Length of Evolutionary Processes
The length of an evolutionary process influences the computational effort required to run a design experiment. This issue is particularly relevant to structural engineering due to the fact that evaluations of structural designs are usually computationally expensive. In yet another series of design experiments with MAs, the impact of the length of design processes (measured by the total number of fitness evaluations) on the quality of produced designs was investigated.
Figure 14 shows the progress of the long-term experiments (10,000 fitness evaluations) with MAES for Problem Ic and compares it to the average fitness obtained after 1,000 evaluations (short-term experiment). The average performance improvement between the long-term processes and short-term processes was equal to about 21,900 lbs., or 4.3 percent. The difference between the average fitness after 10,000 fitness evaluations and the average fitness of the initial parents was equal to more than 55,500 lbs., or 10.2 percent. Similar results were produced by MA-ES for Problem II. Figure 15 shows a similar comparison of the long – and short-term experiments. In this case, the average fitness of the designs produced in the long-term experiments equaled 502,879 lbs. and was more than 21,500 lbs., or 4.1 percent, better than the average fitness obtained in the short-term experiments. The overall performance improvement in the long-term experiments was, on average, equal to more than 58,000 lbs., or 10.3 percent, compared to about 36,000 lbs., or 6.4 percent, achieved in the short-term optimization experiments.
The long-term experiments took ten times more computational time than the short-term experiments. Considering the fact that each evaluation took about 1 minute on a Pentium IV processor, the total time required for a single run of a long-term experiments was equal to almost 7 days compared to about 16 hours necessary for a short-term experiment. Thus, there is always a strong trade-off between the length of a design process and available computational resources. In this case, we can utilize the fact that after a certain number of fitness evaluations the best-so-far fitness curves tend to level-off (see Figure 15) and significant performance improvements are no longer obtained. This threshold, however, needs to be defined empirically for individual problems.
performance of MA-ES for Problem Ic performance of MA-ES for Problem II
Figure 16:Best designs produced in design experiments with MAs for each problem 4.6. Optimal Designs
In extensive design experiments, (sub)optimal designs were produced for each class of structural problems and subproblems investigated in the paper. The final designs generated by MAs for each problem differed not only in the total weight but also in structural shaping patterns produced by configurations of wind bracings. The best designs produced in these experiments are shown in Figure 16. The values located below designs indicate their total weight in lbs. (top) and their maximum horizontal displacement (bottom) in inches.
Figure 16 shows significant differences in fitness (total weight) corresponding to best solutions found for each problem. For example, steel structural systems with the wind bracing system composed solely of X bracing elements (Problem Ia) have generally higher total weight by about 40,000-50,000 lbs. than best designs for other problems. On the other hand, they typically exhibit smaller horizontal displacements (better stiffness). They also show emerging structural shaping pattern of crossed macrodiagonal bracings in the lower and middle part of the structural system (see Figure 16). On the contrary, best solutions for Problems Ib-c and II exhibit random-looking configurations of wind bracing elements.
This paper reported results of a large number of design experiments with memetic algorithms. It also formulated initial recommendations regarding optimal parameter settings for these algorithms when applied to complex structural design problems. These heuristics can be used by researchers and practitioners to quickly set up their design experiments without conducting expensive parameter optimization.
In the reported studies, key evolutionary computation parameters and their values were identified. It was discovered that the type of evolutionary algorithm, the rate of mutation operator, and the size of the parent population were critical for the success of structural design processes. Specifically, evolution strategies produced on average significantly better results than genetic algorithms for the design problems considered in the paper. Also, low mutation rates, i. e. 0.025, resulted in better performance of memetic algorithms. Furthermore, small parent population sizes were generally preferred to large populations.
The experiments also produced significant body of structural design knowledge, both quantitative and qualitative in nature. Specifically, they identified regions of the design spaces in which high – performance solutions could be found. They also determined the ranges of the total weight of structural systems associated with high-performance solutions for both classes of design problems and showed that these ranges are significantly different. Finally, qualitative differences between high – performance solutions have been identified in terms of structural shaping patterns exhibited by configurations of wind bracing elements.
The research presented in this paper will be continued, including the extension of the scope of the empirical studies to other structural design problems. Also, other local search algorithms will be combined with evolutionary algorithms and applied to several structural engineering problems. Another promising direction of future research includes more advanced representations of structural systems and their impact on the performance of memetic algorithms.