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A Comparative Study of Cuckoo Algorithm and Ant Colony

Algorithm in Optimal Path Problems

Guanyu Wang1, a

1Internet of Things Engineering, Beijing University of Posts and Telecommunications, 100876, Beijing, China

Abstract. Finding the optimal path can be realized through a wide range of algorithms, which is demanded

in many fields. Among countless algorithms that are used for solving the optimal path problem, the ant

colony optimization (ACO) is one of the algorithms used to solve the approximate optimal path solution,

while the cuckoo search (CS) algorithm is a swarm intelligence algorithm featuring Levy flight, whose core

idea is derived from the cuckoo nesting property. In order to provide more ideas and directions for future

research on optimal path problems, this paper discusses in detail the advantages and disadvantages of the

two algorithms for solving the optimal path problem and their scopes of application by comparing principles

and flows of the two algorithms.

1 Introduction

1.1. Principles and Difficulties of the Optimal

Path Problem

The path is the path in the undirected graph satisfying the

condition that all the vertices (except for the starting

point and the ending point) and all the edges are different.

The optimal path is the one that best meets certain

requirements in the path. For example, a path will be

called the shortest path if the sum of the weights of its

constituent edges is minimized. The optimal path

algorithm refers to the calculation method for finding the

optimal path of a graph. Many scholars both at home and

abroad have done numerous research on how to solve the

optimal path problem, such as the classic Dijkstra

algorithm, A* algorithm, ant colony optimization, and

simulated annealing algorithm. These classical

algorithms are studied from different angles, and are

continuously supplemented and improved to obtain an

efficient optimal path algorithm. This paper focuses on

the comparison between the cuckoo search and the ant

colony optimization in the optimal path problem.

1.2. Common Algorithms for Solving the

Optimal Path Problem

1.2.1. Dijkstra's algorithm

Dijkstra's algorithm is an algorithm for finding the

shortest paths between nodes in a graph, whose core idea

can be described in this way: Starting from the path, by

constructing the node queue, the path node spreads out

from each possible path until the end. As for

single-source shortest paths which are directed and

weighted, the algorithm uses breadth-first search to

obtain the nodes of the shortest path.

1.2.2. A* Algorithm

Because of its flexibility, the A* algorithm has become

one of the popular algorithms for searching the shortest

path, and can be used in many different situations. Like

other graph search algorithms, A* algorithm searches a

large area of the graph. A* can not only be used to search

for the shortest path like Dijkstra, but also guide itself

with heuristic functions like BFS.

1.2.3. Ant Colony Optimization Algorithm

The ant colony optimization algorithm (ACO) is inspired

by the group behavior of ants searching for food, during

which the ant colony gradually tends to the shortest path.

Marco Dorigo discovered this feature of ant colonies in

nature and applied this inspiration to the calculation of

the shortest path. In 1992, he proposed the ant colony

optimization algorithm in his doctoral thesis. ACO

belongs to a probabilistic algorithm, which is essentially

a heuristic global optimization algorithm featuring

distributed computing, positive feedback of information

and heuristic search.

1.2.4. Particle Swarm Optimization

Particle swarm optimization (PSO) is very popular for

solving practical problems because of its easy

implementation, highly precise solution and fast

convergence. PSO belongs to an evolutionary algorithm,

whose principle lies in finding the optimal solution by

iteration, and evaluating the solution quality by the

fitness of the optimal solution.

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0

(http://creativecommons.org/licenses/by/4.0/).

MATEC Web of Conferences 232, 03003 (2018) https://doi.org/10.1051/matecconf/201823203003

EITCE 2018

2 Cuckoo Search (CS)

2.1. Principles of Cuckoo Search

The cuckoo search algorithm is an emerging

metaheuristic algorithm developed by Xin-she Yang and

Suash Deb of Cambridge University in 2009, which is

based on simulating the nest parasitism of cuckoos in

nature and also combined with the characteristics of

Levy flight to solve the optimization problem.

2.1.1. Brood parasitism

According to long-term observations by biologists,

cuckoos breed young birds in a parasitic manner. Brood

parasitism refers to a special reproductive behavior in

which some birds have eggs in other birds’ nests and rely

on others to hatch and raise their offspring, which can be

summarized in three characteristics. First, cuckoos

usually lay their own eggs in host nests in advance of the

hosts’ hatching period, each cuckoo laying one egg at a

time. Second, cuckoos push the existing eggs out of host

nests or remove them when laying eggs there. Third, as

soon as they hatch, cuckoo chicks will also have the habit

of pushing other host chicks of the same nest out so as to

enjoy exclusive fostering of host birds due to their

nature.

2.1.2. Levy Flight

Apart from cuckoos’ unique approach to hatching, Levy

flight also plays an important part in cuckoo search

algorithm. Numerous studies have shown that many

animals and insects demonstrate regular characteristics of

Levy flight in their flight paths. The discovery of Levy

flight was made in 1996 by Viswanathan and his

colleagues using satellite positioning systems to study the

albatrosses’ foraging behavior. They found that the

albatrosses’ flight intervals obeyed the power-law

distribution and they proved this finding through

consistent spatial distribution of the food quantity on

ocean surface. In addition, a recent study by Reynolds

and Frye showed that fruit flies would fly at right angles

from time to time, which also demonstrated the

intermittent scale-free searching feature of Levy flight.

Figure 1 Levy flight path [1]

By abstracting the cuckoo nesting method into a

theory, a new algorithm is derived. Cuckoo search is

based on three idealized rules:

1) Each cuckoo lays one egg at a time, and dumps its

egg in a randomly chosen nest;

2) The best nests with high quality of eggs will carry

over to the next generation;

3) The number of available host nests is fixed, and

the egg laid by a cuckoo is discovered by the host bird

with a probability Pa. On discovering parasitic eggs, host

birds either push them out or give up existing nests and

build a new one. To simplify the model, let this idealized

rule approximate to the probability Pa of the host birds

building new nests.

The above rules can be summarized as follows:

1) Each egg in a host nest represents a solution;

2) Each cuckoo represents a new solution, whose

purpose is to iteratively replace inferior solutions in the

original scheme with a better one;

3) There is only one egg in each nest.

Based on the above three idealized rules, the formula

for updating the path and position of cuckoo nesting is as

follows:

()

1

i

xt+=

()

i

XT+

α

⊕

()

L

λ

()

i1,2n=…

The steps are as follows:

Step 1: The objective function

()

fx

()

1

T

d

xxx=…

is established to initialize the group. Then the initial

positions

()

i1,2n

i

X=…

of n nests are randomly

generated, and the algorithm parameters are set;

Step 2: The objective function value of each nest is

calculated and the current optimal solution is recorded;

Step 3: The last optimal solution is kept and the

positions of other nest are updated according to the

formula for position updating;

Step 4: The position of the current nest is compared

with that of the previous nest. If the former solution is

better, it will be recorded as the current optimal solution;

Step 5: A random number R as the probability of the

host bird finding parasitic eggs is selected compared with

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MATEC Web of Conferences 232, 03003 (2018) https://doi.org/10.1051/matecconf/201823203003

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Pa. If R > Pa, the position of the nest will be randomly

changed to obtain a new set of nest positions;

Step 6: If the condition for ending is not met, the

operation will return to Step 2;

Step 7: The globally optimal position is output.

2.2. Problems and Drawbacks of Cuckoo Search

Cuckoo search is a new meta-heuristic algorithm for

swarm intelligence optimization, which is established by

combining the brood parasitism of cuckoos and the

model simulation of Levy flight. By continuously

iterating the optimal solution until the globally optimal

solution is obtained, this algorithm provides an idea for

finding the optimal path in some manner, namely,

continuously iterating the optimal path until the globally

optimal path is obtained. Therefore, compared with other

optimal path algorithms, cuckoo search is more

advantageous in some cases. However, the algorithm is

still undergoing rapid development and improvement,

and the processing of some links still needs continuous

optimization.

2.2.1. Convergence

Cuckoo search has been proved to be effective in

practical applications, whose convergence is mainly

based on the establishment of the Markov chain model

through analyzing the homogeneity of the chain to prove

that it satisfies the two conditions of the global

convergence in a random search algorithm. However, the

internal implementation of the cuckoo search is not

comprehensive enough and still needs detailed study and

analysis, which calls for further understanding of the

convergence speed and quality of the algorithm. It is

believed that the convergence of cuckoo search will be

more thoroughly understood, which also plays a

profound role in better applying this algorithm.

2.2.2. Limitations

The original cuckoo search can only be used to solve

continuous problems. As for discrete problems and

multi-objective problems, however, it fails to perform

well. Therefore, the algorithm has certain limitations

when processing some other problems, which is exactly

what we need to further study and improve. As for the

continuous problems that the algorithm can deal with,

there has been much advancement in terms of step size,

parameter, intercoupling with other algorithms and other

factors used to improve the performance of related

indicators. Meanwhile, this algorithm also has problems

in adaptability and obtaining ideal search results, and its

ability to solve complex problems is limited. Future

research should be directed towards studying how to

explore new methods and strategies to improve the

performance of high coupling functions between

variables.

2.2.3. Balance Between global Search and Local

Search

The two indicators of local search ability and global

convergence ability are common standards for measuring

intelligent optimization algorithms. The so-called local

search refers to the ability of the algorithm to gradually

approach the optimal solution, while the global

convergence ability is the ability to decide the

approximate range of the global optimal solution. [2]

These two metrics play a significant role in evaluating

optimization algorithms, which determine the merits of

an algorithm from two different aspects.

The significance of local search lies in that it offers

an idea to solve complex problems, that is, to exchange

time for precision. At present, NP-complete problems

still accounts for a large part in optimal solutions. The

time required for the optimal solution to this problem

exponentially increases over time, which is very

troublesome. Therefore, local search can be used to find

the most approximate (second-best) solution within a

reasonable time span, achieving the balance between

time and solution.

How to balance the local search and global

convergence in cuckoo search is also one of the

directions to be further studied, so it is necessary to

understand the three existing improved algorithms of the

local search algorithm so as to further understand the

internal principles and improving directions of local

search. The improvement of local search from three

aspects is described as follows:

a) Local Optimum

The local optimal refers to an algorithm falling into

the local extremum point in the continuous iterative

search process and then ending the operation. This is

similar to climbing the summit of mountains which has

many small peaks. Due to the lack of knowledge about

the mountains, a small peak reached by the algorithm

might be wrongly judged to be the highest. However, this

local peak is likely to much lower than the summit of the

mountains. Therefore, regarding such a local extremum

point as the optimal solution is unwelcome.

b) Step Size

The step size also affects whether the final solution of

the problem can the optimal one. For some problems, the

mapping function N value that can obtain the optimal

solution must be some fixed values. Therefore, if the

selection range of the step size does not cover those

specific values, chances are that the optimal solution

cannot be obtained. The root cause of this situation lies in

that the step size is fixed and can only search a few fixed

points, which can be solved by changing the step size

during the search.

c) Initial Point

The selection of the initial point is, at heart, similar to

the step size problem, because different step sizes can

also be understood as different initial points. When the

optimal solution can’t be obtained due to the initial point,

it is necessary to randomly select the initial point, and

finally obtain the optimal solution by comparison, thus

avoiding falling into the local optimal solution caused by

the fixed initial point.

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MATEC Web of Conferences 232, 03003 (2018) https://doi.org/10.1051/matecconf/201823203003

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The current research shows that although cuckoo

search is a new type of optimization algorithm, it can be

optimized by following the three improving directions

mentioned above. The convergence and applying

limitations of cuckoo search are still problems to be

solved. By studying ant colony optimization (ACO) and

making a horizontal comparison between CS and ACO in

the following, this paper provides further understanding

of advantages and disadvantages and scopes of

application of these two algorithms, and helps learn how

to analyze different algorithms from various dimensions.

3 Ant Colony Optimization (Aco)

3.1. Principles of Ant Colony Optimization (ACO)

The ant colony algorithm was first proposed in 1991

when Marco Dorigo and his colleagues researched into

new algorithms. They found that ant colonies in nature

secreted pheromones to exchange information when

foraging so as to quickly find the target, inspired by

which, they put forward the ant colony optimization

algorithm based on positive feedback of information.

The basic idea of the ACO is derived from the

shortest foraging path gradually formed by the ant

colonies in nature, which is benefited from the biological

characteristics of the ants. Although ants are visually

undeveloped, they can keep secreting a biological

hormone called pheromone when foraging. The

pheromones secreted by ants are left in paths they pass.

Such paths are random, but because the ants that first

obtain food will leave first, so constant accumulation will

increasingly raise the concentration of pheromones on

the shortest path, which attracts the following ants to

choose this path, resulting in more and more pheromone

on this path. Higher concentration of pheromones in turn

attracts more ants, thus an optimal path will be chosen by

ant colonies in the long run. This selection process is

called the autocatalytic behavior of ants. While

individual ants just randomly select their foraging path,

an ant colony will gradually form the optimal path due to

the secretion of pheromones, objectively reaching the

effect of finding the optimal solution. This phenomenon

is called swarm intelligence.

Problems that are properly solved through swarm

intelligence algorithms are numerous. The TSP problem

is classical case, in which requires the travelling

salesman to find what the shortest possible route that

visits each city and returns to the origin city is under the

condition that each city can only be visited once. Due to

its similarity to the ant colony foraging, ACO is suitable

for modeling and solving this kind of problem. The

solving steps of the ACO are as follows:

Step 1: Parameters are initialized. The ant colony

optimization algorithm involves many parameters are

more, so it’s necessary to initialize the problem scale,

pheromone factor, pheromone volatilization factor,

pheromone constants, heuristic function factor and the

maximum number of iterations and then preprocess the

data information.

Step 2: The initial point of the ant is randomly

positioned and the set point the each ant passes is

calculated until all the set points are passed.

Step 3: The path length of each ant is calculated, the

current optimal solution is recorded, and the pheromone

concentration on the path is updated. [3]

Step 4: Steps from Step 2 to this step are repeated

until the maximum number of hits is reached.

Step 5: The result is output.

k

[ ( )] [ ( )]

[()][()]

P() if

0,

k

ij ij

is is

ij k

s allowed

tt

tt

t

αβ

αβ

τη

τη

⊂

=∈

，

j allowed

Calculation formula

There are many parameters in the formula of ACO, so

difference in the selection of parameters will lead to

difference in the overall calculation process. Selecting

different initial values will also randomly affect the

optimization process. As mentioned above, improper

selection of the initial value will also cause a local

optimum. Because the final aim is to convert the local

optimum into a globally optimal solution, the setting of

parameters is particularly important, the key of which

lies in balancing the relation between local and global

optimum. This is also the key issue gradually studied and

summarized at present. Nevertheless, the optimization

formula and steps described above are followed to solve

problems relating to ACO. The solution, namely the local

or global optimum, can be finally obtained by iterating

according to the formula.

3.2. Problems and Drawbacks of ACO

The ant colony optimization algorithm is fairly popular

among path optimization problems because of its strong

robustness, parallelism and positive feedback. However,

no matter how excellent the algorithm is, it still has

limitations. In the following, some limitations of ACO

are analyzed.

1) Ant colony algorithm is not efficient for dealing

with large-scale combinatorial problems. Because the

time complexity of ACO is O (n*(n-1)*m*T/2), it takes

longer time to process large-scale problems.

2) Because there are many parameters involved in

ACO, improper selection of initial values may lead to

local optimum, stagnation around the range of the local

optimum, immature converge prematurely, eventually

resulting in the convergence of the solution of the entire

ant colony and the failure to obtain the optimal solution.

3) The ant colony algorithm cannot properly handle

continuation problems.

4) Like the situation described in the limitation of

time complexity, when the problem size is too large,

individual ants will move randomly, which takes longer

time to operate the whole algorithm, lowering the

efficiency of finding the optimum.

5) The selection of pheromone-related parameters in

the formula is largely empirical, so if the pheromone

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MATEC Web of Conferences 232, 03003 (2018) https://doi.org/10.1051/matecconf/201823203003

EITCE 2018

parameters are selected without experience and

theoretical arguments, chances are that the results

obtained are greatly dispersed due to the inefficiency and

poor convergence of the algorithm.

4 Comparative Analysis of Cuckoo

Algorithm and Ant Colony Algorithm in

Optimal Path Problems

4.1. Metrics that Measure Algorithm

Performance

It is widely acknowledged that the most important

indicators for measuring algorithms are time complexity

and space complexity. The so-called time complexity

refers to the amount of time required by the algorithm,

and the space complexity refers to the size of memory

space needed by the algorithm. Calculation of these two

complexities is similar, usually represented by the

asymptotic behavior of complexity. In addition to time

complexity and space complexity, there are other metrics,

including correctness, readability, and robustness. The

so-called correctness regulates that the algorithm must be

able to reproduce the same problem stably. At the same

time, the readability of the code also demonstrates

quality of an algorithm because if the logic of an

algorithm is difficult to understand, it is unfriendly for

others to use and improve in the future. Finally, the

robustness of an algorithm is also an important

measuring index. An excellent algorithm should be

equipped with corresponding logic for any abnormal

situation, thus avoiding easy termination or abnormal

situations.

Therefore, it can be seen from the metrics mentioned

above that the ACO has disadvantages of slow

convergence and easily falling into local optimum.

Because the ACO involves too many parameters, it easily

leads to the lack of initial pheromone, and high

complexity. Moreover, the algorithm is prone to stagnate

and falls into local optimum of the algorithm, which is

not conducive to finding the global optimal solution.

Although it can be calculated for multiple times through

random initial values, it is time-consuming to do so.

Therefore, how to find the balance point is also a

problem that needs to be dealt with.

In contrast, cuckoo search is a new meta-heuristic

search algorithm enjoying extensive research prospects.

[4] By measuring it with the above performance metrics,

it can be conclude that the cuckoo search has the

advantages in terms of great robustness on the whole,

readable logics, portability and platform independence.

The algorithm also demonstrates advantages in strong

global search ability, few selected parameters, excellent

search path, and strong ability of solving multi-objective

problems.

4.2. Comparative Analysis and Conclusions of

the Two Algorithms

As a path optimization algorithm for finding approximate

solutions, ACO is not quite suitable for dealing with

problems requiring precision but usually applicable to the

problems whose accurate solutions can’t be obtained

within time demanded. In other words, ACO can be

adopted to obtain the second-best solutions of NP

problems, such as the classic TSP problem. The TSP

problem can also be said to be the shortest path problem,

but the problem is to find the minimum Hamiltonian path

of a complete graph. Therefore, this kind of problem

belongs to the NPC problems. In other words, when the

scale of the problem has a great impact on the time of

processing the problem, the time cost of using the

existing algorithms for finding the exact solution for the

large-scale TSP problem is enormous. Therefore, similar

algorithms like the ACO are usually adopted to find the

second-best solution of such problems, trying to achieve

a balance between time consumed and the precision of

the solution.

While as a new type of optimization algorithm,

cuckoo search solve the continuance problems which

cannot be handled by ACO. Meanwhile, since the cuckoo

search algorithm requires fewer parameters in the

formula of, it is less affected by other factors and more

robust. In the following, a comparative analysis of

specific performances of these two algorithms is made.

4.2.1. Analysis of the Ant Colony Optimization

Algorithm

In a sense, the ACO can be considered as a

self-organizing type in system theory. The so-called

self-organization means that organization instructions

come from the system. [5] Abstractly, self-organization is

the process of increasing the entropy of the system

without external influences (that is, the process of the

system changing from being disorderly to orderly),

which is exactly the case for ACO. In the initial stage of

the algorithm, individual ants search for the solution

disorderly, and the algorithm goes through a series of

optimization calculation. Then those individual ants

spontaneously tend to find the second-best solutions

individual through pheromones, shifting from being

disorderly to orderly. Therefore, the ACO is

self-organizing.

Meanwhile, the ant colony optimization is also a

parallel algorithm. The foraging process of each ant is

independent of each other, and only communicating with

others through the pheromones left by them. Therefore,

practically, a distributed system can be established to

ensure independent process of finding the optimal

solution, which not only enhances the robustness of the

algorithm, but also improve its global search ability.

Another obvious feature of ACO is its positive

feedback. The positive feedback lies in that individual

ants mainly rely on the pheromones generated by other

ants to communicate with each other in the process of

foraging. A large number of individual ants gradually

accumulate pheromones, and finally find the optimal

path. In this process, it can be seen that the essence lies

in the process of positive feedback. The pheromone

concentration in different paths is different because each

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MATEC Web of Conferences 232, 03003 (2018) https://doi.org/10.1051/matecconf/201823203003

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ant leaves pheromones in its own search path, and the

positive feedback reflects the phenomenon that more

pheromones are left in the path of better solutions, which

directs many other ants to find this path. The local or

global optimum which has the most pheromones can be

found by repeating this. The process of positive feedback

is also an important feature of ACO, which is the basis of

its operation.

4.2.2. Analysis of Cuckoo Search

As a relatively new type of path finding algorithm, the

cuckoo search demonstrates obvious advantages,

including its excellent global search ability. Due to the

characteristics of Levi flight, cuckoo search can better

avoid falling into the local optimum. Meanwhile,

compared with ACO, cuckoo search uses fewer

parameters, which contributes to its excellent robustness.

Also, cuckoo search shows good applicability when

dealing with multi-objective problems. What’s more, the

Levi flight demonstrating in cuckoo search is an efficient

global random search algorithm and has been proven to

perform better than other algorithms in optimal problems.

The rise of the cuckoo algorithm over the past few years

has confirmed that based on the advantages mentioned

above, it can be said to be a new star among swarm

intelligence algorithms.

4.2.3. Comparison of Main Advantages and

Disadvantages between Cuckoo Search and Ant

Colony Optimization

Cuckoo Search Ant Colony

O

p

timization

Advantages The algorithm

depends on fewer

parameters and is

robust;

It has the trait of

Levi flight, which

enhances its global

search ability;

It’s easy to couple

with other

algorithms,

demonstrating

strong versatility.

It has a positive

feedback

mechanism and is

effective in

dealing with

distributed

problems;

It’s easy to

combine with

other algorithms,

featuring great

robustness and

reliabilit

y

.

Disadvantages Convergence rate

is affected by Levi

flight and may be

slightly slower.

Initial information

of the algorithm is

scared and it is

easy to fall into

local optimum;

It usually takes

longer time to

search, not

suitable for

large-scale

p

roblems.

Scope of

application

Continuous

optimization

Discrete

optimization；

Small-scale

p

roblems

5 Conclusion

Cuckoo search algorithm is a new type of meta-heuristic

swarm intelligent algorithm for path searching, which

has a good research potential for continuous path finding

problems. At the same time, ant colony optimization is

also a very good algorithm for dealing with discrete NP

problems, demonstrating great research value. Both of

the two algorithms show good robustness when solving

problems in different fields. In addition, the powerful

features such as portability and platform independence of

the cuckoo algorithm are also of profound significance.

The cuckoo search displays the feature of Levi flight

and has strong global search ability. It is suitable for

solving continuous problems and multi-objective

problems. However, the cuckoo search algorithm also has

its limitations, such as slower convergence, and future

research should be focused on improving its

shortcomings. By analyzing the principles, advantages

and disadvantages of cuckoo search and ant colony

optimization, this comparative study inspires future

studies.

Ant colony optimization is a classical algorithm for

finding the approximately optimal solutions. Although it

has disadvantages of slower convergence and easily

falling into the second-best solution because of large

number of parameters, it has a positive feedback

mechanism, strong global search ability, strong

robustness, high reliability, and can be easily combined

with other algorithms. Therefore it is still a good

meta-algorithm researched and improved by scholars.

Studying how to improve the above two algorithms, or

maybe combine advantages of other algorithms into the

two algorithms is also the direction and goal of the

author’s future research plan. Hopefully, the discussion

of cuckoo search and ant colony optimization algorithms

for solving optimal path problems can help future

research. Finally, the author welcomes any suggestion for

improving or correcting this paper.

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[J]. Computer Engineering and Applications,

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MATEC Web of Conferences 232, 03003 (2018) https://doi.org/10.1051/matecconf/201823203003

EITCE 2018