Engineering Optimization: Theory and Practice, Fourth Edition

698 Modern Methods of Optimization

Roulette wheel

Fitness

values

3

2

1

6

5

4 Pointer

16%

8% 36%

24%

4% 12%

String

numbers

Figure 13.1 Roulette-wheel selection scheme.

In Fig. 13.1, the population size is assumed to be 6 with fitness values of the strings

1, 2, 3, 4, 5, and 6 given by 12, 4, 16, 8, 36, and 24, respectively. Since the fifth

string (individual) has the highest value, it is expected to be selected most of the time

(36% of the time, probabilistically) when the roulette wheel is spunntimes (n=6 in

Fig. 13.1). The selection scheme, based on the spinning of the roulette wheel, can be

implemented numerically during computations as follows.

The probabilities of selecting different strings based on their fitness values are

calculated using Eq. (13.11). These probabilities are used to determine the cumulative

probability of stringibeing copied to the mating pool,Pi, by adding the individual

probabilities of strings 1 throughias

Pi=

∑i

j= 1

pj (13.13)

Thus the roulette-wheel selection process can be implemented by associating the cumu-

lative probability range(Pi− 1 −Pi) o thet ith string. To generate the mating pool of

Table 13.1 Roulette-Wheel Selection Process for Obtaining the Mating Pool

Probability of Cumulative Range of

selecting stringi probability value cumulative

for the mating of stringi, probability of

String numberi Fitness valueFi pool,pi Pi=

∑i

j= 1

pj stringi, (Pi− 1 , Pi)

1 12 0.12 0.12 0.00–0.12

2 4 0.04 0.16 0.12–0.16

3 16 0.16 0.32 0.16–0.32

4 8 0.08 0.40 0.32–0.40

5 36 0.36 0.76 0.40–0.76

6 24 0.24 1.00 0.76–1.00