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 

The Design and Application of Quantum-Behaved Particle 
Swarm Optimization Based on Levy Flight 

Yaya Liu 
Shangluo University 
Lyy_yyl@163.com 

Both Particle Swarm Optimization (PSO) and its improved version of Quantum-behaved Particle Swarm 
Optimization (QPSO) are the novel swarm intelligence optimization algorithms. However, the above two 
algorithm makes the search process easy to fall into local optimum and premature convergence because of 
the existence of particle waiting effect. For overcoming the shortcoming of QPSO and improving its search 
ability, considering that the feature of Levy flight is different that of Brown motion used by QPSO, we proposed 
a method of based on Levy flight quantum-behaved particle swarm optimization (LFQPSO). In order to test the 
performance of the proposed algorithm in our work, we apply it to the benchmark function test and compare it 
with standard PSO and QPSO algorithm, which shows that the LFQPSO outperforms the standard PSO and 
QPSO algorithm. 

1. Introduction 

Inspired by the movement of birds, Kennedy and Eherhart (1995) proposed the Particle Swarm Optimization 
(PSO) which is a kind of evolutionary computation technology based on swarm intelligence. PSO takes a 
habitat of bird swarm motion analogous to the space to be optimized, takes each bird to position of feasible 
solution, and thus guides the bird swarm to move toward to a better location or solution through the 
information transfer between individuals. For convenience, the bird or individual is abstracted as the particle 
which is not of the size and quality. PSO has the following features: The particle is moving in the form of a 
track which is determined by the position and speed of the particle at the same time, this is, when a particle 
moves at a certain speed, its trajectory is determined. So the search range of PSO is limited and not covers
the total space of feasible solution, which makes it impossible that the global optimal solution can be 
searched. Considering that the quantum motion has great uncertainty and makes the range of motion enlarge 
since quantum motion obeys the two laws of wave-particle dualism and the uncertainty principle in the 
quantum phase-space. Based on the above feature, Sun et al. (2004) proposed the method of Quantum-
behaved Particle Swarm Optimization (QPSO) in which its evolution equation is different that of PSO, and it 
uses the iterative equation with completely randomization which overcomes the shortcoming of PSO and is a 
novel swarm intelligence optimization algorithm.  
However, both the PSO and the QPSO algorithm have particle waiting effect which makes the search process 
easy to fall into local optimum and premature convergence. Due to the appearance of the particle waiting 
effect, the result is that the individual is close to the better optimal solution according to the Brown motion. In 
fact, in nature the flight or foraging mode of the individual is not entirely random, but is subject to Levy flight 
(Yang and Deb, 2009). The so-called Levy flight is a random walk step size obeys Levy distribution, in detail, 
most of the individual flies or forages only in a small area, but a small part of the individual can suddenly fly far 
away, so this behavior is very conducive to the search process and used in the CS algorithm (Yang and Deb, 
2010). 
For improving the performance of QPSO, combined with the feature of Levy flight, we proposed a method of 
based on Levy flight quantum-behaved particle swarm optimization (LFQPSO), and its evolution equation is 
completely different that of QPSO and its convergence did not happen much change. We will first test it 
against a benchmark function. In order to test the performance of the proposed algorithm in our work, we 
apply it to the benchmark function test and compare it with standard PSO and QPSO algorithm. 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1651084

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Liu Y.Y., 2016, The design and application of quantum-behaved particle swarm optimization based on levy flight, 
Chemical Engineering Transactions, 51, 499-504  DOI:10.3303/CET1651084   

499

mailto:Lyy_yyl@163.com


2. Description of LFQPSO 

2.1 QPSO 

The basic principle of the QPSO algorithm can be described as: let the dimension of the solution space of an 
optimization problem is assumed to 𝑑, in this the solution space, there exists a group of particles 𝑋 =
{𝑋1, 𝑋2, ⋯ , 𝑋𝑚 } in which are composed of M particles, and each particle represents a potential solution to the 
optimization problem. Let at a certain moment t, the location of the ith particles 
is𝑋𝑖 (𝑡) = (𝑋𝑖,1(𝑡), 𝑋𝑖,2(𝑡), ⋯ , 𝑋𝑖,𝑑 (𝑡)), the best location of the 𝑖th particles is 𝑃𝑖 (𝑡) = (𝑃𝑖,1(𝑡), 𝑃𝑖,2(𝑡), ⋯ , 𝑃𝑖,𝑑 (𝑡)) 
and the global best location of the 𝑋 is 𝐺(𝑡) = (𝐺1(𝑡), 𝐺2(𝑡), ⋯ , 𝐺𝑑 (𝑡)) = 𝑃𝑔 (𝑡), 𝑔 ∈ {1,2, ⋯ , 𝑚} in which 𝑔 is the 
subscript of global best location in the 𝑋. QPSO only considers the positions of the particle and does not 
consider the velocity of the particle, which is reason that it is easy to understand and solve. 
For the minimization problem, the objective function value is smaller, the corresponding adaptation value is 
better. The best location Pi(t)of the ith particlesPbest is determined as following: 

𝑃𝑖 (𝑡) = {
𝑋𝑖 (𝑡), 𝑖𝑓 𝑓(𝑋𝑖 (𝑡)) < 𝑓(𝑃𝑖 (𝑡 − 1)))

𝑃𝑖 (𝑡 − 1), 𝑖𝑓 𝑓(𝑋𝑖 (𝑡)) ≥ 𝑓(𝑃𝑖 (𝑡 − 1)))
   (1) 

The global best position of the group is determined by the following formula: 

𝑔 = 𝑎𝑟𝑔𝑚𝑖 𝑛{𝑓(𝑃𝑖 (𝑡))} , 𝑖 ∈ {1,2, ⋯ , 𝑚     (2) 

𝐺(𝑡) = 𝑃𝑔 (𝑡)    (3) 

The particle position update equation of the QPSO algorithm is: 

𝑃𝑖,𝑗 (𝑡) = 𝜑𝑗 (𝑡) ∙ 𝑃𝑖,𝑗 (𝑡) + (1 − 𝜑𝑗 (𝑡)) ∙ 𝐺𝑗 (𝑡),  𝑎𝑛𝑑 𝜑𝑗 (𝑡) ∼ 𝑈(0,     (4) 

𝑋𝑖,𝑗 (𝑡 + 1) = 𝑃𝑖,𝑗 (𝑡) + 𝛼 ∙ |𝐶𝑗 (𝑡) − 𝑋𝑖,𝑗 (𝑡)| ∙ 𝑙𝑛 (
1

𝑢𝑖,𝑗(𝑡)
) 𝑎𝑛𝑑 𝑢𝑖,𝑗 (𝑡)~𝑈(     (5) 

𝐶(𝑡) = (𝐶1(𝑡), 𝐶2(𝑡), ⋯ , 𝐶𝑑 (𝑡) =
∑ 𝑃𝑖 (𝑡)

𝑚
𝑖=1

𝑚⁄ .    (6) 

Based on the above analysis, QPSO algorithm process is as follows: 
Step 1: Let 𝑡 = 0, initializing the particles 𝑋(𝑡) = {𝑋1, 𝑋2, ⋯ , 𝑋𝑚 }in the particle swarm in the problem space and 
use 𝑃(𝑡) = {𝑃1, 𝑃2, ⋯ , 𝑃𝑚 } as the best particles of each particle which are initialized by 𝑋(𝑡). In addition, let 𝐺(𝑡) 
is the global best swarm and is initialized by computing the subscript g of the best swarm according to (2). 
Step 2: Let 𝑡 = 𝑡 + 1, according to (4), (5) and (6), computing the fitness of 𝑓(𝑋(𝑡)) and for 𝑖 = 1 to 𝑚, if 
𝑓(𝑋𝑖 (𝑡 + 1)) < 𝑓(𝑃𝑖 (𝑡)) then 𝑃𝑖 (𝑡) = 𝑋𝑖 (𝑡 + 1). 
Step 3: computing the subscript g of the best swarm according to (2), if 𝑓(𝑃𝑔 (𝑡) < 𝐺(𝑡) then 𝐺(𝑡) = 𝑃𝑔 (𝑡). 
Step 4: updating the 𝑃𝑖 (𝑡) and 𝑋(𝑡) according to (4) and (5). 
Step 5: if no meeting the stop condition then go to step 2. 

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 and Walton, 2011): 

𝐿(𝑠)~
𝑢

|𝑣|
1
𝛽

    (7) 

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

2.3 The description of LFQPSO 

For further improving the performance of QPSO, we use levy flight to change the evolution equation of each 
particles. So the equation (7) is introduced in (4), (5) and (6), then the following evolution equation is following: 

𝑃𝑖,𝑗 (𝑡) = 𝜑𝑗 (𝑡) ∙ 𝑃𝑖,𝑗 (𝑡) + (1 − 𝜑𝑗 (𝑡)) ∙ 𝐺𝑗 (𝑡),  𝑎𝑛𝑑 𝜑𝑗 (𝑡) ∼ 𝑙𝑒𝑣𝑦(𝛽)     (8) 

𝑋𝑖,𝑗 (𝑡 + 1) = 𝑃𝑖,𝑗 (𝑡) + 𝛼 ∙ |𝐶𝑗 (𝑡) − 𝑋𝑖,𝑗 (𝑡)| ∙ 𝑙𝑛 (
1

𝑢𝑖,𝑗(𝑡)
) 𝑎𝑛𝑑 𝑢𝑖,𝑗 (𝑡)~𝑙𝑒𝑣𝑦(𝛽)     (9) 

500



In which 𝑙𝑒𝑣𝑦(𝛽) ∼ 𝑢
|𝑣|

1
𝛽

(𝐺(𝑡) − 𝑃𝑖 (𝑡)) and 𝑢 ∼ 𝑁(0, 𝜎𝑢2), 𝑣 ∼ 𝑁(0,1), 𝜎𝑢 =
𝒯(1+𝛽)𝑠𝑖𝑛 (𝛽𝜋/2)

𝒯(
1+𝛽

2
)𝛽2(𝛽−1)/2

1/𝛽

. Based on the 

above step, Figure 1 presents the flowchart of LFQPSO. 
 

Begin

Initializing the 
population

Computing the fitness 
of the population

Updating the global 
best particles

updating the 
population

Stop

No

End

Yes

 

Figure 1: The flowchart of LFQPSO 

3. Function Optimization 

Function Optimization is often expressed in the following form: 

 min f(X)   subject to     L ≤ X ≤ U     (10) 

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 10 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    (11) 

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   (12) 

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    (13) 

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     (14) 

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    (15) 

501



Its domains is −600 ≤ 𝑥𝑖 ≤ 600, 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) = 20 + 𝑒 − 20𝑒
−

1

5
√

1

𝑛
∑ 𝑥𝑖

2𝑛
𝑖=1

− 𝑒
1

𝑛
∑ 𝑐𝑜𝑠 (2𝜋𝑥𝑖)

𝑛
𝑖=1    (16) 

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) = 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, … , 𝑛), 𝑓(𝑥∗) = 0 
(8) The definition of f8 is following:  

𝑓8(𝑥) = 1 − 𝑐𝑜𝑠 (2𝜋√∑ 𝑥𝑖
2𝑛

𝑖=1 ) + 0.1√∑ 𝑥𝑖
2𝑛

𝑖=1     (18) 

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

𝑓9(𝑥) = 0.1𝑛 − (0.1 ∑ 𝑐𝑜𝑠(5𝜋𝑥𝑖 ) − ∑ 𝑥𝑖
2𝑛

𝑖=1
𝑛
𝑖=1 )    (19) 

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

𝑓10 (𝑥) = − ∑ (exp (
−(𝑥𝑖

2 + 𝑥𝑖+1
2 + 0.5𝑥𝑖 𝑥𝑖+1)

8
⁄ ) cos (4√𝑥𝑖

2 + 𝑥𝑖+1
2 + 0.5𝑥𝑖 𝑥𝑖+1)) + (𝑛 − 1)

𝑛−1
𝑖=1         (20) 

 
Its domains is−5 ≤ 𝑥𝑖 ≤ 5, 𝑖 = 1,2,3, … , 𝑛 
Its argument and optimal value are 𝑥∗ = (0,0,0, … , 𝑛), 𝑓(𝑥∗) = 0 

4. Experiment analysis 

The experimental environment is as follows: software is MATLAB and hardware environment is windows 7 of 
PC which uses 3.3GHz Core Duo processor and 4GB of RAM. The simulation experiment was run for 100 
times and the average experiment results were statistically. The parameters of LFQPSO are α=0.6，β=1.5 
m=100. Each function is iterative 1000 times for each function and the total iterative process is divided into 6 
sub interval, the initial iteration times of the sub intervals and terminates the iterations respectively 
1~100,100~200, 200~400, 400~600, 600~800 and 800~1000. Table 1, Table 2 and Table 3 respectively give 
the result of LFQPSO, QPSO and PSO which reflect their search ability. Table 4 presents the runtime of 1000 
iteration for 10 benchmark function which measure the time performance of LFQPSO, QPSO and PSO. 
 
Table 1: The result of 10 benchmark function obtained by PSO on the different iteration 
function 1 100 200 400 600 800 1000 

𝑓1 277.39 188.54 136.32 48.36 18.32 13.20 1.01 
𝑓2 2987.21 2213.31 1001.47 621.58 284.19 152.43 17.81 
𝑓3 6856.45 3853.82 2103.59 412.25 262.88 144.12 56.36 
𝑓4 424.94 389.45 330.31 713.11 266.21 124.74 117.59 
𝑓5 728.13 704.21 205.19 139.15 87.21 21.25 8.21 
𝑓6 30.85 28.66 25.36 14.23 6.32 4.21 3.01 
𝑓7 163.18 155.21 146.28 136.11 80.96 60.32 7.11 
𝑓8 44.85 40.52 30.48 21.36 12.77 7.29 3.23 
𝑓9 90.22 65.32 58.85 40.74 18.97 20.23 12.66 
𝑓10  24.63 23.75 23.75 20.84 19.52 19.02 18.89 

 

502



As shown table 1, Table 2 and Table 3, the LFQPSO performs better than QPSO and PSO for all 10 
benchmark functions, such as, for the benchmark function 𝑓2 the result of LFQPSO is 3.68, of QPSO is 12.87 
and of PSO is 17.81 after 1000 iteration, this is, the final result of LFQPSO is far better than that of QPSO and 
PSO. The above situation is applicable to other 9 benchmark functions. The simulations for these benchmarks 
suggest that LFQPSO is a very efficient algorithm for function optimization. It can deal with highly nonlinear 
problems and multi peak problem with complex constraints. 
 

Table 2: The result of 10 benchmark function obtained by QPSO on the different iteration 

function 1 100 200 400 600 800 1000 

𝑓1 234.66 107.66 87.96 29.36 11.22 9.36 0.02 

𝑓2 2831.00 2099.39 988.62 455.35 241.87 93.61 12.87 

𝑓3 6841.25 3769.33 1999.45 522.62 313.45 193.75 66.52 

𝑓4 486.14 408.94 353.19 225.91 179.88 144.37 106.71 

𝑓5 709.58 689.14 311.84 110.25 50.69 14.27 6.36 

𝑓6 30.12 28.55 24.32 11.25 4.55 3.00 0.89 

𝑓7 162.36 150.23 140.33 135.36 75.69 58.32 4.21 

𝑓8 41.25 36.32 28.21 19.27 10.27 5.68 2.34 

𝑓9 85.26 61.77 56.21 39.88 20.71 19.31 9.63 

𝑓10  23.56 22.52 21.75 20.11 19.01 18.62 17.98 

  

Table 3: The result of 10 benchmark function obtained by LFQPSO on the different iteration 

function 1 100 200 400 600 800 1000 

𝑓1 144.86 97.73 68.95 18.88 3.14 2.09 0.00 

𝑓2 2520.21 2068.39 966.62 385.53 184.19 52.43 3.68 

𝑓3 6230.88 3440.11 1690.14 412.25 262.88 144.12 56.36 

𝑓4 424.94 389.45 330.31 214.29 168.81 124.74 97.41 

𝑓5 618.13 505.32 205.19 38.16 11.94 1.64 0.00 

𝑓6 20.77 20.52 19.04 5.49 1.88 0.14 0.01 

𝑓7 131.53 131.53 131.53 131.53 55.39 41.53 1.85 

𝑓8 24.83 24.32 22.03 11.55 2.93 1.61 0.82 

𝑓9 83.25 60.72 51.85 38.74 15.77 12.02 8.36 

𝑓10  23.63 21.12 19.51 16.44 14.02 12.22 9.91 

 
 
As shown table 4, the runtime of LFQPSO is higher than that of QPSO which is higher than that of PSO. 
However, the income of new proposed algorithm LFQPSO, its time cost is worth. Take f1 as an example, 
relative QPSO, although the time of LFQPSO increased by 40.01%, the performance increased by 90%. So, 
the above analysis shows that as far as the time performance and the solution quality is concerned, the new 
algorithm is far superior to the other two algorithms. 
 
 

503



Table 4: The runtime of 10 benchmark function  

function PSO QPSO LFQPSO 

f1 2.48 5.24 7.34 

f2 2.56 5.40 7.57 

f3 2.63 5.56 7.78 

f4 2.78 5.88 8.23 

f5 2.74 5.79 8.10 

f6 2.74 5.79 8.11 

f7 2.61 5.52 7.73 

f8 2.87 6.06 8.49 

f9 2.68 5.66 7.93 

f10 2.71 5.73 8.02 

5. Conclusion 

A new improved quantum-behaved particle swarm optimization based on the levy flight (LFQPSO) has been 
presented. For all of the benchmark function that has been used in this work, the LFQPSO has been shown to 
outperform the QPSO and PSO. The superior performance of the LFQPSO is due to its ability to improve a 
local search while improve global search at the same time which is mainly because that LFQPSO uses the 
levy flight instead of the Brown motion so that it overcoming the particle wait effect of QPSO. Experiment 
result of 10 benchmark functions shows our proposed algorithm performs better than QPSO and PSO which 
shows that LFQPSO proposed in this work is not only suitable for the linear and single peak function but also 
for the nonlinear and multi peak function. 

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