A novel quantum inspired cuckoo search for Knapsack problems
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
International Journal of Bio-Inspired Computation Vol. x, No. x, 2011 1
A novel quantum inspired cuckoo search for Knapsack
problems
Abdesslem Layeb
MISC Lab., Computer Science Department
Mentouri University Constantine, Algeria.
layeb@umc.edu.dz
Abstract. This paper presents a new inspired algorithm called Quantum Inspired Cuckoo Search
Algorithm (QICSA). This one is new framework relying on Quantum Computing principles and
Cuckoo Search algorithm. The contribution consists in defining an appropriate representation scheme
in the cuckoo search algorithm that allows applying successfully on combinatorial optimization
problems some quantum computing principles like qubit representation, superposition of states,
measurement, and interference. This hybridization between quantum inspired computing and
bioinspired computing has led to an efficient hybrid framework which achieves better balance between
exploration and exploitation capabilities of the search process. Experiments on knapsack problems
show the effectiveness of the proposed framework and its ability to achieve good quality solutions.
Keywords: Cuckoo Search Algorithm, Quantum Computing, Hybrid Algorithms
Biographical notes: Abdeslem Layeb is Associate Professor in the department of computer science at
the University of Constantine. He is a member of MISC laboratory. He received his PhD degree in
computer science from the University Mentouri of Constantine, Algeria. Dr. Layeb is interested to the
combinatorial optimization methods and its application to solve several problems from different
domains like Bioinformatics, imagery, formal methods,etc. Dr. Layeb has many publications in
theoretical computer science, combinatorial optimization, and bioinformatics.
1 Introduction computation fields. There are many algorithms based swarm
intelligence like Ant Colony optimization [3, 4], eco-
The combinatorial optimization plays a very important role systems optimization [5, 6], etc. The main algorithm for
in operational research, discrete mathematics and computer swarm intelligence is Particle Swarm Optimization (PSO)
science. The aim of this field is to solve several [7, 8], which is inspired by the paradigm of birds grouping.
combinatorial optimization problems that are difficult to PSO was used successfully in various hard optimization
solve. Several methods were developed to solve such problems. The simplicity of implementation and use are the
problems that can be classified into two classes [1]. Exact main features of PSO compared to other evolutionary
methods, like branch and bound, dynamic programming, computing algorithms. In fact, the updating mechanism in
can give the exact solutions. However, in the worst case, the the algorithm relies only on two simple PSO self-updating
computation time required for the execution of such equations.
methods, increases exponentially with the size of the
instance to solve. The second class contains metaheuristic One of the most recent variant of PSO algorithm is Cuckoo
methods which give a sub-optimal solution but in Search algorithm (CS). CS is an optimization algorithm
reasonable times compared to the exact methods [1, 2]. developed by Xin-She Yang and Suash Deb in 2009 [9]. It
was inspired by the obligate brood parasitism of some
Evolutionary computation has been proven to be an cuckoo species by laying their eggs in the nests of other host
effective way to solve complex engineering problems. It birds (of other species). Some bird’s host can involve direct
presents many interesting features such as adaptation, conflicts with the intruding cuckoos. For example, if a
emergence and learning [2]. Artificial neural networks, bird’s host discovers that the eggs are strange eggs, it will
genetic algorithms and swarm intelligence are examples of either throw these alien eggs away or simply abandon its
bio-inspired systems used to this end. In recent years, nest and build a new nest elsewhere [10]. The cuckoo’s
optimizing by swarm intelligence has become a research behavior and the mechanism of Lévy flights [11, 12] have
interest to many research scientists of evolutionary leading to design of an efficient inspired algorithm
Copyright © 200x Inderscience Enterprises Ltd.performing optimization search. The recent applications of decision making, budget controlling, and projects selection
Cuckoo Search for optimization problems have shown its and so on. This problem was proved to be NP-Hard [20, 21].
promising effectiveness. For example, a promising discrete The knapsack problem can be defined as follows:
cuckoo search algorithm is recently proposed to deal with We have a knapsack with maximum capacity equal to C and
nurse scheduling problem [13]. a set of N items. Each item i has a profit pi and a weight wi.
The problem consists in choosing a subset of items that
Quantum Computing (QC) is a new research field that maximize the knapsack profit, and having a total weight less
induced intense researches in the last decade, and that than or equal to C. By using the binary decision variable xi,
covers investigations on quantum computers and quantum with xi = 1 if item i is selected, and xi = 0 otherwise. The
algorithms [14]. QC relies on the principles of quantum problem can be formulated as follows:
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 N (1)
to optimization due to its underlying concepts. Recently, a Maximize ∑ pi xi
i =1
growing theoretical and practical interest is devoted to
researches on merging evolutionary computation and
quantum computing [15, 16]. The aim is to get benefit from N
quantum computing capabilities to enhance both efficiency Subject ∑ wi xi ≤ C (2)
and speed of classical evolutionary algorithms. This has led i =1
to the design of several quantum inspired algorithms such as xi ∈ {0,1}
quantum inspired genetic algorithm [15], quantum
differential algorithm [17], quantum inspired scatter search
[18], etc. Unlike pure quantum computing, quantum The multidimensional knapsack problem (MKP) is a
inspired algorithms don’t require the presence of a quantum generalization of the knapsack problem [22]. In the MKP,
machine to work. Quantum inspired algorithms have been each item xi has a profit pi like in the simple knapsack
used to solve successfully many combinatorial optimization problem. However, instead of having a single knapsack to
problems [15, 19]. fill, we have a number m of knapsack of capacity Cj (j = 1 ...
m). Each item xi has a weight wij that depends of the
Within this issue, we propose in this paper a new hybrid knapsack j (example: an item can have a weight 3 in
algorithm called Quantum Inspired Cuckoo Search knapsack 1, 5 in knapsack 2, etc.). A selected item must be
Algorithm (QICSA) to cope with combinatorial in all knapsacks. The objective in MKP is to select a subset
optimization problems. The proposed algorithm combines of items that maximizes profit and that this subset must be
Cuckoo Search algorithm and quantum computing in new in all m knapsacks. MKP is an important problem because
one. The features of the proposed algorithm consist in many practical problems can be formulated as a MKP, such
adopting a quantum representation of the search space. The as the capital budgeting problem, project selection and
other feature of QICSA is the integration of the quantum cargo loading, and cutting stock problems [23]. MKP can be
operators in the cuckoo search dynamics in order to stated as follows:
optimize a defined objective function. The QICSA
N (3)
Maximize ∑ pi xi
distinguishes with other evolutionary algorithms in that it
offers a large exploration of the search space through i =1
intensification and diversification. We have tested our
algorithm on Knapsack problems [20] and the results are
promising. N
Subject ∑ wi j xi ≤ C j j = 1...m (4)
The remainder of the paper is organized as follows. Section i =1
2 presents the knapsack problems formulation. In section 3, xi ∈ {0,1}
some basic concepts of quantum computing are presented.
Section 4 presents an overview of the cuckoo search
algorithm. In section 5, the proposed algorithm is described. Clearly, there are 2N potential solutions for these problems.
Experimental results are discussed in section 6. Finally, It is obviously that knapsack problem and its variants are
conclusions and future work are drawn. combinatorial optimization problems. Several techniques
have been proposed to deal with knapsack problems [20].
However, it appears to be impossible to obtain exact
2 Knapsack problems solutions in polynomial time. The main reason is that the
required computation grows exponentially with the size of
Knapsack problem (KP) is a well-known optimization the problem. Therefore, it is often desirable to find near
problem. Several problems are encoded like Knapsack optimal solutions to these problems. Efficient heuristic
problem involving resource distribution, investment algorithms offer a good alternative to accomplish this goal.
Exploitation of the optimization philosophy and the parallelgreat ability of quantum computing is an attractive way to Quantum Algorithms consist in applying successively a
probe complex problems like the knapsack problems. series of quantum operations on a quantum system.
Within this perspective, we are interested in applying Quantum operations are performed using quantum gates and
quantum computing principles and a cuckoo search quantum circuits. It should be noted that designing quantum
algorithm to solve these problems. 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
3 Overview of Quantum Computing some properties of quantum computing in the classical
algorithms. Since the late 1990s, merging quantum
Quantum computing is a new theory which has emerged as computation and evolutionary computation has been proven
a result of merging computer science and quantum to be a productive issue when probing complex problems.
mechanics. Its main goal is to investigate all the possibilities Like any other EA, a Quantum Evolutionary Algorithm
a computer could have if it followed the laws of quantum (QEA) relies on the representation of the individual, the
mechanics. The origin of quantum computing goes back to evaluation function and the population dynamics. The
the early 80 when Richard Feynman observed that some particularity of QEA stems from the quantum representation
quantum mechanical effects cannot be efficiently simulated they adopt which allows representing the superposition of
on a computer. During the last decade, quantum computing all potential solutions for a given problem. It also stems
has attracted widespread interest and has induced intensive from the quantum operators it uses to evolve the entire
investigations and researches since it appears more powerful population through generations. QEA has been successfully
than its classical counterpart. Indeed, the parallelism that the applied on many problems [15, 16].
quantum computing provides reduces obviously the
algorithmic complexity. Such an ability of parallel
processing can be used to solve combinatorial optimization 4 Cuckoo Search
problems which require the exploration of large solutions
spaces. The basic definitions and laws of quantum In order to solve complex problems, ideas gleaned from
information theory are beyond the scope of this paper. For natural mechanisms have been exploited to develop
in-depth theoretical insights, one can refer to [14]. heuristics. Nature inspired optimization algorithms has been
extensively investigated during the last decade paving the
The qubit is the smallest unit of information stored in a two- way for new computing paradigms such as neural networks,
state quantum computer. Contrary to classical bit which has evolutionary computing, swarm optimization, etc. The
two possible values, either 0 or 1, a qubit will be in the ultimate goal is to develop systems that have ability to learn
superposition of those two values. The state of a qubit can incrementally, to be adaptable to their environment and to
be represented by using the braket notation: be tolerant to noise. One of the recent developed bioinspired
algorithms is the Cuckoo Search (CS) [9] which is based on
style life of Cuckoo bird. Cuckoos use an aggressive
|Ψ⟩ = α |0⟩+ β |1⟩ (5)
strategy of reproduction that involves the female hack nests
of other birds to lay their eggs fertilized. Sometimes, the egg
where |Ψ⟩ denotes more than a vector Ψ in some vector
→
of cuckoo in the nest is discovered and the hacked birds
discard or abandon the nest and start their own brood
space. |0⟩ and |1⟩ represent the classical bit values 0 and 1
elsewhere. The Cuckoo Search proposed by Yang and Deb
respectively; a and b are complex numbers such that:
2009 [9] is based on the following three idealized rules:
• Each cuckoo lays one egg at a time, and dumps it
| a |2 + | b |2 = 1 (6) in a randomly chosen nest;
• The best nests with high quality of eggs (solutions)
a and b are complex number that specify the probability will carry over to the next generations;
amplitudes of the corresponding states. When we measure • The number of available host nests is fixed, and a
the qubit’s state we may have ‘0’ with a probability | a |2 host can discover an alien egg with a probability pa
and we may have ‘1’ with a probability | b |2. A system of ∈ [0, 1]. In this case, the host bird can either throw
m-qubits can represent 2m states at the same time. Quantum the egg away or abandon the nest so as to build a
computers can perform computations on all these values at completely new nest in a new location.
the same time. It is this exponential growth of the state
space with the number of particles that suggests exponential The last assumption can be approximated by a fraction
speed-up of computation on quantum computers over pa of the n nests being replaced by new nests (with new
classical computers. Each quantum operation will deal with random solutions at new locations). The generation of new
all the states present within the superposition in parallel. solutions x(t+1) is done by using a Lévy flight (eq.7). Lévy
When observing a quantum state, it collapses to a single flights essentially provide a random walk while their
state among those states. 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 5 The proposed Algorithm
essentially form a random walk process which obeys a
power-law step-length distribution with a heavy tail [9]. In section, we present the proposed algorithm called
Quantum Inspired Cuckoo Search (QICSA) which integers
xit +1 = xit + α ⊕ Lévy (λ ) (7) the quantum computing principles such as qubit
representation, measure operation and quantum mutation, in
Lévy ~ u = t -λ (8) the core the cuckoo search algorithm. This proposed model
will focus on enhancing diversity and the performance of
the cuckoo search algorithm.
where α > 0 is the step size which should be related to the
scales of the problem of interest. Generally we take The QICSA architecture, which has been developed to solve
α = O(1). The product ⊕ means entry-wise multiplications. combinatorial problems, is explained in the Figure 2. Our
This entry-wise product is similar to those used in PSO, but architecture contains three essential modules. The first
here the random walk via Lévy flight is more efficient in module contains a quantum representation of cuckoo
exploring the search space as its step length is much longer swarm. The particularity of quantum inspired cuckoo search
in the long run. algorithm stems from the quantum representation it adopts
which allows representing the superposition of all potential
The main characteristics of CS algorithm it’s its solutions for a given problem. Moreover, the generation of
simplicity. In fact, comparing with other population or a new cuckoo depends on the probability amplitudes a and b
agent-based metaheuristic algorithms such as particle swarm of the qubit function Ψ (eq.5). The second module contains
optimization and harmony search, there are few parameters the objective function and the selection operator. The
to set. The applications of CS into engineering optimization selection operator is similar to the elitism strategy used in
problems have shown its encouraging efficiency. For genetic algorithms. Finally, the third module, which is the
example, a promising discrete cuckoo search algorithm is most important, contains the main quantum cuckoo
recently proposed to solve nurse scheduling problem [17]. dynamics. This module is composed of 4 main operations
An efficient computation approach based on cuckoo search inspired from quantum computing and cuckoo search
has been proposed for data fusion in wireless sensor algorithm: Measurement, Mutation, Interference, and Lévy
networks [18]. In more details, the proposed cuckoo search flights operations. QICSA uses these operations to evolve
algorithm can be described as follow: the entire swarm through generations.
Objective function f(x), x =(x1,.., xd)T ; In the following, we will explain in more detail each
Initial a population of n host nests xi (i component of QICSA architecture.
= 1, 2, ..., n);
while (t < MaxGeneration) or (stop
criterion);
• Get a cuckoo (say i) randomly by Lévy Problem
flights;
• Evaluate its quality/fitness Fi;
• Choose a nest among n (say j)
randomly; Quantum
• if (Fi > Fj), representation
Replace j by the new solution; of cuckoo swarm
end
• Abandon a fraction (pa) of worse nests Quantum Cuckoo
• build new ones at new locations via dynamics
Evaluation &
Lévy flights; Selection
Measurement
• Keep the best solutions (or nests with Mutation
quality solutions); Interference
No
• Rank the solutions and find the Stop Lévy flights
current best;
end while
Fig. 1. Cuckoo Search Schema. Best solution
Fig. 2. Architecture of the QICSA algorithm5.1 Quantum representation of cuckoo solution several different solutions for the knapsack problem, which
gives a great diversity to the cuckoo search algorithm.
To successfully apply quantum principles on binary
problems like the knapsack problem, we have needed to a1 a 2 a n Measure
map potential solutions into a quantum representation that
b b ... b →(0,1,...,1)
could be easily manipulated by quantum operators. In terms 1 2 n
of quantum computing, each binary solution is represented
as a quantum register of length N (N is the solution size) as Fig. 4. Quantum measurement.
shown in Figure 3. Each column represents a single qubit
and corresponds to the binary digit 1 or 0. For each qubit, a 5.2.2 Quantum interference
binary value is computed according to its probabilities ai 2
and bi 2 , which can be interpreted as the probabilities to This operation amplifies the amplitude of the best solution
and decreases the amplitudes of the bad ones. It primarily
have respectively 0 or 1. Consequently, all potential
consists in moving the state of each qubit in the direction of
solutions can be represented by a Quantum Vector QV
the corresponding bit value in the best solution in progress.
(figure 3) that contains the superposition of all possible
The operation of interference is useful to intensify research
solutions. This quantum vector can be viewed as a
around the best solution and it plays the role of local search
probabilistic representation of all the problem solutions. It
method. This operation can be accomplished by using a unit
plays the role of a quantum cuckoo in the QICSA algorithm.
transformation which achieves a rotation whose angle is a
A quantum representation offers a powerful way to
function of the amplitudes ai, bi (figure 5). The rotation
represent the solution space and reduces consequently the
angle’s value δθ should be well set in order avoid
required number of cuckoos. Only one cuckoo is needed to
premature convergence. A big value of the rotation angle
represent the entire swarm.
can lead to premature convergence or divergence; however
a1 a 2 an a small value to this parameter can increase the convergence
b b ... b time. Consequently, the angle is set experimentally and its
1 2 n
direction is determined as a function of the values of ai, bi
Fig. 3. Quantum representation of the cuckoo solution. and the corresponding element’s value in the binary vector
(table 1). In our algorithm, we have set the rotation angle
δθ= pi/20. However, we can use a dynamic value of the
5.2 Quantum operators rotation angle in order to increase the performance of the
interference operation.
We have integrated in the cuckoo search algorithm, some of
quantum operations. This integration helps to increase the
optimization capacities of the cuckoo search. ±δθ
bi
5.2.1 Measurement
a) This operation transforms by projection the quantum ai
vector into a binary vector (figure 4). 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 Fig. 5. Quantum interference
following iterations knowing that we operate on traditional
machines. The binary values for a qubit are computed Table 1. Lookup table of the rotation angle
according to its probabilities ai 2 and bi 2 . This Reference bit
a b Angle
operation is accomplished as follows: for each qubit, we value
generate a random number Pr between 0 and 1; the value of >0 >0 1 +δθ
the corresponding bit is 1 if the value bi 2 is greater than >0 >0 0 -δθ
Pr, and otherwise the bit value is 0. Moreover, the >0 05.2.3 Mutation operator cuckoo. For that, we apply the measure operation in order to
get a binary solution which represents a potential solution.
This operator is inspired from the evolutionary mutation. It After this step, we apply the interference operation
allows moving from the current solution to one of its according to the best current element. We replace some
neighbours. This operator allows exploring new solutions worst nests by the current cuckoo if it is better or by new
and thus enhances the diversification capabilities of the random nests generated by the Lévy flight. The selection
search process. In each generation, the mutation is applied phase in QICSA of the best nests or solutions is comparable
with some probability. We distinguish two types of quantum to some form of elitism selection used in genetic algorithms,
mutations: which ensures the best solution is kept always in the next
• Inter-qubit Mutation: this operator performs iteration. Finally, the global best solution is then updated if
permutation between two qubits. It consists first in a better one is found and the whole process is repeated until
selecting randomly a register in the quantum matrix. reaching a stopping criterion. In more details, the proposed
Then, pairs of qubits are chosen randomly according to QICSA can be described as in figure 8.
a defined probability (figure 6).
• Intra-qubit Mutation: it consists in selecting randomly The particularity of QICSA algorithm stems from the
a qubit according to a defined probability, next we quantum representation it adopts which allows representing
make a permutation between the qubit amplitudes ai et the superposition of all potential solutions for a given
bi as it is shown on Figure 7. problem. Moreover, the position of a nest depends on the
probability amplitudes a and b of the qubit function. The
View that the quantum representation offers a great
probabilistic nature of the quantum measure offers a good
diversity; it is recommended to use small values for the of
diversity to the cuckoo search algorithm, while the
mutation probability in order to keep good performance of
interference operation helps to intensify the search around
the quantum inspired cuckoo search.
the good solutions.
0.44 0.77 0.44 0.00 1.00 Input: Problem data
0.99 0.77 − 0.99 1.00 0.00 Output: problem solution
Construct an initial population of p host
Exchange nests
while (stop criterion)
Fig. 6. inter-qubit quantum mutation.
• Apply Lévy flights operator to get cuckoo
randomly;
• Apply randomly a quantum mutation
• Apply measurement operator;
0.44 0.77 0.44 0.00 1.00
• Evaluate the quality/fitness of this
0.99 0.77 − 0.99 1.00 0.00
cuckoo;
• Apply Interference operator;
• replace some nests among n randomly by
0.44 0.77 − 0.99 0.00 1.00 the new solution according to its
0.99 0.77 0.44 1.00 0.00 fitness;
• A fraction (pa) of the worse nests are
abandoned and new ones are built via Lévy
Fig. 7. intra-qubit quantum mutation. flights;
• Keep the best solutions (or nests with
quality solutions);
5.3 Outlines of the proposed algorithm • Rank the solutions and find the current
best;
Now, we describe how the representation scheme including end while
quantum representation and quantum operators has been
embedded within a cuckoo search algorithm and resulted in
Fig. 8. Quantum Inspired Cuckoo Search Schema.
a hybrid stochastic algorithm. Firstly, a swarm of p host nest
is created at random positions to represent all possible
solutions. The algorithm progresses through a number of 6 Implementation and Validation
generations according to the QICSA dynamics. During each
iteration, the following main tasks are performed. A new QICSA for knapsack problems is implemented in Matlab 7
cuckoo is built using the Lévy flights operator followed by and tested on a micro-computer 3 GHz and 1 GB of
the quantum mutation which is applied with some memory. To assess the efficiency and accuracy of our
probability prm. The next step is to evaluate the current approach several experiments were performed. In the firstexperiment, we have used small tests taken from [21]. We
have compared our results by those given by a harmony
search algorithm NGHS [21]. The results are given in the
Table 2. In this experiment, QICSA is better than NGHS in
two tests (f1 and f6) and has the same results in the other
tests (table 2).
Table 2. Results for small test size
test size QICSA NGHS
f1 10 340 295
f2 20 1024 1024
f3 4 35 35
f4 4 23 23
f5 15 481.0694 481.0694
f6 10 52 50 Fig. 9. Friedman test compares initial solution and best solutions.
f7 7 107 107
f8 23 9761 9761 To evaluate the performance of QICSA for
f9 5 130 130
multidimensional knapsack problem, several tests are used
taken from the benchmark library OR-Library
f10 20 1025 1025 (http://people.brunel.ac.uk/~mastjjb/jeb/orlib/mknapinfo.ht
ml). We have made two experiments. In the first
experiment, we have used easy tests of the benchmark
In the second experiment we have used knapsack tests mknap1. In this experiment, QICSA is able to find the best
generated using a random generator. In these tests, the solution for all the tests of mknap1 (table 4).
knapsack capacity is calculated using the following formula:
Table 4. Results MKP tests (mknap1)
Test Number Number of QICSA best
3 N
C= ∑ wi
4 i =0
(9) of items constraints solution known
solution
mknap1 6 10 3800 3800
Where wi is a random weight of item i and N is the number mknap2 10 10 8706,1 8706,1
of items. We have used different size of N varying from 50 mknap3 15 10 4015 4015
to 30000. mknap4 20 10 6120 6120
mknap5 28 10 12400 12400
The found results are summarized in table 3. As we can see, mknap6 39 5 10618 10618
the results of the experiments are encouraging and prove the mknap7 50 5 16537 16537
feasibility and the efficiency of our approach. There is clear
enhancement of the initial solutions as it is confirmed by the The second experiment contains hard MKP instances which
statistical Friedman test (figure 9). are considered to be rather difficult for optimization
approaches. We have used 5 tests of the benchmarks
Table 3. Results for random tests
maknapcb1 (5.100) which have 5 constraints and 100 items,
Test size Initial Best and we have used 5 tests of the benchmarks maknapcb4
solution solution (10.100) which have 10 constraints and 100 items.
50 721 1045 Moreover, we have compared our results with two methods
100 1945 2245 based on PSO algorithms. The first method, denoted as
200 2680 4266 PSO-P, is based on a standard PSO with penalty function
300 3853 6096 technique [23]. The second method, called PSO-R, is based
500 7607 10021 on the structure of PSO-P, in combination with a problem-
700 10927 14571
specific repair operator to guarantee feasible solutions [23].
900 12996 17596
Finally, statistical tests of Freidman were carried out to test
1000 16613 20209
the significance of the difference in the accuracy of each
1200 19620 23603
method in this experiment.
1500 26594 29362
1700 22274 32973
The found results of this experiment are summarized in
2000 36718 39953
Table 5. The found results are encouraging. QICSA
2500 38814 48864
outperforms PSO-P in all the tested instances but it is less
3000 48842 58079competitive than PSO-R because PSO-R uses specific repair quantum operations such as the interference, the quantum
operator based on pseudo-utility ratios derived from the mutation and measurement. We have explained the basis of
surrogate duality approach [23]. Our program ranks third in QICSA through the resolution of two variants of knapsack
the Freidman test (Figure 10) after best-known and PSO-R. problems: the 0-1 knapsack and the multidimensional
However, the performances of QICSA can be increased by knapsack problems. The approach has been thoroughly
the introduction of some specified knapsack heuristic assessed with different instance types and problem sizes.
operators utilizing problem-specific knowledge like the The experimental studies prove the feasibility and the
repair operator used in the method PSO-R. effectiveness of our approach. The proposed algorithm
reduces efficiently the population size and the number of
Table 5. Results MKP tests (mknapcb1, mknapcb4)
iterations to have the optimal solution. Thanks to
Best superposition, interference, and quantum mutation
Test QICSA PSO-P PSO-R
known
operators, better balance between intensification and
5.100.00 24381 23416 22525 24381
diversification of the search is achieved. However, there are
5.100.01 24274 22880 22244 24258 several issues to improve our algorithm. Firstly, in order to
5.100.02 23551 22525 21822 23551 improve the effectiveness of our algorithm, it is better to
integrate a local search method like Tabu Search in the core
5.100.03 23534 22727 22057 23527
of the algorithm. Secondly, to raise the speed of our
5.100.04 23991 22854 22167 23966 program, it is better to use parallels machines because it was
10.100.00 23064 21796 20895 23057 verified effectively that quantum inspired algorithms can
work better on parallels machines. Finally, the performance
10.100.01 22801 21348 20663 22781
of the algorithm may be improved by using a clever startup
10.100.02 22131 20961 20058 22131 solution.
10.100.03 22772 21377 20908 22772
10.100.04 22751 21251 20488 22751
References
1. Jourdan, L., Basseur, M. and Talbi, E.G., Hybridizing exact
methods and metaheuristics: a taxonomy. European Journal of
Operational Research. Vol. 199. 620-629, 2009.
2. Fogel, D.B., An Introduction to Evolutionary Computation,
Tutorial, Congress on Evolutionary Computation (CEC’2001),
Seoul, Korea, 2001.
3. Dorigo M. and Gambardella, L.M. Ant Colony System : A
Cooperative Learning Approach to the Traveling Salesman
Problem, IEEE Transactions on Evolutionary Computation,
Vol.1, N. 1, pp. 53-66, 1997.
4. Hu, X, Zhang, J. and Li, Y. Orthogonal methods based ant
colony search for solving continuous optimization problems.
Journal of Computer Science and Technology, 23(1), pp.2-18,
Fig. 10. Friedman test compares QICSA, PSO-P, PSO-R and 2008.
best known solutions. 5. Nabti, S. and Meshoul, S. Predator Prey Optimisation for
Snake Based Contour Detection, International Journal of
The effectiveness of our approach is explained by the good Intelligent Computing and Cybernetics - IJICC, Emerald
combination between the exploitation of the local random Publishers, 2(2), pp.228-242, 2009.
walk and the exploration of Lévy flights, which leads the 6. Zheng, S. An intensive restraint topology adaptive snake model
algorithm to effectively explore the search space and locate and its application in tracking dynamic image sequence. Inf.
a good solution. Sci. Vol.180, N.16, pp. 2940-2959, 2010.
7. Clerc, M., and Kennedy, J. The particle swarm - explosion,
stability, and convergence in a multidimensional complex
7 Conclusion space. IEEE Trans. Evo. Comp., 6(1) 58–73, 2002.
8. Eberhart, R. C., Shi, Y. and Kennedy, J: Swarm Intelligence.
In this work, we have presented a new inspired algorithm The Morgan Kaufmann Series in Artificial Intelligence. Morgan
based on hybridization between cuckoo search algorithm Kaufmann, San Francisco, CA, USA, 2001.
and quantum computing called Quantum Inspired Cuckoo 9. Yang, X.-S., and Deb, S., Engineering Optimisation by Cuckoo
Search Algorithm (QICSA). The quantum representation of Search, Int. J. Mathematical Modelling and Numerical
the solutions allows the coding of all the potential solutions Optimisation, Vol. 1, No. 4, 330–343, 2010.
with a certain probability. The optimization process consists 10.Barthelemy, P., Bertolotti, J., Wiersma, D. S., 2008. A Lévy
of the application of a cuckoo search dynamics enhanced by flight for light. Nature, 453, 495-498.11.Payne, R. B., Sorenson, M. D., and Klitz, K. The Cuckoos, Oxford University Press, 2005. 12.Pavlyukevich, I. Lévy flights, non-local search and simulated annealing, J. Computational Physics, 226, 1830-1844, 2007. 13.Tein L. H. and Ramli R., Recent advancements of nurse scheduling models and a potential path, in: Proc. 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA 2010), pp. 395-409, 2010. 14.Jaeger, G. Quantum Information: An Overview. Berlin: Springer. 2006 15. Han, K.H. and Kim, J.H.. "Quantum-inspired Evolutionary Algorithms with a New Termination Criterion, Hε Gate, and Two Phase Scheme," IEEE Transactions on Evolutionary Computation, IEEE Press, vol. 8, no. 2, pp. 156-169, April 2004. 16.Layeb, A., Saidouni, D.E. A New Quantum Evolutionary Local Search Algorithm for MAX 3-SAT Problem. In Proceedings of the 3rd international Workshop on Hybrid Artificial intelligence Systems. Lecture Notes in Artificial Intelligence, vol. 5271. Springer-Verlag, Berlin, Heidelberg, (2008)172-179. 17.Draa A., Meshoul S., Talbi H., Batouche A Quantum-Inspired Differential Evolution Algorithm for Solving the N-Queens Problem. Int. Arab J. Inf. Technol. 7(1):pp. 21-27, 2010. 18.Layeb, A. Hybrid Quantum Scatter Search Algorithm for Combinatorial Optimization Problems. In the journal of Annals. Computer Science Series , ISSN: 1583-7165, Vol. 8, No.2, pp.227-244, 2010 19.Layeb, A., Saidouni, D.E. A New Quantum Evolutionary Algorithm with Sifting Strategy for Binary Decision Diagram Ordering Problem, in the International Journal of Cognitive Informatics and Natural Intelligence. ISSN 1557-3958, Vol. 4. No 4, pp. 47-61, December 2010. 20.Pisinger, D.: Where are the hard knapsack problems? Computers and Operations Research, Vol.32, N°. 9, pp. 2271- 2284, 2005. 21.Dexuan Zou, Liqun Gao, Steven Li, Jianhua Wu, Solving 0-1 knapsack problem by a novel global harmony search algorithm, Applied Soft Computing, Volume 11, Issue 2, The Impact of Soft Computing for the Progress of Artificial Intelligence, Pages 1556-1564, March 2011. 22.M. Vasquez and Y. Vimont, Improved results on the 0–1 multidimensional knapsack problem, European Journal of Operational Research 165 (1), pp. 70–81, 2005. 23.Kong, M and Tian, P. Apply the Particle Swarm Optimization to the Multidimensional Knapsack Problem. In the proceedings of ICAISC, LNAI 4029, pp. 1140–1149, 2006. 24.Tein L. H. and Ramli R., Recent advancements of nurse scheduling models and a potential path, in: Proc. 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA 2010), pp. 395-409, 2010. 25.Dhivya, M. Energy Efficient Computation of Data Fusion in Wireless Sensor Networks Using Cuckoo Based Particle Approach (CBPA), Int. J. of Communications, Network and System Sciences, Vol. 4, No. 4, 249-255, 2011.
You can also read
