Engineering Optimization: Theory and Practice, Fourth Edition

13.2 Genetic Algorithms 699

sizenduring numerical computations,nrandom numbers, each in the range of zero
to one, are generated (or chosen). By treating each random number as the cumulative
probability of the string to be copied to the mating pool,nstrings corresponding to then
random numbers are selected as members of the mating pool. By this process, the string
with a higher (lower) fitness value will be selected more (less) frequently to the mating
pool because it has a larger (smaller) range of cumulative probability. Thus strings with
high fitness values in the population, probabilistically, get more copies in the mating
pool. It is to be noted that no new strings are formed in the reproduction stage; only
the existing strings in the population get copied to the mating pool. The reproduction
stage ensures that highly fit individuals (strings) live and reproduce, and less fit indi-
viduals (strings) die. Thus the GAs simulate the principle of “survival-of-the-fittest”
of nature.

Example 13.2 Consider six strings with fitness values 12, 4, 16, 8, 36, and 24 with
the corresponding roulette wheel as shown in Fig. 13.1. Find the levels of contribution
of the various strings to the mating pool using the roulette-wheel selection process with
the following 12 random numbers: 0.41, 0.65, 0.42, 0.80, 0.67, 0.39, 0.63, 0.53, 0.86,
0.88, 0.75, 0.55.

Note:(1) These random numbers are taken from Ref. [13.20]. (2) Although the original
population consists of only 6 strings, the mating pool is assumed to be composed of
12 strings to illustrate the roulette-wheel selection process.

SOLUTION If the given random numbers are assumed to represent cumulative
probabilities, the string numbers to be copied to the mating pool can be determined
from the cumulative probability ranges listed in the last column of Table 13.1 as
follows:

Random number
(cumulative probability of
the string to be copied)

0.41 0.65 0.42 0.80 0.67 0.39 0.63 0.53 0.86 0.88 0.75 0.55

String number to be copied
to the mating pool

5 5 5 6 5 4 5 5 6 6 5 5

This indicates that the mating pool consists of 1 copy of string 4, 8 copies of string 5,
and 3 copies of string 6. This shows that less fit individuals (strings 1, 2, and 3) did
not contribute to the next generation (or died) because they could not contribute to the
mating pool. String 4, although has a small fitness value, contributed 1 copy to the
mating pool based on the random selection process used.

Crossover.After reproduction, the crossover operator is implemented. The purpose
of crossover is to create new strings by exchanging information among strings of the
mating pool. Many crossover operators have been used in the literature of GAs. In most
crossover operators, two individual strings (designs) are picked (or selected) at random
from the mating pool generated by the reproduction operator and some portions of the
strings are exchanged between the strings. In the commonly used process, known as a