CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 51, 2016 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Tichun Wang, Hongyang Zhang, Lei Tian 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-43-3; ISSN 2283-9216 

An Improved Ant Colony Algorithm for Continuous 
Optimization Based on Levy Flight 

Hongzhang Wang*, Conggang Liang 
School of Mathematics and Information Science, Pingdingshan University, Pingdingshan, Henan 467000, China  
13903759297@126.com 

Ant colony optimization (ACO) is proposed on the study of the foraging behavior of ants on the basis of the 
proposed and widely used in the optimization. However, it has some shortcoming such as longer time, hardly 
implement and local optimal etc. For overcoming the above shortcoming, combined with the characteristic of 
Levy flight, based on Levy flight ant colony optimization is proposed which used Levy flight instead of local 
search for improving the searching efficiency. In order to test the performance of the new algorithm, we apply 
it to 20 benchmark function test and compare it with GA, PSO, ACO and LFACO algorithms. The comparison 
result shows that LFACO is far better than the other three algorithms in quality.  

1. Introduction 

With the development of computing technology, it is possible to make random search in the solution space. 
Random search according to the different search methods can be divided into random search and blind 
random search. Random search method is a feasible solution to complete random search in the solution 
space, and the method is very efficient when the scale of the problem is large. A guided random search rule is 
according to certain strategy in solution space to focus on the implementation of random search, and 
accumulated experience in the process of random search, makes the search is intelligent, commonly used 
intelligent methods such as genetic algorithm (Goldberg, 2002), particle swarm optimization (Kennedy and 
Eberhard, 1995). 
Biologists found that by a lot of research, ant colony foraging in the process is the reason why to find the 
shortest path between a food source and their nest is due to ant individuals can on the way after the leave a 
special substance pheromones and become other ants searching for food or nest of clues. Subsequent ants 
encounter pheromone, not only can detect the content of pheromone, but also according to the concentration 
of pheromone to determine the direction of its progress. Information over time will gradually evaporate, so the 
length of the path and the number of residues in the number of ants on the choice of a greater impact. In a 
certain period of time the path by more ants, ants select the path of the greater the probability; ant chooses 
the path, the corresponding path information hormone concentration strengthened, to promote more ants 
select the path. Therefore, a large number of ants through this simple information exchange, to achieve a 
positive feedback of the information learning mechanism, and can quickly find the shortest path from the food 
source to the nest. 
A typical representative of the biological colony intelligent algorithm is the ant colony algorithm. Ant Colony 
Optimization (ACO) is proposed on the study of the foraging behaviour of ants on the basis of the proposed 
(Dorigo and Caro, 1999). After years of development, the model construction algorithm relatively mature and 
perfect, with the characteristics of strong robustness positive feedback, distributed computing etal, has been 
successfully applied to discrete combinatorial optimization field, become one of the most effective means of 
optimization. However, ACO also has the following shortcomings: 
(1) The algorithm only through the guidance of pheromone search optimization, search for a longer time; 
(2) When solving the practical problem by ant colony algorithm, it is necessary to describe the problem firstly, 
and the algorithm is not powerful enough to describe the complex problem; 
(3) In the process of searching, the algorithm is prone to search stagnation phenomenon; 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1651082

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Wang H.Z., Liang C.G., 2016, An improved ant colony algorithm for continuous optimization based on levy flight, 
Chemical Engineering Transactions, 51, 487-492  DOI:10.3303/CET1651082   

487

mailto:13903759297@126.com


In fact, it is not entirely random, but in the nature, that the individual's flight or foraging methods are not 
entirely random, but it obeys the Levy flight (Yang and Deb, 2009). The so-called Levy flight is a step size 
obeys Levy distribution random walk. Specifically, individuals generally only in a small area of flight or 
foraging, but there are a small fraction of an individual will suddenly fly to the distant place. This behavior is 
very conducive to the search process and widely used in the CS algorithm (Yang and Deb, 2010) and CSB 
algorithm (Yin and Liu 2015). In order to improve the search time and search stagnation, this paper proposes 
an ant colony algorithm based on Levy flight. 
In order to Levy flight applications in more practical problems, this paper for continuous space optimization of 
function optimization problems, combined with the ACO algorithm, we proposed a based on Levy flight ant 
colony optimization algorithm (LFACO). This algorithm use Levy flight to perform local search algorithm and 
unlike ACO algorithm and random search, so local search ability of the algorithm has been greatly improved. 
In order to test the performance of the new algorithm, we apply it to 20 benchmark function test and compare 
it with GA, PSO, ACO and LFACO algorithms.  

2. The description of LFACO 

2.1 ACO algorithm 

Assume that solving the problem of the scale is 𝑛, the total number of ant colony is 𝑚, the path (I, J) in time t 
amount of information concentration is 𝜏𝑖𝑗 (𝑡). An ant of K in the process of moving path selection will accord 
the information on each path in the pigment concentration and path of the heuristic information such as factors 
to calculate the state transition probability by which the ant selects the path. Here we assume that 
𝑝𝑖𝑗

𝑘 (𝑡) represents the probability that in the time t ant K is transferred from the node I to the node J, and 
computing the probability calculation is: 

 𝑝𝑖𝑗
𝑘 (𝑡) = {

[𝜏𝑖𝑗(𝑡)]
𝛼∗[𝜇𝑖𝑗(𝑡)]

𝛽

∑ [𝜏𝑖𝑠(𝑡)]
𝛼∗[𝜇𝑖𝑠(𝑡)]

𝛽
𝑠⊂𝑎𝑙𝑙𝑜𝑤𝑒𝑑𝑘

, 𝑖𝑓 𝑗 ∈ 𝑎𝑙𝑙𝑜𝑤𝑒𝑑𝑘

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
                                      (1) 

In the above equation, 𝑎𝑙𝑙𝑜𝑤𝑒𝑑𝑘  represents a collection of the ant K step allowing the selection of node. 𝛼 is 
the pheromone heuristic factor, 𝛽 is expected heuristic factor, 𝜇𝑖𝑗 (𝑡) is the heuristic function. In order to avoid 
earlier information residues too much and cause residue information flooded the heuristic information of the 
phenomenon, in each ant finish or complete traversal of all n cities, to update operation treatment of residual 
information. So in the t+1 time the pheromone concentration of the path (I, J) can be adjusted and updated as 
the following rules:  

𝜏𝑖𝑗 (𝑡 + 1) = (1 − 𝜌) ∗ 𝜏𝑖𝑗 (𝑡) + ∆𝜏𝑖𝑗 (𝑡)                                           (2) 

 ∆𝜏𝑖𝑗 (𝑡) = ∑ ∆𝜏𝑖𝑗
𝑘𝑚

𝑘=1 (𝑡)                                                (3) 

In the above equations, 𝜌 is the pheromone evaporation coefficient, and for preventing unlimited information 
accumulation, the range of P is [0, 1). ∆𝜏𝑖𝑗 (𝑡) is said as the path (I, J) pheromone increment, ∆𝜏𝑖𝑗

𝑘 (𝑡) is said the 
information concentration the 𝑘th ant after path (I, J). So the pseudo code of ACO is as Figure 1. 

 

Figure 1: The pseudo code of ACO 

2.2 Levy Flight 

Animals foraging path was considering as a random or quasi-random manner in nature. However, various 
studies have shown that the flight behavior of many animals and insects obeys the typical characteristics of 
Levy flights. Broadly speaking, Levy flights are a random walk whose step length is drawn from the Levy 
distribution, often in terms of a simple power-law formula L(s)~|s|^(-1-β) where 0<β<2. Obviously, the 
generation of step sizes samples is not trivial using Levy flights. A simple scheme discussed in detail can be 
summarized as following (Yang, 2010): 

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𝐿(𝑠)~
𝑢

|𝑣|
1
𝛽

                                                                   (4) 

In which 𝑢 ∼ 𝑁(0, 𝜎𝑢2), 𝑣 ∼ 𝑁(0, 𝛿𝑣 ) are normal distribution. 

2.3 The description of LFACO 

For improving the performance of ACO, a possible method is to change the pheromone volatilization 
coefficient ρ using Levy distribution while ρ is a constant in ACO, so we propose an new improved ACO 
algorithm based on Levy Flight (LFACO) in which each movement of each ant individual obey the levy 
distribution. In LFACO, the equation (4) is introduced into the equation (2) and gets the following equation for 
updating the location of each ant individual: 

𝜏𝑖𝑗 (𝑡 + 1) = (1 − 𝜌𝑖,𝑗 ) ∗ 𝜏𝑖𝑗 (𝑡) + ∆𝜏𝑖𝑗 (𝑡)                                         (5) 

In the eq(5), 𝜌𝑖,𝑗 ∼ 𝑙𝑒𝑣𝑦(𝛽) , 𝑙𝑒𝑣𝑦(𝛽) ∼
𝑢

|𝑣|
1
𝛽

( 𝜏𝑖𝑗 (𝑡 + 1) −     𝜏𝑖𝑗 (𝑡))  and 𝑢 ∼ 𝑁(0, 𝜎𝑢
2) , 𝑣 ∼ 𝑁(0,1) , 𝜎𝑢 =

𝒯(1+𝛽)𝑠𝑖𝑛 (𝛽𝜋/2)

𝒯(
1+𝛽

2
)𝛽2(𝛽−1)/2

1/𝛽

.Combined the above analysis, the step of LFACO is following as figure 2. 

 
 

Begin

Initializing the oopuation 

Constructing the matrix of 
information

Performing the moving 
operator for each individual

Performing the Levy flight as 
local search  for each individual

Selecting the best individual 
according the fitness

Updating the matrix of 
information

stop

N

End

Y

 

Figure 2: The description of LFACO. 

Function Optimization 

Function Optimization is often expressed in the following form: 

𝑚𝑖𝑛 𝑓(𝑋)   𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜     𝐿 ≤ 𝑋 ≤ 𝑈                                                                     (6) 

In which X = {x1, x2, x3, … , xn}  is a vector, and L = {l1, l2, l3, … , ln} is the lower bound of X , while 
U = {u1, u2, u3, … , un} is the upper bound of X. In this work, we test and verify the effectiveness of the proposed 
LFQPSO algorithm in this chapter by using the following 15 benchmark functions of which its definition, 
domains and optimal value are respectively defined as: 
(1)The definition of f1 is following: 

 𝑓1(𝑥) = ∑ 𝑥𝑖
2𝑛

𝑖=1                                                       (7) 

Its domains is −5.12 ≤ xi ≤ 5.12, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(2)The definition of f2 is following:  

𝑓2(𝑥) = ∑ (𝑖𝑥𝑖
2)𝑛𝑖=1                                                   (8) 

489



Its domains is −5.12 ≤ xi ≤ 5.12, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(3)The definition of f3 is following: 

𝑓3(𝑥) = ∑ [100(𝑥𝑖
2 − 𝑥𝑖+1)

2
+ (𝑥𝑖 − 1)

2]𝑛−1𝑖=1                                     (9) 

Its domains is −2.048 ≤ xi ≤ 2.048, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (1,1,1, … ,1), f(x∗) = 0 
(4)The definition of f4 is following: 

𝑓4(𝑥) = 10𝑛 + ∑ (𝑥𝑖
2 − 10𝑐𝑜𝑠 (2𝜋𝑥𝑖 ))

𝑛
𝑖=1                                                        (10) 

Its domains is −5.12 ≤ xi ≤ 5.12, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(5)The definition of f5 is following: 

𝑓5(𝑥) = ∑
𝑥𝑖

2

4000

𝑛
𝑖=1 − ∏ 𝑐𝑜𝑠 (

𝑥𝑖

√𝑖
)𝑛𝑖=1 + 1                                                        (11) 

Its domains is −1 ≤ 𝑥𝑖 ≤ 1, 𝑖 = 1,2,3, … , 𝑛 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0    
(6)The definition of f6 is following: 

𝑓6(𝑥) = ∑ |𝑥𝑖 |
𝑖+1𝑛

𝑖=1                                                                         (12) 

Its domains is −32.768 ≤ xi ≤ 32.768, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(7)The definition of f7 is following: 

𝑓7(𝑥) = 20 + 𝑒 − 20𝑒
−

1

5
√

1

𝑛
∑ 𝑥𝑖

2𝑛
𝑖=1

− 𝑒
1

𝑛
∑ 𝑐𝑜𝑠 (2𝜋𝑥𝑖)

𝑛
𝑖=1                                                    (13) 

Its domains is −32.768 ≤ xi ≤ 32.768, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(8)The definition of f8 is following: 

𝑓8(𝑥) = 𝑚𝑎𝑥(|𝑥𝑖 |) , 𝑖 = 1,2,3, … , 𝑛                                                                  (14) 

Its domains is −100 ≤ xi ≤ 100, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(9)The definition of f9 is following: 

𝑓9(𝑥) = 𝑚𝑎𝑥(|𝑥𝑖 |) , 𝑖 = 1,2,3, … , 𝑛                                                                  (15) 

Its domains is −100 ≤ xi ≤ 100, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(10)The definition of f10 is following: 

𝑓10 (𝑥) = ∑ 𝑥𝑖
2𝑛

𝑖=1 + (∑ 0.5𝑖𝑥𝑖
𝑛
𝑖=1 )

2 + (∑ 0.5𝑖𝑥𝑖
𝑛
𝑖=1 )

4                                                 (16) 

Its domains is −5 ≤ xi ≤ 10, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(11)The definition of f11 is following: 

𝑓11 (𝑥) = 𝑠𝑖𝑛
2(𝜋𝑦1) + ∑ [((𝑦𝑖 − 1)

2(1 + 10𝑠𝑖𝑛2(𝜋𝑦𝑖 + 1)))]
𝑛−1
𝑖=1 + (𝑦𝑛 − 1)

2(1 + 𝑠𝑖𝑛2(2𝜋𝑦𝑛 )), 𝑦𝑖 = 1 +
𝑥𝑖−1

4
, 𝑖 =

1,2,3, … , 𝑛                                          (17) 

Its domains is −10 ≤ xi ≤ 10, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (1,1,1, … ,1), f(x∗) = 0 
(12)The definition of f12 is following: 

𝑓12 (𝑥) = ∑ (∑ 𝑥𝑗
2𝑖

𝑗=1 )
𝑛
𝑖=1                                                                (18) 

Its domains is −65.536 ≤ xi ≤ 65.536, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(13)The definition of f13 is following: 

490



𝑓13 (𝑥) = 1 − 𝑐𝑜𝑠 (2𝜋√∑ 𝑥𝑖
2𝑛

𝑖=1 ) + 0.1√∑ 𝑥𝑖
2𝑛

𝑖=1                                           (19) 

Its domains is −100 ≤ xi ≤ 100, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(14)The definition of f14 is following: 

𝑓14 (𝑥) = 0.1 + ∑ (𝑠𝑖𝑛 (𝑥𝑖 ))
2 − 0.1 ∏ (𝑒𝑥𝑝 (−𝑥𝑖

2))𝑛𝑖=1
𝑛
𝑖=1                                             (20) 

Its domains is −10 ≤ xi ≤ 10, i = 1,2,3, … , n 
Its argument and optimal value are x∗ = (0,0,0, … ,0), f(x∗) = 0 
(15)The definition of f15 is following: 

𝑓15 (𝑥) = ∑ (𝑥𝑖 − 1)
2𝑛

𝑖=1 − ∑ 𝑥𝑖 𝑥𝑖−1
𝑛
𝑖=2 +

𝑛(𝑛+4)(𝑛−1)

6
                                       (21) 

Its domains is −𝑛2 ≤ 𝑥𝑖 ≤ 𝑛2, 𝑖 = 1,2,3, … , 𝑛 
Its argument and optimal value are 𝑥∗ = 𝑖(𝑛 − 𝑖 + 1), i = 12,3, … , n, 𝑓(𝑥∗) = 0 

4. Experiment Result 

In order to compare the results of LFACO algorithm, GA, PSO, ACO and LFACO algorithm use the R 
programming language implementation, and all the program results in 3.3GHz Core Duo processor, 4GB of 
ram and windows 7 operating system of PC operation. For the 15 benchmark functions, the four algorithms 
are run 100 times respectively, and the average experimental results are shown in Table 1 and table 2. 
As Table 1 shown, for all 20 benchmark test functions, the convergence speed of PSO is the best, and of 
LFACO is better than that of GA and ACO, which shows that LFACO can indeed faster than ACO in optimizing 
the optimal searching process. 
Next we focus on analysis the result of LFACO and the rest of the three algorithms. Table 2 shown the result 
is the best LFACO optimization, and although the execution time of the PSO algorithm best as Table 1 shown, 
but the analysis table 2, we can see that PSO result is the worst in four algorithms, which shows the no an 
algorithm in which some indexes can reflect the best. 

Table 1: Runtime of GA, PSO, ACO and LFACO on the 15 benchmark functions 

Function GA PSO ACO LFACO 
f1 10.41 2.21 10.21 7.34 
f2 10.74 2.28 10.53 7.57 
f3 11.04 22.64 10.82 7.78 
f4 11.67 2.48 11.45 8.23 
f5 11.49 23.58 11.27 8.10 
f6 10.39 2.44 10.73 8.10 
f7 11.84 2.51 11.62 8.35 
f8 10.71 2.27 10.50 7.55 
f9 10.84 2.30 10.63 7.64 
f10 11.18 2.37 10.97 7.89 
f11 11.50 2.44 11.28 8.11 
f12 10.82 2.30 10.61 7.63 
f13 10.96 2.33 10.75 7.73 
f14 11.49 2.44 11.27 8.10 
f15 11.42 2.42 11.20 8.05 

5. Conclusion  

In order to improve performance ACO algorithm in function optimization, in this paper the Levy flight 
introduced to ACO algorithm and get the new update of the individual equations. Based on the above discuss, 
we propose the new algorithm of based on Levy Flight of the ACO algorithm (LFACO) and detailed describe 
its steps and processes. In order to verify the LFACO algorithm, 15 benchmark functions are used to test the 

491



LFACO which is compared with GA, PSO algorithm and ACO algorithm. The comparison results show that for 
the function optimization problem LFACO in effectiveness as well as run time is far better than other three 
algorithms. 

Table 2: the average and standard variation of GA, PSO, ACO and LFACO on the 15 benchmark functions 

Function GA PSO ACO LFACO 
𝑓1 0.00±0.06 0.00±8.06 2.05±1.29 0.00±0.00 
𝑓2 0.16±0.37 5.43±16.97 87.13±20.54 0.00±0.00 
𝑓3 28.71±0.96 44.12±34.10 84.24±32.06 12.67±2.97 
𝑓4 15.80±4.69 97.41±32.9 99.27±13.52 3.01±1.48 
𝑓5 0.67±0.19 0.03±2.67 208.24±39.01 0.00±0.00 
𝑓6 0.00±0.00 0.00±0.00 0.00±0.00 0.00±0.00 
𝑓7 1.86±0.53 0.01±3.41 8.25±0.84 0.04±0.03 
𝑓8 12.43±2.46 17.91±5.35 24.08±2.52 4.90±1.11 
𝑓9 0.25±0.30 80.00±23.41 46.99±8.60 0.00±0.00 
𝑓10 27.32±4.18 86.19±14.73 94.91±101.01 0.68±0.60 
𝑓11 0.01±0.35 1.85±5.09 4.17±13.83 0.00±0.00 
𝑓12 11.32±77.17 258.88±28.96 139.01±31.99 0.00±0.00 
𝑓13 3.30±0.50 0.82±2.06 6.00±0.97 1.80±0.32 
𝑓14 0.13±0.05 3.36±0.75 0.23±0.08 0.10±0.00 
𝑓15 222.33±163.57 192.27±125.16 122.36±43.15 82.83±33.85 

Acknowledgment 

Fund Project: 2011 Henan provincial science and Technology Department of science and technology 
foundation project; Exact solutions of higher order nonlinear partial differential equation; Project approval 
number: 112300410199. 

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