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  • 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) www.jifactor.com IJCET © I A E M E
  • 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 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
  • 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 204 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…
  • 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 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*/ for i = 1 to pop_size apply mutation on each chromosome of C1 /*Temporary population C2 is created*/ 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.
  • 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 206 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*/ 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*/ for i = 1 to pop_size apply mutation on each chromosome of M end if /*Temporary population C1 is created*/ 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.
  • 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 207 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
  • 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 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 [1] D. Beasley, R.B. David and R.R. Martin. An overview of genetic algorithms: Part 1, fundamentals. University Computing, 15(2): 58-69, 1933. [2] D. Beasley, R.B. David and R.R. Martin. An overview of genetic algorithms: Part 2, research topics. University Computing, 15(4): 170-181, 1933. [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 – Wesley, 1989. [5] D.E. Goldberg and K. Deb. A Comparative analysis of selection schemes used in genetic algorithms. In G. J. E. Rawlins, editor, Foundations of Genetic Algorithms, pages 69-93, California, 1991. Morgan Kaufmann. [6] J.H. Holland, Adaptation in natural and artificial systems. MIIT Press, second edition, 1992 [7] H. Muhlenbein. How Genetic algorithms really work: I. mutation and hillclimbing. In R. Manner and B. Manderick, editors, Problem Solving from Nature – PPSN II, pages 15-25, Amsterdam, 1992. [8] K.A. De Jong and J. Sharma. Generation gaps revisited. In Darrell Whiteley, editor, Foundations of Genetic Algorithms 2, pages 19-28. Morgan Kauffmann, 1992. [9] K.A. De Jong. And W.M. Spears. A formal analysis of the role of multi-point crossover in genetic algorithms. Annals of mathematics and artificial intelligence, 5:1-26, 1992. [10] K.A. De Jong. Genetick Algorithms are not function optimizers. In L. Darrell Whiteley, editor, Foundations of Genetic Algorithms 2, pages 5-17, San Mateo, CA, 1993. Morgan Kaufmann. [11] Z. Michalewicz. Genetic Algorithms + Data Structures = Evolution Programs. Springer- Verlag, Berlin, second edition, 1994.
  • 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 209 [12] D.B. Fogel. Evolutionary Computation: Toward a New Philosophy of Machine Intelligence. IEEE Press, 1995. [13] T. Back, D. Fogel, Z. Michalewicz and S. Pidgeon, editors. Handbook of Evolutionary Computation. Oxford University Press, 1997. [14] L.D. Chambers, editor. Practical Handbook of Genetic Algorithms, volume 3, Complex Coding Systems. CRC Press, Boca Raton, 1999. [15] M. Gen and R. Cheng. Genetic Algorithms and Engineering Optimization. Engineering Design and Automation. Wiley Interscience Publication, John Wiley & Sons. Inc., New York, 2000. [16] M. Mitchell. An Introduction to Genetic Algorithms. Prentice-Hall, New Delhi, India, 2002. [17] P.M. Pardalos and E. Romeijn, editors. Handbook of Global Optimization – Volume 2: Heuristic Approaches. Kluwer Academic Publishers, 2002. [18] T.P. Patalia, Dr. G.R. Kulkarni, Behavioral Analysis of Genetic Algorithm for Function Optimization, published at IEEE International Conference, Coimbatore, 2010. [19] Meera Kapoor, Vaishali Wadhwa, Optimization of DE Jong’s Function Using Genetic Algorithm Approach, IJARECE, Volume 1, Issue 1, July 2012. [20] Kapil Juneja, Nasib Singh Gill, Optimization of Dejong Function using GA under Different Selection Algorithms, International Journal of Computer Applications (0975 – 8887) Volume 64– No.7, February 2013. [21] Sugandhi Midha, “Comparative Study of Remote Sensing Data Based on Genetic Algorithm”, International journal of Computer Engineering & Technology (IJCET), Volume 5, Issue 1, 2014, pp. 141 - 152, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375. [22] Rakesh Kumar, Dr Piush Verma and Dr Yaduvir Singh, “A Review and Comparison of Manet Protocols with Secure Routing Scheme Developed using Evolutionary Algorithms”, International Journal of Computer Engineering & Technology (IJCET), Volume 3, Issue 2, 2012, pp. 167 - 180, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375.