After developing the optimization problem and adapting the code into the problem, the code has been run for several times for various combinations of genetic algorithms operators to obtain the optimal combinations of operators that will be used in the study.
First of all, generation number has been changed for various xed population numbers ranging between 500 to 10000 and the results are shown in Figure 4.4. Other parameters are used as the default values of the code. It is seen from the gure that, increasing the generation number has a positive eect and decreases the tness value. However, when the generation number exceeds 200, for all populations considered, there exists no improvement in the tness value. Therefore, it is decided that taking generation number as 200 will be enough to get the optimum tness value and also it will be more practical rather than dealing with higher generation numbers for the problem under consideration.
It can also be seen from Figure 4.4 that, the minimum tness value is obtained when population number is equal to 9000. This can be seen more clearly, if the tness versus population is drawn for each generation number used, which is shown in Figure 4.5. This
gure shows the results of the same runs of Figure 4.4, having population range between 500 and 10000 and generation number range of 10 to 1000. Unfortunately, there exists no asymptotic behavior as in Figure 4.4, and the optimum results are obtained when population number equals to 9000. When the population number increases to 9500 slightly higher value of tness value is obtained. Therefore it is decided to take population number as 9000 for
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Figure 4.4: Eect of Generation Number on Fitness
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0 2000 4000 6000 8000 10000 12000
Population
Figure 4.5: Eect of Population Number on Fitness
the further runs.
While optimum generation and population numbers are obtained, the code is run for the default values of the remaining parameters and also limitation to the values of CW has not been considered yet. Therefore, the minimum tness value obtained from these runs are smaller than the following runs. The parameters found here is used for the following runs and after all parameters are found, these will be checked once more. It can be seen at the end of this Section.
Then, the eect of cross-over type on the tness is observed. From this observation till the end of the study, the decision variables are limited as explained in Subsection 3.3.1.
As mentioned in Chapter 2, two cross-over types exists as SBX-η and BLX-α, and in the code used, η can take any number but α is de ned as 0.5. Therefore, the runs are made for SBX-η and BLX-0.5. The runs are made for these two types for three dierent population numbers which give the smaller tness values on previous runs. For generation number selection, since a relationship between generation number and tness function is obtained in previous runs, trying dierent generation numbers is not required and the selected generation number (200) is used. Figure 4.6 is obtained, as a result of these runs. The eect of change of η on tness value is observed from the gure. The runs made for same combinations for BLX-0.5, gives tness values higher than the minimum of the one obtained by choosing sbx
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Figure 4.6: Eect of η on Fitness
type of cross-over. Therefore, blx type of cross-over excluded from the runs. SBX component of 2 will be in use as it gives the minimum tness value. Also, the minimum is obtained when the population number is chosen as 9000, as can be seen from Figure 4.6.
Subsequently, the in uence of cross-over rate on tness is investigated. As seen from Figure 4.7, the minimum tness value is obtained when cross-over ratio equals to 0.75. For this set of runs, population number, generation number and sbx component are chosen as 9000, 200 and 2, respectively.
Afterwards, the eect of mutation ratio on tness value is investigated. For that purpose, by using the selected values for the operators observed previously, the code is run for various mutation ratio ranging from 0.01 to 0.15. Figure 4.8 is gathered from the study. It is seen from the gure that, for mutation number equals to 0.06, the tness value is minimum.
Therefore, this value is selected to be in use for the study.
As mentioned in Chapter 2, tournament selection operator is recommended for mini-mization problems. Therefore, it is selected as the selection operator of this problem. To decide on the tournament size, code runs are made to get the relation between tournament size and tness function value. The resultant gure is presented in Figure 4.9. It is obtained from the gure that, for small values of tournament size, relatively small tness values are obtained and the minimum is obtained when tournament size equals to 3.
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0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
X-over ratio
Fitness
Figure 4.7: Eect of Crossover Ratio on Fitness
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Mutation Ratio
Fitness
Figure 4.8: Eect of Mutation Ratio on Fitness
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2 3 4 5 6 7 8 9 10 11 12
Tournament Size
Fitness
Figure 4.9: Eect of Tournament Size on Fitness
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0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Population
Fitness
Figure 4.10: Check for the Selected Population Number
Finally, as a result of the studies to get the appropriate genetic algorithms operators and the numerical values of them, the operators are chosen as shown below;
Generation Number = 200, Population Number = 9000, Cross-over Type : sbx-2, Cross-over Ratio = 0.75, Mutation Ratio = 0.06,
Selection Operator : Tournament selection, Tournament Size = 3.
Then nally, the check for the generation and population numbers are done. As stated above, generation number change does not have a big in uence on tness after generation number exceeds 200. Therefore, for the following runs generation number is xed on 200 and the in uence of population number change on tness is observed, as can be seen on Figure 4.10. Again, for the Genetic Algorithm operators and the numerical values used, the optimum result is obtained when population is equal to 9000.