Evolutionary Computation Parameters

The major goal of the reported research is determination of optimal experimental settings for MAs applied to complex structural design problems. In order to achieve it, two classes of design problems were selected (described above) and an extensive evolutionary computation parameter search (sensitivity analysis) was conducted. It involved the following parameters and their values: the length of the design process (specified by the number of fitness evaluations), parent and offspring population sizes, the rate of mutation operator, and the rate of crossover operator.

The experiments were divided into two major groups depending on the termination criterion used in individual runs: short-term experiments (up to 1,000 fitness evaluations) and long-term experiments (up to 10,000 fitness evaluations). This distinction is important from the structural design point of view because evaluations of generated designs are usually very expensive (more than 99% of computational time).

Sensitivity analyses were conducted during short-term processes. The optimal combination of parameters’ values found in the short-term processes was subsequently used in the long-term experiments. The performance analysis of MAs was conducted for both short – and long-term experiments. It included the following performance criteria:

• performance improvement of the best design at the end of the experiment compared to the best design from an initial population

• performance improvement of the average design at the end of the experiment compared to the average design from an initial population

Both improvements were measured by the reduction of the total weight of a structural system being designed.

An extensive parameter search was conducted during short-term experiments. For all combinations of parent and offspring population sizes shown in Table 3, a search for optimal rates of mutation and crossover was conducted. In each case, 12 combinations of mutation and crossover rates were considered, i. e. (mutation rate 0.025, crossover rate 0), (mutation rate 0.025, crossover rate 0.2), etc. The design processes were repeated 5 times for each combination of parameter values using a different value of a random seed each time.

Two types of memetic algorithms were tested: MA-GA and MA-ES. The former utilizes GAs while the latter uses ES. Both MAs were investigated in order to determine which produces better results when combined with the local search for the two classes of structural design problems. The initial population of parents was generated randomly for each experiment. As discussed above, the fitness of a design was determined by the total weight of the steel structural system calculated using the first-order structural analysis. Whenever an infeasible design was generated, it was assigned the fitness value of 0. In other words, the death penalty method was used to handle infeasible solutions (Coello Coello, 2002).

Table 3: Evolutionary computation parameters and their values

EC Parameter

Value(s)

EC Parameter

Value(s)

Type of MA

MA-ES, MA-GA

Initialization

method

random

Pop. sizes (parent, offspring)

(1,5), (1,25), (5,25), (5,125) or (50,250) for MA-ES(p+X)

(5,25), or (50,50) for MA-GA (5,25) for MA-ES(p, X)

Fitness

Total weight of the structural system (determined by the 1st-order analysis)

Generational

model

Overlapping for MA-ES(p+X), Nonoverlapping for MA-ES(p, X) and MA-GA

Constraint handling method

death penalty (infeasible designs assigned 0 fitness)

Selection (parent, survival)

(uniform stoch., truncation) for MA-ES, (fitness prop., uniform stoch.) for MA-

GA

Termination

criterion

1,000 evaluations (short­term), or 10,000 evaluations (long-term)

Mutation rate

0.025, 0.1, 0.3, or 0.5

Number of runs

5 (in each experiment)

Uniform crossover rate

0, 0.2, or 0.5

4. Experimental Results

This section reports the results of design experiments introduced in the previous section. They are grouped with respect to the parameter being investigated.