Discrete Particles Swarm Optimization Algorithm For Uniform Design Construction

In our day to day endeavors we perform experiments in all the fields in our lives,either to find new ideas or to develop on the already existing ideas.An experirent can be defined as a test or series of tests in which purposeful changes are made to the input variables of a process or system so that we may observe,identify,analyze and make objective conclusions that are valid about the changes that may be observed in the output response.Experimental designs are the tools for planning experiments in order to economically obtain reliable information and valid conclusions.An experiment may involve one or more factors with the factors having different levels.An experiment where all the possible level combinations are considered is called a full factorial experiment.In such a case the total number of runs is equal to product sum of thenumber of levels of the individual factors = ?i=1 s=qi where s is the number of factors and qiis the number of levels of factori.With the increase in number of factors and the factor levels.the value of N results to be very large such that it sometimes become difficult to run all the possible combinations.The experimenter in this case has to decide which level combinations are to be used in the experiment.This has led to many types of fractional factorial experiments.A good space filling design is one which points are uniformly scattered throughout the experimental region.Examples of such designs include Latin hypercube designs,minimax and maximin distance designs,randomized orthogonal arrays,digital net designs and uniform designs.Discrepancy is a most popular used measurement of uniformity.A design that has the smallest discrepancy among all the possible designs is called the uniform design.The process of obtaining the design with the smallest discrepancy out of the many is what is called optimization process.The major challenge in uniform design is to generate the design with the lowest measure of uniformity especially when the number of factors and the number of run increases since the number of feasible designs increases polynomiously.The importance of uniform designs surpasses its generation optimization challenges hence making it important to come up with ideas on how to overcome the optimization problem.In this thesis we have studied construction of symmetric u-type designs(all the s,factors have the same number of levels,qi for i=1,2,...,s for qi=n or qi<n and a divisor of n)and the asymmetric u-type designs(some or all the factors have unequal number of levels).We have used the DPSO to optimize the designs.Via this optimization algorithm,we have obtained designs with much lower discrepancy than currently used design.It means DPSO may be another good alternative algorithm for uniform design searching to some other giant optimization methods like TA.Chapter one:In this chapter,we have given a brief introduction of experiments and the importance of statistical design of experiments.We have gone further to discuss space filling design and have concentrated on the uniform design as an example.The main reason why we considered the topic our area of research is given in the section titled " justification of study".Our goals to achieve by the end of the study are made clear in the section of objectives.Chapter two:In this chapter,some basic definitions related to experimental design are reviewed.The basic principles of the experimental designs and the discrepancy as a measure of uniformity are stated.We have also looked at what is optimization and different powerful methods used for optimization.Chapter three:In this chapter,Discrete Particle Swarm Optimization method is discussed in details under its algorithm,the flow chart and the functions we have used to implement it.Different methodology have been applied on how to generate the initial designs for the swarm and the learning process for the particles have also been implemented differently for different swarm.Chapter four:In this chapter,the numerical results are given showing how the parameters such as number of matrices,number of iterations and the number of elements changed in a single iteration in the moves used for the discrete particle swarm optimization affects the results for the swarm.We have gone further and given some comparison result of the optimization method with other methods such as the threshold accepting to prove that DPSO is a strong optimization method and therefore have gone further to use this method to generate uniform designs under the mixture discrepancy.Chapter five:In this chapter,we have revisited the designs generated in the previous chapter and we are able to show that DPSO is more powerful than the TA in terms of generating better designs that possess lower discrepancies and in lesser time and have therefore gone further to generate new uniform designs under the mixture discrepancy.On the other hand we have tried to look at the points that we did not achieve yet,for example the time consumed in optimization process can further be reduced by considering parallel R programming which we still need to research on.