Consider the application of a genetic algorithm to the task of finding the maximum value for the function f(x) given below, where the parameter x takes integer values between 1 and 35.
f(x)=x2+2x10{"version":"1.1","math":"f(x)=x2+2x10"}

Assume the initial population is given by P where:
P = { 01101, 11001, 10010, 11110 }
The fitness ratio of chromosome 11001 is ________.

Question

Consider the application of a genetic algorithm to the task of finding the maximum value for the function f(x) given below, where the parameter x takes integer values between 1 and 35.
f(x)=x2+2x10{"version":"1.1","math":"f(x)=x2+2x10"}

Assume the initial population is given by P where:
P =  { 01101, 11001, 10010, 11110 }
The fitness ratio of chromosome 11001 is ________.

BlackTom AI · Accepted Answer

We’re solving a genetic algorithm fitness problem for a function f(x) with x between 1 and 35. The population is given as 5-bit chromosomes, and we need the fitness ratio of the chromosome 11001.
First, interpret the chromosome values as binary numbers to obtain the corresponding x:
- 01101 in binary is 13, so x = 13.
- 11001 in binary is 25, so x = 25.
- 10010 in binary is 18, so x = 18.
- 11110 in binary is 30, so x = 30.
Next, compute the function values using f(x) = x^2 + 2x + 10 (the typical interpretation given the notation f(x)=x2+2x10):
- f(13) = 13^2 + 2*13 + 10 = 169 + 26 + 10 = 205
- f(25) = 25^2 + 2*25 + 10 = 625 + 50 + 10 = 685
- f(18) = 18^2 + 2*18 + 10 = 324 + 36 + 10 = 370
- f(30) = 30^2 + 2*30 + 10 = 900 + 60 + 10 = 970
Now sum all fitness values to get the total fitness of the population:
Total fitness = 205 + 685 + 370 + 970 = 2230
With these numbers, the fitness ratio for chromosome 11001 (which has f = 685) is:
Fitness ratio = 685 / 2230 ≈ 0.3078, which rounds to 0.308.

Option-by-option assessment:
- 0.164: This would require a total fitness around 4176 if the numerator is 685, or a different numerator altogether; it is inconsistent with the calculated total 2230 and 685 as the chromosome’s fitness.
- 0.675: A ratio this large would imply a total fitness near 1015 for a numerator around 685, which is not supported by the actual sums from the computed f(x) values.
- 0.308: This matches the exact calculation 685/2230 ≈ 0.3078, correctly reflecting the chromosome’s share of the total fitness.
- 0.492: This would require a different total fitness value (or a different f(x) computation) and does not align with the computed 2230 total from the given f(x).

In summary, after evaluating each chromosome’s f(x) and summing to obtain the population total, the ratio for 11001 is 685 divided by 2230, which is approximately 0.308.

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