Engineering Optimization: Theory and Practice, Fourth Edition

Problems 735

(a)x(l)= 0 , x(u)= 5

(b)x(l)= 0 , x(u)= 10

(c)x(l)= 0 , x(u)= 20

13.4 A design variable, with lower and upper bounds 2 and 13, respectively, is to be repre-

sented with an accuracy of 0.02. Determine the size of the binary string to be used.

13.5 Find the minimum off=x^5 − 5 x^3 − 20 x+5 in the range (0, 3) using the ant colony

optimization method. Show detailed calculations for 2 iterations with 4 ants.

13.6 In the ACO method, the amounts of pheromone along the various arcs from nodei

are given byτij= 1 , 2 , 4 , 3 , 5 ,2 forj= 1 , 2 , 3 , 4 , 5 ,6, respectively. Find the arc (ij)

chosen by an ant based on the roulette-wheel selection process based on the random

numberr= 0 .4921.

13.7 Solve Example 13.5 by neglecting pheromone evaporation. Show the calculations for 2

iterations.

13.8 Find the maximum of the functionf= −x^5 + 5 x^3 + 20 x−5 in the range− 4 ≤x≤ 4

using the PSO method. Use 4 particles with the initial positionsx 1 = − 2 , x 2 = 0 ,

x 3 =1, andx 4 =3. Show detailed calculations for 2 iterations.

13.9 Solve Example 13.4 using the inertia term whenθvaries linearly from 0.9 to 0.4 in

Eq. (13.23).

13.10 Find the minimum of the following function using simulated annealing:

f (X)= 6 x^21 − 6 x 1 x 2 + 2 x^22 −x 1 − 2 x 2

Assume suitable parameters and show detailed calculations for 2 iterations.

13.11 Consider the following function for maximization using simulated annealing:f (x)=
x( 1. 5 −x)in the range (0, 5). If the initial point isx(^0 )= 2 .0, generate a neighboring
point using a uniformly distributed random number in the range (0, 1). If the temperature
is 400, find the pbobability of accepting the neighboring point.

13.12 The population of binary strings in a maximization problem is given below:

String Fitness

0 0 1 1 0 0 8

0 1 0 1 0 1 12

1 0 1 0 1 1 6

1 1 0 0 0 1 2

0 0 0 1 0 0 18

1 0 0 0 0 0 9

0 1 0 1 0 0 10

Determine the expected number of copies of the best string in the above population in

the mating pool using the roulette-wheel selection process.

13.13 Consider the following constrained optimization problem:

Minimizef=x^31 − 6 x^21 + 11 x 1 +x 3