I.J. Information Technology and Computer Science, 2012, 5, 58-67
Published Online May 2012 in MECS (http://www.mecs-press.org/)
DOI: 10.5815/ijitcs.2012.05.08
Copyright © 2012 MECS I.J. Information Technology and Computer Science, 2012, 5, 58-67
A Novel Quantum Inspired Cuckoo Search
Algorithm for Bin Packing Problem
Abdesslem Layeb
MISC Lab, Computer science department, Mentouri university of Constantine,
Constantine, Algeria.
Email:layeb@umc.edu.dz
Seriel Rayene Boussalia
Computer science department, Mentouri University of Constantine,
Constantine, Algeria.
Email: seriel.rayene@gmail.com
Abstract— The Bin Packing Problem (BPP) is one of
the most known combinatorial optimization problems.
This problem consists to pack a set of items into a
minimum number of bins. There are several variants of
this problem; the most basic problem is the one-
dimensional bin packing problem (1-BPP). In this paper,
we present a new approach based on the quantum
inspired cuckoo search algorithm to deal with the 1-
BPP problem. The contribution consists in defining an
appropriate quantum representation based on qubit
representation to represent bin packing solutions. The
second contribution is proposition of a new hybrid
quantum measure operation which uses first fit heuristic
to pack no filled objects by the standard measure
operation. The obtained results are very encouraging
and show the feasibility and effectiveness of the
proposed approach.
Index Terms— Bin Packing Problem, Heuristics,
Cuckoo Search Algorithm, Quantum Computing,
Hybrid Algorithms
1. Introduction
The combinatorial optimization plays a very
important role in operational research, discrete
mathematics and computer science. The aim of this
field is to solve several combinatorial optimization
problems that are difficult to solve. Bin packing
problem (BPP) is very known NP-Hard optimization
problem. There are three main variants of BPP
problems: one, two and three dimensional Bin Packing
Problems. They have several real applications such as
container loading, cutting stock, packaging design and
resource allocation, etc. In this paper, we deal with the
one-dimensional Bin Packing Problem (1-BPP) [1, 2, 3].
The 1-BPP consists to pack a set of items having
different weights into a minimum number of bins which
may have also different capacities. Although, this
problem seems to be quite simple to define, it has been
shown to be NP-hard, because it cannot be solved
accurately and optimally in a reasonable time. This is
the reasons, why several approximate methods have
been proposed to solve this problem, which are
generally based on heuristics or metaheuristics. Among
the most popular heuristics used to solve the bin
packing problem, the First Fit algorithm (FF) which
places each item into the first bin in which it will fit.
The second popular heuristic algorithm is the Best Fit
(BF) which puts each element into the filled bin in
which it fits. Moreover, the FF and BF heuristics can be
improved by applying a specific order of items like in
First Fit Decreasing (FFD) and Best Fit Decreasing
(BFD), etc [4,5,6]. Moreover, many kinds of
metaheuristics have been used to solve the bin packing
problems like genetic algorithms [7], Ant colony [8],
etc.
Evolutionary computation has been proven to be
an effective way to solve complex engineering
problems. It presents many interesting features such as
adaptation, emergence and learning [9]. Artificial neural
networks, genetic algorithms and swarm intelligence are
examples of bio-inspired systems used to this end [10].
In recent years, optimizing by swarm intelligence has
become a research interest to many research scientists
of evolutionary computation fields. There are many
algorithms based swarm intelligence like Ant Colony
optimization [11, 12], eco-systems optimization [13],
etc. The main algorithm for swarm intelligence is
Particle Swarm Optimization (PSO) [14, 15], which is
inspired by the paradigm of birds grouping. PSO was
used successfully in various hard optimization problems.
One of the most recent variant of PSO algorithm is
Cuckoo Search algorithm (CS). CS is an optimization
algorithm developed by Xin-She Yang and Suash Deb
in 2009 [16]. It was inspired by the obligate brood
parasitism of some cuckoo species by laying their eggs
in the nests of other host birds (of other species). Some
bird’s host can involve direct conflicts with the
intruding cuckoos. For example, if a bird’s host
discovers that the eggs are strange eggs, it will either
throw these alien eggs away or simply abandon its nest
and build a new nest elsewhere [17]. The cuckoo’s
behavior and the mechanism of Lévy flights [18, 19]
have leading to design of an efficient inspired algorithm
performing optimization search [20, 21]. The recent
applications of Cuckoo Search for optimization
problems have shown its promising effectiveness.
Moreover, a promising discrete cuckoo search
algorithm is recently proposed to deal with knapsack
A Novel Quantum Inspired Cuckoo Search Algorithm for Bin Packing Problem 59
Copyright © 2012 MECS I.J. Information Technology and Computer Science, 2012, 5, 58-67
problems [22].
Quantum Computing (QC) is a new research field
that induced intense researches in the last decade, and
that covers investigations on quantum computers and
quantum algorithms [23]. QC relies on the principles of
quantum mechanics like qubit representation and
superposition of states. QC is able of processing huge
numbers of quantum states simultaneously in parallel.
QC brings new philosophy to optimization due to its
underlying concepts. Recently, a growing theoretical
and practical interest is devoted to researches on
merging evolutionary computation and quantum
computing [24]. The aim is to get benefit from quantum
computing capabilities to enhance both efficiency and
speed of classical evolutionary algorithms. This has led
to the design of several quantum inspired algorithms
such as quantum inspired genetic algorithm [25],
quantum differential algorithm [26], quantum inspired
scatter search [27], etc. Unlike pure quantum computing,
quantum inspired algorithms don’t require the presence
of a quantum machine to work. Quantum inspired
algorithms have been used to solve successfully many
combinatorial optimization problems [24, 28]. Recently
a new hybrid algorithm called Quantum Inspired
Cuckoo Search algorithm (QICSA) is proposed to cope
with combinatorial optimization problems [29].The
proposed algorithm combines Cuckoo Search algorithm
and quantum computing in new one. The features of the
proposed algorithm consist in adopting a quantum
representation of the search space. The other feature of
QICSA is the integration of the quantum operators in
the cuckoo search dynamics in order to optimize a
defined objective function.
The present study was designed to investigate the
use of the QICSA algorithm to deal with the one
dimensional bin packing problem. The main features of
the proposed approach are the use of the qubit
representation to represent the search space and a set of
quantum operators operating on this search space.
Moreover, we have proposed a new hybrid quantum
measure operation based on both the standard quantum
measure operation and the First Fist heuristic. We have
tested our algorithm on some popular data sets [6] and
the results are promising.
The remainder of the paper is organized as follows.
In section 2, a formulation of the tackled problem is
given. In section 3, on overview of quantum computing
is presented. In section 4, the cuckoo search algorithm
search presented. The proposed method is described in
section 5. Experimental results are discussed in section
6. Finally, conclusions and future work are drawn.
2. Problem formulation
Bin packing problem is an important task in
solving many real problems such as the loading of
tractor trailer trucks, cargo airplanes and ships, etc. For
example, the container loading problem consists to
design an affective loading plan for containers. It
consists of finding the most economic storage of articles
that have all the same dimensions but with different
weights in containers (also called bins) of equal
capacity. The constraint is that the bins do not exceed
its capacity. This problem can be modeled as a one-
dimensional bin packing problem. The principal
objective is to minimize the number of bins used for
storing the set of all items. Formally, the bin packing
problem can be stated as follows:
n
j
j
yyzMin
1
)(
(1)
Subject to constraints:
jij
n
j
i
cyxw
1
*Nj
(2)
1
1
n
i
ij
x
*Nj
(3)
1,0 ,
iji
xy
*, Nji
(4)
With :
y
i
= 1 if the bin i is used; else 0
x
ij
= 1 if the item j is stocked in bin i.
In the above model the objective function is to
minimize the total number of bins used to pack all items
which have the same capacity ( eq.1). The first
constraint guarantees that the weights of items (w
i
)
filled in the bin j do not exceed the bin capacity. The
second constraint ensures that each item is placed only
in one bin. It appears to be impossible to obtain exact
solutions in polynomial time. The main reason is that
the required computation grows exponentially with the
size of the problem. Therefore, it is often desirable to
find near optimal solutions to these problems. Efficient
heuristic algorithms offer a good alternative to
accomplish this goal. Within this perspective, we are
interested in applying a QICSA algorithm.
3. Overview of Quantum Computing
Quantum computing is a new theory which has
emerged as a result of merging computer science and
quantum mechanics. Its main goal is to investigate all
the possibilities a computer could have if it followed the
laws of quantum mechanics. The origin of quantum
computing goes back to the early 80 when Richard
Feynman observed that some quantum mechanical
During the last decade, quantum computing has
attracted widespread interest and has induced intensive
investigations and researches since it appears more
powerful than its classical counterpart. Indeed, the
parallelism that the quantum computing provides
reduces obviously the algorithmic complexity. Such an
60 A Novel Quantum Inspired Cuckoo Search Algorithm for Bin Packing Problem
Copyright © 2012 MECS I.J. Information Technology and Computer Science, 2012, 5, 58-67
ability of parallel processing can be used to solve
combinatorial optimization problems which require the
exploration of large solutions spaces. The basic
definitions and laws of quantum information theory are
beyond the scope of this paper. For in-depth theoretical
insights, one can refer to [23].
The qubit is the smallest unit of information stored
in a two-state quantum computer. Contrary to classical
bit which has two possible values, either 0 or 1, a qubit
will be in the superposition of those two values. The
state of a qubit can be represented by using the bracket
notation:
|Ψ
= α |0
+ β |1
(5)
where |Ψ denotes more than a vector
in some
vector space. |0 and |1 represent the classical bit
values 0 and 1 respectively; a and b are complex
numbers such that:
| a |
2
+ | b |
2
= 1 (6)
a and b are complex number that specify the
probability amplitudes of the corresponding states.
When we measure the qubit’s state we may have ‘0’
with a probability | a |
2
and we may have ‘1’ with a
probability | b |
2
. A system of m-qubits can represent 2
m
states at the same time. Quantum computers can
perform computations on all these values at the same
time. It is this exponential growth of the state space
with the number of particles that suggests exponential
speed-up of computation on quantum computers over
classical computers. Each quantum operation will deal
with all the states present within the superposition in
parallel. When observing a quantum state, it collapses
to a single state among those states.
Quantum Algorithms consist in applying
successively a series of quantum operations on a
quantum system. Quantum operations are performed
using quantum gates and quantum circuits. It should be
noted that designing quantum algorithms is not easy at
all. Yet, there is not a powerful quantum machine able
to execute the developed quantum algorithms.
Therefore, some researchers have tried to adapt some
properties of quantum computing in the classical
algorithms. Since the late 1990s, merging quantum
computation and evolutionary computation has been
proven to be a productive issue when probing complex
problems. Like any other EA, a Quantum Evolutionary
Algorithm (QEA) relies on the representation of the
individual, the evaluation function and the population
dynamics. The particularity of QEA stems from the
quantum representation they adopt which allows
representing the superposition of all potential solutions
for a given problem. It also stems from the quantum
operators it uses to evolve the entire population through
generations. QEA has been successfully applied on
many problems [24, 28, 29].
4. Cuckoo Search Algorithm
In order to solve complex problems, ideas gleaned
from natural mechanisms have been exploited to
develop heuristics. Nature inspired optimization
algorithms has been extensively investigated during the
last decade paving the way for new computing
paradigms such as neural networks, evolutionary
computing, swarm optimization, etc. The ultimate goal
is to develop systems that have ability to learn
incrementally, to be adaptable to their environment and
to be tolerant to noise. One of the recent developed
bioinspired algorithms is the Cuckoo Search (CS) [16]
which is based on style life of Cuckoo bird. Cuckoos
use an aggressive strategy of reproduction that involves
the female hack nests of other birds to lay their eggs
fertilized. Sometimes, the egg of cuckoo in the nest is
discovered and the hacked birds discard or abandon the
nest and start their own brood elsewhere. The Cuckoo
Search proposed by Yang and Deb 2009 [16] is based
on the following three idealized rules:
Each cuckoo lays one egg at a time, and dumps
it in a randomly chosen nest;
The best nests with high quality of eggs
(solutions) will carry over to the next
generations;
The number of available host nests is fixed,
and a host can discover an alien egg with a
probability pa [0, 1]. In this case, the host
bird can either throw the egg away or abandon
the nest so as to build a completely new nest in
a new location.
The last assumption can be approximated by a
fraction pa of the n nests being replaced by new nests
(with new random solutions at new locations). The
generation of new solutions x(t+1) is done by using a
Lévy flight (eq.7). Lévy flights essentially provide a
random walk while their random steps are drawn from a
Lévy distribution for large steps which has an infinite
variance with an infinite mean (eq.8). Here the
consecutive jumps/steps of a cuckoo essentially form a
random walk process which obeys a power-law step-
length distribution with a heavy tail [16].
(7)
(8)
where α > 0 is the step size which should be related to
the scales of the problem of interest. Generally we take
α = O(1). The product means entry-wise
multiplications. This entry-wise product is similar to
those used in PSO, but here the random walk via Lévy
flight is more efficient in exploring the search space as
its step length is much longer in the long run.
A Novel Quantum Inspired Cuckoo Search Algorithm for Bin Packing Problem 61
Copyright © 2012 MECS I.J. Information Technology and Computer Science, 2012, 5, 58-67
The main characteristics of CS algorithm it’s its
simplicity. In fact, comparing with other population or
agent-based metaheuristic algorithms such as particle
swarm optimization and harmony search, there are few
parameters to set. The applications of CS into
engineering optimization problems have shown its
encouraging efficiency. For example, a promising
discrete cuckoo search algorithm is recently proposed to
solve nurse scheduling problem [19]. Another binary
version of cuckoo search is proposed in [22] to sole the
knapsack problems. An efficient computation approach
based on cuckoo search has been proposed for data
fusion in wireless sensor networks [20]. In more details,
the proposed cuckoo search algorithm can be described
as follow:
Objective function f(x), x =(x
1
,.
.
,
x
d
)
T
;
Initial a population of n host nests x
i
(i = 1, 2, ..., n);
while (t < MaxGeneration) or (stop
criterion);
Get a cuckoo (say i) randomly
b
y
Lévy flights;
Evaluate its quality/fitness F
i
;
Choose a nest among n (say j)
randomly;
if (F
i
> F
j
),
Replace j by the new solution;
end
Abandon a fraction (p
a
) of worse
nests
build new ones at new locations
via Lévy flights;
Keep the best solutions (or
n
ests
with quality solutions);
Rank the solutions and find the
current best;
end while
Fig. 1. Cuckoo Search Schema.
4.1 Quantum Inspired Cuckoo Search
In section, we present the Quantum Inspired
Cuckoo Search (QICSA) which integers the quantum
computing principles such as qubit representation,
measure operation and quantum mutation, in the core
the cuckoo search algorithm. This proposed model will
focus on enhancing diversity and the performance of the
cuckoo search algorithm [29].
The QICSA architecture, which has been
developed to solve combinatorial problems, is explained
in the Figure 2. Our architecture contains three essential
modules. The first module contains a quantum
representation of cuckoo swarm. The particularity of
quantum inspired cuckoo search algorithm stems from
the quantum representation it adopts which allows
representing the superposition of all potential solutions
for a given problem. Moreover, the generation of a new
cuckoo depends on the probability amplitudes a and b
of the qubit function Ψ (eq.5). The second module
contains the objective function and the selection
operator. The selection operator is similar to the elitism
strategy used in genetic algorithms. Finally, the third
module, which is the most important, contains the main
quantum cuckoo dynamics. This module is composed of
4 main operations inspired from quantum computing
and cuckoo search algorithm: Measurement, Mutation,
Interference, and Lévy flights operations. QICSA uses
these operations to evolve the entire swarm through
generations [29].
Fig. 2. Architecture of the QICSA algorithm [29]
5. The proposed approach for solving the bin
packing problem
The development of the suggested approach called
QICSABP is based mainly on a quantum representation
of the searchspace associated with the problem and a
QICSA dynamic used to explore this space by operating
on the quantum representation by using quantum
operations. In order to show how quantum computing
concepts have been tailored to the problem at hand, we
need first to derive a representation scheme which
includes the definition of an appropriate quantum
representation of potential pin packing solutions and the
definition of quantum operators. Then, we describe how
these defined concepts have been integrated in cuckoo
search algorithm
Problem
Quantum
representation
of
cuckoo swarm
Sto
p
Quantum Cuckoo
dynamics
Measurement
Mutation
Interference
Lévy flights
Best solution
Evaluation &
Selection
No
62 A Novel Quantum Inspired Cuckoo Search Algorithm for Bin Packing Problem
Copyright © 2012 MECS I.J. Information Technology and Computer Science, 2012, 5, 58-67
5.1 Quantum representation of variable order
In order to easily apply quantum principles on bin
packing problem, we need to map potential solutions
into a quantum representation that could be easily
manipulated by quantum operators. The bin packing
solution is represented as binary matrix (Figure.3)
satisfying the following criteria:
For N objects, the size of the binary matrix is
N*N. The columns represent the bins and the
rows represent the objects.
The presence of 1 in the position (i,j) indicates
that the object i is filled in the bin i
In each row there is a single 1, i.e. the object is
filled in one bin.
The following example shows a binary solution for
bin packing instance of 4 objects. According to the
example the objects 2 and 3 are packed in the bin 2, the
object 2 is filled in the bin 1, and the object 4 is filled in
the bin 3. So the number of bins used is 3, the last bin is
not used.
0100
0010
0001
0010
Fig. 3. Binary representation of bin packing solution.
In terms of quantum computing, each variable
emplacement is represented as a quantum register as
shown in Figure 4. One quantum register contains a
superposition of all possible variable positions. Each
column
i
i
b
a
represents a single qubit and corresponds
to the binary digit 1 or 0. The probability amplitudes a
i
and b
i
are real values satisfying 1
22
ii
ba . For each
qubit, a binary value is computed according to its
probabilities
2
i
a
,
2
i
b
, and the bin capacity.
2
i
a
and
2
i
b can be interpreted as the probabilities to have
respectively 0 or 1. Consequently, all feasible variable
orders can be represented by a quantum matrix QM
(Figure 5) that contains the superposition of all possible
variable permutations. This quantum matrix can be
viewed as a probabilistic representation of all potential
bin packing solutions. When embedded within a cuckoo
search algorithm framework, it plays the role of a nest.
A quantum representation offers a powerful way to
represent the solution space and reduces consequently
the required number of cuckoo. Only one quantum
matrix is needed to represent the entire swarm.
Fig.4. Quantum register encoding a row in the binary matrix.
nm
nm
n2
n2
n1
n1
2m
2m
22
22
21
21
1m
1m
12
12
11
11
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
Fig. 5. Quantum representation of variable ordering.
5.2 Quantum operators
The quantum inspired cuckoo search algorithm
uses some of quantum inspired operations like
measurement, interference, and mutation. This
integration helps to increase the optimization capacities
of the cuckoo search.
5.2.1 Measurement
This operation transforms by projection the
quantum vector into a binary vector (figure 6).
Therefore, there will be a solution among all the
solutions present in the superposition. But contrary to
the pure quantum theory, this measurement does not
destroy the superposition. That has the advantage of
preserving the superposition for the following iterations
knowing that we operate on traditional machines. The
binary values for a qubit are computed according to its
probabilities
2
i
a
and
2
i
b
.
For the bin packing problem, this operation is
accomplished as follows: for each qubit, we generate a
random number Pr between 0 and 1; the value of the
corresponding bit is 1 if the value
2
i
b
is greater than
Pr, and otherwise the bit value is 0. However, the use
the standard measure operation defined in [29] can lead
to infeasible solutions and then increase the
computational time of the algorithm to find good
solutions. By using the standard measure, we can get an
overloaded bin or unpacked item (we can get a zero
vector for variable emplacement). In order to delete this
kind of solutions, we have introduced bin capacity in
the measure operation. So, the value 1 is obtained if the
m
m
2
2
1
1
b
a
...
b
a
b
a
