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Cuckoo Search and Fireﬂy Algorithm:

Overview and Analysis

Xin-She Yang

Abstract Fireﬂy algorithm (FA) was developed by Xin-She Yang in 2008, while

cuckoo search (CS) was developed by Xin-She Yang and Suash Deb in 2009. Both

algorithms have been found to be very efﬁcient in solving global optimization prob-

lems. This chapter provides an overview of both cuckoo search and ﬁreﬂy algorithm

as well as their latest developments and applications. We analyze these algorithms

and gain insight into their search mechanisms and ﬁnd out why they are efﬁcient.

We also discuss t he essence of algorithms and its link to self-organizing systems. In

addition, we also discuss important issues such as parameter tuning and parameter

control, and provide some topics for further research.

Keywords Algorithm

· Cuckoo search · Fireﬂy algorithm · Metaheuristic ·

Optimization ·Self-organization

1 Introduction

Optimization and computational intelligence are active research areas with rapidly

expanding literature. For most applications, time, money and resources are always

limited, and thus their optimal use becomes increasingly important. In modern design

applications, it requires a paradigm shift in thinking and design to ﬁnd energy-

saving and greener design s olutions. However, to obtain optimal solutions to design

problems are non-trivial, and many real-world optimization problems can be really

hard to solve.For example, it is well-knownthatcombinatorialoptimizationproblems

such as the travelling salesman problem (TSP) are NP-hard, which means that there

are no efﬁcient algorithms in general.

X.-S. Yang (

)

School of Science and Technology, Middlesex University, London NW4 4BT, UK

e-mail: xy227@cam.ac.uk; x.yang@mdx.ac.uk

X.-S. Yang (ed.), Cuckoo Search and Fireﬂy Algorithm,1

Studies in Computational Intelligence 516, DOI: 10.1007/978-3-319-02141-6_1,

2 X.-S. Yang

Even without efﬁcient algorithms, we still have to solve these challenging prob-

lems in practice. This leads to the development of various exotic techniques and

problem-speciﬁc methods so that problem-speciﬁc knowledge(such as gradients) can

be used to guide better search procedures. In general, algorithms that use problem-

related knowledge should perform better than black-box-type methods that do not

use problem knowledge at all. But the incorporation of problem-speciﬁc knowl-

edge often limits the use of a method or an algorithm. In addition, it is not easy or

too computationally extensive to incorporate such knowledge. Sometimes, it may

be impossible to incorporate such knowledge, either because we may not have any

insight into the problem or because we do not know how to. In this case, the only

option is to use black-box-type algorithms that do not assume any knowledge of the

problem of interest.

In most cases, for NP-hard problems, the only alternative is to use heuristic or

metaheuristic methods by trial and error. Heuristic methods are to search for solu-

tions by trial and error, while metaheuristic methods can be considered as higher-

level heuristic methods that use information and selection of the solutions to guide

the search process. Optimization algorithms are the tools and techniques for solving

optimization problems with the aim to ﬁnd its optimality, though such optimality is

not always reachable. This search for optimality is complicated further by the fact

that uncertainty is almost always present in the real-world systems. Therefore, we

seek not only the optimal design but also robust design in engineering and industry.

Optimal solutions, which are not robust enough, are not practical in reality. Subopti-

mal solutions or good robust solutions are often the choice in such cases. In the last

twenty years, nature-inspired metaheuristic algorithms have gained huge popularity

in optimization, artiﬁcial intelligence, machine learning, computational intelligence,

data mining and engineering applications [40, 65, 78].

The aim of this chapter is to review both cuckoo search and ﬁreﬂy algorithm and

their applications. Therefore, the chapter is organized as follows: Sect. 2 outlines

brieﬂy the basic formulation of optimization problems, while Sect.3 discusses the

essence of an optimization algorithm. Section4 provides a detailed introduction to

cuckoo search, and Sect. 5 introduces the ﬁreﬂy algorithm in greater detail. Section 6

highlights the importance of the right amount of randomization, while Sect. 7 touches

the challenging issues of parameter tuning and parameter control. Finally, Sect.8

draw conclusions brieﬂy.

2 Optimization Problems

Optimization problems can be formulated in many ways. For example, the commonly

used method of least-squares is a special case of maximum-likelihood formulations.

By far the most widely used formulation is to write a nonlinear optimization problem

minimize f

(x), (m = 1, 2, ..., M), (1)

Cuckoo Search and Fireﬂy Algorithm: Overview and Analysis 3

subject to the constraints

(x) = 0,(j = 1, 2, ..., J ), (2)

(x) ≤ 0,(k = 1, 2, ..., K ), (3)

where f

, h

and g

are in general nonlinear functions. Here the design vector

or design variables x = (x

, x

, ..., x

) can be continuous, discrete or mixed in

d-dimensional space. The functions f

are called objective or cost functions, and

when M > 1, the optimization problem is called multiobjective or multicriteria [65].

It is worth pointing out that here we write the problem as minimization, it can

also be written as maximization by simply replacing f

(x) by − f

(x). When all

functions are nonlinear, we are dealing with nonlinear constrained problems. In some

special cases when f

, h

and g

are linear, the problem becomes linear, and we can

use the widely linear programming techniques such as the simplex method. When

some design variables can only take discrete values (often integers), while other

variables are real continuous, the problem is of mixed type, which is often difﬁcult

to solve, especially for large-scale optimization problems.

In general, multiobjective optimization requires multiobjective techniques to

obtain the Pareto fronts of a problem of interest. However, it is possible to com-

bine different objectives into a single objective by using weighted sum

f (x) =



m=1

, α

> 0,



m=1

= 1. (4)

This is a simple approach that allows us to use algorithms for single-objective opti-

mization without major modiﬁcations. However, this approach does havesome disad-

vantages as it can only deal with convex Pareto fronts. In addition, true multiobjective

approaches to multiobjective optimization can give far more information and insight

into the problem.

3 The Essence of an Optimization Algorithm

Before we go any further to discuss nature-inspired algorithms such as cuckoo search

and ﬁreﬂy algorithm, let us analyze the essence of an optimization algorithm.

In essence, optimization is a process of searching for the optimal solutions to a par-

ticular problem of interest, and this search process can be carried out using multiple

agents which essentially form a system of evolving agents. This system can evolve

by iterations according to a set of rules or mathematical equations. Consequently,

such a system will show some emergent characteristics, leading to self-organizing

states which correspond to some optima of the objective landscape. Once the self-

organized states are reached, we say the system converges. Therefore, to design

4 X.-S. Yang

an efﬁcient optimization algorithm is equivalent to mimicking the evolution of a

self-organizing system [4, 38].

3.1 The Essence of an Algorithm

Mathematically speaking, an algorithm is a procedure to generate outputs for a given

set of inputs. From the optimization point of view, an optimization algorithm gener-

ates a new solution x

t+1

to a given problem from a know solution x

at iteration or

time t. That is

t+1

= A(x

, p(t)), (5)

where A is a nonlinear mapping from a given solution (i.e., a d-dimensional vector) x

to a new solution vector x

t+1

. The algorithm A has k algorithm-dependent parameters

p(t) = ( p

, ..., p

) which can be time-dependent and can thus be tuned.

To ﬁnd the optimal solution x

∗

to a given optimization problem S with an often

inﬁnitely number of states is to select some desired states φ from all states ψ, accord-

ing to some predeﬁned criterion D.Wehave

S(ψ)

A(t)

−→ S(φ(x

∗

)), (6)

where the ﬁnal converged state φ corresponds to an optimal solution x

∗

of the problem

of interest. However, the converged states can be multiple, and each corresponds to

a local optimum. Among all the optima, there should be the global optimality. This

self-organization process does not give an indication that it will converge to the global

optimal solution. The aim is to tune the algorithm and control its behaviour so that

true global optima (states) can be selected out of many possible states.

The selection of the system states in the design space is carried out by running

the optimization algorithm A. The behavior of the algorithm is controlled by the

parameters p, the initial solution x

t=0

and the stopping criterion D. We can view the

combined S + A(t) as a complex system with a self-organizing capability.

The change of states or solutions of the problem of interest is through the

algorithm A. In many classical algorithms such as hill-climbing, gradient informa-

tion of the problem S is used so as to select states, say, the minimum value of the

landscape, and the stopping criterion can be a given tolerance or accuracy, or zero

gradient, etc.

An algorithm can act like a tool to tune a complex system. If an algorithm does

not use any state information of the problem, then the algorithm is more likely to be

versatile to deal with many types of problems. However, such black-box approaches

can also imply that the algorithm may not be efﬁcient as it could be for a given

type of problem. For example, if the optimization problem is convex, algorithms

that use such convexity information will be more efﬁcient than the ones that do not

use such information. In order to be efﬁcient to select states/solutions efﬁciently, the

information from the search process should be used to enhance the search process.

In many case, such information is often fed into the selection mechanism of an

Cuckoo Search and Fireﬂy Algorithm: Overview and Analysis 5

algorithm. By far the most widely used selection mechanism to select or keep the

best solution found so far. That is, the simplest form of ‘survival of the ﬁttest’.

From the schematic representation (6) of the optimization process, we can see

that the performance of an algorithm may also depend on the type of problem S it

solves. On the other hand, the ﬁnal, global optimality is achievable or not (within

a given number of iterations) will also depend on the algorithm used. This may be

another way of stating the so-called no-free-lunch theorems.

Optimization algorithms can very diverse with several dozen widely used algo-

rithms. The main characteristics of different algorithms will only depend on the

actual, nonlinear, often implicit form of A(t) and its parameters p(t).

3.2 An Ideal Algorithm?

The number of iterations t needed to ﬁnd an optimal solution f or a given accu-

racy largely determines the overall computational efforts and the performance of an

algorithm. A better algorithm should use less computation and fewer iterations.

In an extreme case for an iterative algorithm or formula (5), an ideal algorithm

should only take one iteration t = 1. You may wonder if such an algorithm exists

in practice. The answer is “yes, it depends”. In fact, for a quadratic function such as

f (x) = ax

where a > 0, the well-know Newton-Raphson method

n+1

= x

−



)



)

, (7)

is the ideal algorithm. If we start the iteration from any random guess x

= b,we

have

= x

−



)



)

= b −

2ab

= 0, (8)

which gives the global optimal solution f

∗

(0) = 0atx

∗

= 0.

Obviously, we can extend this to all class of quadratic functions. Even it is possible

to extend to more generalized convex functions. However, many problems are not

convex, and certainly not quadratic. Therefore, the so-called ideal algorithm does

not exist in general. As we have mentioned earlier, there is no good algorithm for

solving NP-hard problems.

There are many optimization algorithms in the literature and no single algorithm

is suitable for all problems [64]. Even so, the search for efﬁcient algorithm still forms

the major efforts among researchers, and this search for the ‘Holy Grail’ continues,

unless some proves analytically otherwise.

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