An Introduction to Nature-inspired

Metaheuristic Algorithms

Dr P. N. Suganthan School of EEE, NTU, Singapore

Workshop on Particle Swarm Optimization and

Evolutionary Computation

Institute for Mathematical Sciences, NUS

Feb 20th, 2018

Some Slides Contributed by Dr Swagatam Das, ISI,

Kolkata, and Dr Kaizhou Gao, NTU.

Benchmark Functions & SurveysResources available from

http://www.ntu.edu.sg/home/epnsugan

IEEE SSCI 2018, Bangaluru, in Nov. 2018

EMO-2019, Evolutionary Multi-Criterion Optimization

10-13 Mar 2019, MSU, USA

https://www.coin-laboratory.com/emo2019

2

Randomization-Based ANN, Pseudo-Inverse Based

Solutions, Kernel Ridge Regression, Random

Forest and Related Topicshttp://www.ntu.edu.sg/home/epnsugan/index_files/RNN-Moore-Penrose.htm

http://www.ntu.edu.sg/home/epnsugan/index_files/publications.htm

Consider submitting to

SWEVO journal dedicated to

the EC-SI fields. SCI

Indexed from Vol. 1, Issue 1.

2 Year IF= 3.8

5 Year IF=7.7

3

Swarm & Evolutionary

Computation Journal

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 4

5

• What is the practical goal of (global)

optimization?

– “There exists a goal (e.g. to find as small a

value of f() as possible), there exist resources

(e.g. some number of trials), and the problem is

how to use these resources in an optimal way.”

– A. Torn and A. Zilinskas, Global Optimisation.

Springer-Verlag, 1989. Lecture Notes in

Computer Science, Vol. 350.

Global Optimization

6

Hard Optimization Problems

• Goal: Find

– where S is often multi-dimensional; real/integer valued

or binary

– And subject to

– Many classes of optimization problems (and algorithms)

exist.

– When might it be worthwhile to consider metaheuristic

approaches?

SffS xxxx ),(*)(such that *

nn SS 1,0or R

( ) 0, 1,...,

, 1,...,

j

i i i

g j M

A x B i N

x

7

• A candidate solution to an optimization problem specifies the values of the decision variables, and therefore also the value of the objective function.

• A feasible solution satisfies all constraints.

• An optimal solution is feasible and provides the best objective function value.

• A near-optimal solution is feasible and provides a superior objective function value, but not necessarily the best.

Types of Solutions

8

• Optimization problems can be continuous (an

infinite number of feasible solutions) or

combinatorial (a finite number of feasible solutions).

• Continuous problem generally maximize or

minimize a function of continuous variables such as

min 4x + 5y where x and y are real numbers

• Combinatorial problems generally maximize or

minimize a function of discrete variables such as

min 4x + 5y where x and y are countable items (e.g.

integer only).

• Mixed decision variables are also possible.

Continuous vs Combinatorial

9

• Combinatorial optimization is the mathematical study of finding an optimal arrangement, grouping, ordering, or selection of discrete objects usually finite in numbers.

- Lawler, 1976

• In practice, combinatorial problems are often more difficult because there is no derivative informationand the surfaces are not smooth.

• In general nature inspired metaheuristics are not effective for solving combinatorial problems compared to heuristics.

• However, hybrids are very effective.

Definition of Combinatorial Optimization

10

• Constraints can be hard (must be satisfied) or soft

(is desirable to satisfy).

Example: In your course schedule a hard constraint

is that no classes overlap. A soft constraint is that no

class be before 10 AM.

Constraints may be formulated as objectives and

multi-objective evolutionary algorithms can be used.

• Constraints can be explicit (stated in the problem) or

implicit (obvious to the problem).

Constraints

11

• TSP (Traveling Salesman Problem)

Given the coordinates of n cities, find the shortest

closed tour which visits each once and only once

(i.e. exactly once).

Example: In the TSP, an implicit constraint is that

all cities be visited once and only once.

Constraints

-12-

– Combinatorial, continuous problem

– Bound constrained problem

– Constrained problem

– Single / Multi-objective problem

– Static / Dynamic optimization problem

– Expensive Problem

– Large Scale problem

– With/out noise

– Multimodal problem (niching, crowding, etc.)

Aspects of an Optimization Problem

13

• Constructive search techniques work by constructing a

solution step by step, evaluating that solution for (a) feasibility

and (b) objective function.

• Improvement search techniques obtain a solution by moving

to a neighboring solution, evaluating that solution for (a)

feasibility and (b) objective function.

---------------------------------------

• Search techniques may be deterministic (always arrive at the

same final solution through the same sequence of solutions,

although they may depend on the initial solution). Examples

are LP (simplex method), tabu search, simple heuristics, and

greedy heuristics.

• Search techniques may be stochastic where the solutions

considered and their order are different depending on random

variables. Examples are metaheuristic methods.

Solution Approaches

14

Simple

Few decision variables

Differentiable

Single modal

Objective easy to calculate

No or light constraints

Feasibility easy to determine

Single objective

deterministic

Hard

Many decision variables

Discontinuous, combinatorial

Multi modal

Objective difficult to calculate

Severely constraints

Feasibility difficult to determine

Multiple objective

Stochastic

15

• For Simple problems, enumeration or exact

methods such as differentiation or mathematical

programming or branch and bound will work the

best.

• For Hard problems, differentiation is not possible

and enumeration and other exact methods such as

math programming are not computationally

practical. For these problems, heuristics are used.

16

Search is the term used for constructing/improving solutions to obtain the optimum or near-optimum.

Solution Encoding (representing the solution)

Neighborhood Nearby solutions (in the encoding or solution space)

Move Transforming current solution to another (usually neighboring) solution

Evaluation The solutions’ feasibility and objective function value

Search Basics

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 17

MetaheuristicsA few points from: https://en.wikipedia.org/wiki/Metaheuristic

I. Heuristic (from Greek εὑρίσκω "I find, discover") is a technique

designed for finding an approximate more quickly when classic methods

are too slow, or fail to find any exact solution. This is achieved by

trading optimality, completeness, accuracy, or precision for speed.

II. Meta – beyond, for example inspiration from nature.

III. Metaheuristics may make few assumptions about the optimization

problem being solved, and so they may be usable for a variety of

problems.

IV. Many metaheuristic methods have been published with claims of

novelty and practical efficacy. While the field also features high-quality

research, unfortunately many of the publications have been of poor

quality; flaws include vagueness, lack of conceptual elaboration, poor

experiments, and ignorance of previous literature.

V. If you wish to be known as a “father of an invention”, nature-

inspired metaheuristics is a good field … ☺18

MetaheuristicsI. When to use: Only when traditional methods with proofs, etc. can’t

offer a satisfactory solution within the available time. – As these two research fields have diverged, cross-checking is not done

frequently, i.e. recent development of traditional methods are ignored …

II. Proofs for Metaheuristics: A little contradictory as metaheuristics

are developed to complement mathematical programming methods

with proofs, etc. – Metaheuristics claims to make no assumption about the problem being solved

while mathematical theories developed for these metaheuristics have several

assumptions, etc.

III. From “Metaheuristics – the Metaphor Exposed”, by Kenneth Sorensen:https://web.archive.org/web/20131102075645/http://antor.ua.ac.be/system/files/mme.pdf

19

Some of the Metaheuristicshttps://en.wikipedia.org/wiki/List_of_metaphor-

based_metaheuristics

Evolution strategy (1960s), Genetic algorithms (1970s)

– 1.1 Simulated annealing (Kirkpatrick et al. 1983)

– 1.2 Ant colony optimization (Dorigo, 1992)

– 1.3 Particle swarm optimization (Kennedy &

Eberhart 1995)

– 1.4 Harmony search (Geem, Kim & Loganathan

2001)

– 1.5 Artificial bee colony algorithm (Karaboga 2005)

– 1.6 Bees algorithm (Pham 2005)

– 1.7 Glowworm swarm optimization (Krishnanand &

Ghose 2005)

– 1.8 Shuffled frog leaping algorithm (Eusuff, Lansey

& Pasha 2006)

– 1.9 Imperialist competitive algorithm (Atashpaz-

Gargari & Lucas 2007)

20

Some of the Metaheuristicshttps://en.wikipedia.org/wiki/List_of_metap

hor-based_metaheuristics

– 1.10 River formation dynamics

(Rabanal, Rodríguez & Rubio 2007)

– 1.11 Intelligent water drops algorithm

(Shah-Hosseini 2007)

– 1.12 Gravitational search algorithm

(Rashedi, Nezamabadi-pour &

Saryazdi 2009)

– 1.13 Cuckoo search (Yang & Deb

2009)

– 1.14 Bat algorithm (Yang 2010)

– 1.15 Spiral optimization (SPO)

algorithm (Tamura & Yasuda

2011,2016-2017)

– 1.16 Flower pollination algorithm (Yang

2012)

21

– 1.17 Cuttlefish optimization

algorithm (Eesa, Mohsin, Brifcani &

Orman 2013)

– 1.18 Artificial swarm intelligence

(Rosenberg 2014)

– 1.19 Duelist Algorithm (Biyanto

2016)

– 1.20 Killer Whale Algorithm (Biyanto

2016)

– 1.21 Rain Water Algorithm (Biyanto

2017)

– 1.22 Mass and Energy Balances

Algorithm (Biyanto 2017)

– 1.23 Hydrological Cycle Algorithm

(Wedyan et al. 2017)

Never possible to have a

complete list of

metaheuristics …

More on Metaheuristics• Dr Xin She Yang’s Scholar page:

https://scholar.google.co.uk/citations?user=fA6aTlAAAAAJ

• Dr Seyedali Mirjalili’s scholar page:

https://scholar.google.com/citations?user=TJHmrREAAAAJ&hl

=en

• Cultural Algorithms, Brainstorming, Fireworks,

Spider Monkey, Egyptian Vulture, Wolf, Grey Wolf,

…

22

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 23

The Genetic Algorithm (GA)

• Directed search algorithms based on the

mechanics of biological evolution

• Developed by John Holland, Ken De Jong,

University of Michigan (1970’s)

– To understand the adaptive processes of natural

systems

– To design artificial simulation software that

retains the robustness of natural systems

24

Evolution in the real world

• Each cell of a living thing contains chromosomes - strings of DNA

• Each chromosome contains a set of genes - blocks of DNA

• Each gene determines some aspect of the organism (like eye colour)

• A collection of genes is sometimes called a genotype

• A collection of aspects (like eye colour) is sometimes called a phenotype

• Reproduction of offspring involves recombination (orcrossover) of genes from parents and then small amounts of mutation (errors) in copying

• The fitness of an organism is how much it can reproduce before it dies

• Evolution based on “survival of the fittest” realized by selection operation. 25

Emulating Evolution: GA

Generate a population of random chromosomes (i.e.

potential solutions to the problem)

Repeat (each generation)

Calculate fitness of each chromosome

Repeat

Select pairs of parents

Generate offspring with crossover and mutation

Until a new population has been produced

Until best solution is good enough26

How do you encode a solution?

• Obviously this depends on the problem!

• GA’s often encode solutions as fixed length bitstrings (e.g.

101110, 111111, …) (although real-coded GAs exist now)

• For the GA to work, we need to be able to “test” any string

and get a “score” (fitness) indicating how “good” that

solution is

• GA is a competitive method if the problem naturally has

binary solution. In this case, each bit represents a decision

variable.

• However, it is also possible to encode integers, floating

values by bitstrings using quatization with prespecified

range for each decision variable. 27

Search Space• By encoding several decision variables into the chromosome

many dimensions can be searched, e.g. two dimensions f(x,y)

• Search space an be visualised as a surface or fitness landscape in which fitness dictates height

• Each possible genotype is a point in the space

• A GA tries to move the points to better places (higher fitness) in the space

• Obviously, the nature of the search space dictates how a GA will perform– A completely random space would be bad for a GA

– Also GA’s can get stuck in local maxima if search spaces contain lots of these

– Generally, spaces in which small improvements get closer to the global optimum are good

28

Fitness landscapes

29

The figures are for 2-dimensional real decision

variables where vertical height represents fitness

of each point.

Above 2 landscapes are suitable for GA to search while the 3rd is not.

Parent Selection, Crossover & Mutation

• Two highly fit parent bit strings are selected and

randomly combined to produce two new offspring

(bit strings).

• Many schemes are possible so long as better scoring chromosomes/parents more likely selected

• “Roulette Wheel” selection can be used:– Add up the fitness's of all chromosomes

– Generate a random number R in that range

– Select the first chromosome in the population that - when all previous fitness’s are added - gives you at least the value R

• A few bits in each offspring may then be changed

randomly (mutation)30

Recombination by One Point Crossover

101000000

100101111

Crossover

single point -

random

1011011111

100000000

Parent1

Parent2

Offspring1

Offspring2

With some high probability (crossover

rate) apply crossover to the parents.

(typical values are 0.8 to 0.95)

Other alternatives are 2-point crossover and uniform crossover.

2-point crossover is a simple extension of one point crossover

Uniform crossover randomly mix bits from two parents to

obtain 2 offspring 31

Mutation

1011011111

100000000

Offspring1

Offspring2

1011001111

101000000

Offspring1

Offspring2

With some small probability (the mutation rate) flip

each bit in the offspring (typical values between 0.1

and 0.001, or just about a couple of bits)

mutated

Original offspring Mutated offspring

32

Many Variants of GA• Different parent selection methods to generate

offspring– Tournament

– Elitism,

– Roulette wheel, etc.

• Different recombination– Multi-point crossover

– Multi-parent crossover, etc.

• Different kinds of encoding other than bitstring– Integer values

– Floating values

– Categorical, symbolic, etc.

• Different kinds of mutation (for non-binary encoding)

33

Many parameters/Operators to set

• Any GA implementation needs to decide on a number of parameters: Population size (N), mutation rate (m), crossover rate (c)

• Typical parameter (arbitrary) values might be: N = 50, m = 0.05, c = 0.9

• Selection of operators such as 1-point / 2-point / uniform crossover, roulette wheel / tournament selection, etc.

• Often these have to be “tuned” based on results obtained - no general theory to deduce good values

• Adaptive methods with ensemble of operators and parameter values can also be used.

34

Why does GA work?

• Some theories about this and some controversy

• Holland introduced Schema theory

• The idea is that crossover preserves “good bits” from different parents, combining them to produce better solutions

• A good encoding scheme would therefore try to preserve “good bits” during crossover and mutation

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 36

-37-

Evolution Strategy (ES, from 1960s)

• Real variable optimizer

• Randomly generate an initial population of M solutions.

Compute the fitness values of these M solutions.

• Use all vectors as parents to create nb offspring vectors by

Gaussian mutation.

– Gaussian mutation: Xnew=Xold+N(0, )

• Calculate the fitness of nb vectors, and prune the

population to M fittest vectors.

• Go to the next step if the stopping condition is satisfied.

Otherwise, go to Step 2.

• Choose the fittest one in the population in the last

generation as the optimal solution.

• CMA-ES is a competitive state of the art variant of ES.

Differential Evolution• A stochastic population-based algorithm for continuous function

optimization (Storn and Price, 1995)

• Finished 3rd at the First International Contest on Evolutionary Computation, Nagoya, 1996 (icsi.berkley.edu/~storn)

• Outperformed several variants of GA and PSO over a wide variety of numerical benchmarks over past several years.

• Continually exhibited remarkable performance in competitions on different kinds of optimization problems like dynamic, multi-objective, constrained, and multi-modal problems held under IEEE Congress on Evolutionary Computation (CEC) conference series.

• Very easy to implement in any programming language.

• Very few control parameters (typically three for a standard DE) and their effects on the performance have been well studied.

• Complexity is very low as compared to some of the most competitive continuous optimizers like CMA-ES. 38

DE is an Evolutionary Algorithm

This Class also includes GA, Evolutionary Programming and Evolutionary Strategies

Initialization Mutation Recombination Selection

Basic steps of an Evolutionary Algorithm

39

Representation

Min

Max

May wish to constrain the values taken in each domain

above and below.

x1 x2 x D-1 xD

Solutions are represented as vectors of size D with each

value taken from some domain.

X

40

Population Size - NP

x1,1 x2,1 x D-1,1 xD,1

x1,2 x2,2 xD-1,2 xD,2

x1,NP x2,NP x D-1,NP xD, NP

We will maintain a population of size NP

1X

2X

NPX

41

Different values are instantiated for each i and j.

Min

Max

x2,i,0 x D-1,i,0 xD,i,0x1,i,0

, ,0 ,min , ,max ,min[0,1] ( )j i j i j j jx x rand x x

0.42 0.22 0.78 0.83

Initialization Mutation Recombination Selection

, [0,1]i jrand

iX

42

Initialization Mutation Recombination Selection

➢For each vector select three other parameter vectors randomly.

➢Add the weighted difference of two of the parameter vectors to the

third to form a donor vector (most commonly seen form of

DE-mutation):

➢The scaling factor F is a constant from (0, 2)

➢Self-referential Mutation

).(,,,,

321 GrGrGrGi iii XXFXV

43

Initialization Mutation Recombination Selection

Components of the donor vector enter into the trial offspring vector in the following way:

Let jrand be a randomly chosen integer between 1,...,D.

Binomial (Uniform) Crossover:

44

Initialization Mutation Recombination Selection

➢“Survival of the fitter” principle in selection: The trial

offspring vector is compared with the target (parent)

vector and the one with a better fitness is admitted to the

next generation population.

1, GiX

,,GiU

)()( ,, GiGi XfUf

if

,,GiX

if )()( ,, GiGi XfUf

➢Importance of parent-mutant crossover & parent-

offspring competition-based selection

45

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 46

-47-

Particle Swarm Optimizer

• Introduced by Kennedy and Eberhart in 1995

• Emulates flocking behavior of birds to solve optimization problems

• Each solution in the landscape is a particle

• All particles have fitness values and velocities

-48-

Particle Swarm Optimizer• Two versions of PSO

– Global version (May not be used alone to solve multimodal

problems): Learning from the personal best (pbest) and the best

position achieved by the whole population (gbest)

– Local Version: Learning from the pbest and the best position

achieved in the particle's neighborhood population (lbest)

• The random numbers (rand1 & rand2) should be generated

for each dimension of each particle in every iteration.

1 2* 1 ( ) * 2 ( )

d d d d d d d

i i i i i i

d d d

i i i

V c rand pbest X c rand gbest X

X X V

1 2* 1 ( ) * 2 ( )

d d d d d d d

i i i i i k i

d d d

i i i

V c rand pbest X c rand lbest X

X X V

i – particle counter & d – dimension counter

+ 𝜔𝑉𝑖𝑑

-49-

Particle Swarm Optimizer

• where and in the equation are the acceleration constants

are two random numbers in the range [0,1];

• represents the position of the ith particle;

• represents the rate of the position change

(velocity) for particle i.

• represents the best previous

position (the position giving the best fitness value) of the ith

particle;

• represents the best previous

position of the population;

• represents the best previous

position achieved by the neighborhood of the particle;

1c 2c

1 and 2d d

i irand rand1 2( , ,..., )D

i i i iX X XX

1 2( , ,..., )D

i i i ipbest pbest pbestpbest

1 2( , ,..., )Dgbest gbest gbestgbest

1 2( , ,..., )D

i i i iV V VV

1 2( , ,..., )D

k k k klbest lbest lbestlbest

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 50

Harmony search algorithm

The following work suggests that HSA can be derived from ES:

“Metaheuristics – the Metaphor Exposed”, by Kenneth Sorensen:https://web.archive.org/web/20131102075645/http://antor.ua.ac.be/system/files/mme.pdf

Geem, Zong Woo, Joong Hoon Kim, and G. V. Loganathan (2001). A new heuristic optimization

algorithm: harmony search. Simulation 76.2: pp. 60-68.

Kai-Zhou Gao, Ponnuthurai N Suganthan, Quan-Ke Pan. Pareto-based grouping discrete

harmony search algorithm for multi-objective flexible job shop scheduling. Information sciences,

289: 76-90, 2014

Harmony search algorithm

Water Cycle Algorithm: Basic Concept

Se

a

Strea

m

River

Sea

Hadi Eskandar, Ali Sadollah, Ardeshir Bahreininejad, Mohd Hamdi. Water cycle algorithm – A novel metaheuristic optimization method for solving constrained engineering optimization problems. Computers & Structures, 110-111: 151-166, 2012.

Kaizhou Gao, Peiyong Duan, Rong Su, Junqing Li. Bi-objective Water Cycle Algorithm for Solving Remanufacturing Rescheduling Problem. Proceedings of Asia-Pacific Conference on Simulated Evolution and Learning (SEAL 2017), 671-683.

Schematic view of Water Cycle Process

Nature Water Cycle Algorithm

Precipitation Initial Population

Stream(s) Individual(s) of population

River(s) Second best solution (a number of best solution)

Sea Best solution (optimum solution)

Surface Runoff Moving streams to rivers, and rivers to sea

Evaporation Evaporation condition

Water cycle process Iteration

Water Cycle Algorithm: Movement Strategy

( 1) ( ) ( ( ) ( ))Stream Stream River StreamX t X t rand C X t X t

( 1) ( ) ( ( ) ( ))Stream Stream Sea StreamX t X t rand C X t X t

( 1) ( ) ( ( ) ( ))River River Sea RiverX t X t rand C X t X t

1

1 1 1 12

1 2 3

3 2 2 2 2

1 2 3

4

5

1 2 36

pop pop pop pop

pop

N

N

N N N N

N

N

Sea

River

Riverx x x x

Riverx x x x

Total Population Stream

Streamx x x x

Stream

Stream

1SR

Sea

N Number of Rivers

Streams Pop SRN N N

Jaya Algorithm

• The definition of Jaya is victory in Sanskrit

• In the Jaya algorithm, the applied strategy always tries to

become victorious by reaching the best solution and hence it

is named Jaya, and it is reported as a simple and applicable

optimization approach in the literature.

1 2( 1) ( ) ( ( ) ( )) ( ( ) ( )), 1,2,3,...,i i i i i i

Best Worst PopX t X t r X t X t r X t X t i N

R. Venkata Rao, Jaya: A simple and new optimization algorithm for solving constrained and unconstrained optimization problems. International Journal of Industrial Engineering Computations, 7: 19–34, 2016.

K Gao, Y Zhang, A Sadollah, A Lentzakis, R Su. Jaya, harmony search and water cycle algorithms for solving large-scale real-life urban traffic light scheduling problem. Swarm and Evolutionary Computation 37, 58-72, 2017.

Jaya algorithm

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 58

59

Bees in nature

ABC Algorithm

Artificial Bee Colony Algorithm

D. Karaboga, AN IDEA BASED ON HONEY BEE SWARM FOR NUMERICAL

OPTIMIZATION,TECHNICAL REPORT-TR06,Erciyes University, Engineering

Faculty, Computer Engineering Department 2005.

60

Bees in Nature1- A colony of honey bees can extend itself over long

distances in multiple directions (more than 10 km)

Flower patches with plentiful amounts of nectar or pollen that

can be collected with less effort should be visited by more

bees, whereas patches with less nectar or pollen should

receive fewer bees.

61

Bees in Nature

2- Scout bees search randomly from one

patch to another

62

Bees in Nature

3- The bees who return to the hive, evaluate

the different patches depending on certain

quality threshold (measured as a

combination of some elements, such as

sugar content)

63

Bees in Nature

4- They deposit their nectar or pollen go to

the “dance floor” to perform a “waggle

dance”

64

Bees in Nature

5- Bees communicate through this waggle

dance which contains the following information:

1. The direction of flower patches

(angle between the sun and the

patch)

2. The distance from the hive

(duration of the dance)

3. The quality rating (fitness)

(frequency of the dance)

65

Bees in Nature

These information helps the colony to send its bees precisely

6- Follower bees go after the dancer bee to the patch to gather

food efficiently and quickly

7- The same patch will be advertised in the waggle dance again

when returning to the hive is it still good enough as a food

source (depending on the food level) and more bees will be

recruited to that source

8- More bees visit flower patches with plentiful amounts of

nectar or pollen

66

Bees in Nature

Thus, according to the fitness, patches can

be visited by more bees or may be

abandoned

Artificial Bee Colony Algorithm

Initialization

• The food sources indicating the probable solutions to the objective

function are represented as where c is the cycle number and i is the

running index that can takes values {1, 2.. SN} (SN being food

number). The j th component of the i th food source (real vector)

are initialized as

max max min

0 (0,1).i j j j jrand

where j is {1,2…D} for a D-dimensional problem.

The food sources continue to improve upon their previous values

till the termination criteria are met. 67

Artificial Bee Colony Algorithm

Employer Bee Phase

• Each food source has an employed forager associated with it and their

population size equals the number of food sources. An employed bee

modifies the position of the food source and searches an adjacent food

source with respect to a single dimension as:

.i j i j i j k j i jv c c c

Now the fitness of the newly produced site is calculated.

1/ 1 ( ) if ( ) 0

1 ( ) if ( ) 0

i i

i

i i

f ffitness

f f

where f (.) denote the objective function.

The fitter one is selected by using greedy selection.68

Artificial Bee Colony Algorithm

Onlooker Bee Phase

• The onlooker bees select the food source based on its nectar

content. The chances of selecting a food source is determined by

its probability value computed as

1

ii SN

i

i

fitnessp

fitness

Identical to the employed bee phase.

A random number is generated in the range [0, 1]

If this value is greater than probability (calculated above) for a

given food source, positional modification is done.

This is carried on till all the onlookers have been allotted a food

source.69

Artificial Bee Colony Algorithm

Scout Bee Phase

• The scout phase was designed to prevent the wastage of

computational resources.

• In case the trial counter for a food source exceeds a pre-

defined limit then it is deemed to be exhausted such that no

further improvement is possible.

• In such a case, it is randomly reinitialized and its fitness value

is re-evaluated and counter is reset to 0.

• This is motivated from the fact that repeated exploitation of

nectar by a forager causes the exhaustion of the nectar content.

70

71

Cuckoo Search Algorithm• Cuckoos have a parasitic breeding behaviour and

they engage in the obligate brood parasitism by

laying their eggs in the nests of other host birds

(often other species).

• Three basic types of brood parasitism: intra-

specific brood parasitism, cooperative breeding,

and nest takeover.

• If a host bird discovers the eggs are not their

owns, they will either throw these alien eggs away

or simply abandon the nest and build a new nest

elsewhere.

• Some cuckoo species such as the New World

brood-parasitic Tapera have evolved in such a

way that female parasitic cuckoos are often very

specialized in the mimicry in color and pattern of

the eggs of a few chosen host species.

• The timing of egg-laying of some species is also

amazing

X.-S. Yang; S. Deb (December 2009). "Cuckoo search via Lévy flights". World Congress

on Nature & Biologically Inspired Computing (NaBIC 2009). IEEE Publications. pp. 210–

214.

Cuckoo search algorithm• In CS algorithm each egg in a nest represents a solution, and a cuckoo egg

represent a new solution, the aim is to use the new and potentially better solutions (cuckoos) to replace a not-so-good solution in the nests.

• The basic steps of the CS algorithm involve:

1) Each cuckoo lays one egg at a time, and dump its egg in randomly chosen nest;

2) The best nests with high quality of eggs will carry over to the next generations;

3) The number of available host nests is fixed, and the egg laid by a cuckoo is discovered by the host bird with a probability [0, 1].

• Levy flight is used in generating new cuckoo solutions:

where α > 0 is the step size which is related to the scales of the problem of interest. In most cases, it was taken as α = 1. The product ⊕ means entry-wise multiplications. Lévy flights essentially provide a random walk while their random steps are drawn from a Lévy distribution:

)()()1( Levyxx t

i

t

i

tuLevy ~

72

Flow-Chart for CS Algorithm

73

Group Search Optimization

(GSO)•GSO is inspired by the food searching

behavior and group living theory of social

animals, such as birds, fish and lions.

•The foraging strategies of these animals

mainly include: producing, e.g., searching

for food; and joining (scrounging), e.g.,

joining resources uncovered by others.

•GSO also employs ‘rangers’ which

perform random walks to avoid entrapment

in local minima.

S. He, Q.H. Wu, J.R. Saunders, Group Search Optimizer: an optimization algorithm

Inspired by animal searching behavior, IEEE Transactions on Evolutionary

Computation 13 (October (5)) (2009) 973–990. 74

Members of GSO In GSO, a group consists of three kinds of members:

• Producers

• Scroungers

• Rangers

The producer, scroungers and rangers do not differ intheir relevant phenotypic characteristics. Therefore, theycan switch among the three roles.

The framework mainly follows the Producer–Scrounger(PS) model, which states that group members searcheither for “finding” (producer) or for “joining” (scrounger)opportunities.

75

• At each iteration, a group member, located in the most promising area,

conferring the best fitness value, is chosen as the producer. It locates in the

most promising area and stays still.

• The other group members are selected as scroungers or rangers by

random. Then, each scrounger makes a random walk towards the producer,

and the rangers make a random walk in arbitrary direction.

Individual Representation in GSO

The i-th member at the k-th iteration is located in a position:nk

i RX

The associated head angle is: 1 1

,...........,n

k k k

i i i

The search direction associated with the i-th member is represented as a

unit vector : 1,...........,

n

k k k k n

i i i iD d d R

The vector can be obtained from Cartesian to Polar transformation as:

1

1

1

cos .q

nk k

i i

q

d

1

1

sin cos ,j qn

nk k k

i i i

q j

d

2,3,..., .j n

1

sin .n n

k k

i id

76

Producer activities

At the k-th iteration the producer behaves as:

• The producer will scan at zero degree and then scan

laterally by randomly sampling three points in the

scanning field: one point at zero degree:

• One point in the left side of the hypercube:

• And one point in the right hand side hypercube:

where r is a normally distributed random number with

mean=0 and standard deviation=1

kk

p

k

pz DlrXX max1

2/max2max1 rDlrXX kk

p

k

pr

2/max2max1 rDlrXX kk

p

k

pl

77

Contd.

• The producer will then find the best point with the best

resource (fitness value). If the best point has a better

resource than its current position, then it will fly to this

point. Or it will stay in its current position and turn its

head to a new angle:

where is the maximum turning angle.

• If the producer cannot find a better area after a iterations,

it will turn its head back to zero degree:

where a is a constant.

max2

1 rkk

1

max R

kak

78

Scrounger dynamics

• At the k-th iteration, the area copying behavior of the i-th

scrounger can be modeled as a random walk towards

the producer:

• where is a uniform random sequence in the

range (0, 1).

k

i

k

p

k

i

k

i XXrXX 3

1

nRr 3

79

Ranger movements

• Besides the producer and the scroungers, a small number of

rangers have been also introduced into the GSO algorithm.

• Random walks, which are thought to be the most efficient

searching method for randomly distributed resources, are

employed by rangers.

• If the i-th member is selected as a ranger, at the k-th

iteration, it produces a random heading angle:

• then it chooses a random distance:

and moves to the point:

max2

1 rkk

max1 lrli

11 kk

ii

k

i

k

i DlXX

80

Behavior of Fireflies

• Most of the two thousand firefly species, and mostfireflies produce short and rhythmic flashes which areunique a particular species.

• The flashing light is produced by a process ofbioluminescence.

• The fundamental functions of such flashes are– to attract mating partners (communication),

– to attract potential prey.

– flashing may also serve as a protective warningmechanism.

• Both sexes of fireflies are brought together via the rhythmic flash, the rate of flashing and the amount of time form part of the signal system.

81

82

• The flashing light produced by fireflies can be

formulated to be associated with the objective

function to be optimized, which makes it possible

to formulate new optimization algorithms.

83

Firefly Algorithm

• For simplicity in describing our new Firefly Algorithm

(FA), we now use the following three idealized rules:– All fireflies are unisex so that one firefly will be attracted to other fireflies

regardless of their sex;

– Attractiveness is proportional to their brightness, thus for any two

flashing fireflies, the less brighter one will move towards the brighter

one. The attractiveness is proportional to the brightness and they both

decrease as their distance increases. If there is no brighter one than a

particular firefly, it will move randomly;

– The brightness of a firefly is affected or determined by the landscape of

the objective function.

• Other forms of brightness can be defined in a similar

way to the fitness function in genetic algorithms.

84

Flow Chart of the FA

85

Intensity and Attractiveness

86

87

88

89

Distance and Movement

90

91

2

Attraction

Randomization

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 92

Conceptual Comparison

93

DE versus ABC

94

Local search/linear movements by varying a single component

DE vs ABC & CS vs EP

95

ABC vs DE &

PSO vs Other Swarm Algorithms

96

• The similarities often stem from

The very objective of designing good

metaheuristics

97

Should have a good trade-off between

exploitation and exploration

Must be able to rapidly converge to the

global optimum

Should not impose serious computational

overheads

Should have very few (or no) algorithmic

control parameters except the general ones

(population size, total no. of iterations,

problem dimensions etc.)

Efficient

metaheuristics

The nature-inspired story lines are different. When the story

is converted to equations, they become too similar !!

Hybridization Aspects• Synergy of two or more SI algorithms may lead to

new global optimizers with great performance on a

set of problems.

• The biggest challenge is to determine

– which algorithms to combine?

– Which component s should be taken?

• Hybridization must perform better than each

component algorithm (state of the art variants).

• Successful in the combinatorial cases: Nature

inspired algorithms + local search heuristics.

• Limited research community & less successful in

real variable optimization. 98

Genetic Programming (from 1960s/70s)

• When the chromosome encodes an entire

program or function itself this is called

genetic programming (GP)

• In order to make this work encoding is often

done in the form of a tree representation

• Crossover entails swapping subtrees

between parents

• Successful in symbolic regression tasks.

Genetic Programming

It is possible to evolve whole programs like this but

only small ones. Large programs with complex

functions present big problems

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 101

Theoretical Studies• Some works are for population-based, though they are based

on simplifications like only one global minimum, smooth

objective function, constant coefficients instead of stochastic.

Bacterial Foraging works assume small population.

• Other works are mostly on (1+1) EAs, very easy binary

problems (like onemax, maximize the sum of bits in a bit string).

• My standard review question: “Can you demonstrate that your

theory can overall improve performance of a state of the art

algorithm?”

• The answer is always “no” as theories are for simplified models

while state of the art methods are too complicated for exact

analysis.

• Proofs & rigorous analysis for metaheuristics: A little contradictory as

metaheuristics are developed to complement mathematical

programming methods with proofs, etc. 102

Theoretical Studies• Sayan Ghosh, Swagatam Das, Athanasios V. Vasilakos, Kaushik

Suresh: On Convergence of Differential Evolution Over a Class of

Continuous Functions With Unique Global Optimum. IEEE Trans.

Systems, Man, and Cybernetics, Part B 42(1): 107-124 (2012).

• Swagatam Das, Arpan Mukhopadhyay, Anwit Roy, Ajith Abraham,

Bijaya K. Panigrahi: Exploratory Power of the Harmony Search

Algorithm: Analysis and Improvements for Global Numerical

Optimization. IEEE Trans. Systems, Man, and Cybernetics, Part B

41(1): 89-106 (2011).

• Sayan Ghosh, Swagatam Das, Debarati Kundu, Kaushik Suresh, Ajith

Abraham: Inter-particle communication and search-dynamics of lbest

particle swarm optimizers: An analysis. Inf. Sci. 182(1): 156-168

(2012).

• Swagatam Das, Sambarta Dasgupta, Arijit Biswas, Ajith Abraham,

Amit Konar: On Stability of the Chemotactic Dynamics in Bacterial-

Foraging Optimization Algorithm. IEEE Trans. Systems, Man, and

Cybernetics, Part A 39(3): 670-679 (2009).

103

Theoretical Studies

• Swagatam Das, Sambarta Dasgupta, Arijit Biswas, Ajith Abraham, Amit

Konar: On Stability of the Chemotactic Dynamics in Bacterial-Foraging

Optimization Algorithm. IEEE Trans. Systems, Man, and Cybernetics, Part

A 39(3): 670-679 (2009).

• Sambarta Dasgupta, Swagatam Das, Arijit Biswas, Ajith Abraham: On

stability and convergence of the population-dynamics in differential

evolution. AI Commun. 22(1): 1-20 (2009).

• Jun He, Xin Yao: Average Drift Analysis and Population Scalability. IEEE

Trans. Evolutionary Computation 21(3): 426-439 (2017).

• J. He, T. Chen, and X. Yao, “On the easiest and hardest fitness functions,”

IEEE Trans. Evol. Comput., vol. 19, no. 2, pp. 295–305, Apr. 2015

• T. Friedrich, P. S. Oliveto, D. Sudholt, and C. Witt, “Analysis of diversity-

preserving mechanisms for global exploration,” Evol. Comput., vol. 17, no.

4, pp. 455–476, 2009.

• Arijit Biswas, Swagatam Das, Ajith Abraham, Sambarta Dasgupta: Stability

analysis of the reproduction operator in bacterial foraging optimization.

Theoretical Computer Science, 411(21): 2127-2139 (2010).104

Theoretical Studies on PSO

• Maurice Clerc’s works: The particle swarm-explosion, stability,

and ... - Semantic Scholar

• V Kadirkamanthan:

https://scholar.google.com/citations?user=JN9QqjEAAAAJ&

hl=en

• Prof Andries Engelbrecht:

https://scholar.google.com/citations?user=h9pOfj0AAAAJ&hl

=en

105

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 10

6

General Thoughts: NFL

(No Free Lunch Theorem)

• Glamorous Name for Commonsense?

– Over a large set of problems, it is impossible to find a single best algorithm

– DE with Cr=0.90 & Cr=0.91 are two different algorithms Infinite algos.

– Practical Relevance: Is it common for a practicing engineer to solve several

practical problems at the same time? NO

– Academic Relevance: Very High, if our algorithm is not the best on all

problems, NFL can rescue us!!

Other NFL Like Commonsense Scenarios

Panacea: A medicine to cure all diseases, Amrita: nectar of immortal perfect life

Silver bullet: in politics … (you can search these on internet)

Jack of all trades, but master of none

If you have a hammer all problems look like nails 107

General Thoughts: Convergence

• What is exactly convergence in the context of EAs & SAs ?

– The whole population reaching an optimum point (within a tolerance)…

– Single point search methods & convergence …

• In the context of real world problem solving, are we going to reject a

good solution because the population hasn’t converged ?

• Good to have all population members converging to the global

solution OR good to have high diversity even after finding the

global optimum ? (Fixed Computational budget Scenario)

What we do not want to have:

For example, in the context of PSO, we do not want to have chaotic oscillations

c1 + c2 > 4.1+

108

General Thoughts: Algorithmic Parameters

• Good to have many algorithmic parameters / operators ?

• Good to be robust against parameter / operator variations ? (NFL?)

• What are Reviewers’ preferences on the 2 issues above?

• Or good to have several parameters/operators that can be tuned

to achieve top performance on diverse problems? YES

• If NFL says that a single algorithm is not the best for a very large set

of problems, then good to have many algorithmic parameters &

operators to be adapted for different problems !!

CEC 2015 Competitions: “Learning-Based Optimization”

Similar Literature: Thomas Stützle, Holger Hoos, …109

General Thoughts: Nature Inspired Methods

• Good to mimic too closely natural phenomena? Lack of freedom to

introduce heuristics due to conflict with the natural phenomenon.

• Honey bees solve only one problem (gathering honey). Can this

ABC/BCO be the best approach for solving all practical problems?

• NFL & Nature inspired methods.

• Swarm inspired methods and some nature inspired methods do not

have crossover operator.

• Dynamics based methods such as PSO and survival of the fitter

method: PSO always moves to a new position, while DE moves

after checking fitness. 110

OverviewI. General Introduction

II. Introduction to Metaheuristics

III. Genetic Algorithms (GA)

IV. Evolution Strategy (ES), Differential Evolution (DE)

V. Particle Swarm Optimization (PSO)

VI. Harmony Search, Water Cycle, Jaya Algorithm

VII. Artificial Bee Colony (ABC), Group Search Algorithm, Cuckoo

Search (CSA) Firefly Algorithm (FA)

VIII.Conceptual Similarities & Hybridization

IX. Theoretical Studies

X. General Thoughts

XI. Future Directions 11

1

-112-

I - Population Topologies• In population based algorithms, population members exchange

information between them.

• Single population topology permits all members to exchange

information among themselves – the most commonly used.

• Other population topologies have restrictions on information

exchange between members – the oldest is island model

• Restrictions on information exchange can slow down the

propagation of information from the best member in the population

to other members (i.e. single objective global optimization)

• Hence, this approach

– slows down movement of other members towards the current best

member(s)

– Enhances the exploration of the search space

– Beneficial when solving multi-modal problems

As global version of the PSO converges fast, many topologies were

Introduced to slow down PSO …

I - PSO with Euclidean Neighborhoods

Presumed to be the oldest paper to consider distance based

dynamic neighborhoods for real-parameter optimization.

Lbest is selected from the members that are closer (w.r.t.

Euclidean distance) to the member being updated.

Initially only a few members are within the neighborhood (small

distance threshold) and finally all members are in the n’hood.

Island model and other static/dynamic neighborhoods did not

make use of Euclidean distances, instead just the indexes of

population members.

Our recent works are extensively making use of distance based

neighborhoods to solve many classes of problems.

113

P. N. Suganthan, “Particle swarm optimizer with neighborhood

operator,” in Proc. Congr. Evol. Comput., Washington, DC, pp.1958–

1962, 1999.

II - Ensemble Methods

• Ensemble methods are commonly used for pattern

recognition (PR), forecasting, and prediction, e.g. multiple

predictors.

• Not commonly used in Evolutionary algorithms ...

There are two advantages in EA (compared to PR):

1. In PR, we have no idea if a predicted value is correct or

not. In EA, we can look at the objective values and make

some conclusions.

2. Sharing of function evaluation among ensembles possible.

114

III - Adaptations

• Self-adaptation: parameters and operators are evolved by

coding them together with decision vectors

• Separate adaptation based on performance: operators

and parameter values yielding improved solutions are

recorded and rewarded.

• 2nd approach is more successful and frequently used with

population-based numerical optimizers.

115

Two Subpopulations with Heterogeneous

Ensembles & Topologies

▪ Proposed for balancing exploration and exploitation capabilities

▪ Population is divided into exploration / exploitation sub-poplns➢ Exploration Subpopulation group uses exploration oriented ensemble

of parameters and operators

➢ Exploitation Subpopulation group uses exploitation oriented ensemble

of parameters and operators.

• Topology allows information exchange only from explorative subpopulation

to exploitation sub-population. Hence, diversity of exploration popln not

affected even if exploitation popln converges.

• The need for memetic algorithms in real parameter optimization: Memetic

algorithms were developed because we were not able to have an EA or SI to be

able to perform both exploitation and exploration simultaneously. This 2-popln

topology allows with heterogeneous information exchange.

116

Two Subpopulations with Heterogeneous

Ensembles & Topologies

▪ Sa.EPSDE realization (for single objective Global):

N. Lynn, R Mallipeddi, P. N. Suganthan, “Differential Evolution with Two

Subpopulations," LNCS 8947, SEMCCO 2014.

▪ 2 Subpopulations CLPSO (for single objective Global)N. Lynn, P. N. Suganthan, “Comprehensive Learning Particle Swarm Optimization with

Heterogeneous Population Topologies for Enhanced Exploration and Exploitation,”

Swarm and Evolutionary Computation, 2015.

▪ Neighborhood-Based Niching-DE: Distance based neighborhood forms local

topologies while within each n’hood, we employ exploration-exploitation

ensemble of parameters and operators.

S. Hui, P N Suganthan, “Ensemble and Arithmetic Recombination-Based Speciation Differential

Evolution for Multimodal Optimization,” IEEE T. Cybernetics, pp. 64-74 Jan 2016.

10.1109/TCYB.2015.2394466

B-Y Qu, P N Suganthan, J J Liang, "Differential Evolution with Neighborhood Mutation for

Multimodal Optimization," IEEE Trans on Evolutionary Computation, DOI:

10.1109/TEVC.2011.2161873. (Supplementary file), Oct 2012. (Codes Available: 2012-TEC-

DE-niching)117

IV - Population Size Reduction

• Evolutionary algorithms are expected to explore the

search space in the early stages

• In the final stages of search, exploitation of previously

found good regions takes place.

• For exploration of the whole search space, we need a

large population while for exploration, we need a small

population size.

• Hence, population size reduction will be effective for

evolutionary algorithms.

118

THANK YOU

Q & A

119