Mutation on a chromosome. 

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... in which N Pop is the population size. The fitness value F k is obtained from either Eq. (9) or (10). Note that subscript k in F k signifies that the fitness value is computed for each respective k th chromosome. It is interesting to note that GAs utilize only the numerical values of the objective function and of its associated constraints for the evaluation of the chromosome fitness, as seen from Eqs. (9) – (12). This advantageous feature makes GAs readily applicable to real-world problems where the performance functions are generally implicit with respect to random variables. In accordance with the binary representation of chromosomes, a simple binary crossover is applied (confer Figure 2). The mutation operation also assists the exploration for potential solutions which may be overlooked by the crossover operation. According to the chromosome representation, a binary mutation is employed for the purpose (confer Figure 3). Simple GAs perform well in locating a single optimum but face difficulties when requiring multiple optima (see e.g. (De Jong, 1975; Mahfoud, 1995a; Mahfoud, 1995b and Miller & Shaw, 1995)). Niching methods can identify multiple solutions with certain extent of diversity (Miller & Shaw, 1995). Among niching methods, Deterministic Crowding Genetic Algorithms (DCGAs) (Mahfoud, 1995a and Mahfoud, 1995b) have been commonly used in multimodal functions optimization. It is noted that DCGAs is originally designed for unconstrained optimization problems. The adaptive penalty described in the previous subsection will be used in conjunction with DCGAs to handle constraints. DCGAs work as follows. First, all members of population are grouped into NPop/2 pairs, where NPop is the population size. The crossover and mutation are then applied to all pairs. Each offspring competes against one of the parents that produced it. For each pair of offspring, two sets of parent-child tournaments are possible. DCGAs hold the set of tournaments that forces the most similar elements to compete. The following provides a pseudo code of DCGAS (Brownlee, 2004). NPop : Population size. d ( x, y ) : Distance between individuals x and y . F ( x ) : Fitness of individual population member. 1. Randomly initialize population. 2. Evaluate fitness of population. 3. Loop until stop condition: a. Shuffle the population. b. Crossover to produce NPop /2 pairs of offspring. c. Apply mutation (optional). d. Loop for each pair of offspring: i. If( d (parent1,child1)+ d (parent2,child2)) ≤ ( d (parent2,child1)+ d (parent1,child2)). 1. If F (child1) > F (parent1), child1 replaces parent1. 2. If F (child2) > F (parent2), child2 replaces parent2. ii. Else 1. If F (child1) > F (parent2), child1 replaces parent2. 2. If F (child2) > F (parent1), child2 replaces parent1. In the following section, the applications of GAs to enhance the capability in PRA for numerous key aspects of risk-based information will be demonstrated. The likelihood of each combination of variable magnitudes in contributing a failure event is an interesting issue in PRA. The focus is particular on the so-called Point of Maximum Likelihood (PML) in failure domain. PML represents the combination of variable magnitudes that most likely contribute to the failure and to the corresponding failure probability. Since PML is the point of highest JPDF in failure domain D F , the PML x * can be obtained from solving the following optimization ...