734 Modern Methods of Optimization
(b)How is an inequality constrained optimization problem converted into an uncon-
strained problem for use in GAs?
(c)What is the difference between a crisp set and a fuzzy set?
(d)How is the output of a neuron described commonly?
(e)What are the basic operations used in GAs?
(f)What is a fitness function in GAs?
(g)Can you consider SA as a zeroth-order search method?
(h)How do you select the length of the binary string to represent a design variable?
(i)Construct the objective function to be used in GAs for a minimization problem with
mixed equality and inequality constraints.
(j)How is the crossover operation performed in GAs?
(k)What is the purpose of mutation? How is it implemented in GAs?
(l)What is the physical basis of SA?
(m)What is metropolis criterion and where is it used?
(n)What is a neural network?
(o)How is a neuron modeled in neural-network-based models?
(p)What is a sigmoid function?
(q)How is the error in the output minimized during network training?
(r)What is the difference between a random quantity and a fuzzy quantity?
(s)Give two examples of design parameters that can be considered as fuzzy.
(t)What is a valuation set?
(u)What is the significance of membership function?
(v)Define the union of two fuzzy setsAandB?
(w)How is the intersection of two fuzzy setsAandBdefined?
(x)Show the complement of a fuzzy set in a Venn diagram.
(y)How is the optimum solution defined in a fuzzy environment?
(z)How is the fuzzy feasible domain defined for a problem with inequality constraints?Problems
13.1 Consider the following two strings denoting the vectorsX 1 andX 2 :X 1 :{
1 0 0 0 1 0 1 1 0 1}X 2 :{
0 1 1 1 1 1 0 1 1 0}Find the result of crossover at location 2. Also, determine the decimal values of the
variables before and after crossover if each string denotes a vector of two variables.
13.2 Two discrete fuzzy sets,AandBare defined as follows:A={
( 60 , 0. 1 ) ( 62 , 0. 5 ) ( 64 , 0. 7 ) ( 66 , 0. 9 ) ( 68 , 1. 0 ) ( 70 , 0. 8 )}B={
( 60 , 0. 0 ) ( 62 , 0. 2 ) ( 64 , 0. 4 ) ( 66 , 0. 8 ) ( 68 , 0. 9 ) ( 70 , 1. 0 )}Determine the union and intersection of these sets.
13.3 Determine the size of the binary string to be used to achieve an accuracy of 0.01 for a
design variable with the following bounds: