50120140504022

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 4, April (2014), pp. 202-209 © IAEME
202
PERFORMANCE & CONVERGENCE ANALYSIS OF A NOVEL MODEL OF
GENETIC ALGORITHM TOWARDS GLOBAL MINIMA
Yatin Patadiya1
, M/s Saroj Hiranwal2
1, 2
Computer Science & Engineering, Sri Balaji College of Engineering & Technology, Jaipur,
Rajsthan, India
ABSTRACT
NP is the set of decision problems where the solution can be found in polynomial time by a
non-deterministic turing machine & can be verified in polynomial time by deterministic turing
machine. The hardest of NP problems are called NP-complete problems. Solving an NP complete
problem in deterministic way takes exponential time. Function optimization problems are a class of
NP-complete problems. Function optimization is the process of finding absolutely best values of the
variables so that value of an objective function becomes optimal. A genetic algorithm (GA) is a
search heuristic that mimics the process of natural evolution. Genetic algorithms belong to the larger
class of evolutionary algorithms (EA), which generate solutions using techniques such as inheritance,
mutation, selection, and crossover. Two most widely used models of genetic algorithm are Holland
model & Common model. Both these models have little difference & generally they work the same
way. In this work, we present performance and analysis of genetic algorithms for optimization of test
functions.
Keywords: Evolutionary Algorithms, Function Optimization, Genetic Algorithm, Global Minima.
I. INTRODUCTION
NP is the set of decision problems where the solution can be found in polynomial time by a
non-deterministic turing machine & can be verified in polynomial time by deterministic turing
machine. NP contains many important practical problems, the hardest of which are called NP-
complete problems. NP hard problems are the problems whose solutions can not even be verified in
polynomial time. Solving an NP problem in deterministic way takes exponential time which can be
too large beyond the human imagination such as like hundreds of thousands of years.
INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &
TECHNOLOGY (IJCET)
ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)
Volume 5, Issue 4, April (2014), pp. 202-209
© IAEME: www.iaeme.com/ijcet.asp
Journal Impact Factor (2014): 8.5328 (Calculated by GISI)
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IJCET
© I A E M E

203
II. FUNCTION OPTIMIZATION
Function optimization is the process of finding absolutely best values of the variables so that
value of an objective function becomes optimal. Global optimization is a process of finding the
absolutely best set of admissible conditions under specified constraints to achieve an objective,
assuming both are formulated in mathematical terms. Global optimization problems are a class of
NP-complete problems[3] so there is not a single algorithm that solves global optimization problems
in polynomial time. Optimization problems can be categorized in several categories depending on the
characteristics of problem [17]. Two general categories are continuous optimization and discrete
optimization depending upon variables of objective functions are continuous or discrete. Basically
function optimization problems are made up of following three parts.
• An objective function: It specifies the objective function for which optimization is required to
be performed. It includes minimization or maximization functions depending upon problem
such as to achieve maximum profit at the minimum cost in organization.
• A set of variables: It specifies all the variables which affect the value of the objective function.
In organization, the variables might include the amounts of different resources used or the time
spent on each activity.
• A set of constraints: It indicates the set of rules. The variables can take certain values and they
cannot take other values depending on the constraints. In the industry, we cannot have
unlimited resources or money, time spent on each activity cannot be negative.
Mathematical Formulation:
max or min F(x)
subject to x ∈ D where D={x : l <= x <= u}
subject to gj(x) <= 0, where j=1….J
• x ∈ Rn
: real n-vector of decision
• f: Rn
-> R : continuous objective function,
• D ⊂ Rn
: non-empty set of feasible decisions (a proper subset of Rn);
• l and u : explicit, finite lower and upper bounds on x,
• g : Rn
-> Rm
: finite collection of continuous constraint functions (J-vector).
The above shown model is called bounded, constraint optimization model. If the 1st
condition
is relaxed then it becomes unbounded means decision variables can take any value. Relaxation of 2nd
condition is known as unconstrained optimization.
III. GENETIC ALGORITHM
Nature has been great source of inspiration in the various fields of human life since ancient
age. Many inventions have been done as per the principals of natural phenomena and models. The
story in the computer field is not much different. Researchers are trying to develop intelligence
machines and to make them more and more intelligence since 1950s. Conventional deterministic
model of von-Neuman fails or gives poor performance in many real world applications like pattern
reorganization, classification, clustering, optimization process, design of complex model, etc… But
in all these applications bio inspired models of computation like artificial neural network, genetic

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algorithm, fuzzy logic, etc… work very well. John Holland along with his colleagues has developed
genetic algorithm at the University of Michigan during early 1960s [6]. Genetic algorithms are
probabilistic, robust and heuristic search algorithms premised on the evolutionary ideas of natural
selection and genetic. Charles Darwin had revealed the process of evolution in the nature during
1850s. According to evolution theory, each organism has to live in highly uncertain environment and
has to adapt to new conditions and constraints to survive. In the natural selection process, the fittest
one survives and others die off. Fittest organisms are selected for the mating purpose and they
produce new child by sexual recombination. Sometimes due to genes deficiency in an offspring, a
new child has some characteristics which are not present in the parents. So main aim of each living
organism is to survive, to mate and to produce as many offspring as possible. Genetic algorithm
follows the same natural phenomenon. More over solving any problem with genetic algorithm, it is
required to design different parameters and operators carefully [1][6]. Components of genetic
algorithm are described subsequently.
• Chromosomes: All living objects are different than other objects of the same type or different
types. These differences are due to genetic structure, which is called chromosomes.
Chromosomes or individuals are consisting of genes. Genes may contain different possible
values depending on the environment & constraints. The encoding process of solution as a
chromosome is most difficult aspect of solving any problem using genetic algorithm.
• Fitness Function: Fitness function is an evaluation function used to measure how good a
chromosome is. Fitness value is assigned to each chromosome by fitness function using their
genetic structure and relevant information of the chromosome. Fitness value plays big role
because subsequent genetic operators use fitness values to select chromosomes.
• Reproduction: During each successive generation, a proportion of the existing population is
selected to breed a new generation. Individual solutions are selected through a fitness-based
process. Reproduction methods are roulette wheel selection, tournament selection.
• Crossover or recombination works as per the principle of sexual recombination. In biological
systems, recombination is a complex process that occurs between male and female of same
type. Two chromosomes are physically aligned, breakage occurs at one or more location on
each chromosome and homologous chromosome fragments are exchanged before the breaks
are repaired. Same concept is also applied in the genetic algorithm. In general, crossover
operator recombines two chromosomes so it is also known as recombination. Crossover
methods are 1-point, n-point, uniform crossover.
• Mutation is a genetic operator used to maintain genetic diversity from one generation of a
population to the next. It is analogous to biological mutation. Mutation alters one or more gene
values in a chromosome from its initial state. Sometimes due to mutation, the solution may
change entirely from the previous solution.
IV. ENCODING
Encoding is the first step towards genetic algorithm. The structure of a solution vector in any
search problem depends on the problem characteristics. It may be possible that, in some problems a
solution is a single real value; in some problems it may be a real valued vector specifying dimensions
to the problem's parameters whereas in some other problems, a solution may be a strategy or an
algorithm for achieving a task. So encoding of solution as a chromosome is generally problem
dependent [2]. First the data is encoded with the help of some encoding technique. Then it is given to
genetic algorithm. Different types of encoding techniques are available such as binary encoding, gray
code encoding, decimal encoding, etc…

205
V. HOLLAND MODEL
This model is originally proposed by John Holland and widely used by many researchers [16]
[6]. In this model, crossover and mutation work independent of each other. First crossover is applied
on the mating pool and temporary population is created then mutation is applied. Crossover
probability Pc decides whether to perform crossover on two randomly selected chromosomes or to
copy them directly in the next generation population set. Mutation probability Pm is per gene
probability, it decides whether to perform mutation on particular gene or not. Generally crossover
probability is high like 0.95, 0.90, 0.8, even more. Mutation probability is commonly low, like 0.01,
0.02, 0.05.
Begin
gen = 0
Initialize P(gen)
While termination_condition not satisfied
Begin
Evaluate each chromosome in P(gen)
/* Reproduction */
for i = 1 to pop_size
select 1 chromosome and place it into mating pool M
/* Mating Pool M is created*/
/*Crossover*/
for i = 1 to (pop_size) / 2
apply crossover on randomly selected chromosomes from M
/*Temporary population C1 is created*/
/*Mutation*/
apply mutation on each chromosome of C1
gen = gen + 1
P(gen) = C2
End
End
Figure 1: Procedure of Holland Model
VI. COMMON MODEL
In Common model, Instead of applying crossover and mutation in sequence, either one is
applied according to probability. It may be possible that many times crossover is applied and then
mutation is applied, so first local evolution is done and then mutation is used to explore new points.
Mutation probability is same as Holland model, it is per gene probability.

206
Begin
gen = 0
Initialize P(gen)
While termination_condition not satisfied
Begin
Evaluate each chromosome in P(gen)
/* Reproduction */
select 1 chromosome and place it into mating pool M
/* Mating Pool M is created*/
temp = random(0,1)
if (temp <= Pc)
/*Crossover*/
for i = 1 to (pop_size) / 2
apply crossover on randomly selected chromosomes from M
else
/*Mutation*/
apply mutation on each chromosome of M
end if
gen = gen + 1
P(gen) = C1
End
End
Figure 2: Procedure of Common Model
VII. NEW MODEL
In this work, we propose new model of genetic algorithm. Holland model and Common
model has little difference. In new model there is no concept of selection or reproduction. Instead of
reproduction phase, it is better to give chance to entire population set to mate. We apply sorting
operator. Crossover is applied between ith
and i+1th
chromosomes in the population set. After
crossover, we apply mutation same way as Holland model and mutation probability is per gene
probability. After completing one generation, we choose chromosomes such a way that generation
gap is less than 1.

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VIII. SUCCESS RATIO ANALYSIS
Table 1: Models wise success ratio
Decimal Encoding
Holland Common New
1-Point Uniform 1-Point Uniform 1-Point Uniform
Lavy Best 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
S.R.% 100 100 100 100 100 100
Easom Best -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000
S.R.% 88 72 52 24 100 100
Figure 3: Success Ratio Chart
IX. CONVERGENCE ANALYSIS
Figure 4: Comparison of convergence rate between Holland model, Common model & new model
for Levy’s function
0
25
50
75
100
Levy Easom
SuccessRatio(%)
Test Problems
Holland Common New

208
Figure 5: Comparison of convergence rate between Holland model, Common model & new model
for Easom’s function
X. CONCLUSION
In this work, we have optimized Lavy & Easom’s function using Holland & Common model.
Then we have introduced a new model & optimized same functions using new model. If we look at
performance & convergence analysis of these three models then we can say that new model works
better than existing Holland & Common models.
XI. REFERENCES
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fundamentals. University Computing, 15(2): 58-69, 1933.
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[3] C.H. Papadimitriou and K. Steiglitz. Combinatorial Optimization. Prentice-Hall, 1982.
[4] D.E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Learning. Addision
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California, 1991. Morgan Kaufmann.
[6] J.H. Holland, Adaptation in natural and artificial systems. MIIT Press, second edition, 1992
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[12] D.B. Fogel. Evolutionary Computation: Toward a New Philosophy of Machine Intelligence.
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