Sea F una función que satisface que para todo e > 0 existe una constante C ( , tal que

The case study presented was used as a flexible foundation for future studies to be conducted. Its formulation allows the ratio of decomposable and non-decomposable constraints to be changed such that the effects on the optimizer may be observed.

For example more non-decomposable constraints may be added to test the limit of the BOA’s increased efficiency as the problem transitions from nearly decomposable to non-decomposable.

Additionally, the problem is currently formulated as a rule based approach, but the objective functions or constraints may be adapted to add numerical simulation results.

The adaptation allows FEA simulations or other numerical modeling approaches to be used provided the results can be condensed into a set of training data. This also allows for the use of heuristic and qualitative terms where an optimizer such as the SOGA might struggle or need adaptation. The added flexibility and capability will become important as the BOA transitions from a tool used purely for its efficiency to one that allows the designer to understand problem dynamics.

In the aforementioned numerical simulations, the results lack simplistic condensed mathematical relationships that can be readily absorbed and applied. This is often a shortcoming of numerical simulation and only through repeated use or trial and error does an engineer build up intuition that can rapidly be used to change respond to and improve designs. The BOA holds promise to return some basic capability to designers who have not spent the years necessary to understand the intimate details of the simulation. Improved designer understanding of the design problem is incredibly important and will become the focus going forward in chapter 5.

3.4 Summary

To demonstrate the BOA’s ability to optimize early stage naval structural design problems, a case study was created and simulated solving for minimum production cost. The problem was designed to exhibit nearly-decomposable behavior, common of structural cost estimation and rules based failure mode evaluation approaches. This problem specific topology was hypothesized to provide an advantage to the BOA as compared to other heuristic optimizers. The BOA was then compared to a SOGA using an effort benchmarking strategy focused on objective function evaluations, the most costly portion of the design optimization simulation.

The presented results were the first demonstrated example of BOA application to practical design of ship structures. It also remains one of the few discrete formulations used to optimize engineering problems beyond the scope of canonical benchmarking problems. The BOA results showed the ability of the optimizer to successfully solve the structural design problem, verifying it as a useful optimization technique

Furthermore when comparing the effort required to reach acceptably fit design solutions the BOA outperforms the SOGA as problem size increases towards sizes comparable to practical ship design problems . The trends indicate that the BOA is indeed an ideal optimization approach for use in solving structural design problems.

Its response to constraint violation shows the value of the Bayesian networks, resolving the constrains much faster that the BOA on average.

CHAPTER 4

Learning

4.1 Overview

While the gains made by the BOA in efficiently solving nearly decomposable struc-tural problems are groundbreaking, it falls into a pattern that seems to exist for a majority of heuristic optimizers. Optimizer performance is incrementally improved using various techniques and addition relevant to unique problem specific differences, to gain efficiency advantage. In some sense the BOA application to the nearly de-composable structure falls exactly into this pattern. This result however makes no attempt to change how the designer understands or comprehends the problem, seek-ing to solely chase speed. It leaves a knowledge gap in designer understandseek-ing, and particularly in early stage design where many of the potential costs are determined, this is incredibly dangerous. As a result, changes need to be made to the design op-timization process not product, in order to have the process capture and learn more information about the problem and relate it to designers in a simple and comprehen-sible manner.

The exciting part of this paradigm shift is the manner in which the BOA learns about and solves the design problem, through its BN. These networks are legacy represen-tation of the relationships being negotiated and explored at various stages of the design problem. Additionally, edges are the exact type of simple relationships that

a designer needs, showing how change or cause and effect might propagate through-out the design. They have rigorous mathematically backbones that can be coupled to the optimization process to help designers better understand large and complex problems.

The BOA can be one potential solution to the shift in design process. The op-timization routine must learn about the problem and provide insight back to the designer. To do this the first most salient place to examine is the Bayesian network.

Through simple network analysis techniques, the design interaction can be identified and quantified. This chapter of the thesis will focus on initial attempts to learn from the Bayesian network, setting the stage for use of the lessons learned in chapter 5.

The results of this analysis will seek to determine importance of variables through in degree and out degree analysis. The networks of the initial optimizations are studied first to see what the algorithm is producing. Then, a structured regimented approach to understand how the designer adds and removes edges to a decomposable problem is presented. Finall, partial derivatives within this network are explored to attempt to connect the mathematical function to the observed results.