This document presents a parametric study aimed at enhancing the performance of genetic algorithms (GA) using a ranked-based roulette wheel selection method. The study investigates the effects of varying population size, mutation rate, and crossover rate on the GA's effectiveness in function optimization, revealing that optimal parameter settings significantly improve solution quality and algorithm reliability. The findings indicate that specific ranges for population size, crossover rate above 0.8, and mutation rate between 0.0035-0.0065 are crucial for achieving better results in solving complex optimization problems.

1. Parametric Study to Enhance Genetic Algorithm's Performance using Ranked based Roulette Wheel Selection method Omar Al Jadaan Dr. C.R. Rao Prof. Lakishmi Rajamani

2. Abstract Selection operator is one of the important aspects in increasing the performance of the GA. There are several ways for selection. Some of these are: Tournament selection, Ranking selection, and Proportional selection. There are many ways for proportional selection. The most popular are Roulette Wheel Selection (RWS), Stochastic Reminder Roulette Wheel Selection (SRRWS), and Stochastic Universal Sampling (SUS). Previous work, using Ranked based Roulette Wheel Selection (RRWS) genetic operator in the domain of function optimization has shown an increase in the gain of resources, reliability and diversity of the population; and a decrease the uncertainty of selection process. Both these consequences are meant to avoid the premature convergence of the population. Furthermore, the solutions obtained have been, in general, superior to the solutions achieved by the GA with standard Roulette Wheel Selection. This paper, presents an extensive empirical study carried out to determine the best parameter settings to be used with RRWS with a view to enhancing the GA's performance. These parameters include the population size, the mutation and crossover rates.

3. RRWS Selection Method The selection operator involves randomly chosen members of the population to enter a mating pool. The operator is carefully formulated to ensure that better members of the population (with higher fitness) have a greater probability of being selected for mating, but that worse members of the population still have a small probability of being selected. Having some probability of choosing worse members is important to ensure that the search process is global and does not simply converge to the nearest local optimum. There are several ways for the selection: some of these are Tournament selection, Ranking selection, and Proportional selection. Refer to [3] for more details Roulette Wheel Selection Method.

4. Ranked Based Roulette Wheel Ranked Based Wheel Selection uses modified roulette wheel selection algorithm where each individual is assigned a fitness value equal to its rank in the population: the highest rank has the highest probability to be selected. The probability is calculated as illustrated in the following equation:

5. Previous Works The selection genetic operator is tested in function optimization domain. The chosen functions are eight well-known functions from the GA literature. The goals are function optimization (minimization). Roulette wheel selection is easy to implement and mimics nature more faithfully and therefore is much more appealing. But it is slower than the ranked based roulette wheel selection in convergence to near the optimum solution. If good solution is discovered early, its fitness value dominates other fitness values. Then it will occupy majority portions of the mating pool. This will reduce the diversity in the mating pool and cause the GAs to converge to wrong solutions. Ranked roulette wheel selection overcomes this problem and increases the diversity. RRWS is outperforming the conventional RWS in convergence, time, reliability, certainty, and more robustness .

6. Experimental settings The GA is implemented with ranked based roulette-wheel selection, elite of size 2, single point crossover, and 10 variables, each variable encoded in 10 bits for 200 generations. The remaining parameters are changed to study their influence on the GA's performance while using RRWS mechanism. First the effect of the population sizes from 50-500 with step of 50 is analyzed. In previous work the population size is fixed to 140 (1.4 * variable numbers (10) * number of bits (10)), and fixing the remaining parameters, mutation rate to 0.01, crossover rate to 1.0. Second the effect of the crossover rate from 0.1-1.0 with step of 0.1 is analyzed. Whereas In previous work the crossover rate is fixed to 1.0, and by fixing the remaining parameters, mutation rate to 0.01, population size to 250. Finally the effect of the mutation rate from 0.0005-0.0095 with step of 0.001 is analyzed. In previous work the mutation rate is fixed to 0.1, and by fixing the remaining parameters, crossover rate to 1.0, population size to 250.

7. Results Table1. Population parameter results.

11. Discussion Population size The variation of this parameter has a great influence on the obtained results. Medium sizes always give better results. Table 1 shows the starting hit to the optimum value from the best solutions for different population sizes for each test function. objfun6: The best values graph does not hit the optimum value with popSize=50, but it hit it at generation 38 when the popSize=300 The average values graph is fluctuating with popSize=50, but the fluctuation is decreased with popSize=400. The average graph becomes closer to the best value graph with popSize=400. From the above observations popSize in the range of 300-400 is enough to solve this test function. From the overall results the population size 300 is recommended for optimizing functions and 350 population sizes can be used for difficult functions like objfun6, objfun7.

12. Crossover The variation of this parameter has great influence on the obtained results. A high rate (0.8) always gives better results. Table 2 shows the starting hit to the optimum value from the best solutions for different crossover rates for each test function. objfun6: This function is difficult so the GA does not hit the optimum value with this set of parameters, but it was close to it. objfun7: The best values graph starts hitting the optimum value with 0.5 crossover rate at generation 60, but it decreases to generation 26 with 0.9 crossover rate. The average values graph becomes closer to the best values graph with crossover rate 0.7 and the fluctuation is decreased with 0.8-1.0 crossover rates. From the above observations crossover rate in the range of 0.6-1.0 is enough to solve this test function, and reach the optimum solution. From the overall results the recommended crossover rate is 0.8 for optimizing functions and 0.9 rate can be used for difficult functions like objfun6, objfun7.

13. Mutation The mutation rate is changed from 0.0005 to 0.0095. In all test functions mutation rate in the range 0.0005-0.0035 allows the GA to achieve the best performances. As the mutation rate increases, the results become worst table 3 illustrates the starting hit of the optimal value. objfun6: The best values graph starts to hit the optimum value at generation 55 with 0.0095 mutation rate, and decreased to generation 30 with 0.0015 mutation rate. The average values graph becomes closer to the best values graph with 0.0055 mutation rate. From the above observations the mutation rate in the range 0.0035-`0.0065 is sufficient. objfun7: The best values graph starts to hit the optimum value at generation 23 with 0.0095 mutation rate, and decreased to generation 18 with 0.0005 mutation rate. The average values graph becomes closer to the best values graph with 0.0035 mutation rate. From the above observations the mutation rate in the range of 0.0035-0.0065 is sufficient. From the overall results the recommended mutation rate is 0.0035 for optimizing functions and 0.0065 rate can be used for difficult functions like objfun6, objfun7

14. Global Influences of the Parameters The right choice of parameters values is an important issue when using ranked roulette wheel selection in the GA. All the parameters influence the results in an expressive way. We run the GA with a new set of parameters and with the old set of parameters and compare the obtained results. The following table shows the values of the parameters derived from the experiments, and reports the optimization results. It is obvious; that choosing the appropriate values of the parameters increases the performance of the GA. From table 4, choosing the right GA parameters within pre-given criteria, improve the performance of the GA in the optimization of the test functions, and these results are quite better than the results achieved in our initial work, where the parameters are chosen based only in a small set of experiments. The average improvement in performance is 25%. If we compare the new results with the ones obtained by the RRWSGA, we can conclude that choosing the correct values of the parameters, makes the GA reaches the optimum solution faster.

15. Conclusions Using ranked based roulette wheel selection operator in the GA shows that RRWS is capable to increase the gain of resources, reliability and diversity; and a decrease in the uncertainty in selection process during the entire evolutionary process. But the choice of the parameters to run the GA with RRWS is always made without any strongly supported criteria. In this paper extensive parametric study carried out to enhance the GA's performance when using RRWS. In this study three parameters varied: the population size, and the mutation and crossover rates. The results show that the right choices of these parameters influence the GA performance. In fact, using an appropriate parameter settings of the GA achieve much better solutions than the ones obtained with the old set of parameters. All the parameters had great influence in the obtained results. As the mutation rate increases results become worst. Studies involving the population's size indicate that medium sizes are sufficient to reach the optimum global solution. Concerning the crossover rate, value of 0.8 allows the RRWSGA to achieve the best solutions. In fact, the algorithm is able of reach the optimum global solution faster and steadier with the application of RRWS. The population size must be chosen in the appropriate range in optimization of the test functions, crossover rate superior to 80% and mutation rate in the range of 0.0035-0.0065 are the correct choices in the optimization of test functions. By combining all those parameters and choosing the correct values obtained results are better than the ones achieved previously by the RRWSGA.

16. objfun7(population size)

26. objfun7(Crossover)

36. objfun7(Mutation)

46. popSize=300 Crossover=0.8 Mutation=.0.0035 objfun7(after applying our results)

47. References AL Jadaan, O. Rao, C. R. Rajamani, L. , Ranked based Roulette Wheel Selection Method, ISRAMA2005,Calcutta mathematical society, 2005 (In press). Holland, J. H., Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975. Goldberg, D. E., Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, New York, NY, 1989.