References and Bibliography 731
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Analysis and Optimization, San Francisco, Sept. 24-26, 1990.
S. S. Rao, T. S. Pan, and V. B. Venkayya, Optimal placement of actuators in actively controlled structures using genetic algorithms, AIAA Journal , Vol. 29, No. 6, pp. 942-943, 1991.
P. Hajela, Genetic search: an approach to the nonconvex optimization problem, AIAA
Journal , Vol. 26, No. 7, pp. 1205-1210, 1990.
P. Hajela and C. Y. Lin, Genetic search strategies in multicriterion optimal design,
Structural Optimization , Vol. 4, pp. 99-107, 1992.
D. E. Goldberg, Computer-aided pipeline operation using genetic algorithms and rule learning, Part I: Genetic algorithms in pipeline optimization, Engineering with Comput- ers , Vol. 3, pp. 35-45, 1987.
D. E. Goldberg and C. H. Kuo, Genetic algorithms in pipeline optimization, ASCE
Journal of Computing in Civil Engineering , Vol. 1, No. 2, pp. 128-141, 1987.
C. Y. Lin and P. Hajela, Genetic algorithms in optimization problems with dis- crete and integer design variables, Engineering Optimization , Vol. 19, pp. 309-327,
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Springer-Verlag, Berlin, 1994.
B. Hajek, Cooling schedules for optimal annealing, Mathematics of Operations
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Science , Vol. 220, pp. 671-680, 1983.
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D. Reidel, Boston, 1987.
A. Corana, M. Marchesi, C. Martini, and S. Ridella, Minimizing multimodal functions of continuous variables with the simulated annealing algorithm, ACM Transactions on
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G.-S. Chen, R. J. Bruno, and M. Salama, Optimal placement of active/passive members in truss structures using simulated annealing, AIAA Journal , Vol. 29, pp. 1327-1334,
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M. Atiqullah and S. S. Rao, Parallel processing in optimal structural design using simulated annealing, AIAA Journal , Vol. 33, pp. 2386-2392, 1995.
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K. M. Ragsdell and D. T. Phillips, Optimal design of a class of welded structure using geometric programming, ASME Journal of Engineering for Industry , Vol. 98, No. 3, pp. 1021-1025, 1976.
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732 Modern Methods of Optimization
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13.32
13.33
13.34
13.35
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A. K. Dhingra and S. S. Rao, A neural network based approach to mechanical design optimization, Engineering Optimization , Vol. 20, pp. 187-203, 1992.
L. Berke and P. Hajela, Applications of artificial neural nets in structural mechanics,
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A. K. Dhingra, S. S. Rao, and V. Kumar, Nonlinear membership functions in the fuzzy optimization of mechanical and structural systems, AIAA Journal , Vol. 30, No. 1, pp. 251-260, 1992.
A. K. Dhingra and S. S. Rao, An integrated kinematic-kinetostatic optimal design of planar mechanisms using fuzzy theories, ASME Journal of Mechanical Design , Vol. 113, pp. 306-311, 1991.
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13.1 Define the following terms:
(a) Fuzzy parameter
(b) Annealing

Review Questions 733
(c) Roulette wheel selection process
(d) Pheromone evaporation rate
(e) Neural network
(f) Fuzzy feasible domain
(g) Membership function
13.2
(h) Multilayer feedforward network
Match the following terms:
(a) Fuzzy optimization
(b) Genetic algorithms
(c) Neural network method
(d) Simulated annealing
(e) Particle swarm optimization
(f) Ant colony optimization
Based on shortest path
Analysis equations not programmed
Linguistic data can be used
Based on the behavior of a flock of birds
Based on principle of survival of the fittest
Based on cooling of heated solids
13.3 Answer true or false:
(a) GAs can be used to solve problems with continuous design variables.
(b) GAs do not require derivatives of the objective function.
(c) Crossover involves swapping of the binary digits between two strings.
(d) Mutation operator is used to produce offsprings.
(e) No new strings are formed in the reproduction stage in GAs.
(f) Simulated annealing can be used to solve only discrete optimization problems.
(g) Particle swarm optimization is based on cognitive and social learning rates of groups of birds.
(h) Particle swarm optimization method uses the positions and velocities of particles.
(i) Genetic algorithms basically maximize an unconstrained function.
(j) Simulated annealing basically solves an unconstrained optimization problem.
(k) GAs seek to find a better design point from a trial design point.
(l) GAs can solve a discrete optimization problem with no additional effort.
(m) SA is a type of random search technique.
(n) GAs and SA can find the global minimum with high probability.
(o) GAs are zeroth-order methods.
(p) Discrete variables need not be represented as binary strings in GAs.
(q) SA will find a local minimum if the feasible space is nonconvex.
(r) The expressions relating the input and output are to be programmed in neural- network-based methods.
(s) Several networks architectures can be used in neural-network-based optimization.
(t) A fuzzy quantity is same as a random quantity.
13.4
(u) Ant colony optimization solves only discrete optimization problems.
(v) Fuzzy optimization involves the maximization of the intersection of the objective function and feasible domain.
Give brief answers:
(a) What is Boltzmann's probability distribution?

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 sets A and B ?
(w) How is the intersection of two fuzzy sets A and B defined?
(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?
13.1 Consider the following two strings denoting the vectors X
1 and X
2
:
X
1
: 1
X
2
:
0
0 0 0 1 0 1 1 0
1 1 1 1 1 0 1 1
_
0
13.2
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.
Two discrete fuzzy sets, A and B are defined as follows:
A = ( 60 , 0 .
1 ) ( 62 , 0 .
5 ) ( 64 , 0 .
7 ) ( 66 , 0 .
9 ) ( 68 , 1 .
0 ) ( 70 , 0 .
8 )
B _ , 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: