metaheuristics: review and application

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Metaheuristics: review and applicationAnupriya Gognaa & Akash Tayalba ECE Department, Northern India Engineering College, GGSIPUniversity, FC-26, Shastri Park, Delhi, 110053, Indiab ECE Department, Indira Gandhi Institute of Technology, GGSIPUniversity, Delhi, IndiaPublished online: 20 May 2013.

To cite this article: Anupriya Gogna & Akash Tayal (2013) Metaheuristics: review andapplication, Journal of Experimental & Theoretical Artificial Intelligence, 25:4, 503-526, DOI:10.1080/0952813X.2013.782347

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Metaheuristics: review and application

Anupriya Gognaa* and Akash Tayalb

aECE Department, Northern India Engineering College, GGSIP University, FC-26, Shastri Park, Delhi110053, India; bECE Department, Indira Gandhi Institute of Technology, GGSIP University, Delhi, India

(Received 24 May 2012; final version received 30 September 2012)

The area of metaheuristics has grown immensely in the past two decades as a solutionto real-world optimisation problems. They are able to perform well in situations whereexact optimisation techniques fail to deliver satisfactory results. For complexoptimisation problems (Nondeterministic polynomial time-hard problems), metaheur-istic techniques are able to generate good quality solution in relatively much less timethan traditional optimisation techniques. Metaheuristics find applications in a widerange of areas including finance, planning, scheduling and engineering design. Thispaper presents a review of various metaheuristic algorithms, their methodology, recenttrends and applications.

Keywords: application of metaheuristics; memetic algorithm; P-metaheuristics;S-metaheuristics

1. Introduction

Most real-world problems have high complexity, non-linear constraints, interdependencies

amongst variables and a large solution space. This warrants the use of a technique that is capable

of solving complex optimisation problems in real time. Metaheuristic algorithms are one such

technique. They are optimisation methods that deliver reasonably good solution in a reasonable

amount of time. Optimisation is concerned with finding the best value of a set of variables in

order to achieve the goal of minimising (or maximising) an objective function subject to given

set of constraints. Need for optimisation is inherent in almost all walks of life ranging from our

daily life to financial and business planning, and from industrial control to engineering design.

Optimisation techniques can be broadly classified into exact and approximate methods

(Talbi, 2009). Exact methods obtain optimal solutions and guarantee their optimality. This

category includes methods such as branch and bound algorithm, dynamic programming,

Bayesian search algorithms and successive approximation methods. Approximate methods are

aimed at providing good-quality solution (instead of guaranteeing a global optimum solution) in

a reasonable amount of time. Approximate methods can be further split into approximation

algorithms and heuristic methods (Talbi, 2009). The former scheme provides provable solution

quality and provable run-time bounds, whereas the latter involves finding reasonably good

solution in a reasonable time. Heuristic algorithms are highly problem specific. Metaheuristics

form a class of algorithms that act like guiding mechanism for the underlying heuristics. They

are not problem or domain specific and can be applied to any optimisation problem. The term

metaheuristics was introduced by Glover (1986).

q 2013 Taylor & Francis

*Corresponding author. Email: [email protected]

Journal of Experimental & Theoretical Artificial Intelligence, 2013

Vol. 25, No. 4, 503–526, http://dx.doi.org/10.1080/0952813X.2013.782347

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http://dx.doi.org/10.1080/0952813X.2013.782347

Metaheuristics is formally defined as an iterative generation process that guides a

subordinate heuristic by combining intelligently different concepts for exploring and exploiting

the search space; learning strategies are used to structure information in order to find efficiently

near-optimal solutions (Osman & Laporte, 1996). Metaheuristic algorithms create a balance

between intensification and diversification of the search space. They can be used to efficiently

solve NP-hard problems with large number of variables and non-linear objective functions.

Development in the field of metaheuristics largely stems from the importance of complex

optimisation problems to the industrial and scientific world. In the past two decades, many

metaheuristic algorithms have been proposed. Some of them are genetic algorithms (GA;

Holland, 1975), memetic algorithm (MA; Moscato, 1989), artificial immune system (AIS;

Farmer, Packard, & Perelson, 1986), simulated annealing (SA; Kirkpatrick, Gellat, & Vecchi,

1983), tabu search (TS; Glover & Laguna, 1997), ant colony optimisation (ACO; Dorigo, 1992),

particle swarm optimisation (PSO; Kennedy & Eberhart, 1995) and differential evolution (DE;

Price, Storn, & Lampinen, 2006). The need for having a large number of metaheuristic

algorithms arise because of the implication of no free lunch (NFL) theorem proposed byWolpert

and Macready (1997). According to the NFL theorem, the averaged performance for all possible

problems is the same for all algorithms. Thus, no global distinction can be made between

performances of any two algorithms apart from conditions where one algorithm is more suited to

an application than the other. Metaheuristics find application in areas such as telecommunica-

tion, engineering design, machine learning, logistics and business planning.

This paper is organised as follows. Section 2 describes classification of metaheuristics.

Section 3 describes single-solution-based metaheuristic algorithms. Population-based

metaheuristic algorithms are discussed in Section 4. Section 5 discusses the applications of

metaheuristics. Finally, concluding remarks are mentioned in Section 6.

2. Classification of metaheuristics

There are different ways to classify metaheuristic algorithms based on characteristics used to

differentiate amongst them (Talbi, 2009). They include:

. Nature inspired versus non-nature inspired

The former ones are inspired by natural phenomena such as evolution (GA) or bird flocking

behaviour (PSO). The non-nature-inspired ones include tabu search and iterated local search

(ILS).

. Population-based versus single-point search

Single-solution-based methods, also called trajectory methods, manipulate a single solution.

Population-based methods iterate and manipulate a whole family of solutions. Single-solution-

based techniques (e.g. TS, SA and local search) are intensification oriented while population-

based techniques (e.g. GA and PSO) are more focused on exploration of search space. This

classification is used in our paper.

. Iterative versus greedy

Iterative algorithms (e.g. PSO and SA) start from a (set of) solution that is manipulated as the

search process proceeds. Greedy algorithms (e.g. ACO) start from an empty solution, and at each

stage, a decision variable is assigned a value, which is added to the solution set.

. Dynamic versus static objective function

A. Gogna and A. Tayal504

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The algorithms with static objective function (e.g. PSO) keep the objective function given in

the problem representation ‘as it is’ while others (e.g. guided local search) modify it during the

search process.

. Memory usage versus memory less

Memory-less algorithms (e.g. SA) use information only about the current state of search,

while others make use of some information gathered during the iterative process (e.g. TS).

. One versus multiple neighbourhood structures

Most metaheuristic algorithms work on a single neighbourhood structure (e.g. ILS). Other

metaheuristics use a set of neighbourhood structures (e.g. variable neighbourhood search).

3. Single-solution-based metaheuristics (S-metaheuristics)

S-metaheuristics apply generation and replacement procedure to a single solution. In the

generation stage, a set of candidate solutions is generated from the current solution. In the

replacement stage, a suitable solution, from the generated set, is chosen to replace the current

solution. This process continues till a satisfactory result (good-quality solution) is obtained.

Main concepts of S-metaheuristics are:

. Generation of initial solution

Initial solution can be generated either randomly or by using a greedy heuristic. Use of the

latter technique is more complex but results in faster convergence.

. Definition of neighbourhood

A neighbourhood function N is a mapping, N: S ! 2S, that assigns to each solution s of S

(search space) a set of solutions, N(s) , S. The neighbourhood N(s) of a solution s in a

continuous space is the ball with centre s and radius equal to r, with r . 0. For a discrete

optimisation problem, the neighbourhood N(s) of a solution s is represented by the set {s0/d(s0, s)#r}, where d represents a given distance that is related to the move operator. A solution s0in the neighbourhood of s (s0 [ N(S)) is called a neighbour of s. A neighbourhood with strong

locality leads to better performance. Choosing a rich neighbourhood improves the chances of

finding good solution and also increases the computation time. Some of the S-metaheuristic

algorithms has been described in this section.

3.1 Simulated annealing

SA algorithm was proposed by Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953). It

was first applied in the field of optimisation by Kirkpatrick et al. (1983). SA is based on the

annealing process in statistical mechanics, which involves heating a substance and then gradually

cooling it to obtain a high-strength crystalline structure. Aim of annealing is to reach the lowest

energy statewhileminimising the total entropy production.Analogy between physical system and

optimisation problem is stated in Table 1. SA is a neighbourhood search, memory-less algorithm

with a capability to escape local optima and hence avoid premature convergence.

3.1.1 Methodology of SA

SA algorithm starts with an initial solution, and at each iteration, a new solution (random

neighbour) is generated using the neighbourhood strategy adopted. If the value of objective

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function for the new solution is less (for a minimising problem), then it is accepted, and existing

solution is replaced. If the cost function of a new neighbour is higher than that of the current

solution, then a probability measure is applied to decide whether to replace the original solution

or not. Probability measure is defined as

P ¼ exp2DE

KT

� �; ð1Þ

where DE is the change in energy (cost function) between old and candidate solutions, K is the

Boltzmann constant and T is the control parameter value at a given iteration.

Probability measure allows selection of non-improving solutions, thereby helping algorithm

to get out of local optima. The algorithm starts with high initial temperature to ensure that large

number of solutions can be accepted in the beginning. As the iterations proceed, temperature (T)

reduces and algorithm begins to converge towards an optimum solution.

3.1.2 Pseudocode for SA

Select algorithm’s parameters

Bounds of solution space, h_b, l_b

Initial temperature T0 $ 0

Cooling schedule CT (t)

Maximum iteration at fixed temperature, N

Objective function f (*)

Begin

Generate initial solution s [ S

s ˆ rand £ (h_b – l_b) þ l_b

Start iteration at initial temperature

T ˆ T0

Calculate fitness value for initial solution

val ˆ f(s)

Initialize repetition counter

n ˆ 0

Table 1. Analogy between Physical System and Optimization Problem

Physical System Optimization Problem

States SolutionEnergy Objective FunctionGround state Optimal SolutionTemperature Control parameterChange of state Neighbour


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Repeat

While n # N

Generate candidate solution in neighbourhood of s, s 0 [ S

Compute fitness value for candidate

val_new ˆ f(s 0)

If val . new_val,

s ˆ s 0

elseif (exp(val_new - val)/KT) , rand,

s ˆ s 0

end if

Increment iteration counter

n ˆ n þ 1

End while

Reduce temperature as per cooling schedule

T ˆ CT (T)

Until stopping criterion is met

The stopping criteria could be minimum temperature reached or no change in fitness over a

given number of iterations, etc. A list of various cooling schedules is given in Section 3.1.3.

3.1.3 Variations and advancements in SA

Cooling schedules affect the total entropy production in the process. Hence, they have a

significant impact on the performance of the algorithm. Because of this, several cooling

strategies have been proposed (Nourani & Andresen, 1998). Most commonly used cooling

strategies are as follows:

. Exponential, T(t) ¼ a t T0: it helps keep the system close to equilibrium.

. Linear, T(t) ¼ T0 – bt: it is the most popular strategy used.

. Logarithmic, T(t) ¼ c/log(d þ t): it gives guaranteed, but very slow, convergence.

. Very slow decrease, T(t) ¼ T(t 2 1)/1 þ bT(t 2 1).

. Non-monotonic (Hu, Kahng, & Tsao, 1995), in which the temperature rises again. It

helps improve algorithm’s finite time performance.. Adaptive, in which cooling rate varies during search based on information gathered

(Ingber, 1996).

3.1.4 Features and considerations for SA

SA can be easily implemented to give good solutions if certain precautions are taken into

account. First, the initial temperature should be set adequately high. Too low a temperature can


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cause the system to get stuck in local optima, and a very high value of T can cause difficulty in

reaching the optimum solution. Also, there should be a gradual reduction in control parameter T.

Iterations at each temperature should be enough to stabilise the system. The employed cooling

schedule greatly influences the quality of solution. SA is well suited to problems with a rough

landscape (large number of local optima), but if the optimisation problem has a smooth

landscape with limited local optima, SA is not of much use and unnecessarily delays

convergence.

3.2 Tabu search

TS algorithm, a local search methodology, was proposed in 1986 by Glover and Laguna (1997).

It uses the information gathered during the iterations (stored in memory) to make the search

process more efficient. As in case of SA, it also accepts non-improving solutions to move out of

local optima. However, TS algorithm searches the whole neighbourhood deterministically

unlike SA that employs a random search. The distinguishing feature of TS is the use of a short-

term memory that prevents previously visited solutions from being accepted again. It speeds up

the attainment of optimum solution.

3.2.1 Methodology of TS

TS algorithm starts with an initial (randomly selected) solution. It maintains a tabu list (short-term

memory) that has a record of previously visited solutions. Anymove (solution) stored in tabu list is

not allowed (accepted). This helps the algorithm from being trapped in cycles. At each iteration,

new candidate solutions are generated. A new solution is accepted, in accordance with the

contents of tabu list, if it is better than the current one. If no better solution exists, the best

neighbour is selected to replace the existing solution. Worse solutions are also accepted to move

out of local optima.Medium- and long-termmemories can also be used to enhance intensification

and diversification of TS algorithm (Talbi, 2009). The former stores the best solution obtained

during the search, and the latter helps in exploring unexplored areas. Aspiration criteria is another

feature that helps override the tabu list and allow tabu moves, if they result in better solutions.

3.2.2 Pseudocode for TS




Maximum size of tabu list, s_t

Begin

Initialize tabu list, short term and long term memory

Tabu list ˆ Ø

Medium term memory ˆ Ø

Long term memory ˆ Ø

Generate initial solution s [ S


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s ˆ rand £ (h_b – l_b) þ l_b

Calculate fitness value for initial solution

val ˆ f(s)

Repeat

Generate candidate solution in neighbourhood of s, s 0 [ S

if s0 � tabu list or s 0 [ aspiration criteria

Compute fitness value for candidate

val_new ˆ f(s 0)

If val . new_val,

s ˆ s0

end if

end if

Update tabu list, aspiration criteria, medium and long term memory

If tabu list size . s_t

Delete earliest entry

end if


The stopping criteria could be minimum value of fitness function, maximum number of

iterations or maximum iterations without improvement, etc.

3.2.3 Variations and advancements in TS

Advancements in original TS scheme are proposed to make the search process more effective.

One of the variations is to store moves or solution attributes rather than actual solutions in the

tabu list. This reduces the time and memory requirement of the algorithm. Also, the size of tabu

list may be static, dynamic or adaptive. The advancements in TS algorithm concentrate on better

exploitation of information gathered, efficient neighbourhood operators, better initial solutions

and application of parallel search strategies. An efficient way of utilising the information

gathered is to use the elite (best) solution found so far to generate new solutions to speed up

convergence. Other methods, such as the reactive TS attempts to find ways of making the search

move away from local optima that have already been visited. An exhaustive description of

advancements in TS can be found in the study by Gendreau (2002).

3.2.4 Features and considerations for TS

The effectiveness of TS algorithm depends on appropriate selection of neighbourhood operator

and search space. This requires in-depth knowledge of problem at hand. In addition,


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diversification should be very carefully handled to avoid getting stuck at local minima. Also,

tabu list should be carefully formulated to make search effective while minimising computation

time and memory requirement. The best feature of TS is its ability to avoid cycles. However,

unlike SA, it is not efficient enough in moving out of the local minima.

4. Population-based metaheuristics (P-metaheuristics)

P-metaheuristics apply generation and replacement procedure to a family of solutions spread

over the search space. A set of solution is initialised. It is then iterated and manipulated to

generate a new set. The new solution set is selected from amongst the existing and newly

generated solution set using a replacement strategy. This process continues until the stopping

criterion is met. Most P-metaheuristics are inspired by nature. P-metaheuristic algorithms vary

amongst themselves in the way they perform generation, selection and how search memory is

organised. Main concepts of such algorithms are:

. Initial population

A well-initialised population leads to an efficient algorithm. Some of the initialisation

techniques are summarised in Table 2.

. Stopping criteria

The stopping criteria can be a fixed number of iterations, minimum value of objective

function or maximum iterations without improvement.

. Some of the prominent P-metaheuristic algorithms have been described in this section.

4.1 Genetic algorithm

GA, a global search heuristic, was proposed by Holland (1975). It is inspired by Darwin’s theory

of evolution. They are characterised by inherent parallelism.

4.1.1 Methodology of GA

GA begins with a set of solution or initial population, which can be generated using any of the

techniques mentioned in Table 2. The individuals can be represented in the form of binary

strings, tree structure, vector of real numbers, etc. based on problem under consideration. Each

encoded solution is called a chromosome, with each decision variable in a chromosome being

called a gene. From the current population, a set of individuals is selected based on a selection

criterion to be parents. The selection criterion is based on the fitness function (objective

Table 2. Methods for Initializing Population

Method Technique Diversity

Random Generation Solution set Initialized using pseudo- randomnumbers

Medium to high

Sequential Diversification Solutions are generated in sequence into optimize diversity

Very High

Parallel Diversification Solutions generated in a parallel, independentway

Very High

Heuristic Initialization Any heuristic can be used forinitialization

Low


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function) value corresponding to each individual. These parents produce offsprings to be a part

of new generation. The selection criteria used can be:

. Roulette wheel selection

In this, the selection probability of an individual is directly proportional to its fitness.

. Rank-based fitness assignment

In this method, relative fitnesses are associated with individuals. It prevents the selection

process from being dominated by highly fit individuals, thus preserving diversity in population.

. Tournament selection

In this, a set of individuals are randomly chosen and the best amongst them is selected for

mating.

. Elitism

In this scheme, a fixed number of best chromosomes are kept and remaining population is

generated using any of selection procedure discussed above. This helps in preserving the best

solution in the population.

Selected individuals are used to produce new solutions (offsprings) using crossover and

mutation operators.

. Crossover operators

The operation of crossover is used to produce offsprings that inherit characteristics from

both parents. Crossover has a high probability ranging from 0.6–1.0. Some of the crossover

techniques are single-point crossover, two-point crossover or uniform crossover. For real-coded

GAs (RCGAs), we can use techniques such as arithmetic crossover, heuristic crossover and

geometrical crossover.

. Mutation operator

Mutation is a rare phenomenon with a low probability of around 0.001–0.1. It is used to

change some information contained in randomly selected chromosomes. It aids in diversification

of population and larger exploration of the search space.

The next task is to produce a new generation. For this, old generation needs to be replaced by

the new one. The extreme replacement strategies are (Talbi, 2009):

. Generational replacement

The replacement will concern the whole population of size m. The offspring population will

systematically replace the parent population.

. Steady-state replacement

At each generation, only one offspring is generated and it may replace the worst individual

of the parent population.

Between these two extreme replacement strategies, many distinct schemes exist that involve

replacement of a given number, l (1 , l , m), of individuals of the population.

4.1.2 Pseudocode for GA



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Population size, NP

Chromosome size, ND


Maximum no. of generations, NG

Crossover probability, pc

Mutation probability, pm

Begin

Generate initial population

pop ˆ rand(NP, ND) £ (h_b – l_b) þ l_b

Calculate fitness value for initial population

for i ¼ 1:NP

val (i) ˆ f (pop(i,:))

end

Repeat

While no of offspring # required

Select parents (pa, ma) from current population based on their fitness

If pc . rand,

Offspring ˆ crossover (ma, pa)

If pm . rand,

Offspring ˆ mutate (offspring)

End while

Generate new population using some replacement strategy


4.1.3 Variations and advancements in GA

In addition to the traditional crossover (e.g. uniform and single-point) and mutation (e.g.

Gaussian and uniform) operators, new operators have been introduced to improve the quality of

solution. Some of the crossover operators are as follows:

. Unimodal normally distributed crossover (Ono & Kobayashi, 1997): in this crossover,

three parents are used to generate two or more children.. Simulated binary crossover (SBX): it is used to generate offspring close to parents (Deb

& Agrawal, 1995).


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. Hybrid crossover with multiple descendants (Sanchez, Lozano, Villar, & Herrera,

2009). Independent component-analysis-based crossover (Takahashi & Kita, 2001): it shows

good search ability.

Variations proposed in mutation operator are as follows:

. Directed mutation operator (Korejo, Yang, & Li, 2010) that introduces a bias towards

promising areas.. Principal component analysis-based mutation operator (Munteanu & Lazarescu, 1999):

it is used to homogenise the components such that a situation where only a few

principal components are important (and rest negligible) can be avoided to ensure

diversity.

4.1.4 Features and considerations for GA

GA exhibits inherent parallelism and is a good technique for solving complex optimisation

problems. However, it can suffer from premature convergence. The effectiveness of algorithm is

based on selection of objective function, population size, and probability of crossover and

mutation. Hence, these parameters must be carefully selected. A larger population size and large

number of generations increase the likelihood of obtaining a global optimum solution, but

substantially increases processing time.

4.2 Differential evolution

DE was proposed by Storn and Price (1995). It resembles evolutionary algorithm (EA) but

differs from the traditional ones in the way candidate solutions are generated and the use of

greedy selection scheme.

4.2.1 Methodology of DE

DE begins with an initial randomly generated population. Each solution is a real number vector;

hence, DE is traditionally used for continuous optimisation. At each generation, a new set of

candidate solutions is generated. DE, instead of the usual crossover operator, uses a

recombination operator based on linear combination. The offsprings replace the parents only if

they are better. A child is created by mutating existing individuals, largely picked at random

from the population. The notation DE/x/y/z is generally used to define a DE strategy (Talbi,

2009), where x specifies the vector to be mutated, y is the number of difference vectors used and

z denotes the crossover scheme. The commonly used crossover variants are binomial and

exponential. The former is similar to uniform crossover and the latter is similar to a two-point

crossover in GA. More popular amongst the two is the binomial crossover that is used in

pseudocode given in section 4.2.2.

4.2.2 Pseudocode for DE



Population size, NP


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Chromosome size, ND


Scaling factor, F


Crossover constant, cr

Begin


pop ˆ rand(NP, ND) £ (h_b – l_b) þ l_b


for i ¼ 1:NP

val (i) ˆ f(pop(i,:))

end

Repeat

Select 3 parents at random

for i ¼ 1:ND

If rand(i) # cr

offspring(i) ˆ parent3(i) þ F £ (parent1(i) 2 parent2(i))

else

offspring (i) ˆ parent(i)

End if

Compute fitness of offspring

val_new ˆ f (offspring)

if val_new , val

parent ˆ offspring

end if

Until stopping criteria is met

4.2.3 Variations and advancements in DE

DE is a simple and efficient algorithm and hence finds use in large number of optimisation

problems. Many improvements have been proposed to the original DE algorithm. DE can be

used with trigonometric mutation operator (Fan & Lampinen, 2003) or a neighbourhood-based

mutation operator. Also, an adaptive crossover operator can be employed. They help in


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balancing exploration and exploitation components. A summary of these advancements is given

in the study by Neri and Tirrone (2010).

4.2.4 Features and considerations for DE

Convergence of DE algorithm can be controlled using the number of parents and scaling factor.

These parameters can be tuned to achieve a trade-off between convergence speed and

robustness. Increase in number of parents and reduction in scaling factor makes convergence

easier but is computationally intensive. Crossover constant is just a fine-tuning parameter in DE

and does not have much impact on convergence speeds.

4.3 Particle swarm optimisation

PSO was developed by Kennedy and Eberhart (1995). It is based on the swarm behaviour (birds

flocking). It is a population-based mechanism, but unlike EA, it maintains a single static

population whose members are tweaked in response to the search history. PSO can be easily

implemented in a wide range of optimisation problems.

4.3.1 Methodology of PSO

PSO also begins with an initial random population that mimics the birds in a flock. Each solution

is called a particle and the population is termed as a swarm. An analogue of directed mutation

moves the particle in space based on the globally best position attained by any particle in the

swarm (gBest or global best) and the best position attained by the concerned particle (pBest or

personal best). The velocity is updated at each stage using the following equation:

vðt þ 1Þ ¼ wðtÞ £ vðtÞ þ c1 £ r1ðgBest2 xðtÞÞc2 £ r2ðpBest2 xðtÞÞ: ð2Þ

The position is updated using the following equation:

xðt þ 1Þ ¼ xðtÞ þ vðt þ 1Þ; ð3Þ

where w is the inertia weight, r1 and r2 are the random numbers, and c1 and c2 are the

acceleration constants.

4.3.2 Pseudocode for PSO



Swarm size, NS

Dimension of particle, ND


Maximum no. of iteration, NMAX

Begin

Generate initial swarm


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swarm ˆ rand(NS, ND) £ (h_b – l_b) þ l_b

Initialize velocity vector, V

V ˆ rand (NS, ND)

Calculate fitness value for initial swarm

for i ¼ 1:NS

val (i) ˆ f(swarm(i,:))

end

Compute gBest and pBest

Repeat

Generate new velocity vector using equation (2)

Compute new position vecto, new_swarm using equation (3)

for i ¼ 1:NS

new_val (i) ˆ f(new_swarm(i,:))

end

Replace pBest and gBest values


4.3.3 Variations and advancements in PSO

Various modifications to the original PSO algorithm have been introduced. To reduce the

possibility of particles flying out of the problem space, a restriction can be imposed on the velocity

increment at each stage. Inertial weight can also be controlled via fuzzy techniques. Apart from

using the entire swarm to compute gBest, local small neighbourhood-based structures can be used.

This is similar to the neighbourhood topologies used with local search methods. A review of

developments in PSO and their application is given in the study by Eberhart and Shi (2001).

4.3.4 Features and considerations for PSO

Inertia weight decides the expanse of search space explored. A high value of inertia weightw, at the

start of iteration helps in exploring a wide search space. As iterations proceed,w is reduced in order

to intensify the search. Acceleration factors are used to make the particles follow the best obtained

solution. Low values of acceleration constants allow particles to roam far from target regions before

being tugged back, while high values result in abrupt movement towards the target regions. The

performance of PSO is relatively insensitive to swarm size, provided it is not too small.

4.4 Ant colony optimisation

ACO proposed by Dorigo et al. in 1982 (Dorigo, 1992) is based on the cooperative behaviour of

ants. Ants are able to find the shortest way to food source using the pheromone trail left by other


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ants foraging for food. The same principle was applied by Dorigo (1992) for solving

optimisation problems. Ants while moving around for food keep depositing pheromones on their

path. As the ants that took shortest path to food start returning, the concentration of pheromone

on that trail increases. Ants that move later, foraging for food, follow this trail. Hence, the

concentration of pheromones further increases. This can lead to shortest path being followed by

all ants. The same principle is employed in optimisation problem to find the best solution.

4.4.1 Methodology of ACO

ACO uses artificial ants as iterative stochastic solution-generating procedures. The process starts

with generation of a random set of ants (solutions). Their fitness is evaluated using an objective

function suited to the concerned problem. Accordingly, pheromone concentration associated

with each possible route (Dorigo, Maniezzo, & Colorni, 1996) is changed in a way to reinforce

good solutions according to the following equation:

GijðtÞ ¼ r £ Gijðt2 1Þ þ DGij; ð4Þ

where Gij(t) is the pheromone concentration at iteration t. DGij is the change in pheromone

concentration between two iterations and r is the pheromone evaporation rate. As the solution

improves, pheromone concentration increases. Based on the updated value of pheromone

concentration, the ant’s path (associated solution) is changed. Evaporation of pheromone is

necessary to avoid getting stuck at local optima.

4.4.2 Pseudocode for ACO



No of ants, NS

Pheromone evaporation rate, r


Maximum no. of iteration, NMAX

Begin

Generate initial solution set

set ˆ rand(NS, ND) £ (h_b – l_b) þ l_b

Calculate fitness value for initial solution set

for i ¼ 1:NS

val (i) ˆ f(set(i,:))

end

Repeat

Generate new solution, set 0


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Compute fitness of new solution

new_val ˆ f(set 0)

For each ant determine its best position;

Determine the best global ant;

Mark better solutions/routes with pheromone, D Gij

Update pheromone concentration using equation (4)

Until stopping criteria is met

4.4.3 Variations and advancements in ACO

Modifications have beenmade in the traditionalACOalgorithm to enhance its performance.An elitist

strategy was introduced in the study by Dorigo et al. (1996). Elitist strategy provides a strong

additional weight to the best route since the beginning. This helps the search process tomove towards

the best solution and also increases the convergence speed.We can also introduce a limit to the value

of pheromone trails.

4.4.4 Features and considerations for ACO

ACO has an inherent parallelism with an in-built positive feedback mechanism that helps in

reaching the optimum solution faster. The evaporation (of pheromones) phenomenon is used to

prevent premature convergence. The value of evaporation constant ranges from 0.01 to 0.2

(Talbi, 2009). ACO can be used to find solutions for complex optimisation problems if the

parameters of the algorithm are carefully chosen.

4.5 Memetic algorithms

The term memetic algorithms (Moscato, 1989), first introduced in 1989, refer to a broad class of

metaheuristic algorithms that exhibit a hybrid approach. The idea behind MAs is to combine the

evolutionary concept of GA (survival of the fittest, and passing on of traits of parents to

offspring) with the idea of every individual (of the population) developing on its own during its

lifetime. It also aims at introducing problem-dependent knowledge into the search process. It is

one of the most powerful techniques of solving NP-hard optimisation problems.

4.5.1 Methodology of MA

MA integrates EA (such as GA) and a local search procedure (e.g. hill climbing) to generate a

hybrid algorithm with features of both its constituents. In MA, agents are processing unit that are

capable of holding multiple solutions (individuals of EA) and methods to improve them. These

agents interact with each other through cooperation and competition.

MA starts with generating an initial population of solutions (individual). This population can

either be randomly generated or, for better result, it could be generated using a heuristic

(problem-specific) approach. Next step is the generation of new population. This involves the

use of several operations to generate offsprings. The first operator is the selection operator that

selects the best individual from the population in a way similar to that used in EAs. It is followed

by the recombination of selected individuals to produce offsprings as in the case of GA.


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Generated offsprings undergo mutation to enhance diversity in the population. In the next step, a

speciality of MA, a local search operator is used to cause random variations in the generated

population. This operation is repeated in order to search, in the neighbourhood of each

individual, for a candidate solution with better fitness than the original solution (as in SA). It

mimics the biological phenomena of an individual learning during its lifetime. This is followed

by updation of population. It follows either of the following two approaches:

. Plus strategy: it uses a union of sets containing existing and new population and selects

best individual from it to form the next generation.. Comma strategy: in this, the new generation is selected from only the newly generated

solution set.

The distinguishing and compulsory feature of MA is the restart population step. If the

population converges, i.e. no further improvement is likely, the entire process restarts. The best

individual from the current population is retained, while others are generated randomly as in the

case of the initial population. This ensures that there is no premature convergence. The process

continues till satisfactory results are achieved.

4.5.2 Pseudocode for MA



Population size, NP

Chromosome size, ND



Crossover probability, pc

Mutation probability, pm

Begin


pop ˆ rand(NS, ND) £ (h_b – l_b) þ l_b


for i ¼ 1:NP

val (i) ˆ f(pop(i,:))

end

Repeat

While no of offspring # required

Select parents (pa, ma) from current population based on their fitness


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If pc . rand,

Offspring ˆ crossover (ma, pa)

If pm . rand,

Offspring ˆ mutate (offspring)

Apply local search procedure

End while

Update population based on chosen strategy

If plus strategy

new_pop [ popS

offspring

end if

if comma strategy

new_pop [ offspring

end if

If population converges, restart

pop ˆ rand(NS, ND) £ (h_b – l_b) þ l_b

end if


4.5.3 Variations and advancements in MA

There is a scope for variation in operators used with MAs. Traditionally uniform crossover is

used. The recombination operator can also be a hybrid or heuristic operator, with problem-

specific design. It will make the search process more efficient. The way and order in which

mutation, local search and recombination operator are applied can also be changed.

Combination of GA with various local search techniques such as SA or TS can also be used.

Techniques have been introduced to alter the neighbourhood definition. A study on these issues

is given in the study by Santos and Santos (2004).

4.5.4 Features and considerations for MA

MA exploits problem knowledge by incorporating heuristics, approximation algorithms, local

search techniques and truncated exact methods. It aims at creating a balance between explorative

feature of EA and exploitation feature of local search algorithms. For a good MA, solution

representation must be carefully selected. Also careful design and use of operators, which are

tuned to problem under consideration, generate better results.

4.6 Artificial immune system

Immune system is a highly robust, adaptive and inherently parallel system with learning

capabilities. These features inspired the design of optimisation approach based on natural


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immune system. The field of AIS applies the concept of immunology to optimisation and

machine learning. Immune system can be divided into two parts: innate immune system and

adaptive immune system. The former is a static mechanism that detects and destroys certain

invading organisms. The latter system shows the capability of responding to unknown foreign

bodies and builds a response to them that can remain in the body over a long period of time. The

biological processes that are simulated to design AIS algorithms (Talbi, 2009) include pattern

recognition, clonal selection for B cells, negative selection of T cells, affinity maturation, danger

theory and immune network theory.

Clonal selection theory: It is a widely used theory for modelling immune system response to

antigens (foreign bodies). It is based on the concept of cloning and affinity (measure of

interaction between system components) maturation. When the body is exposed to an antigen,

B cell-producing antibodies that best bind to antigen clone themselves. This leads to production

of a large amount of required antibodies, thus fighting the infection rapidly. This process is a

kind of selection (only desired B-cell clone) based on fitness criteria (affinity measure). In

addition to this, a high-rate mutation (somatic mutation) is applied to diversify the population.

One of the main algorithms based on clonal selection theory, the CLONALG algorithm (de

Castro & Zuben, 2000), was proposed in 2000.

Negative selection mechanism: It is a mechanism by which self-cells (body’s own cells) are

protected from attack by antibodies. It allows only those cells (lymphocytes) to survive that do

not respond to self-cells. The antibodies that attach to self-cells are eliminated. Negative

selection algorithm (Forrest, Perelson, Allen, & Cherukuri, 1994) is based on the principles of

the maturation of the T cells and their ability to distinguish between self-/non-self cells. It was

put forth in 1994.

Immune network theory: According to Immune network theory, immune cells interact not

only with the antigens but also amongst themselves by means of simulating, activating and

suppressing each other. aiNET (de castro & Zuben, 2001) is one the most popular algorithm

based on immune network theory.

4.6.1 Methodology of AIS (CLONALG)

Implementation of AIS requires representation of solution, description of affinity measure,

selection of cells to be cloned, mutation to increase diversity and elimination of undesired cells.

Representation of solution can be done in any of traditional ways (e.g. binary or real-valued

strings) based on the optimisation problem. Affinity measure can be described in terms of a

distance metric (e.g. Euclidean distance, Manhattan distance or Hamming distance). For

CLONALG, an initial population of solutions is generated randomly. A set of solutions from the

population is selected (based on the selection criteria used) and then cloned. The clone size is

directly proportional to the affinity measure. It is followed by mutation, whose rate is inversely

proportional to the affinity measure. It helps in escaping out of local minima. Fitness of new

population is evaluated, and fitter members of the new population replace the weakest ones in

the existing population. Alongside, some random solution(s) is also added to increase diversity.

4.6.2 Pseudocode for CLONALG



Antibody population size, NP


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Begin


antibody ˆ rand(NP)


for i ¼ 1:NP

val (i) ˆ f(antibody(i,:))

end

Repeat

Select a predetermined number of antibodies with highest affinities

Clone the selected antibodies

Maturate the cloned antibodies

Evaluate all cloned antibodies

Add some of best cloned antibodies to the pool of antibodies

Remove worst members of the antibodies pool

Add new random antibodies into the population

Until Stopping criteria is met

4.6.3 Variations and advancements in AIS

Several new features and immune theories have been incorporated in AIS in the past decade.

One of them is the Danger theory (Matzinger, 2001). Idea behind this theory is that the immune

system does not respond to foreign elements but to danger. Thus, there is a need to discriminate

between internal (self) and foreign. A summary of many changes proposed in AIS is presented in

the study by Dasgupta (2006).

4.6.4 Features and considerations for AIS

Implementation of an AIS algorithm should take note of the nature of the problem’s data, and

choose a representation that intuitively maps the data’s characteristics. Also affinity measure and

distance metric used can have some sort of bias reflecting the problem definition. However, care

should be taken not to introduce undesirable bias.

5. Application of metaheuristics

Metaheuristic algorithms are used in a range of real-world optimisation problems. The

application areas of metaheuristics include communication, image and signal processing,

scheduling problems, very-large-scale integration design and financial planning. Table 3 lists the

application areas of metaheuristic algorithms reviewed in this paper.


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A comprehensive list of various metaheuristic techniques and their application is given in the

study by Osman and Laporte (1996).

6. Conclusion

Metaheuristic algorithms have garnered immense interest in the past two decades because of

their applicability to real-world optimisation problems. Metaheuristics is well suited to solving

NP hard as well as multi-objective optimisation problems. Traditional optimisation methods fail

to deliver satisfactory result in such cases because of the complexity of problem structure and the

size of problem instance.

The salient features of metaheuristic algorithms that make them better suited to real-world

optimisation problems are mentioned below.

. General applicability

Table 3. Application area(s) of Metaheuristic

Algorithm Area (s) of Application

Simulated Annealing (SA) † Job shop scheduling in production system† Vehicle routing in transport and logistics

management† Communication: mobile network design, routing, channel

allocationTabu Search (TS) † Resource allocation in industry, university etc.

† Engineering technology: cell placement in VLSI,power distribution, structural design

† Artificial Intelligence: pattern recognition, data mining,clustering

Genetic Algorithm (GA) † Financial planning, stock predictions† Image processing: compression, segmentation† Sequencing in FMS (flexible management systems)

Differential Evolution (DE) † Signal and image processing† Product design: aerodynamic† Chemical engineering

Particle Swarm Optimization (PSO) † Telecommunications: network design, routing† Neural network training† System simulation and identification† Decision making and planning† Signal processing

Ant Colony Optimization (ACO) † Set problems: set partitioning and covering,maximum independent set, bin packing

† Scheduling: flow shop scheduling, process planning† Bioinformatics: protein folding, DNA sequencing

Memetic Algorithm (MA) † Machine learning: neural network training, patternclassification

† System engineering† Scheduling: production planning, rostering† Set problems: bin packing, set covering

Artificial Immune System (AIS) † Data mining and data analysis† Clustering and classification† Network and computer security† Engineering design optimization


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Exact methods and heuristic techniques are problem-specific methodologies. For each

problem, a new mechanism needs to be adopted. Also, learning from one application cannot be

easily applied to another problem. However, in case of metaheuristics the task is to just adapt the

algorithm to specific problem rather than starting from the scratch.

. Reasonable computation time

Even for highly complex problem structures, metaheuristic algorithms can generate near-

optimal solutions in a reasonable amount of time for real-time applications.

. Find global optima

They are more suited to problems with multiple local minima than exact methods that have

higher chances of being stuck at local optima. Metaheuristic algorithms create a good balance

between intensification and diversification of search space.

However, metaheuristic algorithms also have their own set of limitations. They are listed

below

. Metaheuristic algorithms do not guarantee optimal solutions.

. It is not easy to theoretically prove the efficiency of algorithms. Studies usually rely on

empirical results to prove the same.. Development times of algorithmic framework are often high.. Each metaheuristic algorithm has its own set of advantages and limitations that makes it

more suitable to a particular kind of application. Finding the best suited algorithm is a

challenging task.

S-metaheuristic algorithms are aimed at intensification. They are good at exploring promising

areas of the search space. However, they are local search algorithms and are thus not able to

efficiently explore the search space.This drawback is overcomebypopulation-basedmethods.They

iterate a set of solutions, which leads to better coverage and exploration of the search space. The

drawbacks of both can be eliminated by the use of hybrid metaheuristics. This approach aims at

combining more than one metaheuristic method for problem solving. Another possibility is the use

of hyper-heuristics, which helps in deciding on an optimum sequence of metaheuristic algorithms,

combining advantages of each, to get the best solution possible. Thus, it can be safely said that

metaheuristics, hybrid metaheuristics and hyper-heuristics provide an efficient way of dealing with

highly complex optimisation problems encountered in industrial and scientific domains.

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metaheuristics: review and application

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