NEW REAL CODED GENETIC ALGORITHMS FOR GLOBAL OPTIMIZATION

Abstract

In Real Coded Genetic Algorithms (RCGAs), crossover and mutation are considered to be the fundamental search operators. In the recent past, a lot of effort has been put into the development of sophisticated real coded crossover and mutation operators to improve the performance of RCGAs for function optimization. Hence, this thesis is based on the introduction of new crossover and mutation operators for RCGAs. Apart from that, comparative study of different combination of existing and new proposed crossover and mutation operators are presented. This thesis is computationally dominant and interdisciplinary in nature. The objectives of this thesis are: 1. To design efficient and reliable computational techniques for obtaining the global optimal solution of unconstrained/ constrained non-linear optimization problems. 2. To test the algorithms on test problems appearing in literature. 3. To use the algorithms for solving real life problems arising in various fields of science and engineering. Chapter 1 is introductory in nature. Besides stating the relevant definitions it presents the existing literature review. In Chapter 2, a new real coded crossover operator, called the Laplace Crossover (LX) is proposed. LX is used in conjunction with two well known mutation operators namely the Non-Uniform Mutation (NUM) (Michalewicz [1992]) and Makinen, Periaux and Toivanen Mutation (MPTM) (Makinen et al. [1999]) to define two new generational genetic algorithms LX-MPTM and LX-NUM. These two genetic algorithms are compared with two existing genetic algorithms (HX-MPTM (Maaranen [2004]) and HX-NUM (Meittinen et al [2003]) which comprise of Heuristic Crossover operator (HX) (Michalewicz [1992]) and the same two mutation operators. A set of 20 test problems available in the global optimization literature are used to test the performance of these four genetic algorithms. To judge the performance of the LX operator, two kinds of analysis is performed. Firstly, a pair-wise comparison is performed between LX-MPTM and HX-MPTM, and then between LX-NUM and HX-NUM. Secondly the overall comparison of performance of genetic algorithms is carried out based on a performance index (PI). In Chapter 3, a new real coded mutation operator called Power Mutation (PM) is introduced for RCGA. The performance of PM is compared with two other existing real coded mutation operators taken from literature namely: Non-Uniform Mutation (NUM) and Makinen, Periaux and Toivanen Mutation (MPTM). Using the various combinations of two crossovers (Laplace Crossover and Heuristic Crossover) and three mutation operators (the newly defined mutation, PM, NUM and MPTM) six generational RCGAs are compared on the same set (as above) of 20 benchmark global optimization test problems. Various performance criterions like computational cost, success rate, solution quality, efficiency and reliability are reported using two kinds of analysis. In Chapter 4, the algorithms developed in Chapters 2 and 3 are extended for obtaining global optimal solution of constrained optimization problems. In addition two new genetic algorithms are proposed which viz. LX-POL and SBX-PM which make use of Laplace Crossover with Polynomial Mutation and Simulated Binary Crossover with Power Mutation. Constraints are handled using the parameter less approach proposed by Deb [2004] and SBX-POL (Deb [2000] Deb and Agrawal [1995]) which originally uses parameter less approach is also included in the above set of algorithms. So, in total, nine RCGAs are used for comparative study. These are as follows LX-POL, LX-PM, LX-MPTM, LX-NUM HX-PM, HX-MPTM, HX-NUM SBX-POL, SBX-PM ii The performance of all the genetic algorithms is compared on a large set of constrained benchmark test problems. All the algorithms are compared based on three criteria: accuracy, efficiency and reliability using two kinds of analysis. In Chapter 5, the RCGAs developed in chapter 2 and 3 are applied to estimate the parameters of a dynamical system governed by a system of differential equations. Two dynamical systems are chosen and the problems are formulated as an unconstrained global optimization problem. The first model, Lotka-Volterra model, consists of six unknowns whereas the second model, which is large scale dynamical system, contains 22 unknown parameters. Both the models are solved using the real coded Genetic Algorithms developed in chapters 2 and 3. The numerical and graphical results are presented and discussed. In Chapter 6, a real life problem from the field of Mechanical Engineering (Ramji [2004]) is taken. The objective of this problem is to determine the optimum design of suspension system of three wheeled motor vehicles. The optimization process of vehicle suspension system involves: (a) Modeling and analyzing the vehicle behaviour. (b) Definition of the optimization objective, and the specifications for any proposed solution. (c) Choosing an appropriate methodology to satisfy the design objectives and (d) using mathematical programming techniques to optimize the vehicle and suspension parameters to satisfy one or more criteria. The primary objective of the three-wheeled vehicle suspension design is to provide sufficient vibration isolation in different directions due to road disturbances, so that the desired level of comfort for the driver and passengers is obtained. In general, vehicle suspension characteristics and other parameters (like mass, inertia and geometrical parameters) have been well known to affect the ride behaviour and are of interest to the vehicle dynamicist. The optimum values of parameters which minimize the discomfort of the driver and passengers are to be determined. . iii This problem is formulated as a multiobjective optimization problem and is solved by two widely used approaches viz: the Min-Max approach and Summation approach. These two methods have been applied for two popular three wheelers in Indian context, namely Bajaj vehicle and Vikram vehicle. The problem turns out to be 13 variables unconstrained optimization. The problem has been solved by using the new RCGAs developed in chapters 2 and 3 and the results obtained are compared with the existing results published. In Chapter 7 an attempt is made to solve a class of real life problems which has its origin in electrical power systems (Birla et al. [2006], Birla [2006]). The problem is to compute the values of the decision variables called "Relays", which control the act of isolation of faulty lines from the system without disturbing the healthy lines. In other words it is required to determine the optimal relay operating times. This problem can be modeled as a non-linear constrained optimization problem, in which the objective function to be minimized is the sum of the operating times of all the primary relays, which are expected to operate in order to clear the faults of their corresponding zones. The constraints are bounds on all decision variables, complexly interrelated times of the various relays (called selectivity constraints) and restrictions on each term of the objective function to be between certain specified limits. Finally, in Chapter 8, based on the present study, conclusions are drawn and future research work in this direction is suggested.