Accepted May 27, 2017
Fuzzy Logic Controller (FLC) has become one of the most frequently utilised
algorithms to adapt the metaheuristics parameters as an artificial intelligence
technique. In this paper, the 𝜉 parameter of Ant Colony System (ACS)
algorithm is adapted by the use of FLC, and its behaviour is studied during
this adaptation. The proposed approach is compared with the standard ACS
algorithm. Computational results are done based on a library of sample
instances for the Traveling Salesman Problem (TSPLIB).
Keyword:
Ant colony algorithm
Fuzzy logic controller
Parameter adaptation
Copyright © 2017 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Abdellatif El Afia,
National School of Computer Science and Systems Analysis (ENSIAS),
Mohammed Ben Abdallah Regragui Avenue, Madinat Al Irfane, BP 713, Agdal Rabat, Morocco.
Email: a.elafia@um5s.net.ma
NOMENCLATURES
J Random proportional rule
Q Random variable uniformly distributed between [0, 1]
Parameter to control exploration and exploitation
Set of cities not yet visited by ant k positioned on city r
length of a nearest neighbour tour and n is the number of cities
length of the globally best tour found from the beginning of the algorithm
length standard deviation
Greek Symbols
pheromone amount between city r and city s
Heuristic information between city r and city s
𝛽 Parameter that determines the relative importance of pheromone versus heuristic value
𝜉 Local pheromone evaporation parameter
Initial value of the pheromones
𝜌
𝜇
Global pheromone evaporation parameter
Mean of lengths
1. INTRODUCTION
The Ant Colony System (ACS) is one of the most effective extensions of the basic Ant System (AS)
algorithm (Dorigo and Gambardella, 1996) [1-3]. The ACS algorithm is described in pseudo-code (Figure 1).
The procedure of ACS technique consists of three steps: Pheromone initialization, Construct Ants Solutions,
and Global pheromone updating.

 ISSN: 2088-8708
IJECE Vol. 7, No. 4, August 2017 : 2161 – 2168
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Procedure-ACS
Set parameters, initialize pheromone trails
While (termination condition not met) do
Place each ant in a randomly chosen node
Construct Ants Solutions
Update global pheromones
End
Figure 1. The Ant Colon System procedure.
The Ant Colony System applied to Transport Salesman Problem can be formulated as follow: m
ants are initially placed on n cities randomly, each ant builds a complete solution in the Construct Ants
Solution using the so called pseudo-random proportional rule Equation (1) and “Equation (2):
[ ][ ] (1)
Where:
: the set of cities not yet visited by ant k positioned on city r.
: the pheromone amount between city r and city s.
: the heuristic information between city r and city s.
𝛽 : the parameter that determines the relative importance of pheromone versus heuristic value.
That is, the best edge is chosen with a probability . Otherwise, with probability (1 - ), an edge is
selected by biased exploration according to the following Equation:
{
[ ] [ ]
∑ [ ][ ] (2)
During the construction of solutions, each ant modifies the amount of pheromone on the visited
edges by applying the local updating rule:
( ) 𝜉 ( ) 𝜉 (3)
Where:
𝜉 : the local pheromone evaporation parameter.
: the initial value of the pheromones.
In the global pheromone step, once all ants have built a solution, the amount of pheromone on edges
is modified again by the best ants, using the global updating rule:
( ) 𝜌 ( ) (4)
Where:
: the length of the globally best tour found from the beginning of the algorithm.
𝜌 : the global pheromone evaporation parameter.
Several approaches have been proposed to adapt the parameters on Ant Colony algorithms [4-8].
Espacially the adaptation using Fuzzy Logic, in this part the most important are presented. In the first
approach, Amir et al., developed in their work, a Fuzzy Logic Controller (FLC) to adapt the parameters and
automatically throughout the run according to definite performance measures which are the error of the
best-so-far tour compared to the best-known tour to the TSP problem and the variance among the solutions
found by the population of ants.
Also, in [9] FLC was used to improve ACO. In their work, a solution is constructed by an ant based
on pheromone trails and heuristic information for solving the reliability problem for a series system, in order

IJECE ISSN: 2088-8708 
The Effect of Updating the Local Pheromone on ACS Performance using Fuzzy Logic (Abdellatif El Afia)
2163
to find one technology for each subsystem. The Fuzzy set of this approach is the heuristic information related
to the subsystem in consideration. In [10] Olivas et al., suggested an improved ACO by dynamically adapting
the responsible parameter for the evaporation of the pheromone with the use of fuzzy logic. In this paper,
authors tried to control the abilities for diversification and intensification of the search space. To do so, they
used two metrics that measure the algorithm performance, which are diversity and iteration as inputs of the
Fuzzy system and the parameter as output. On the other hand, Fuzzy Logic was used to vary other
metaheuristic parameters. Valdez et al [11] described a hybridization of PSO and GA using Fuzzy Logic for
decision making and parameters adaptation. Thus, three Fuzzy system were proposed; the first one is
responsible for deciding which are the best results of the FPSO + FGA, and the two second ones are in
charge of changing the values of the crossover the mutation , the social acceleration , and the
cognitive acceleration . In [12] a Fuzzy Logic approach was proposed to dynamically adapt the cognitive
and the social factors and to improve the convergence and diversity of the population in PSO algorithm.
Three Fuzzy Systems were modelled for adapting the and parameters, with respect to three performance
measures, which are the diversity of the swarm, the average error, and the iterations of the algorithm. The
performance of ACS depends strongly on the values set to parameters. Thus, varying parameters while
solving a problem can enhance the performance of the algorithm. In this paper, the Fuzzy Logic was used to
dynamically adapt the pheromone parameters of ACS based on the approach proposed by Olivas et al [10].
To do this, first we applied the Fuzzy Logic to the 𝜉 parameter, then we compared the obtained results with
the standard ACS algorithm.
This paper is constructed as follows: in Section 2 we outline the proposed method. The experimental
part is discussed in the third section. From our experimental results, we give some conclusions and
suggestions for future work in Section fourth.
2. RESEARCH METHOD
To dynamically adapt the local pheromone parameter in ACS, we have used a Fuzzy Logic System
[14] that is characterized by two inputs and one output. In fact, as inputs in each case we used a proportion of
elapsed iterations and a measure of the diversity of the colony compared to the best ant, while the output is
the parameter to be adapted, in our case the parameter. Where,
(5)
∑ √∑ ( ̅ ) (6)
where, Current iteration is the number of past iterations, and total of iteration is the entire number of
iterations needed for testing the algorithm, m is the population size, i is the number of the ant, n is the entire
number of dimensions, j is the number of the dimension, is the j dimension of the ant i, ̅ is the j
dimension of the current best ant of the colony.
Procedure CAS-FLC
1. Calculate Iteration from “Eq.(5)” and Diversity from “Eq.(6)”
2. Fuzzify the Iteration and Diversity using the membership functions described in figures 3 and 4.
3. Evaluate the fuzzy rules using “Eq. (7)”.
4. Defuzzify the results of evaluation phase and output the crisp value for 𝜉 using “Eq. (8)”
For each ant in the population do
If q < then with probability choose the node to move to using “Eq. (1)”
Else with Probability (1- choose the node to step to using “Eq. (2)”
Update pheromone amount using “Eq. (3)”
Until all ants build a solution
Figure 2. Adaptive local parameter using FLC.
In Figure 2, we described the adaptation of 𝜉 parameter using the FLC after one established
iteration. In facts, in each iteration ants build solutions incrementally to the problem, by moving through
neighbour components solution of the problem, using the transition rule described by Equations (1) and (2).
While moving, an ant modifies the amount of pheromone using the local pheromone updating procedure
described by Equation (3). Once all ants have terminated a solution the amount of pheromone is modified

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again using the global pheromone updating procedure. The obtained information from the first iteration are
used to calculate the first inputs of the FLC algorithm, and so on.
2.1. Fuzzification
The Fuzzification process is the first step in the FLC system, which corresponds to transforming the
crisp values into fuzzy grades using the membership functions. This process facilitates the application of the
rule set. For each input variable, we define three membership functions (MF), to define a qualitative category
for each one: {Low, Medium, High}. In practice, there are various shapes of membership functions that can
be used in the fuzzification process, for example: Triangular MFs, Trapezoidal MFs, Gaussian MFs,
Generalized bell MFs, 𝜋-Shaped Membership Function, S-Shaped Membership Function [15]. In this work,
we used the triangular membership functions that are very popular, easy to implement, and respond to our
needs. The input variables with their membership functions are illustrated in Figure 3 and Figure 4.
Figure 3. Iteration as input variable. Figure 4. Diversity as input variable
Figure 3 illustrates the three triangular membership functions of the iteration input variable with a
range from 0 to 1. In Figure 4 the three triangular membership functions of the Diversity input variable are
shown with a range from 0 to 1. Let x the crisp value of Iteration and y the crisp value of Diversity. For each
input we calculate the degree of membership: 𝜇 , 𝜇 , and 𝜇 . Where, Low=[0, 0.5],
Medium=[0, 1], and High=[0.5, 1]
2.2. Rule Evaluation
In fuzzy logic, there are various ways to define a fuzzy rule. Indeed, they can be divided into three
main classes: the fuzzy conjunction, the fuzzy disjunction, and the fuzzy implication [16]. In this work, we
used a Mamdani’s fuzzy conjunction fuzzy rule. Once the variables and membership functions are designed,
we define the rule base which is composed by IF- Then rules. In fact, the rule base is developed according to
some knowledge about the ACS algorithm and the chosen metrics. The rules (Figure 5) of the proposed fuzzy
system with Iteration and Diversity as input and 𝜉 as output are as follows:
If (Iteration is Low) and (Diversity is Low) then (𝜉 is Low)
If (Iteration is Low) and (Diversity is Medium) then (𝜉 is MediumLow)
If (Iteration is Low) and (Diversity is High) then (𝜉 is Medium)
If (Iteration is Medium) and (Diversity is Low) then (𝜉 is MediumLow)
If (Iteration is Medium) and (Diversity is Medium) then (𝜉 is Medium)
If (Iteration is Medium) and (Diversity is High) then (𝜉 is MediumHigh)
If (Iteration is High) and (Diversity is Low) then (𝜉 is Medium)
If (Iteration is High) and (Diversity is Medium) then (𝜉 is MediumHigh)
If (Iteration is High) and (Diversity is High) then (𝜉 is High)
Figure 5. IF-THEN rules of our fuzzy system
To evaluate the fuzzy rules we used the Min fuzzy set operation, assuming that we are using the
Mamdani’s conjunction operator (AND). For each rule we return the lowest value from the calculated
degrees of membership of the two inputs.

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𝜇 {𝜇 𝜇 } (7)
Where, i is the index of the rule, j and k are indexes for the fuzzy sets {Low, Medium, High} for x
and y. Then, the results of all rules are summed together to produce a set of fuzzy outputs.
2.3. Defuzzification
After the evaluation of the rules, we obtain a fuzzy output that’s need to be transformed to a crisp
value using one of the defuzzification methods. In fact, the commonly used techniques for defuzzification
are: Mean of Maximum (MOM) method, Center of Gravity method, and the height method [16], [17].
Figure 6 shows the 𝜉 parameter as output variable, with a range from 0 to 1, and granulated into five
triangular membership functions. Where, output set is: S= { }.
Figure 6. 𝜉 as output variable
In order to obtain the output variable in crisp value, we defuzzify the obtained results from the
inference process using the Center of Gravity (COG) algorithm described by Equation 8:
𝜉
∑ [ ]
∑ [ ]
(8)
Where, p=9 is the number of output membership function, is the singleton of output membership
function, and 𝜇 the result of all rule evaluation as shown in Table 1.
Table 1. The singleton of the output membership function
X
Low Medium High
Y
Low = 1/6 = 2/6 =3/6
Medium =2/6 =3/6 =4/6
High =3/6 =4/6 =5/6
Where, corresponds to Low membership function of the output, corresponds to Medium Low
membership function of the output, corresponds to Medium membership function of the output and so on.
3. RESULTS AND DISCUSSION
To test the effectiveness of the proposed algorithm, we compared it to the standard ACS with a set
of benchmark TSP instances from the TSPLIB [18]. Table 2 tabulates the size and the o(ptimal tour length of
each instance. We run the ACS algorithm with the following parameters: 𝛽 = 2, = 0.9, 𝜌 = 0.1 and m=10,
which were proven to be the best setting parameters for ACS performance [19]. Programming by Matlab
R2013a, the instances will be run 30 times separately, 1000 iterations each time. The stop condition is: 100 as
the maximum number of ACS iterations without improving. The initial position of ants is set randomly on all
experiments. The best comparison results are shown in Table 2.

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Table 2. Chararteristics of TSP benchmark instances.
TSP att48 berlin52 ch130 d198 eil51 eil76 eil101 kroA100 lin105 Pr226
Number of cities 48 52 130 98 51 76 101 100 105 226
est known solutions 10628 7542 6110 15780 426 538 629 21282 14379 80369
3.1. Comparison on the Solution Accuracy
Comparing the running of ACS with a fixed set of parameters against the proposed method shows
better results for both minimum and average length in most of the instances, especially when dynamically
adapting the parameter. In Table 3, it can also be observed that in instances of small sizes the best found
solutions for the two algorithms are almost the same with a privilege in the average solutions for the
proposed algorithm. But in large size instances there is a big difference between the best solutions that are
found for each of the two algorithms. Thus, it can be concluded that as the size of problem becomes larger as
the proposed algorithm can offer better results.The experimental results reflect the role of learning to provide
the best solutions.
Table 3. Summary of results using ACSFL algorithm for TSP instances
TSP
ACSFL ACS
Min Avg CPUtime Min Avg CPUtime
att48 33523.70 33715.49 42.99 33523.70 33692.67 38.75
berlin52 7544.36 7546.53 32.43 7544.36 7556.46 29.45
ch130 6246.24 6348.35 1093.7 6234.56 6371.55 1676.9
d198 16032.71 16327.04 2590.2 16147.38 16414.42 1739.1
eil51 428.98 432.94 116.38 428.98 435.48 111.1
eil76 548.49 556.31 328.6 552.92 558.02 388.08
eil101 646.44 662.76 480.9 657 669.03 387.8
kroA100 21285.44 21611.54 410.18 21355.28 21748.18 438.56
lin105 14382.99 14524.95 254.9 14382.99 14559.82 460
Pr226 80468.49 81853.71 1353.7 80763.10 82127.78 3448.2
3.2. Comparison on the Convergence Speed
It can be noted from the Table 3, that the time of finding the best length for the proposed algorithm
when adapting the parameter outperforms both the conventional ACS and the proposed algorithm when
varying the parameter. However, it can be seen that the time of finding the same best length is less in the
proposed algorithm when adapting the parameter than the two others. The Figures 7(a), 7(b), 7(c) and 7(d)
below show the result of running both algorithms on four chosen instances of TSP benchmarks which are:
eil51, kroA100, eil101, and lin105.
The Figures actually go in line with these observations, since a very big difference is noticed
between the solutions found by the three algorithms, the proposed algorithm when varying parameter
converges to a better solution than the conventional one and the proposed algorithm when adapting the
parameter in all the figures. In addition to the quality of solution, there is a faster convergence in the
proposed algorithm when dynamically adapting the parameter.

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(a) (b)
(c) (d)
Figure 7. (a) Sample run on eil51 instance, (b) Sample run on kroA100 instance, (c) Sample run on eil101
instance, (d) Sample run on lin105 instance
3.3. Statistical Test
To compare the proposed method with the standard one, we have used the T-Test as a statistical test
that compares the means of lengths returned by the proposed method and the standards ACS according to the
following Equation:
√
(9)
Where, n is the lengths size.
The results obtained from applying this test are illustrated in "Table 4". The 30 experiments for each
instance are the used parameters for the tests, the null hypothesis 𝜇 𝜇 ) says that the proposed
method returns greater lengths of averages when compared with the other method, while the alternative
hypothesis ( 𝜇 𝜇 ) says that the proposed algorithm returns better average when compared with the
other method, the level of significance is 5 percent, and the critical value = 1.699, so the rejection region is
for all values of T-Test lowers than . From the results in table 4, the proposed method fail to reject the null
hypothesis only in 2 instances, and this is for the smallest problems which are the easiest ones, however the
proposed method can achieve better results with level of significance of 5 percent in all other results.

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Table 4. Results of comparison using T-Test
TSP att48 berlin52 ch130 d198 eil51 eil76 eil101 kroA100 lin105 Pr226
ACS -1.306 -0.611 -3.634 -2.490 -3.002 -3.823 -4.678 -2.273 -3.487 -3.157
Fuzzyglobal 0.590 -2.225 -2.126 -3.432 -3.385 -5.292 -3.959 3.990 -5.663 -4.938
4. CONCLUSIONS
This paper proposed a new evolved Ant Colony System Algorithm based on a fuzzy system so as to
dynamically adapt the local pheromone parameter that has a crucial impact in avoiding falling in the local
best optimum during the construction solution phase. The simulation result on TSP showed that the proposed
method that dynamically adapt the local pheromone parameter has higher convergence speed and better
quality of optimal solution. In other words, the effect of local updating is to make a better use of pheromone
information to explore new best solutions.The proposed method when adapting parameter gives a flexible
control of pheromone information which balances between the exploration search and exploitation then
finding better solutions.
REFERENCES
[1] M. Dorigo, L. M. Gambardella, “Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman
Problem”, IEEE Transactions on Evolutionary Computation, 1(1), 53-66, 1997.
[2] M. Dorigo, C. Blum, “Ant Colony Optimization Theory: A Survey”, Theoretical Computer Science, 344(2),
243-278, 2005.
[3] M. Dorigo, L. M. Gambardella, “Ant Colonies for the Travelling Salesman Problem”, BioSystems, 43(2), 73-81,
1997.
[4] H. Yu, “Optimized Ant Colony Algorithm by Local Pheromone Update”, Vol. 12, No. 2, pp. 984 ~ 990, 2014.
[5] F. Xiao-rong, F. Xing-jie, “A Dynamic Multi-nest Ant Colony Algorithm for Aircraft Landing Problem”,
TELKOMNIKA Indonesian Journal of Electrical Engineering, Vol. 12, No. 3, pp. 2196 ~ 2202, 2014.
[6] M. Jafari, H. Khotanlou, “A Routing Algorithm Based on Ant Colony, Local Search and Fuzzy Inference to
Improve Energy Consumption in Wireless, Sensor Networks”, International Journal of Electrical and Computer
Engineering (IJECE), Vol. 3, No. 5, pp. 640~650, 2013.
[7] C. Qi, “Vehicle Routing Optimization in Logistics Distribution using Hybrid Ant Colony Algorithm”,
TELKOMNIKA, Vol. 11, No. 9, pp. 5308~5315, 2013.
[8] D.N. Le, “PSO and ACO Algorithms Applied to Optimization Resource Allocation to Support QoS Requirements
in NGN”, International Journal of Information & Network Security (IJINS), Vol.2, No.3, June 2013, pp. 216~228.
[9] F. Ahmadizar, H. Soltanpanah, “Reliability Optimization of a Series System with Multiple-Choice and Budget
Constraints using an Efficient Ant Colony Approach”, Expert Systems with Applications, 38(4), 3640-3646, 2011.
[10] F. Olivas, F. Valdez, O. Castillo, “Ant Colony Optimization with Parameter Adaptation Using Fuzzy Logic for TSP
Problems”, In Design of Intelligent Systems Based on Fuzzy Logic, Neural Networks and Nature-Inspired
Optimization (pp. 593-603). Springer International Publishing, 2015.
[11] F. Valdez, P. Melin, O.Castillo, “An Improved Evolutionary Method with Fuzzy Logic for Combining Particle Swarm
Optimization and Genetic Algorithms”, Applied Soft Computing, 11(2), 2625-2632, 2011.
[12] P. Melin, F. Olivas, O. Castillo, F. Valdez, J. Soria, M. Valdez, “Optimal Design of Fuzzy Classification Systems
Using PSO with Dynamic Parameter Adaptation Through Fuzzy Logic”, Expert Systems with Applications, 40(8),
3196-3206, 2013.
[13] A. Sombra, F. Valdez, P. Melin, O. Castillo, “A New Gravitational Search Algorithm using Fuzzy Logic to Parameter
Adaptation", IEEE Congress on Evolutionary Computation, 2013.
[14] L. A. Zadeh, “Fuzzy sets”, Information and Control, 8(3), 338-353, 1965.
[15] Y. Bai, D. Wang, “Fundamentals of Fuzzy Logic Control-Fuzzy Sets, Fuzzy Rules and Defuzzifications”, In
Advanced Fuzzy Logic Technologies in Industrial Applications (pp. 17-36). Springer London, 2006.
[16] C. C. Lee, “Fuzzy Logic in Control Systems: Fuzzy Logic Controller”, II. IEEE Transactions on Systems, Man, and
Cybernetics, 20(2), 419-435, 1990.
[17] Sheikh, M. R. I., Junji, T. (2011), “Smoothing Control of Wind Farm Output Fluctuations by Fuzzy Logic
Controlled SMES”, International Journal of Electrical and Computer Engineering, 1(2), 119.
[18] G. Reinelt, “Tsplib Discrete and Combinatorial Optimization”, 1995.
[19] T. Stützle et al., “Parameter Adaptation in Ant Colony Optimization”, In Autonomous search (pp. 191-215), Springer
Berlin Heidelberg. 2011.

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