3。1。 Genetic Algorithms

GAs are a stochastic (probabilistic) optimization approach whose search method is based on the theory of “survival of the fittest”。 It uses term inpiduals (or genotype, chromosomes) in a population。 GAs spur  the  parallel  search  between  many peaks and exploit the search space。 This characteristic makes GA more robust  as  compared to  other  point-to-point  search methods。

The GA input data are population size, number of generations, number of bits per variable, crossover probability and mutation probability。 The population P(t) cue be expressed in a vector form,

where / is the generation number, I( is the set of design variables, and d is the number of  design variables。

During each generation, chromosome I; is evaluated for fitness according to the ability of attaining the satisfactory result。 The fitness function F is calculated according  to

1 if  any constraint is violated

I

0 otherwise

where f is the objective function, and Z is a value greater than the objective function。 The selection of this fitness function heavily penalizes chromosomes that violate any of the constraints。 A chromosome that violates any of the constraints becomes very small as compared to the chromosome that satisfies all constraints。 This is because the term (I — /)Z in the objective function evaluation goes to zero in case of any constraint violation。 GAs are characterized by the genetic operations of selection, crossover and mutation [23]。

Selection。 Using roulette wheel fit inpiduals  are selected for genetic   operations。

Inpiduals having beneficial fitness are at higher chances for   selection。

Crossover。    For each chromosome in population, if the randomly generated number r < Pc (probability of crossover), then that chromosome is selected for crossover。 Two such selected vectors exchange their string information in hope of better children。 Crossover occurs at a randomly generated number within number of bits and this point is called point of crossover。

Mutation。 For each chromosome in the population, if the  randomly  generated number r Pin (probability of mutation), then the chromosome is selected for mutation。 Mutation is a process that changes the binary structure of a randomly selected bit: from 0 to I or vice versa。 This offers persity to the search direction and prevents convergence to local minima。

These genetic operations generate a new  population  with  higher  average fitness。 The newly created population replaces the old one and the iterative procedure  repeats till the specific convergence criteria are met, usually, the number of generations。 The control parameters for the SGA and GAE approaches  are  listed  in  Table  I。  The values for these parameters were selected after experimentation using the developed tools。

The flow of operations for optimization using GAs is shown in Fig。 4。

The SGA verifies newly generated children during the crossover operation in order to prevent a bad solution from entering into the new population。 If the fitness of a child is lower than that of the parent, then the child is not permitted into the new population。 The SGA could loose good members during the genetic operation of selection。

The GAE is similar to the SGA with an additional feature called “elitist  strategy”。 The elitism prevents loosing the best results from the current generation due to the genetic operations of  selection,  crossover  and mutation  [24]。 The  objective  of GAE is to freeze or forward the best solution of the previous generation to the next genera- tion。 The GAE fitness function F is calculated according to Eg。 (I I)。